ENTRY ARTIFICIAL INTELLIGENCE Authors: Oliver Knill: March 2000 Literature: Peter Norvig, Paradigns of Artificial Intelligence Programming Daniel Juravsky and James Martin, Speech and Language Processing +------------------------------------------------------------ | Adaptive Simulated Annealing +------------------------------------------------------------ Adaptive Simulated Annealing A language interface to a neural net simulator. +------------------------------------------------------------ | artificial intelligence +------------------------------------------------------------ artificial intelligence (AI) is a field of computer science concerned with the concepts and methods of symbolic knowledge representation. AI attempts to model aspects of human thought on computers. Aspectrs of AI: computer vision language processing pattern recognition expert systems problem solving roboting optical character recognition artificial life grammars game theory +------------------------------------------------------------ | Babelfish +------------------------------------------------------------ Babelfish Online translation system from Systran. +------------------------------------------------------------ | Chomsky +------------------------------------------------------------ Chomsky Noam Chomsky is a pioneer in formal language theory. He is MIT Professor of Linguistics, Linguistic Theory, Syntax, Semantics and Philosophy of Language. +------------------------------------------------------------ | Eliza +------------------------------------------------------------ Eliza One of the first programs to feature English output as well as input. It was developed by Joseph Weizenbaum at MIT. The paper appears in the January 1966 issue of the "Communications of the Association of Computing Machinery". +------------------------------------------------------------ | Google +------------------------------------------------------------ Google A search engine emerging at the end of the 20'th century. It has AI features, allows not only to answer questions by pointing to relevant webpages but can also do simple tasks like doing arithmetic computations, convert units, read the news or find pictures with some content. +------------------------------------------------------------ | GPS +------------------------------------------------------------ GPS General Problem Solver. A program developed in 1957 by Alan Newell and Herbert Simon. The aim was to write a single computer program which could solve any problem. One reason why GPS was destined to fail is now at the core of computer science. There are a large set of problems which are NP hard and where finding a solution becomes exponentially hard in dependence of the size of the problem. Nonetheless, GPS has been a useful tool for exploring AI programming. +------------------------------------------------------------ | HAL +------------------------------------------------------------ HAL The HAL 9000 computer was the main character in Stanley Kurbrick's film 2001: a Space Odyssey. HAL is an AI agent capable to understand advanced language processing behavior as speaking and understanding language and even reading lips. +------------------------------------------------------------ | Lisp +------------------------------------------------------------ Lisp Lisp is one of the oldest programming languages still in widespread use today. "Common Lisp" is the most widely accepted standard. Other dialects like "Franz Lisp" MacLisp, InterLisp, ZetaLisp or "Standard Lisp" are considered obsolete. Lisp is the most popular language for AI programming. Lisp programs are concise and are uncluttered by low-level detail. +------------------------------------------------------------ | Loebner Prize +------------------------------------------------------------ Loebner Prize A competition attempted to put various computer programs to the Turing test. A consistent result over the years has been that even the crudest programs can fool some of the judges some of the time. +------------------------------------------------------------ | MIT ai lab +------------------------------------------------------------ MIT ai lab Massachusetts Institute of Technology AI laboratory. +------------------------------------------------------------ | neural network +------------------------------------------------------------ neural network Artificial neural networks try to simulate biological neural networks as found in the brain. Such a network consists of many simple processors called neurons, each possibly having some local memory. These neurons are connected and evolve depending to their local data and on the inputs they receive via the connections. A neural network can either be an algorithm, or be realized as actual hardware. Neural networks typically allow training. They learn by adjusting the weights of the connections on the basis of presented patterns. The individual neurons are elementary non-linear signal processors. Neural networks are distinguished from other computing devices by a high degree of interconnection allowing parallelism. There is no idle memory containing data and programs. Each neuron is pre-programmed and continuously active. +------------------------------------------------------------ | pattern recognition +------------------------------------------------------------ pattern recognition A branch of artificial intelligence concerned with the classification or description of observations. The classification uses either statistical, syntactic or neural aproches. +------------------------------------------------------------ | pilot +------------------------------------------------------------ pilot Programmed Inquiry Learning Or Teaching. +------------------------------------------------------------ | prolog +------------------------------------------------------------ prolog A popular AI programming language used in Europe and Japan. Prolog shares most of Lisp's advantages in terms of flexibility and conciseness. +------------------------------------------------------------ | regular expression +------------------------------------------------------------ regular expression is a language for specifying text search strings. It is used in UNIX programs like vi, perl, emacs or grep. It is also used in Microsoft word or web search engines. +------------------------------------------------------------ | scheme +------------------------------------------------------------ scheme A dialect of Lisp which is gaining popularity, primarily for teaching and experimenting with programming language design and techniques. +------------------------------------------------------------ | Shrdlu +------------------------------------------------------------ Shrdlu Terry Winograd's SHRDLU system of 1972 simulated a robot embedded in a world of toy blocks. The program was able to accept natural language text commands. +------------------------------------------------------------ | Student +------------------------------------------------------------ Student Student was an early language understanding program written by Daniel Bubrow in 1964. It was designed to read and solve the kind of word problems found in high school algebra books. Unlike Eliza, "Student" must process and understand a great deal of input as well as be able to solve algebraic equations. +------------------------------------------------------------ | toy problem +------------------------------------------------------------ toy problem A deliberately oversimplified case of a challenging problem used to investigate, prototype, or test algorithms for a real problem. +------------------------------------------------------------ | Turing test +------------------------------------------------------------ Turing test A test introduced in 1950 by Alan Turing. There are three participants. Two people and a computer. One person plays the role of an interrogator who has to find out, which of the two others is a machine. This interrogator is connected to the two other participants through teletype. The task of the machine is to fool the interrogator into believing it is a person. The task of the other participant is to convince the interrogator that he is human. Turing predicted that in 2000 a machine with 10 Gig memory would have a 30 percent change of fooling a human interrogator after 5 minutes of questioning. +------------------------------------------------------------ | Weizenbaum +------------------------------------------------------------ Weizenbaum Joseph Weizenbaum was the principal developer of Eliza, one of the first programs to feature English output as well as input. This file is part of the Sofia project sponsored by the Provost's fund for teaching and learning at Harvard university. There are 22 entries in this file. COUNT: 22 ENTRY ABSTRACT ALGEBRA Authors: started Oliver Knill: September 2003 Literature: Lecture notes +------------------------------------------------------------ | additive +------------------------------------------------------------ A function f: G to H from a semigroup G to a semigroup H is additive if f(a+b) = f(a) + f(b). A group-valued function on sets is additive if f(Y cup Z) = f(Y) + f(Z) if Y and Z are disjoint. +------------------------------------------------------------ | algebra +------------------------------------------------------------ An algebra over a field K is a ring with 1 which is also a vector space over K and whose multiplication is bilinear with respect to K. Examples: the complex numbers C is an algebra over the field of real numbers K=R. The quaternion algebra H is an algebra over the field of complex numbers. The matrix algebra M(n,R) is an algebra over the field R. +------------------------------------------------------------ | An algebraic number field +------------------------------------------------------------ An algebraic number field is a subfield of the complex numbers that arises as a finite degree algebraic extension field over the field of rationals. +------------------------------------------------------------ | alternating group +------------------------------------------------------------ The alternating group G is the subgroup of the symmetric group of n objects given by the elements which can be written as a product of an even number of transpositions. +------------------------------------------------------------ | Artinian module +------------------------------------------------------------ An Artinian module is a module which satisfies the descending chain condition. Every Artinian module is a Noetherian module but the integers for example are a Noetherian module which is not an Artinian module. +------------------------------------------------------------ | Artinian ring +------------------------------------------------------------ An Artinian ring is a ring which when considered as a R-module is an Artinian module. +------------------------------------------------------------ | Artinian ring +------------------------------------------------------------ Two elements of an integral domain that are unit-multipliers of each other are called associate numbers. +------------------------------------------------------------ | Cayley's theorem +------------------------------------------------------------ Cayley's theorem assures that every finite group is isomorphic to a permutation group. +------------------------------------------------------------ | center +------------------------------------------------------------ The center of a group (G,*) is the set of all elements g which satisfy g h = h g for all h in G. The center is a subgroup of G. +------------------------------------------------------------ | commutator +------------------------------------------------------------ The commutator of two elements g,h in a group (G,*) is defined as g,h=g*h*g^-1*h^-1. +------------------------------------------------------------ | commutator subgroup +------------------------------------------------------------ The commutator subgroup of a group (G,*) is the set of all commutators g,h in G. It is a subgroup of G. +------------------------------------------------------------ | factor group +------------------------------------------------------------ A factor group G/N is defined when N is a normal subgroup of the group G. It is the group, where the elements are equivalent classes g N and operation (g N) (h N) = (g h) N which is defined because N was assumed to be normal. For example, if G is the group of additive integers and N=k N with an integer k, then G/N = Z_k is finite group of integers modulo k. +------------------------------------------------------------ | finite group +------------------------------------------------------------ A group is called a finite group if G is a set with finitely many elements. For example, the set of all permutations of a finite set form a finite group. The set of all operations on the Rubik cube form a finite group. +------------------------------------------------------------ | group +------------------------------------------------------------ A group (X,+,0) is a set X with a binary operation + and a zero element 0 (also called neutral element or identity) with the following properties (a+b)+c = a+(b+c) associativity a+0 = a zero element forall a exists b a+b=0 inverse Examples: the real numbers form a group under addition 5+2.34=7.34, 3-3=0. the set GL(n,R) of real matrices with nonzero determinant form a group under matrix multiplication the nonzero integers form a group under multiplication 4*7=28. all the invertible linear transformations of the plane plane form a group under composition. The "zero element" is the identity transformation T(x)=x. all the continuous functions on the unit interval form a group with addition (f+g)(x) = f(x)+g(x). all the permutations on a finite set form a group under composition. the set of subsets Y of a set X with the operation A Delta B = (A cup B) setminus (A cap B) form a group. The inverse of A is A itself because A Delta A = emptyset, the zero element is emptyset. +------------------------------------------------------------ | normal subgroup +------------------------------------------------------------ a normal subgroup of a group (G,*) is a subgroup (H,*) of (G,*) which has the property that for all g in H and all g in G one has g^-1 h g is in H. For an abelian group all subgroups are normal. The subgroup Sl(n,R) of Gl(n,R) is a normal subgroup. +------------------------------------------------------------ | ring +------------------------------------------------------------ A ring (X,+,*,0) is a set X with a binary operation + and a binary operation * such that (X,+,0) is a commutative group and (X,*) is a semigroup and such that the distributivity laws a*(b+c) = a*b + a*c, (a+b)*c - a*c+b*c hold. Examples: the integers Z form a ring with addition and multiplication the set of rational numbers Q, the set of real numbers R or the complext numbers C form a ring with addition and multiplication. the set of 3x3 matrices with real entries form a ring with addition and matrix multiplication. the set P of polynomials with real coefficients form a ring with addition and multiplication. the set of subsets Y of a set X with addition Delta and multiplication cap forms a ring. the set of continuous functions on an interval 0,1 with addition (f+g)(x) = f(x)+g(x) and multiplication f*g(x) = f(x) g(x). +------------------------------------------------------------ | commutative group +------------------------------------------------------------ A commutative group is a group (X,+,0) which is commutative: a+b=b+a. the set of real numbers R forms a commutative group under addition. the set of permutations S of a set X form a noncommutative group under composition. +------------------------------------------------------------ | commutative ring +------------------------------------------------------------ A commutative ring is a ring (X,+,*,0) for which the multiplicative semigroup (X,*) is commutative: a*b = b*a. Examples: the integers form a commutative ring. the set of 2 x 2 matrices form a noncommutative ring the set of polyomials with real coefficients (x^2+pi x+2) * (x+5x) = 6 x^3 + 6 pi x^2+12x. +------------------------------------------------------------ | function field +------------------------------------------------------------ A function field is a finite extension of the field C(z) of rational functions in the variable z. +------------------------------------------------------------ | homomorphism +------------------------------------------------------------ An homomorphism phi between two groups G,H is a map f: G to H which has the property phi(g*h) = phi(g) * phi(h) and phi(0)=0 for all elements g,h in G. Examples: if G is the multiplicative group (R^+,*) of positive real numbers and H is the additive group (R,+) of all positive real numbers then phi(x)=log(x) is a homomorphism: if G is the group of matrices with nonzero determinant and H is the group of nonzero real numbers and phi(A)= det(A), we have phi(x*y) = phi(x) phi(y). +------------------------------------------------------------ | isomorphism +------------------------------------------------------------ An isomorphism phi between two groups G,H is a homomorphism between groups which is also invertible. +------------------------------------------------------------ | number field +------------------------------------------------------------ A number field is a finite extension of Q, the field of rational numbers. It is a field extension of Q which is also a vector space of finite dimension over Q. Since the elements of a number field are algebraic numbers, roots of a fixed polyonomial a_0+a_1 z+... + z^n with integer coefficitients, one calls them also algebraic number fields. The study of algebraic number fields is part of algebraic number theory. Examples: quadratic fields: Q(sqrtd), where d is a rational number. It is in general a field extension of degree 2 over the field of rational number. cyclotimic fields: Q(xi), where xi is a n'th root of 1. It is a field extension of degree phi(n), where phi(n) is the Euler function. +------------------------------------------------------------ | octonions +------------------------------------------------------------ The octonions can be written as linear combinations of elements e_0,e_1,e_2,...,e_7. The multiplication is determined by the multiplication table * 1 e_1 e_2 e_3 e_4 e_5 e_6 e_7 1 1 e_1 e_2 e_3 e_4 e_5 e_6 e_7 e_1 e_1 -1 e_4 e_7 -e_2 e_6 -e_5 -e_3 e_2 e_2 -e_4 -1 e_5 e_1 -e_3 e_7 -e_6 e_3 e_3 -e_7 -e_5 -1 e_6 e_2 -e_4 e_1 e_4 e_4 e_2 -e_1 -e_6 -1 e_7 e_3 -e_5 e_5 e_5 -e_6 e_3 -e_2 -e_7 -1 e_1 e_4 e_6 e_6 e_5 -e_7 e_4 -e_3 -e_1 -1 e_2 e_7 e_7 e_3 e_6 -e_1 e_5 -e_4 -e_2 -1 Octonions are also called Cayley numbers. The multiplication of octonions is not associative. Octonions have been discovered by John T. Graves in 1843 and independently by Arthur Cayley. +------------------------------------------------------------ | order +------------------------------------------------------------ The order of a finite group is the set of elements in the group. +------------------------------------------------------------ | p-group +------------------------------------------------------------ A p-group is a finite group with order p^n, where p is a prime integer and n>0. +------------------------------------------------------------ | quaterions +------------------------------------------------------------ The quaterions can be written as linear combinations of elements 1,i,j,k. The multiplication is determined by the multiplication table * 1 i j k 1 1 i j k i i -1 k -j j j -k -1 i k k j -i -1 Quaternions are useful to compute rotations in three dimensions. +------------------------------------------------------------ | semigroup +------------------------------------------------------------ A semigroup (X,+) is a set X with a binary operation + which satisfies the associativity law (a+b)+c = a+(b+c). Examples: a group is a semigroup. the set of finite words in an alphabet with composition form a semigroup word1 + word2 = word1word2 the natural numbers form a semigroup under addition. +------------------------------------------------------------ | commutative semigroup +------------------------------------------------------------ A commutative semigroup is a semigroup (X,+) which is commutative. a+b=b+a. the natural numbers form a commutative semigroup under addition. composition of words over a finite alphabet form a noncommutative semigroup +------------------------------------------------------------ | kernel +------------------------------------------------------------ The kernel of a homomorphism between two groups G,H is the set of all elements in G which are maped to the zero element of H. For example, SL(n,R) is the kernel of the homomorphism from GL(n,R) to R setminus 0 defined by phi(A) = det(A). +------------------------------------------------------------ | subgroup +------------------------------------------------------------ A subgroup of a group G is a subset of G which is also a group. Examples: the set of n x n matrices with determinant 1 is a subgroup of the set of n x n matrices with nonzero determinant. the trivial subgroup 0 is always a subgroup of a group (G,*,0). +------------------------------------------------------------ | Theorem of Cauchy +------------------------------------------------------------ The Theorem of Cauchy in group theory states that every finite group whose order is divisible by a prime number p contains a subgroup of order p. +------------------------------------------------------------ | sedenions +------------------------------------------------------------ sedenions form a zero Divisor Algebra. By a theorem of Frobenius (1877), there are three and only three associative finite division algebras: the real numbers R, the complex numbers C and the quaternions Q. Similar algebras in higher dimensions have zero divisors: sedenions are examples. +------------------------------------------------------------ | field +------------------------------------------------------------ A field is a commutative ring (R,+,*,0,1) such that (R,+,0) and (R setminus 0,*,1) are both commutative groups. +------------------------------------------------------------ | theorem of Zorn +------------------------------------------------------------ By a theorem of Zorn (1933), every alternative, quadratic, real non-associative algebra without zero divisors is isomorphic to the 8-dimensional octonions O. +------------------------------------------------------------ | Theorem of Hurwitz +------------------------------------------------------------ Theorem of Hurwitz: the normed composition algebras with unit are: real numbers, complex numbers, quaternions; and octonions. +------------------------------------------------------------ | Theorem of Kervaire and Milnor +------------------------------------------------------------ Theorem of Kervaire and Milnor In 1958, Kervaire and Milnor proved independently of each other that the finite-dimensional real division algebras have dimensions 1,2,4, or 8. +------------------------------------------------------------ | Theorem of Adams +------------------------------------------------------------ Theorem of Adams In 1960, Adams proved that a continuous multiplication in R^n+1 with two-sided unit and with norm product exists only for n+1 = 1,2,4, or 8. +------------------------------------------------------------ | Theorem of Hurwitz +------------------------------------------------------------ Theorem of Hurwitz: the normed composition algebras with unit are: real numbers complex numbers quaternions octonions +------------------------------------------------------------ | Theorems of Sylov +------------------------------------------------------------ Theorems of Sylov If G is a finite group of order |G|=p^n q, where p is a prime number, then G has a subgroup of order p^n. Such groups are called Sylov groups and all of them are isomorphic. Furthermore, the number N of different p-Sylov groups in G satisfies N =1 mod (p). This file is part of the Sofia project sponsored by the Provost's fund for teaching and learning at Harvard university. There are 39 entries in this file. COUNT: 39 AMS FIELDS Authors: Oliver Knill: September 2003 Literature: AMS Website +------------------------------------------------------------ | AMS CLASSIFICATION +------------------------------------------------------------ AMS CLASSIFICATION 00-xx General 01-xx History and biography 03-xx Mathematical logic and foundations 05-xx Combinatorics 06-xx Order, lattices, ordered algebraic structures 08-xx General algebraic systems 11-xx Number theory 12-xx Field theory and polynomials 13-xx Commutative rings and algebras 14-xx Algebraic geometry 15-xx Linear and multilinear algebra; matrix theory 16-xx Associative rings and algebras 17-xx Nonassociative rings and algebras 18-xx Category theory; homological algebra 19-xx K-theory 20-xx Group theory and generalizations 22-xx Topological groups, Lie groups 26-xx Real functions 28-xx Measure and integration 30-xx Functions of a complex variable 31-xx Potential theory 32-xx Several complex variables and analytic spaces 33-xx Special functions 34-xx Ordinary differential equations 35-xx Partial differential equations 37-xx Dynamical systems and ergodic theory 39-xx Difference and functional equations 40-xx Sequences, series, summability 41-xx Approximations and expansions 42-xx Fourier analysis 43-xx Abstract harmonic analysis 44-xx Integral transforms, operational calculus 45-xx Integral equations 46-xx Functional analysis 47-xx Operator theory 49-xx Calculus of variations and optimal control; optimization 51-xx Geometry 52-xx Convex and discrete geometry 53-xx Differential geometry 54-xx General topology 55-xx Algebraic topology 57-xx Manifolds and cell complexes 58-xx Global analysis, analysis on manifolds 60-xx Probability theory and stochastic processes 70-xx Mechanics of particles and systems 74-xx Mechanics of deformable solids 76-xx Fluid mechanics 78-xx Optics, electromagnetic theory 80-xx Classical thermodynamics, heat transfer 81-xx Quantum theory 82-xx Statistical mechanics, structure of matter 83-xx Relativity and gravitational theory 85-xx Astronomy and astrophysics 86-xx Geophysics 90-xx Operations research, mathematical programming 91-xx Game theory, economics, social and behavioral sciences 92-xx Biology and other natural sciences 93-xx Systems theory; control 94-xx Information and communication, circuits 97-xx Mathematics education This file is part of the Sofia project sponsored by the Provost's fund for teaching and learning at Harvard university. There are 1 entries in this file. COUNT: 1 ENTRY MATH CITATIONS Collected by Oliver Knill: 2000-2002 +------------------------------------------------------------ | solution +------------------------------------------------------------ solution Every problem in the calculus of variations has a solution, provided the word solution is suitably understood. -- David Hilbert +------------------------------------------------------------ | enhusiast +------------------------------------------------------------ enhusiast The real mathematician is an enthusiast per se. Without enthusiasm no mathematics. -- Novalis +------------------------------------------------------------ | royal +------------------------------------------------------------ royal There is no royal road to geometry. -- Euclid +------------------------------------------------------------ | computer +------------------------------------------------------------ computer One may be a mathematician of the first rank without being able to compute. It is possible to be a great computer without having the slightest idea of mathematics -- Novalis +------------------------------------------------------------ | analysis +------------------------------------------------------------ analysis Geometry may sometimes appear to take the lead over analysis, but in fact precedes it only as a servant goes before his master to clear the path and light him on the way. -- James Joseph Sylvester +------------------------------------------------------------ | freedom +------------------------------------------------------------ freedom The essence of mathematics lies in its freedom. -- Georg Cantor +------------------------------------------------------------ | fantasy +------------------------------------------------------------ fantasy Fantasy, energy, self-confidence and self-criticism are the characteristic endowments of the mathematician. -- Sophus Lie +------------------------------------------------------------ | magacian +------------------------------------------------------------ magacian Pure mathematics is the magician's real wand. -- Novalis +------------------------------------------------------------ | axiomatics +------------------------------------------------------------ axiomatics When a mathematician has no more ideas, he pursues axiomatics. -- Felix Klein +------------------------------------------------------------ | turbulence +------------------------------------------------------------ turbulence The paper "On the nature of turbulence" with F. Takens was eventually published in a scientific journal. (Actually, I was an editor of the journal, and I accepted the paper by myself for publication. This is not a recommended procedure in general, but I felt that it was justified in this particular case). -- D. Ruelle, in Chance and Chaos +------------------------------------------------------------ | hairy-ball +------------------------------------------------------------ hairy-ball A good topological theorem to mention any time is the theorem which, in essence, states that however you try to comb the hair on a hairy ball, you can never do it smoothly - the so-called 'hairy-ball' theorem. You can make snide comments about the grooming of the hosts' dog or cat in the meantime as you pick hairs off your trouser leg. -- R. Ainsley in Bluff your way in Maths, 1988 +------------------------------------------------------------ | large +------------------------------------------------------------ large LARGE NUMBERS: (10^n means that 10 is raised to the n'th power) 10^4 One "myriad". The largest numbers, the Greeks were considering. 10^5 The largest number considered by the Romans. 10^10 The age of our universe in years. 10^22 Distance to our neighbor galaxy Andromeda in meters. 10^23 Number of atoms in two gram Carbon (Avogadro). 10^26 Size of universe in meters. 10^41 Mass of our home galaxy "milky way" in kg. 10^51 Archimedes's estimate of number of sand grains in universe. 10^52 Mass of our universe in kg. 10^80 The number of atoms in our universe. 10^100 One "googol". (Name coined by 9 year old nephew of E. Kasner). 10^153 Number mentioned in a myth about Buddha. 10^155 Size of ninth Fermat number (factored in 1990). 10^(10^6) Size of large prime number (Mersenne number, Nov 1996). 10^(10^7) Years, ape needs to write "hound of Baskerville" (random typing). 10^(10^(33)) Inverse is chance that a can of beer tips by quantum fluctuation. 10^(10^(42)) Inverse is probability that a mouse survives on sun for a week. 10^(10^51)) Inverse is chance to find yourself on Mars (quantum fluctuations) 10^(10^100) One "Gogoolplex", Decimal expansion can not exist in universe. -- from R.E. Crandall, Scient. Amer., Feb. 1997 +------------------------------------------------------------ | analytic +------------------------------------------------------------ analytic The statement sometimes made, that there exist only analytic functions in nature, is in my opinion absurd. -- F. Klein, Lectures on Mathematics, 1893 +------------------------------------------------------------ | violence +------------------------------------------------------------ violence The introduction of numbers as coordinates ... is an act of violence... -- H. Weyl, Philosophy of Mathematics and Natural Science 1949 +------------------------------------------------------------ | beauty +------------------------------------------------------------ beauty Mathematics possesses not only truth but supreme beauty - a beauty cold and austere, like that of a sculpture -- Bertrand Russell +------------------------------------------------------------ | geometry +------------------------------------------------------------ geometry Geometry is magic that works... -- R. Thom. Stability Structurelle et Morphogenese, 1972 +------------------------------------------------------------ | Zermelo +------------------------------------------------------------ Zermelo Ernst Zermelo, who created a system of axioms for set theory, was a Privatdozent at Goettingen when Herr Geheimrat Felix Klein held sway over the fabled mathematics department. As Pauli told it, "Zermelo taught a course on mathematical logic and stunned his students by posing the following question: All mathematicians in Goettingen belong to one of two classes. In the first class belong those mathematicians who do what Felix Klein likes, but what they dislike. In the second class are those mathematicians who do what Felix Klein likes, but what they dislike. To what class does Felix Klein belong?" Jordan, having listened intently, broke into roaring laughter. Pauli paused, took a sip of wine and said disapprovingly, "Herr Jordan, you have laughed too soon". He continued: "None of the awed students could solve this blasphemous problem. Zermelo then crowed in his high-pitched voice, 'But, meine Herren, it's very simple. Felix Klein isn't a mathematician.'" Jordan laughed again. Pauli drained his wine glass approvingly and concluded with "Zermelo was not offered a professorship at Goettingen". -- E.L. Schucking, in 'Jordan, Pauli,Politics, Brecht and a variable gravitational constant' Physics Today, Oct. 1999 +------------------------------------------------------------ | Conway +------------------------------------------------------------ Conway In the beginning, everything was void, and J.H.W.H.Conway began to create numbers. Conway said, "Let there be two rules which bring forth all numbers large and small. This shall be the first rule: Every number corresponds to two sets of previously created numbers, such that no member of the left set is greater than or equal to any member of the right set. And the second rule shall be this: One number is less than or equal to another number if and only if no member of the first number's left set is greater than or equal to the second number, and no member of the second number 's right set is less than or equal to the first number." And Conway examined these two rules he had made, and behold! they were very good. And the first number was created from the void left set and the void right set. Conway called this number "zero", and said that it shall be a sign to separate positive numbers from negative numbers. Conway proved that zero was less than or equal to zero, and he saw that it was good. And the evening and the morning were the day of zero. On the next day, two more numbers were created, one with zero as its left set and one with zero as its right set. And Conway called the former number "one", and the latter he called "minus one". And he proved that minus one is less than but not equal to zero and zero is less than but not equal to one. And the evening... -- D. Knuth, Surreal numbers, 1979 +------------------------------------------------------------ | obvious +------------------------------------------------------------ obvious Mathematics consists essentially of : a) proving the obvious b) proving the not so obvious c) proving the obviously untrue For example, it took mathematicians until the 1800'ies to prove that 1+1=2 and not before the late 1970 were they confident of proving that any map requires no more than four colors to make it look nice, a fact known by cartographers for centuries. There are many not-so-obvious things which can be proved true too. Like the fact that for any group of 23 people, there is an even chance two or more of them share birthdays. (With groups of twins this becomes almost certain. Not quite certain as you will of course point out: they might all have been born either side of midnight). Mathematicians are also fond of proving things which are obviously false, like all straight lines being curved, and an engaged telephone being just as likely to be free if you ring again immediately after, as if you wait twenty minutes. -- R. Ainsley in Bluff your way in Maths, 1988 +------------------------------------------------------------ | infimum +------------------------------------------------------------ infimum There exists a subset of the real line such that the infimum of the set is greater then the supremum of the set. -- Gary L. Wise and Eric B. Hall, Counter examples in probability and real analysis, 1993, First Example in book +------------------------------------------------------------ | transcendental +------------------------------------------------------------ transcendental Transcendental number : A number which is not the root of any polynomial equation, like pi and e, and which can only be understood after several hours meditation in the lotus position. -- R. Ainsley in Bluff your way in Maths, 1988 +------------------------------------------------------------ | illiteracy +------------------------------------------------------------ illiteracy There are great advantages to being a mathematician: a) you do not have to be able to spell b) you do not have to be able to add up The illiteracy of mathematicians is taken for granted. There still persists a myth that mathematics somehow involves numbers. Many fondly believe that university students spend their time long dividing by 173 and learning their 39 times table; in fact, the reverse is true. Mathematicians are renowned for their inability to add up or take away, in much the same way as geographers are always getting lost, and economists are always borrowing money off you. -- R. Ainsley in Bluff your way in Maths, 1988 +------------------------------------------------------------ | prime +------------------------------------------------------------ prime In this note we would like to offer an elementary 'topological' proof of the infinitude of the prime numbers. We introduce a topology into the space of integers S, by using the arithmetic progressions (from -infinity to +infinity) as a basis. It is not difficult to verify that this actually yields a topological space. In fact, under this topology, S may be shown to be normal and hence metrisable. Each arithmetic progression is closed as well as open, since its complement is the union of the other arithmetic progressions (having the same difference). As a result, the union of any finite number of arithmetic progressions is closed. Consider now the set A which is the union of A(p), where A(p) consists of primes greater or equal to p. The only numbers not belonging to A are -1 and 1, and since the set -1,1 is clearly not an open set, A cannot be closed. Hence A is not a finite union of closed sets, which proves that there is an infinity of primes. -- H. Fuerstenberg, On the infinitude of primes, American Mathematical Montly, 62, 1955, p. 353 +------------------------------------------------------------ | barber +------------------------------------------------------------ barber The barber in a certain town shaves all the people who don't shave themselves. Who shaves the barber? This is meant to be a clever little paradox with no solution but you can annoy the asker intensely by saying it's easy and that the barber is a women. You can then ask the following (a version of Russell's Paradox, - point this out too): in a library there are some books for the catalogue section which is a list of all books which don't list themselves. Shold he or she include this book in its own list? If so, then it becomes a book which lists itself, so it shouldn't be in the list of books which don't and vice versa. This should keep the most determined assailant at bay while you attack the wine. -- R. Ainsley in Bluff your way in Maths, 1988 +------------------------------------------------------------ | Hadamard +------------------------------------------------------------ Hadamard Hadamard, trying to find a job in a US university, came to a small university and was received by the chairman of the department of mathematics. He explained who he was and gave his curriculum vitae. The chairman said: 'our means are very limited and I can not promise that we shal take you'. Then Hadamard noticed that among the portraits on he wall was his own. 'That's me!' he said. 'Well, come again in a week, we shal think about this'. On his next visit, the answer was negative and his portrait had been removed. -- Vladimir Mazya and Tatyana Shaposhnikova, in Jacques Hadamard, a universal Mathematician, AMS History of Mathematics Volume 14 +------------------------------------------------------------ | Cantor +------------------------------------------------------------ Cantor The appropriate object is known as the Cantor set, because it was discovered by Henry Smith in 1875. (The founder of set theory, Georg Cantor, used Smith's invention in 1883. Let's fact it, 'Smith set' isn't very impressive, is it?) -- Ian Stewart, in Does God Play Dice, 1989 p. 121 +------------------------------------------------------------ | jouissance +------------------------------------------------------------ jouissance ... Thus the erectile organ comes to symbolize the place of jouissance, not in itself, or even in the form of an image, but as a part lacking in the desired image: that is why it is equivalent to the (-1)^(1/2) of the signification produced above, of the Jouissance that it restores by the coefficient of its statement to the function of lack of signifier (-1). -- Lacan, Ecrits, Paris 1966 (cited in 'Fashionable nonsense' by Alan Sokal and Jean Bricmont) +------------------------------------------------------------ | Mandelbrot +------------------------------------------------------------ Mandelbrot Mandelbrot made quite good computer pictures, which seemed to show a number of isolated "islands" (in the Mandelbrot set M). Therefore, he conjectured that the set M has many distinct connected components. (The editors of the journal thought that his islands were specks of dirt, and carefully removed them from the pictures). -- John Milnor, in Dynamics in one complex variable, 1991 +------------------------------------------------------------ | sin +------------------------------------------------------------ sin sin, cos, tan, cot, sec, cosec - Formulae derived from the sides of triangles but which crop up in completely unexpected places. Sins are extremely common, but rarely do you encounter secs in mathematics. -- R. Ainsley in Bluff your way in Maths, 1988 +------------------------------------------------------------ | Moser +------------------------------------------------------------ Moser This reminds me of the Hilbert story, which I learned from my teacher Franz Rellich in Goettingen: When Hilbert - who was old and retired - was asked at a party by the newly appointed Nazi-minister of education: "Herr Geheimrat, how is mathematics in Goettingen, now that it has been freed of the Jewish influences" he replied: "Mathematics in Goettingen? That does not EXIST anymore". -- Jurgen Moser, in Dynamical Systems-Past and Present, Doc. Math. J. DMV I p. 381-402, 1998 +------------------------------------------------------------ | wine +------------------------------------------------------------ wine There are two glasses of wine, one white and one red. A teaspoonful of wine is taken from the red and mixed in with the white. Then a teaspoonful of this mixture is taken and mixed in with the red. Which is bigger, the amount of red in the white or the amount of white in the red? The answer is that the're both the same, because there's the same volume in each glass, so whatever quantity of red is in the white must be equal to the quantity of white in the red. However in practice it is impossible to do this because the white always runs out first at parties and the red always gets spilt on someone's white trousers. -- R. Ainsley in Bluff your way in Maths, 1988 +------------------------------------------------------------ | Monty-Hall +------------------------------------------------------------ Monty-Hall "Suppose you're on a game show and you are given a choice of three doors. Behind one door is a car and behind the others are goats. You pick a door-say No. 1 - and the host, who knows what's behind the doors, opens another door-say, No. 3-which has a goat. (In all games, the host opens a door to reveal a goat). He then says to you, "Do you want to pick door No. 2?" (In all games he always offers an option to switch). Is it to your advantage to switch your choice?" -- The three doors problem, also known as Monty-Hall Problem +------------------------------------------------------------ | sex +------------------------------------------------------------ sex Pure mathematician - Anyone who prefers set theory to sex. -- R. Ainsley in Bluff your way in Maths, 1988 +------------------------------------------------------------ | mad +------------------------------------------------------------ mad There was a mad scientist ( a mad ...social... scientist ) who kidnaped three colleagues, an engineer, a physicist, and a mathematician, and locked each of them in separate cells with plenty of canned food and water but no can opener. A month later, returning, the mad scientist went to the engineer's cell and found it long empty. The engineer had constructed a can opener from pocket trash, used aluminum shavings and dried sugar to make an explosive, and escaped. The physicist had worked out the angle necessary to knock the lids off the tin cans by throwing them against the wall. She was developing a good pitching arm and a new quantum theory. The mathematician had stacked the unopened cans into a surprising solution to the kissing problem; his dessicated corpse was propped calmly against a wall, and this was inscribed on the floor in blood: Theorem: If I can't open these cans, I'll die. Proof: assume the opposite... +------------------------------------------------------------ | induction +------------------------------------------------------------ induction Proof by induction - A very important and powerful mathematical tool, because it works by assuming something is true and then goes on to prove that therefore it is true. Not surprisingly, you can prove almost everything by induction. So long as the proof includes the following phrases: a) Assume true for n; then also true for n+1 because.. (followed by some plausible but messy working out in which n, n+1 appear prominently). b) But is true for n=0 (a little more messy working out with lots of zeros sprayed at random through the proof). c) So is true for all n. Q.E.D. -- R. Ainsley in Bluff your way in Maths, 1988 +------------------------------------------------------------ | horse +------------------------------------------------------------ horse LEMMA: All horses are the same color. Proof (by induction): Case n=1: In a set with only one horse, it is obvious that all horses in that set are the same color. Case n=k: Suppose you have a set of k+1 horses. Pull one of these horses out of the set, so that you have k horses. Suppose that all of these horses are the same color. Now put back the horse that you took out, and pull out a different one. Suppose that all of the k horses now in the set are the same color. Then the set of k+1 horses are all the same color. We have k true => k+1 true; therefore all horses are the same color. THEOREM: All horses have an infinite number of legs. Proof (by intimidation): Everyone would agree that all horses have an even number of legs. It is also well-known that horses have fore-legs in front and two legs in back. But 4 + 2 = 6 legs is certainly an odd number of legs for a horse to have! Now the only number that is both even and odd is infinity; therefore all horses have an infinite number of legs. However, suppose that there is a horse somewhere that does not have an infinite number of legs. Well, that would be a horse of a different color; and by the Lemma, it doesn't exist. QED +------------------------------------------------------------ | dean +------------------------------------------------------------ dean Dean, to the physics department. "Why do I always have to give you guys so much money, for laboratories and expensive equipment and stuff. Why couldn't you be like the maths department - all they need is money for pencils, paper and waste-paper baskets. Or even better, like the philosophy department. All they need are pencils and paper." +------------------------------------------------------------ | astronomer +------------------------------------------------------------ astronomer An astronomer, a physicist and a mathematician were holidaying in Scotland. Glancing from a train window, they observed a black sheep in the middle of a field. "How interesting," observed the astronomer, "all Scottish sheep are black!" To which the physicist responded, "No, no! Some Scottish sheep are black!" The mathematician gazed heavenward in supplication, and then intoned, "In Scotland there exists at least one field, containing at least one sheep, at least one side of which is black." -- J. Steward in 'Concepts of Modern Mathematics' +------------------------------------------------------------ | coffee +------------------------------------------------------------ coffee An engineer, a chemist and a mathematician are staying in three adjoining cabins at an old motel. First the engineer's coffee maker catches fire. He smells the smoke, wakes up, unplugs the coffee maker, throws it out the window, and goes back to sleep. Later that night the chemist smells smoke too. He wakes up and sees that a cigarette butt has set the trash can on fire. He says to himself, "Hmm. How does one put out a fire? One can reduce the temperature of the fuel below the flash point, isolate the burning material from oxygen, or both. This could be accomplished by applying water." So he picks up the trash can, puts it in the shower stall, turns on the water, and, when the fire is out, goes back to sleep. The mathematician, of course, has been watching all this out the window. So later, when he finds that his pipe ashes have set the bed-sheet on fire, he is not in the least taken aback. He says: "Aha! A solution exists!" and goes back to sleep. +------------------------------------------------------------ | logs +------------------------------------------------------------ logs Taking logs - Broadly speaking, any equation which looks difficult will look much easier when logs are taken on both sides. Taking logs on one side only is tempting for many equations, but may be noticed. -- R. Ainsley in Bluff your way in Maths, 1988 +------------------------------------------------------------ | cat +------------------------------------------------------------ cat Theorem: A cat has nine tails. Proof: No cat has eight tails. A cat has one tail more than no cat. Therefore, a cat has nine tails. +------------------------------------------------------------ | chocolate +------------------------------------------------------------ chocolate Prime number - A number with no divisors. Boxes of chocolates always contain a prime number so that, whatever the number of people present, somebody has to have that one left over. -- R. Ainsley in Bluff your way in Maths, 1988 +------------------------------------------------------------ | aleph +------------------------------------------------------------ aleph Aleph-null bottles of beer on the wall, Aleph-null bottles of beer, You take one down, and pass it around, Aleph-null bottles of beer on the wall. +------------------------------------------------------------ | qed +------------------------------------------------------------ qed At the end of a proof you write Q.E.D, which stands not for Quod Erat Demonstrandum as the books would have you believe, but for Quite Easily Done. -- R. Ainsley in Bluff your way in Maths, 1988 +------------------------------------------------------------ | 1+1 +------------------------------------------------------------ 1+1 1+1 = 3, for large values of 1 +------------------------------------------------------------ | painting +------------------------------------------------------------ painting Group theory - An exceedingly beautiful branch of pure mathematics used for showing in how many ways blocks of wood can be painted. -- R. Ainsley in Bluff your way in Maths, 1988 +------------------------------------------------------------ | engeneer +------------------------------------------------------------ engeneer Mathematician: 3 is prime,5 is prime,7 is prime, by induction - every odd integer higher than 2 is prime. Physicist: 3 is prime, 5 is prime, 7 is prime, 9 is an experimental error, 11 is prime,... Engineer: 3 is prime, 5 is prime, 7 is prime, 9 is prime, 11 is prime,... Programmer: 3's prime, 5's prime, 7's prime, 7's prime, 7's prime,... Salesperson: 3 is prime, 5 is prime, 7 is prime, 9 -- we'll do for you the best we can,... Software seller: 3 is prime, 5 is prime, 7 is prime, 9 will be prime in the next release,... Biologist: 3 is prime, 5 is prime, 7 is prime, 9 -- results have not arrived yet,... Advertiser: 3 is prime, 5 is prime, 7 is prime, 11 is prime,... Lawyer: 3 is prime, 5 is prime, 7 is prime, 9 -- there is not enough evidence to prove that it is not prime,... Accountant: 3 is prime, 5 is prime, 7 is prime, 9 is prime, deducing 10 percent tax and 5 percent other obligations. Statistician: Let's try several randomly chosen numbers: 17 is prime, 23 is prime, 11 is prime... Psychologist: 3 is prime, 5 is prime, 7 is prime, 9 is prime but tries to suppress it,... +------------------------------------------------------------ | pi +------------------------------------------------------------ pi PI= 3.14159265358979323846264338327950288419716939937510582097494459230781640628 +------------------------------------------------------------ | e +------------------------------------------------------------ e Euler E= 2.71828182845904523536028747135266249775724709369995957496696762772407663035 +------------------------------------------------------------ | cancel +------------------------------------------------------------ cancel THEOREM: The limit as n goes to infinity of sin x/n is 6. PROOF: cancel the n in the numerator and denominator. +------------------------------------------------------------ | coffee +------------------------------------------------------------ coffee A mathematician is a device for turning coffee into theorems. -- P. Erdos +------------------------------------------------------------ | stupider +------------------------------------------------------------ stupider Finally I am becoming stupider no more. -- Epitaph, P. Erdos wrote for himself +------------------------------------------------------------ | Erdoes +------------------------------------------------------------ Erdoes epsilon child bosses women slaves men captured married liberated divorced recaptured remarried trivial beings nonmathematicians noise music poison alcohol preaching giving a lecture supreme fascist god died stopped doing mathematics preach lecture Joedom UDSSR Samland USA on the long wave length communists on the short wave length fashists -- from the vocabulary of P. Erdos 'the man who loved only numbers' +------------------------------------------------------------ | Chebyshev +------------------------------------------------------------ Chebyshev Chebyshev said it, and I say it again There is always a prime between n and 2n -- P. Erdos +------------------------------------------------------------ | Outrage +------------------------------------------------------------ Outrage Outrage, disgust, the characterization of group theory as a plague or as a dragon to be slain - this is not an atypical physist's reaction in the 1930s-50s to the use of group theory in physics. -- S. Sternberg +------------------------------------------------------------ | digits +------------------------------------------------------------ digits Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin. -- J. von Neumann +------------------------------------------------------------ | poet +------------------------------------------------------------ poet The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas, like the colors or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics... It may be very hard to define mathematical beauty, but that is just as true of beauty of any kind - we may not know quite what we mean by a beautiful poem, but that does not prevent us from recognizing one when we read it. -- G.H. Hardy +------------------------------------------------------------ | melancholy +------------------------------------------------------------ melancholy It is a melancholy experience for a professional mathematician to find himself writing about mathematics. -- G.H. Hardy +------------------------------------------------------------ | Hilbert +------------------------------------------------------------ Hilbert There is a much quoted story about David Hilbert, who one day noticed that a certain student had stopped attending class. When told that the student had decided to drop mathematics to become a poet, Hilbert replied, "Good- he did not have enough imagination to become a mathematician". -- R. Osserman +------------------------------------------------------------ | refreree +------------------------------------------------------------ refreree Referee's report: This paper contains much that is new and much that is true. Unfortunately, that which is true is not new and that which is new is not true. -- H. Eves 'Return to Mathematical Circles', 1988. +------------------------------------------------------------ | weapons +------------------------------------------------------------ weapons Structures are the weapons of the mathematician. -- N. Bourbaki +------------------------------------------------------------ | undogmatic +------------------------------------------------------------ undogmatic Mathematics is the only instructional material that can be presented in an entirely undogmatic way. -- M. Dehn +------------------------------------------------------------ | solve +------------------------------------------------------------ solve Each problem that I solved became a rule which served afterwards to solve other problems -- R. Decartes +------------------------------------------------------------ | tool +------------------------------------------------------------ tool For a physicist mathematics is not just a tool by means of which phenomena can be calculated, it is the main source of concepts and principles by means of which new theories can be created. -- F. Dyson +------------------------------------------------------------ | sheet +------------------------------------------------------------ sheet If the entire Mandelbrot set were placed on an ordinary sheet of paper, the tiny sections of boundary we examine would not fill the width of a hydrogen atom. Physicists think about such tiny objects; only mathematicians have microscopes fine enough to actually observe them. -- J. Eving +------------------------------------------------------------ | recommendation +------------------------------------------------------------ recommendation Sample letter of recommendation: Dear Search Committee Chair, I am writing this letter for Mr. Still Student who has applied for a position in your department. I should start by saying that I cannot recommend him too highly. In fact, there is no other student with whom I can adequately compare him, and I am sure that the amount of mathematics he knows will surprise you. His dissertation is the sort of work you don't expect to see these days. It definitely demonstrates his complete capabilities. In closing, let me say that you will be fortunate if you can get him to work for you. Sincerely, A. D. Advisor (Prof.) -- from MAA Focus Newsletter +------------------------------------------------------------ | cube +------------------------------------------------------------ cube To divide a cube into two other cubes, a fourth power or in general any power whatever into two powers of the same denomination above the second is impossible, and I have assuredly found an admirable proof of this, but the margin is too narrow to contain it. -- P. de Fermat +------------------------------------------------------------ | reality +------------------------------------------------------------ reality Mathematics is not only real, but it is the only reality. That is that entire universe is made of matter, obviously. And matter is made of particles. It's made of electrons and neutrons and protons. So the entire universe is made out of particles. Now what are the particles made out of? They're not made out of anything. The only thing you can say about the reality of an electron is to cite its mathematical properties. So there's a sense in which matter has completely dissolved and what is left is just a mathematical structure. -- M. Gardner +------------------------------------------------------------ | arithmetic +------------------------------------------------------------ arithmetic God does arithmetic. -- K.F. Gauss +------------------------------------------------------------ | hypothesis +------------------------------------------------------------ hypothesis Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis? -- P.R. Halmos +------------------------------------------------------------ | dice +------------------------------------------------------------ dice God not only plays dice. He also sometimes throws the dice where they cannot be seen. -- S.W. Hawking +------------------------------------------------------------ | wissen +------------------------------------------------------------ wissen 'Wir muessen wissen. Wir werden wissen.' (We have to know. We will know.) -- D. Hilbert (engraved in tombstone) +------------------------------------------------------------ | physics +------------------------------------------------------------ physics Physics is much too hard for physicists. -- D. Hilbert +------------------------------------------------------------ | Hofstadter +------------------------------------------------------------ Hofstadter Hofstadter's Law: It always takes longer than you expect, even when you take into account Hofstadter's Law. -- D.R. Hofstadter, Goedel-Escher-Bach +------------------------------------------------------------ | experience +------------------------------------------------------------ experience The science of mathematics presents the most brilliant example of how pure reason may successfully enlarge its domain without the aid of experience. -- E. Kant +------------------------------------------------------------ | doughnut +------------------------------------------------------------ doughnut A topologist is one who doesn't know the difference between a doughnut and a coffee cup. -- J. Kelley +------------------------------------------------------------ | Kovalevsky +------------------------------------------------------------ Kovalevsky Say what you know, do what you must, come what may. -- S. Kovalevsky +------------------------------------------------------------ | god +------------------------------------------------------------ god God made the integers, all else is the work of man. -- L. Kronecker +------------------------------------------------------------ | abstract +------------------------------------------------------------ abstract There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world. -- N. Lobatchevsky +------------------------------------------------------------ | medicine +------------------------------------------------------------ medicine Medicine makes people ill, mathematics make them sad and theology makes them sinful. -- M. Luther +------------------------------------------------------------ | intelligence +------------------------------------------------------------ intelligence The mathematician who pursues his studies without clear views of this matter, must often have the uncomfortable feeling that his paper and pencil surpass him in intelligence. -- E. Mach +------------------------------------------------------------ | flesh +------------------------------------------------------------ flesh I tell them that if they will occupy themselves with the study of mathematics they will find in it the best remedy against the lusts of the flesh. -- T. Mann +------------------------------------------------------------ | philosophers +------------------------------------------------------------ philosophers Today, it is not only that our kings do not know mathematics, but our philosophers do not know mathematics and - to go a step further - our mathematicians do not know mathematics. -- J.R. Oppenheimer +------------------------------------------------------------ | obvious +------------------------------------------------------------ obvious Mathematics consists of proving the most obvious thing in the least obvious way. -- G. Polya +------------------------------------------------------------ | whispers +------------------------------------------------------------ whispers However successful the theory of a four dimensional world may be, it is difficult to ignore a voice inside us which whispers: "At the back of your mind, you know a fourth dimension is all nonsense". I fancy that voice must have had a busy time in the past history of physics. What nonsense to say that this solid table on which I am writing is a collection of electrons moving with prodigious speed in empty spaces, which relative to electronic dimensions are as wide as the spaces between the planets in the solar system! What nonsense to say that the thin air is trying to cursh my body with a load of 14 lbs. to the square inch! What nonsense that the star cluster which I see through the telescope, obviously there NOW, is a glimpse into a past age 50'000 years ago! Let us not be beguiled by this voice. It is discredited... -- Sir Arthur Eddington +------------------------------------------------------------ | decimal +------------------------------------------------------------ decimal The first million decimal places of pi are comprised of: 99959 0's 99758 1's 100026 2's 100229 3's 100230 4's 100359 5's 99548 6's 99800 7's 99985 8's 100106 9's --David Blatner, the joy of pi +------------------------------------------------------------ | historians +------------------------------------------------------------ historians Math historians often state that the Egyptians thought pi = 256/81. In fact, there is no direct evidence that the Egyptians conceived of a constant number pi, much less tried to calculate it. Rather, they were simply interested in finding the relationship between the circle and the square, probably to accomplish the task of precisely measuring land and buildings. --David Blatner, the joy of pi +------------------------------------------------------------ | pi +------------------------------------------------------------ pi 2000 BC Babilonians use pi=25/8, Egyptians use pi=256/81 1100 BC Chinese use pi=3 200 AC Ptolemy uses pi=377/120 450 Tsu Ch'ung-chih uses pi=255/113 530 Aryabhata uses pi=62832/20000 650 Brahmagupta uses pi=sqrt(10) 1593 Romanus finds pi to 15 decimal places 1596 Van Ceulen calculates pi to 32 places 1699 Sharp calculates pi to 72 places 1719 Tantet de Lagny calculates pi to 127 places 1794 Vega calculates pi to 140 decimal places 1855 Richter calculates pi to 500 decimal places 1873 Shanks finds 527 decimal places 1947 Ferguson calculates 808 places 1949 ENIAC computer finds 2037 places 1955 NORC computer computes 3089 places 1959 IBM 704 computer finds 16167 places 1961 Shanks-Wrench (IBM7090) find 100200 places 1966 IBM 7030 computes 250000 places 1967 CDC6600 computes 500000 places 1973 Guilloud-Bouyer (CDC7600) find 1 Mio places 1983 Tamura-Kanada (HITACM-280H) compute 16 Mio places 1988 Kanada (HITAC M-280H) computes 16 Mio digits 1989 Chudnovsky finds 1000 Mio digits 1995 Kanada computes pi to 6000 Mio digits 1996 Chudnovsky computes pi to 8000 Mio digits 1997 Kanada determines pi to 51000 Mio digits --David Blatner, the joy of pi +------------------------------------------------------------ | FBI +------------------------------------------------------------ FBI The following is a transcript of an interchange between defence attorney Robert Blasier and FBI Special Agent Roger Martz on July 26, 1995, in the courtroom of the O.J. Simpson trial: Q: Can you calculate the area of a circle with a five-millimeter diameter? A: I mean I could. I don't...math I don't ... I don't know right now what it is. Q: Well, what is the formula for the area of a circle? A: Pi R Squared Q: What is pi? A: Boy, you ar really testing me. 2.12... 2.17... Judge Ito: How about 3.1214? Q: Isn't pi kind of essential to being a scientist knowing what it is? A: I haven't used pi since I guess I was in high school. Q: Let's try 3.12. A: Is that what it is? There is an easier way to do... Q: Let's try 3.14. And what is the radius? A: It would be half the diameter: 2.5 Q: 2.5 squared, right? A: Right. Q: Your honor, may we borrow a calculator? pause Q: Can you use a calculator? A: Yes, I think. Q: Tell me what pi times 2.5 squared is. A: 19 Q: Do you want to write down 19? Square millimeters, right? The area. What is one tenth of that? A: 1.9 Q: You miscalculated by a factor of two, the size, the minimum size of a swatch you needed to detect EDTA didn't you? A: I don't know that I did or not. I calculated a little differently. I didn't use this. Q: Well, does the area change by the different method of calculation? A: Well, this is all estimations based on my eyeball. I didn't use any scientific math to determine it. --David Blatner, the joy of pi +------------------------------------------------------------ | beauty +------------------------------------------------------------ beauty To those who do not know Mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty of nature. ... If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in. -- Richard Feynman in "The Character of Physical Law" +------------------------------------------------------------ | Bacon +------------------------------------------------------------ Bacon All science requires Mathematics. The knowledge of mathematical things is almost innate in us... This is the easiest of sciences, a fact which is obvious in that no one?s brain rejects it; for laymen and people who are utterly illiterate know how to count and reckon. -- Roger Bacon +------------------------------------------------------------ | deductions +------------------------------------------------------------ deductions Pure mathematics consists entirely of such asseverations as that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing... It's essential not to discuss whether the proposition is really true, and not to mention what the anything is of which it is supposed to be true... If our hypothesis is about anything and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. -- Bertrand Russell +------------------------------------------------------------ | ambitious +------------------------------------------------------------ ambitious The more ambitious plan may have more chances of success -- G. Polya, How To Solve It +------------------------------------------------------------ | fourteen +------------------------------------------------------------ fourteen THEOREM: Every natural number can be completely and unambiguously identified in fourteen words or less. PROOF: 1. Suppose there is some natural number which cannot be unambiguously described in fourteen words or less. 2. Then there must be a smallest such number. Let's call it n. 3. But now n is "the smallest natural number that cannot be unambiguously described in fourteen words or less". 4. This is a complete and unambiguous description of n in fourteen words, contradicting the fact that n was supposed not to have such a description! 5. Since the assumption (step 1) of the existence of a natural number that cannot be unambiguously described in fourteen words or less led to a contradiction, it must be an incorrect assumption. 6.Therefore, all natural numbers can be unambiguously described in fourteen words or less! +------------------------------------------------------------ | 1=2 +------------------------------------------------------------ 1=2 THEOREM: 1=2 PROOF: 1. Let a=b. 2. Then a^2 = ab, 3. a^2 + a^2 = a^2 + ab, 4. 2 a^2 = a^2 + ab, 5. 2 a^2 - 2 ab = a^2 + ab - 2 ab, 6. and 2 a^2 - 2 ab = a^2 - ab 7. Writing this as 2 (a^2 - a b) = 1 (a^2 - a b), 8. and cancelling the (a^2 - ab) from both sides gives 1=2. +------------------------------------------------------------ | primes +------------------------------------------------------------ primes II III V VII XI XIII XVII XIX XXIII XXIX ... +------------------------------------------------------------ | Queen +------------------------------------------------------------ Queen "Can you do addition?" the White Queen asked. "What's one and one and one and one and one and one and one and one and one and one?" "I don't know," said Alice, "I lost count.". -- Lewis Carrol alias Charles Lutwidge Dodgson, Alice's Adventures in Wonderland +------------------------------------------------------------ | subtraction +------------------------------------------------------------ subtraction "She can't do Subtraction", said the White Queen. "Can you do Division? Divide a loaf by a knife -- what's the answer to that?" "I suppose --" Alice was beginning, but the Red Queen answerd for her. "Bread and butter, of course ..." -- Lewis Carrol alias Charles Lutwidge Dodgson, Alice's Adventures in Wonderland +------------------------------------------------------------ | subtraction +------------------------------------------------------------ Theorem: the square root x of 2 is irrational. Proof: x=n/m with gcd(n,m)=1 implies 2=n^2/m^2 which is 2 m^2=n^2 so that n must be even and n^2 a multiple of 4. Therefore m is even. This contradicts gcd(n,m)=1. +------------------------------------------------------------ | blackboard +------------------------------------------------------------ blackboard It is still an unending source of surprise for me to see how a few scribbles on a blackboard or on a sheet of paper could change the course of human affairs. -- Stanislaw Ulam. +------------------------------------------------------------ | ephermeral +------------------------------------------------------------ ephermeral Of all escapes from reality, mathematics is the most successful ever. It is a fantasy that becomes all the more addictive because it works back to improve the same reality we are trying to evade. All other escapes- sex, drugs, hobbies, whatever - are ephemeral by comparison. The mathematician's feeling of triumph, as he forces the world to obey the laws his imagimation has created, feeds on its own success. The world is premanently changed by the workings of his mind, and the certainty that his creations will endure renews his confidence as no other pursuit. -- Gian-Carlo Rota +------------------------------------------------------------ | joke +------------------------------------------------------------ joke A good mathematical joke is better, and better mathematics than a dozen mediocre papers. -- John Edensor Littlewood +------------------------------------------------------------ | Leibniz +------------------------------------------------------------ Leibniz pi/4 = 1-1/3+1/5-1/7+1/9 .... -- Wilhelm von Leibniz +------------------------------------------------------------ | war +------------------------------------------------------------ war It has been said that the First World War was the chemists' war because mustard gas and chlorine were empolyed for the first time, and that the Second World War was the physicists war, because the atom bomb was detonated. Similarly, it has been argued that the Third World War would be the mathematicians' war, because mathematics will have control over the next great weapon of war - information. -- Simon Singh, in 'The code book' +------------------------------------------------------------ | clearly +------------------------------------------------------------ clearly Never speak more clearly than you think. -- Jeremy Bernstein +------------------------------------------------------------ | Piaget +------------------------------------------------------------ Piaget What, in effect are the conditions for the construction of formal thought? The child must not only apply operations to objects - in other words, mentally execute possible actions on them - he must also 'reflect' those operations in the absence of the objects which are replaced by pure propositions. Thus 'reflection' is thought raised to the second power. Concrete thinking is the representation of a possible action, and formal thinking is the representation of a representation of possible action... It is not surprising, therefore, that the system of concrete operations must be completed during the last years of childhood before it can be 'reflected' by formal operations. In terms of their function, formal operations do not differ from concrete operations except that they are applied to hypotheses or propositions whose logic is an abstract translation of the system of 'inference' that governs concrete operations. -- Jean Piaget +------------------------------------------------------------ | Mersenne +------------------------------------------------------------ Mersenne An integer 2^n-1 is called a Mersenne number. If it is prime, it is called a Mersenne prime. In that case, n must be prime. Known examples are n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377. It is not known whether there are infinitely many Mersenne primes. +------------------------------------------------------------ | Mersenne +------------------------------------------------------------ A positive integer n is called a perfect number if it is equal to the sum of all of its positive divisors, excluding n itself. Examples are 6=1+2+3, 28=1+2+4+7+14. An integer k is an even perfect number if and only if it has the form 2^(n-1)(2^n-1) and 2^n-1 is prime. In that case 2^n-1 is called a Mersenne prime and n must be prime. It is unknown whether there exists an odd perfect number. +------------------------------------------------------------ | Wilson +------------------------------------------------------------ Wilson WILSON'S THEOREM: p prime if and only if (p-1)!==-1 ( mod p) PROOF. 1,2, ..., p-1 are roots of x^(p-1)==0 ( mod p). A congruence has not more roots then its degree, hence x^(p-1) -1 == (x-1)(x-2) ... (x-(p-1)) mod p. For x=0, this gives -1 == (-1)^(p-1) (p-1)! == (p-1)! which is also true for p=2. -- from P. Ribenboim, 'The new book of prime number records' +------------------------------------------------------------ | twin +------------------------------------------------------------ twin There is keen competition to produce the largest pair of twin primes. On October 9, 1995, Dubner discovered the largest known pair of twin primes p,p+2, where p=570918348*10^5120 - 1. It took only one day with 2 crunchers. The expected time would be 150 times longer! What luck! -- from P. Ribenboim, 'The new book of prime number records' +------------------------------------------------------------ | lion +------------------------------------------------------------ lion How to catch a lion: THE HILBERT METHOD. Place a locked cage in the desert. Set up the following axiomatic system. (i) The set of lions is non-empty (ii) If there is a lion in the desert, then there is a lion in the cage. Theorem. There is a lion in the cage THE PEANO METHOD. There is a space-filling curve passing through every point of the desert. Such a curve may be traversed in as short a time as we please. Armed with a spear, traverse the curve faster than the lion can move his own length. THE TOPOLOGICAL METHOD. The lion has a least the connectivity of a torus. Transport the desert into 4-space. It can now be deformed in such a way as to knot the lion. He is now helples. THE SURGERGY METHOD. The lion is an orientable 3-manifold with boundary and so may be rendered contractible by surgery. THE UNIVERSAL COVERING METHOD. Cover the lion by his simply-connected covering space. Since this has no holes, he is trapped. THE GAME THEORY METHOD. The lion is a big game, hence certainly a game. There exists an optimal strategy. Follow it. THE SCHROEDINGER METHOD. At any instant there is a non-zero probability that the lion is in the cage. Wait. THE ERASTOSHENIAN METHOD. Enumerate all objects in the desert: examine them one by one; discard all those that are not lions. A refinement will capture only prime lions. THE PROJECTIVE GEOMETRY METHOD. The desert is a plane. Project this to a line, then project the line to a point inside the cage. The lion goes to the same point. THE INVERSION METHOD. Take a cylindrical cage. First case: the lion is in the cage: Trivial. Second case: the lion is outside the cage. Go inside the cage. Invert at the boundary of the cage. The lion is caught. Caution: Don't stand in the middle of the cage during the inversion! +------------------------------------------------------------ | Euler +------------------------------------------------------------ Euler Euler's formula: A connected plane graph with n vertices, e edges and f faces satisfies n - e + f = 2 Proof. Let T be the edge set of a spanning tree for G. It is a subset of the set E of edges. A spanning tree is a minimal subgraph that connects all the vertices of G. It contains so no cycle. The dual graph G* of G has a vertex in the interior of each face. Two vertices of G* are connected by an edge if the correponding faces have a common boundary edge. G* can have double edges even if the original graph was simple. Consider the collection T* of edges E* in G* that correspond to edges in the complement of T in E. The edges of T* connect all the faces because T does not have a cycle. Also T* does not contain a cycle, since otherwise, it would seperate some vertices of G contradicting that T was a spanning subgraph and edges of T and T* don't intersect. Thus T* is a spanning tree for G*. Clearly e(T)+e(T*)=2. For every tree, the number of vertices is one larger than the number of vertices. Applied to the tree T, this yields n = e(T)+1, while for the tree T* it yields f=e(T*)+1. Adding both equations gives n+f=(e(T)+1)+(e(T*)+1)=e+2. -- from M.Aigner, G. Ziegler "Proofs from THE BOOK" +------------------------------------------------------------ | irrational +------------------------------------------------------------ irrational Theorem: e = sum(k) 1/k! is irrational. Proof. e=a/b with integers a,b would imply N = n! (e - sum(kb because n! e and n!/k! were both integers. However, 0n n!/k!=1/(n+1) + 1/(n+1)(n+2) + ...<1/(n+1) + 1/(n+1)^2 + ...=1/n (second sum is a geometric series) for every n is not possible. -- from M.Aigner, G. Ziegler "Proofs from THE BOOK" +------------------------------------------------------------ | Wiener +------------------------------------------------------------ Wiener After a few years at MIT, the Mathematician Norbert Wiener moved to a larger house. His wife, knowing his nature, figured that he would forget his new address and be unable to find his way home after work. So she wrote the address of the new home on a piece of paper that she made him put in his shirt pocket. At lunchtime that day, the professor had an inspiring idea. He pulled the paper out of his pocket and used it to scribble down some calculations. Finding a flaw, he threw the paper away in disgust. At the end of the day he realized he had thrown away his address, he now had no idea where he lived. Putting his mind to work, he came up with a plan. He would go to his old house and await rescue. His wife would surely realize that he was lost and go to his old house to pick him up. Unfortunately, when he arrived at his old house, there was no sign of his wife, only a small girl standing in front of the house. "Excuse me, little girl" he said "but do you happen to know where the people who used to live here moved to?" "It's okay, Daddy," said the little girl, "Mommy sent me to get you". Moral 1. Don't be surprised if the professor doesn't know your name by the end of the semester. Moral 2. Be glad your parents aren't mathematicians. if your parents are mathematicians, introduce yourself and get them to help you through the course. - From the introduction of "How to ace calculus" by C. Adams, A. Thompson and J. Hass +------------------------------------------------------------ | funeral +------------------------------------------------------------ funeral David Hilbert was one of the great European mathematicians at the turn of the century. One of his students purchased an early automobile and died in one of the first car accidents. Hilbert was asked to speak at the funeral. "Young Klaus" he said, "was one of my finest students. He had an unusual gift for doing mathematics. He was insterested in a great variety of problems, such as..." There was a short pause, follwed by "Consider the set of differentiable functions on the unit interval and take their closure in the ..." Moral 1. Sit near the door. Moral 2. Some mathematicians can be a little out of touch with reality. If your professor falls in this category, look at the bright side. You will have lots of funny stories by the end of the semester. - From the introduction of "How to ace calculus" by C. Adams, A. Thompson and J. Hass +------------------------------------------------------------ | rabbit +------------------------------------------------------------ rabbit In a forest a fox bumps into a little rabbit, and says, "Hi, junior, what are you up to?" "I'm writing a dissertation on how rabbits eat foxes," said the rabbit. "Come now, friend rabbit, you know that's impossible!" "Well, follow me and I'll show you." They both go into the rabbit's dwelling and after a while the rabbit emerges with a satisfied expression on his face. Along comes a wolf. "Hello, what are we doing these days?" "I'm writing the second chapter of my thesis, on how rabbits devour wolves." "Are you crazy? Where is your academic honesty?" "Come with me and I'll show you." ...... As before, the rabbit comes out with a satisfied look on his face and this time he has a diploma in his paw. The camera pans back and into the rabbit's cave and, as everybody should have guessed by now, we see an enormous mean-looking lion sitting next to the bloody and furry remains of the wolf and the fox. The moral of this story is: It's not the contents of your thesis that are important -- it's your PhD advisor that counts. - Unknown Usenet Source +------------------------------------------------------------ | poet +------------------------------------------------------------ poet It is true that a mathematician who is not also something of a poet will never be a perfect mathematician. - K. Weierstrass, Quoted in D MacHale, Comic Sections (Dublin 1993) +------------------------------------------------------------ | equilateral +------------------------------------------------------------ equilateral THEOREM: All triangles are equilateral. PROOF: 1) Given an arbitrary triangle ABC. Construct the middle orthogonal on AB in D and cut it with the line dividing the angle at C. Call the intersection E. Form the normal from E to AC in F and from E to BC in G. Draw the lines AE und BE. C * / / *F *G / E* / | / | / |D A*---------*------------*B 2. The angles ECF and ECG are gleich. The angles EFC and EGC are both right angles. Because the triangles ECF and ECG have also EC common, they must be congruent. Therefore CF=CG and EF=EG. 3. The sides DA and DB are equal. The angle EDA and EDB are both right angles. Because the triangles EDA and EDB have also ED in common, they have to be congruent and EA=EB. 4. The angle EGB and EFA are both right angle. Also, EF=EG and EA=EB. Therefore both triangles EGB and EFA are congruent. Therefore FA=GB. 5. Since CF=CG and FA=GB, addition of the sides gives also CA=CB. 6. Having proved that two arbitrary sides are equal, all are equal. +------------------------------------------------------------ | widow +------------------------------------------------------------ widow I married a widow, who had an adult stepdaughter. My father, a widow and who often visited us, fell in love with my stepdaughter and married her. So, my father became my son-in-law and my stepdaughter became my stepmother. But my wife became the mother-in-law of her father-in-law. My stepmother, stepdaughter of my wife had a son and I therefore a brother, because he is the son of my father and my stepmother. But since he was in the same time the son of our stepdaughter, my wife became his grandmother and I became the grandfather of my stepbrother. My wife gave me also a son. My stepmother, the stepsister of my son, is in the same time his grandmother, because he is the son of her stepson and my father is the brother-in-law of my child, because his sister is his wife. My wife, who is the mother of my stepmother, is therefore my grandmother. My son, who is the child of my grandmother, is the grandchild of my father. But I'm the husband of my wife and in the same time the grandson of my wife. This means: I'm my own grandfather. +------------------------------------------------------------ | dots +------------------------------------------------------------ dots I never could make out what those damned dots meant. -- Lord Randolph Churchill (1849-1895) Brittish conservative politician, referring to decimal points. +------------------------------------------------------------ | ladder +------------------------------------------------------------ ladder The mathematician has reached the highest rung on the ladder of human thought. -- Havelock Ellis +------------------------------------------------------------ | ignorant +------------------------------------------------------------ ignorant Let no one ignorant of mathematics enter here. -- Plato, Inscription written over the entrance to the academy +------------------------------------------------------------ | god +------------------------------------------------------------ god I knew a mathematician, who said 'I do not know as much as God. But I know as much as God knew at my age'. -- Milton Shulman, Candian writer +------------------------------------------------------------ | english +------------------------------------------------------------ english English professor: In English, a double negative makes a positive. In other languages such as Russian, a double negative is still a negative. There are, however, no languages in which a double positive makes a negative. Student in back of class: "Yea, right" This file is part of the Sofia project sponsored by the Provost's fund for teaching and learning at Harvard university. There are 124 entries in this file. COUNT: 124 ENTRY COMPUTABILITY Authors: Oliver Knill: nothing real yet Literature: not yet, some lectures of E.Engeler on computation theory +------------------------------------------------------------ | Church's theses +------------------------------------------------------------ The generally accepted Church's theses tells that everything which is computable can be computed using a Turing machine. In that case, the problem to determine, whether a Turing machine will halt, is not computable. +------------------------------------------------------------ | cipher +------------------------------------------------------------ A cipher is a secret mode of writing, often the result of subsituting numbers of letters and then carrying out arithmetic operations on the numbers. +------------------------------------------------------------ | Coding theory +------------------------------------------------------------ Coding theory is the theory of encryption of messages employed for security during the transmission of data or the recovery of information from corrupted data. +------------------------------------------------------------ | Cooks hypothesis +------------------------------------------------------------ Cooks hypothesis P=NP. A proof or disproof is one of the millenium problems. +------------------------------------------------------------ | Graph isomorphism problem +------------------------------------------------------------ Graph isomorphism problem It is not known whether graph isomorphism can be decided in deterministic polynomial time. It is an open problem in computational complexity theory. +------------------------------------------------------------ | Inductive structure +------------------------------------------------------------ Inductive structure A set U with a subset A and operations g_1,...,g_n define an inductive structure (U,A,g_1,...,g_n) If all elements of U can be generated by repeated applications of the operations g_i on elements of A. Examples: (N,A=0,1,g_1(a,b)=a+b, g_2(a,b)=a*b defines an inductive structure. If U=N is the set of natural numbers, A=1,2,3 g_1(x,y)=3x-4,g_2(x,y,z)=7x+5y-z, then (U,A,g_1,g_2) define an inductive structure. +------------------------------------------------------------ | syntactic structure +------------------------------------------------------------ An inductive structure (U,A,g_1,...,g_n) is called a syntactic structure if it is uniquely readable that is if g_1(u_1,...,u_k) = g_2(v_1,...,v_l), then g_1=g_2,k=l and u_1=v_1,...,u_k=v_k. Example: if X is the set of finite words in the alphabet p,q,r,K,N and A=p,q,r. Define g_1(x,y) = Kxy and g_2(x,y)=Nx and U the set of words generated from A. The structure is the language of elementary logic in polnic notation. It is a syntactic structure. Syntactic structures are in general described by grammers. +------------------------------------------------------------ | grammar +------------------------------------------------------------ A grammar (N,T,G) is given by two sets of symbols N,T and a finite set G of pairs (n_i,t_i) which define transitions n_i to t_i. For example: N=S,T=K,N,p,q,r, G= S to p, S to q,S to r, S to KSS, S to NS . Acoording to Chomsky, one classifies grammers with additional conditions like context sensitivity or regularity. +------------------------------------------------------------ | context sensitive +------------------------------------------------------------ A grammer (N,T,G) is called context sensitive if (n,t) in G then |t| geq |n|. +------------------------------------------------------------ | context sensitive +------------------------------------------------------------ This file is part of the Sofia project sponsored by the Provost's fund for teaching and learning at Harvard university. There are 10 entries in this file. COUNT: 10 ENTRY COMPUTER Authors: Oliver Knill: May 2001 Literature: for video stuff: http://www.doom9.org, foldoc +------------------------------------------------------------ | AAC +------------------------------------------------------------ AAC Advanced Audio Coding will be the successor of AC3 audio. It is based on AC3 while adding a number of improvements in various areas. Currently player and hardware support for this upcoming audio format is still very limited. +------------------------------------------------------------ | acrobat +------------------------------------------------------------ acrobat A product from Adobe for manipulating documents stored in the PDF (Portable Document Format). +------------------------------------------------------------ | amd +------------------------------------------------------------ amd Daemon which enables the NFS automount. +------------------------------------------------------------ | AMD +------------------------------------------------------------ AMD Advanced Micro Devices, Chip company. +------------------------------------------------------------ | arpwatch +------------------------------------------------------------ arpwatch Daemon to log and buil a database of Ethernet address/IP address pairings it sees on a LAN interface. +------------------------------------------------------------ | ASCII +------------------------------------------------------------ ASCII American Standard Code for Information Interexchange, an industry standard, which assigns letters, numbers and other characters within the 256 slots available in the 8-bit code. +------------------------------------------------------------ | AC3 +------------------------------------------------------------ AC3 Initially known as Audio Coding 3 AC3 is a synonym for Dolby Digital these days. Dolby Digital is an advanced audio compression technology allowing to encode up to 6 separate channels at bitrates up to 448kbit/s. For more information please check out the Dolby website. +------------------------------------------------------------ | ASF +------------------------------------------------------------ ASF Advanced Streaming Format. Microsoft's answer to Real Media and streaming media in general. +------------------------------------------------------------ | AT +------------------------------------------------------------ AT keyboard The standard keyboard used with the IBM compatible computer. +------------------------------------------------------------ | backdoor +------------------------------------------------------------ A backdoor is a "mechanism surreptitiously introduced into a computer system to facilitate unauthorized access to the system". An example of a backdoor is "bindshell". +------------------------------------------------------------ | AVI +------------------------------------------------------------ AVI Audio Video Interleave. The video format most commonly used on Windows PC's. It defines how video and audio are attached to each other, without specifying a codec. +------------------------------------------------------------ | Bandwidth +------------------------------------------------------------ Bandwidth Bandwidth measures how much information can be carried in a given time period over a wired or wireless communications link. A typical broadband speed is 1270 Kbps (kilo bit per second) which is 155.6 KBytes/sec Technology Speed mbit/s 56k modem 0.056 DSL varies cable varies T1 1.544 Ethernet 10.000 T3 44.736 OC-3 155.520 OC-12 622.080 OC-48 2,488.320 OC-96 4,976.640 OC-192 9,953.280 OC-255 13,219.200 see http://home.cfl.rr.com/cm3/speedtest7.htm http://jetstreamgames.co.nz/speed/ADSLdownload1MB.html http://home.cfl.rr.com/eaa/Bandwidth.htm +------------------------------------------------------------ | BUP file +------------------------------------------------------------ BUP file A bup file is a Back UP file of an IFO file. These files are commonly found on DVDs. +------------------------------------------------------------ | Byte +------------------------------------------------------------ One Byte is an information unit of a sequence of 8 bits. +------------------------------------------------------------ | CASE +------------------------------------------------------------ CASE: Computer Aided Software Engineering. +------------------------------------------------------------ | Cell (ID) +------------------------------------------------------------ Cell (ID) A cell is the smallest video unit on a DVD. Normally used to contain a chapter it can also be used to contain a smaller unit in case of multiangles or seamless branching titles. +------------------------------------------------------------ | certificate +------------------------------------------------------------ A certificate is a digital identifcation of a physical or abstract object, a person, business, computer, program or document. A digital certificate is much like a passport. It is issued by a certificate authority, which vouches for its authenticity. +------------------------------------------------------------ | Codec +------------------------------------------------------------ Codec COder/DECoder. A codec is a piece of software that allows to encode something - usually audio or video - to a specific format and can decode media encoded in this specific format again. Popular Codecs are MPEG1, MPEG2, MPEG-4 (=divx=xvid), realvideo, wmv, dv Indeo, etc. MPEG, AVI, ASF, Quicktime is not a codec but a container format - that can be encoded using different codecs. In avi container files, there's mostly mpeg4 video content and mp3 audio content but this is not obligatory. For DVD, the video should be in mpg2, the audio in mp2 and both of these will be in a mpeg-ps (program stream aka "vob") container. +------------------------------------------------------------ | Container +------------------------------------------------------------ Container A container is, like the name says, a construct to contain data - in this case video and audio date and possibly subtitles and navigational information. For instance, you would like to put a soundless video stream and the audio track together in one file. To do that you need a container format. Examples of container formats are: AVI, ASF, OGM, Quicktime, VOB and MPG. In avi container files, there's mostly mpeg4 video content and mp3 audio content but this is not obligatory. For DVD, the video should be in mpg2, the audio in mp2 and both of these will be in a mpeg-ps (program stream aka "vob") container. % A cookie is a block of information recorded and stored within the client's browser. +------------------------------------------------------------ | CSS +------------------------------------------------------------ CSS Cascading Style Sheets is a simple mechanism for adding style (e.g. fonts, colors, spacing) to Web documents. For example: body, table font-family: verdana, arial, geneva, sans-serif; +------------------------------------------------------------ | CSS +------------------------------------------------------------ CSS Content Scrambling System. Prioprietary scrambling system for video DVDs. Designed to stop people from making copies of DVDs, most commercial DVDs are encrypted using CSS. During playback, DVDs are then decrypted on the fly. Only parts of the DVD are encrypted (for instance all IFO and BUP files are not encrypted, and VIDEOTS.VOB often isn't encrypted either) and the encryption scheme is rather weak and was quickly defeated. If you want to know what CSS does, insert a DVD video disc into your PC, start playing the disc using a software DVD player, then close the player. Now copy a 0.99GB VOB file from the disc to your harddisk and try to play back that VOB file in your software DVD player. You'll see a lot of funny colored blocks all over the picture making the movie unwatchable. But you'll also see parts of the movie (the parts that are not encrypted). +------------------------------------------------------------ | DAR +------------------------------------------------------------ DAR Display Aspect Ratio. Indicates the dimension of a screen. Most PC screens have a DAR of 4:3, meaning that the horizontal size is 4/3 as large as the vertical size. For TVs we have a lot of old 4:3 displays and more and more 16:9 displays. As you can guess from the numbers 16:9 displays are broader than 4:3 displays having the same diagonal size. 16:9 screens are more suited to display Hollywood movies which are usually shot with an aspect ratio of 1:2.35 or 1:1.85 (meaning that the horizontal size of the picture is 1.85 times as wide as the vertical size). +------------------------------------------------------------ | Deinterlace +------------------------------------------------------------ Deinterlace The process of restoring a progressive video stream out of an interlaced one is called deinterlacing. +------------------------------------------------------------ | Demultiplexing +------------------------------------------------------------ Demultiplexing The opposite of multiplexing. In this process a combined audio/video stream will be separated into the number of streams it consists of (a video stream, at least one audio stream and a navigational stream). Every VOB encoder demultiplexes the VOB files before encoding (FlaskMpeg, mpeg2avi, dvd2mpg, ReMpeg2) and every DVD player does the same (audio and video are being treated by different circuits, or decoded by different filters on a PC). +------------------------------------------------------------ | Descrambling +------------------------------------------------------------ Descrambling DVDs are usually CSS scrambled - imagine you decide to give a number to each letter, starting with 1 for a, etc. A sentence would become a couple of digits - that's what we call scrambled. Of course CSS is much better than that but it's still quite easy to crack. Descrambling means reversing the scrambling process, rendering our digits to a sentence again, or making our movie playable again - you can try to copy a movie to your hard disk when you've authenticated your DVD drive and play it, you'll get a garbled picture because it's still scrambled. Common CSS descramblers either use a pool of known descrambling keys (DeCSS or DODSrip - they contain a large number of keys but not all of them) or try to derive the key by a cryptographic attack (VobDec - that's why it works on most disc since it's not dependent on a pool of discs). +------------------------------------------------------------ | Digital Video +------------------------------------------------------------ Digital Video Digital video is usually compressed. Since standard loss less compression is insufficient for video, the video codecs have to get rid of unimportant information - stuff the human eye won't see or is unlikely to see. Since that is still not enough modern compression algorithms use keyframes, I and P frames in order to save space. +------------------------------------------------------------ | DivX +------------------------------------------------------------ DivX There are 2 flavors of DivX today: DivX is the name of the hacked Microsoft MPEG4 codecs (Windows Media Video V3). Those codecs were developed by Microsoft for use in its proprietary Windows Media architecture and initially supported encoding AVIs and ASFs but all non-beta versions included an AVI lock, making it impossible to use them to encode to the AVI format - and only a few tools support ASF today. What the makers of DivX did is remove that AVI lock making it possible to encode to AVI again, and changed the name to DivX video in order to prevent confusion of codecs, since it's possible to have both the unhacked and hacked codecs on the same computer if you use the Windows Media Encoder. The latest releases of DivX also include a hacked Windows Media Audio Codec called DivX audio - the hack of that codec is not perfect yet and its use is limited for higher bitrates. This codec is also known as DivX3. The other DivX is a brand-new MPEG-4 video codec developed by DivXNetworks. It offers much advanced encoding controls and 2 pass encoding. Furthermore the codec can play the old DivX3 movies. The codec is commonly called DivX4. +------------------------------------------------------------ | DHCPD +------------------------------------------------------------ DHCPD Daemon to service which can dynamically assign IP addresses to its client hosts. +------------------------------------------------------------ | DOM +------------------------------------------------------------ DOM The Document Object Model is a platform- and language-neutral interface that will allow programs and scripts to dynamically access and update the content, structure and style of documents. +------------------------------------------------------------ | DOS +------------------------------------------------------------ DOS is a Disk operating system, based on a command line user interface. MS-DOS 1.0 was released in 1981 for IBM computers. While MS-DOS is not much used by itself today, it still can be accessed from Windows 95, Windows 98 or Windows ME by clicking Start/Run and typing command or CMD in Windows NT, 2000 or XP. +------------------------------------------------------------ | DRC +------------------------------------------------------------ DRC Dynamic Range Compression. AC3 Tracks contain a much larger dynamic range that most audio equipment can handle, therefore most standalone and software DVD player will compress the dynamic range somewhat, according to the actual dynamic range. In layman terms the volume will be augmented dynamically, e.g. explosions won't become louder or only a bit louder, whereas in normal dialogues the volume will be augmented quite a bit. Since your player will do the same this is the way to go to have augmented volume. +------------------------------------------------------------ | DTML +------------------------------------------------------------ DTML document template markup language. +------------------------------------------------------------ | DTP +------------------------------------------------------------ DTP Desktop publishing. +------------------------------------------------------------ | Dynamic HTML +------------------------------------------------------------ Dynamic HTML is a term used by some vendors to describe the combination of HTML, style sheets and scripts that allows documents to be animated. +------------------------------------------------------------ | Elementary Stream (ES) +------------------------------------------------------------ Elementary Stream (ES) An elementary stream is a single (video or audio) stream without container. For instance a basic MPEG-2 video stream (.m2v or .mpv) is an MPEG-2 ES, and on the audio side we have AC3, MP2, etc files that are ES. Most DVD authoring program require ES as input. +------------------------------------------------------------ | EULA +------------------------------------------------------------ EULA End user licence agreement. +------------------------------------------------------------ | FAT +------------------------------------------------------------ FAT File allocation table. Filesystem used by Windows. Example: Windows 95 users rely on the FAT 16, In 1996 Microsoft introduced the FAT 32 file system, which is still very widely used today besides NTFS on the windows platform. +------------------------------------------------------------ | FUD +------------------------------------------------------------ FUD stands for Fear, Uncertainty, Doubt. It is a marketing technique used when a competitor launches a product that is both better than yours and costs less, i.e. your product is no longer competitive. Unable to respond with hard facts, scare-mongering is used via 'gossip channels' to cast a shadow of doubt over the competitors offerings and make people think twice before using it. +------------------------------------------------------------ | GUI +------------------------------------------------------------ GUI - Graphical User Interface; A desktop-like interface usually containing icons, menus and windows. Invented by Xerox, later "borrowed" by Microsoft and Apple. +------------------------------------------------------------ | HTML +------------------------------------------------------------ HTML Hypertext markup language. Will be replaced by XHTML, and XHTML 2.0 in particular. +------------------------------------------------------------ | HTTP +------------------------------------------------------------ HTTP Hypertext Transfer Protocol. +------------------------------------------------------------ | HTTPD +------------------------------------------------------------ HTTPD Daemon to Apache webserver. +------------------------------------------------------------ | Hypertext +------------------------------------------------------------ Hypertext - shortcuts or links between different parts of a document, article, website or world wide web. While early hypertext formats were already Apples Hypercard, it is now common in HTML (Hypertext markup language). +------------------------------------------------------------ | inetd +------------------------------------------------------------ inetd Daemon which is at the heart of providing network services like telnet or ftp. +------------------------------------------------------------ | IFO file +------------------------------------------------------------ IFO file InFOrmation file commonly found on DVDs. Such files contain navigational information for DVD players. +------------------------------------------------------------ | Interlaced +------------------------------------------------------------ Interlaced Interlaced is a video storage mode. An interlaced video stream doesn't contain frames but fields with each field containing either even or odd lines of one frame. +------------------------------------------------------------ | IP +------------------------------------------------------------ IP Internet Protocol. Standard which defines the structure of a message sent between two computers over the network. +------------------------------------------------------------ | IPFW +------------------------------------------------------------ IPFW IP firewall. +------------------------------------------------------------ | ICMP +------------------------------------------------------------ ICMP Internet Control Message Protocol. ICMP messages contain information about communication between two computers. +------------------------------------------------------------ | Java +------------------------------------------------------------ Java is a true compiler-based, low level programming language. % Javascript is a scripting programming language. It was developed by Netscape and used to create interactive Web sites. JavaScript is a popular client-side scripting language because it is supported by virtually all browsers. +------------------------------------------------------------ | KISS +------------------------------------------------------------ KISS - Keep It Simple Stupid. Rule of thumb for software designers. Keep design small to minimize confusion. +------------------------------------------------------------ | LDAP +------------------------------------------------------------ LDAP Lightweight Directory Access Protocol. A network directory which can substitute DNS and much more. Not to be confused with a database. A directory is mostly looked up and not written often into. +------------------------------------------------------------ | LDIF +------------------------------------------------------------ LDIF LDAP interchange Format is a standard text file for storing LDAP configuration information and directory contents. +------------------------------------------------------------ | MathML +------------------------------------------------------------ MathML is a low-level specification for describing mathematics as a basis for machine to machine communication. It provides a foundation for the inclusion of mathematical expressions in Web pages. +------------------------------------------------------------ | miniDVD +------------------------------------------------------------ miniDVD Basically a DVD on a CD. A miniDVD can contain bitrates up to 10mbit/s (audio and video combined). Video is MPEG2, preferably VBR and audio can be MPEG1 audio layer 2, raw uncompressed PCM or AC3. Video quality can be up to an actual DVD level if a limited playtime is accpted. +------------------------------------------------------------ | MPEG +------------------------------------------------------------ MPEG MPEG means Motion Picture Expert Group and it's the resource for video formats in general. This group defines standards in digital video, among it the MPEG1 standard (used in Video CDs), the MPEG2 standard (used on DVDs and SVCDs), the MPEG4 standard and several audio standards - among them MP3 and AAC. Files containing MPEG-1 or MPEG-2 video often use either the .mpg or .mpeg extension. +------------------------------------------------------------ | MPEG4 +------------------------------------------------------------ MPEG4 Is pretty much a collection of standards defined by the MPEG Group, and it should become the next standard in digital video. MPEG4 allows the use of different encoding methods, for instance a keyframe can be encoded using ICT or Wavelets resulting in different output qualities. +------------------------------------------------------------ | MPG +------------------------------------------------------------ MPG MPG can be either an abbreviation for MPEG or is used as a file extension for MPEG-1 and MPEG-2 video data. It is a container to contain MPEG-1/2 video stream and MPEG1 layer 2 audio (aka mp2 files). MPG containers are also refered to as program streams (PS). +------------------------------------------------------------ | MM4 +------------------------------------------------------------ MM4 Multiple MPEG 4: A combination of different bitrate encoded files. For instance you could take a 2000kbit/s encode, a 910kbit/s encode and combine the files together, use the lower bitrate file and replace scenes where the quality gets too bad due to a lot of action with the parts taken from the 2000kbit/s one. +------------------------------------------------------------ | NAT +------------------------------------------------------------ NAT Network Address translation. A typical home user with broadband access and router performs Network Address Translation, or NAT allowing multiple computers to share a single fast Internet connection. +------------------------------------------------------------ | .Net +------------------------------------------------------------ .Net A collectionof technologies pushed by Microsoft. It contains C# programming language (an alternative to Java). Part of the .Net initiative. It builds on standards like XML and SOAP. +------------------------------------------------------------ | Network layers +------------------------------------------------------------ Network layers Application layer: Client and server programs. Transport layer: TCP and UDP protocols, service ports Network layer: IP packets, IP addresses, ICMP messsages Data link layer: Ethernet frames and MAC addresses Physical layer: Copper wire, fiberoptic cable, radio +------------------------------------------------------------ | Newbie +------------------------------------------------------------ Newbie - (Also n00b and newb) a newcomer to a certain computer topic or program asking help from experienced user. +------------------------------------------------------------ | NFS +------------------------------------------------------------ NFS Network file system. +------------------------------------------------------------ | OGM +------------------------------------------------------------ OGM OGM stands for OGg Media which is the name of the Ogg container implementation by Tobias Waldvogel. OGM can be used as an alternative to the AVI container and it can contain Ogg Vorbis, MP3 and AC3 audio, all kinds of video formats, chapter information and subtitles. +------------------------------------------------------------ | Perl +------------------------------------------------------------ Perl Perl is a high-level programming language. It derives from the C programming language and to a lesser extent from sed, awk, the Unix shell, and at least a dozen other tools and languages. Perl's process, file, and text manipulation facilities make it particularly well-suited for tasks involving quick prototyping, system utilities, software tools, system management tasks, database access, graphical programming, networking, and world wide web programming. These strengths make it especially popular with system administrators and CGI script authors, but mathematicians, geneticists, journalists, and even managers also use Perl. +------------------------------------------------------------ | PHP +------------------------------------------------------------ PHP PHP is a widely-used general-purpose scripting language that is especially suited for Web development and can be embedded into HTML. +------------------------------------------------------------ | Pocket PC +------------------------------------------------------------ Pocket PC Operating system for handhelds. Usually running Microsoft CE or the Palm OS. +------------------------------------------------------------ | PNG +------------------------------------------------------------ PNG is graphics file format for the lossless, portable, well-compressed storage of raster images. Indexed-color, grayscale, and truecolor images are supported, plus an optional alpha channel for transparency. Sample depths range from 1 to 16 bits per component (up to 48bit images for RGB, or 64bit for RGBA). +------------------------------------------------------------ | Python +------------------------------------------------------------ Python is an interpreted, high-level, object-oriented programming language. +------------------------------------------------------------ | QNX +------------------------------------------------------------ QNX is a realtime, microkernel, preemptive, prioritized, message passing, network distributed, multitasking, multiuser, fault tolerant operating system. +------------------------------------------------------------ | Qt +------------------------------------------------------------ Qt ("kjut") Multi platform toolkit and graphics library. Developed by Trolltech. Runs on Windows systems including XP, all unix derivates with X windows as well as Mac OS X. +------------------------------------------------------------ | PCMCIA +------------------------------------------------------------ PCMCIA Personal Computer Memory Card International Association. +------------------------------------------------------------ | RDBM +------------------------------------------------------------ RDBM relational data base manager. +------------------------------------------------------------ | rff/tff +------------------------------------------------------------ rff/tff RFF means repeat first frame, it's a technique used to make the necessary 29.97 frames per second out of a 24 frames per second source - the movie like it was recorded with a traditional movie camera used by Hollywood. The rff flag tells the player to repeat one field of the video stream. Tff means top field first and is also used to perform a telecine to make a 24fps movie into 29.97fps. +------------------------------------------------------------ | Real time operating systems +------------------------------------------------------------ Real time operating systems Operating systems which are used in handhelds, robots, telphone switches. Examples: QNX, VxWorks (as used in Mars rovers), Windows CE (as used in handhelds), Nucleus RTX. Realtime systems must function reliably in event of failures. It is said that the three most important things in Realtime system design are timing, timing and timing. +------------------------------------------------------------ | Ripping +------------------------------------------------------------ Ripping Ripping means copying a DVD movie to the hard disk of the computer. This includes the authentication process for the DVD Drive and the actual CSS Descrambling. CSS (Content Scrambling System) is a copy protection scheme designed to prevent unauthorized copying of DVD movies, although many argue that it was also designed to control where DVD movies can be played since without a CSS license you essentially have to crack the encryption to play a DVD movie. The term "ripping" is also often used to describe the whole process of descrambling a DVD, then convert the audio and video into another format. +------------------------------------------------------------ | RSS +------------------------------------------------------------ RSS is a method of distributing links to content in a web site so that others can use use it. It's a mechanism to "syndicate" the content. The original RSS, version 0.90, was designed by Netscape as a format for building portals of headlines to mainstream news sites. RSS is an acronym for Really Simple Syndication. +------------------------------------------------------------ | RTFM +------------------------------------------------------------ RTFM - Read the fucking manual. Common answer to basic and often repeated questions, that could be avoided in the first place just by looking at the manual. +------------------------------------------------------------ | RSS +------------------------------------------------------------ Really Simple Syndication RSS is an XML-based format for content distribution. For example, News.com offers several RSS feeds with headlines, descriptions and links back to News.com for the full story. % ROUTER a machine designed to direct packets from their source host to their destination. +------------------------------------------------------------ | RDBMS +------------------------------------------------------------ RDBMS A Relational Database Management System. Stores data related tables. A single database can be spread across several tables unlike flat-file databases where each database is self-contained in a single table. +------------------------------------------------------------ | SBC +------------------------------------------------------------ SBC Smart Bitrate Control. A new kind of DivX encoder called Nandub can modify many internal codec parameters on the fly during compression, giving you better quality and a lot more control over the encoding session. More information can be found in the SBC guide in the DivX guides section. +------------------------------------------------------------ | SGML +------------------------------------------------------------ SGML The standard Generalized Markup Language (SGML) is a meta Markup Languages like XML. They are used for defining markup languages. A markup language defines using SGML or XML has a specific vocabulary. +------------------------------------------------------------ | SMIL +------------------------------------------------------------ SMIL Synchronized Multimedia Integration Language. It enables simple authoring of interactive audiovisual presentations. SMIL is typically used for "rich media"/multimedia presentations which integrate streaming audio and video with images, text or any other media type. SMIL is an easy-to-learn HTML-like language, and many SMIL presentations are written using a simple text-editor. +------------------------------------------------------------ | SOAP +------------------------------------------------------------ SOAP Simple object access protocol. +------------------------------------------------------------ | SQL +------------------------------------------------------------ SQL structured query language. +------------------------------------------------------------ | Sun ONE +------------------------------------------------------------ Sun ONE Sun Open net initiative. Answer to .Net initiative of Microsoft. Has Java, XML and SOAP as foundation. +------------------------------------------------------------ | SVG +------------------------------------------------------------ SVG The Scalable Vector Graphics is a language for describing two-dimensional graphics in XML. SVG allows for three types of graphic objects: vector graphic shapes, images and text. Graphical objects can be grouped, styled, transformed and composited into previously rendered objects. The feature set includes nested transformations, clipping paths, alpha masks, filter effects and template objects. +------------------------------------------------------------ | TCP +------------------------------------------------------------ TCP Transmission control protocol. An IP message type. Most network services run over TCP. A typical TCP connection is visiting a remote web site. +------------------------------------------------------------ | TCPA +------------------------------------------------------------ TCPA (Trusted Computing Platform Architecture) belongs to DRM (Digital rights management). TCPA aims at integrity of kernel and system components - to assure you that your system can be trusted. Palladium, on the other hand, uses similar technology to make sure that the user does not do anything else than what is allowed by content owners. +------------------------------------------------------------ | TLD +------------------------------------------------------------ TLD Top level domain. The last entry in a webaddress. The TLD of www.w3c.org is "org". In the 1980s, seven TLDs (.com, .edu, .gov, .int, .mil, .net, and .org) were created. Later four of the new TLDs (.biz, .info, .name, and .pro) as well as sponsered TLD's .aero, .coop, and .museum) were created. TLDs with two letters (such as .de, .mx, and .jp) have been established for over 240 countries and external territories and are referred to as "country-code" TLD. +------------------------------------------------------------ | UDP +------------------------------------------------------------ UDP User Datagramm Protocol. Sendes transport-level data between two network-based programs. For example, internet-time servers are assigned UDP services. +------------------------------------------------------------ | UML +------------------------------------------------------------ UML Unified modelling language is a language for specifying, visualizing, constructing, and documenting software systems, as well as for business modeling and modeling of other non-software systems. +------------------------------------------------------------ | VCD +------------------------------------------------------------ VCD Video CD, works on many DVD players, there are software players on almost every operating systems, doesn't need a fast computer but the image is VHS-like. Video is MPEG1 at 1150kbit/s and audio MPEG1 audio layer 2 at 224kbit/s. +------------------------------------------------------------ | VLDB +------------------------------------------------------------ VLDB Very large data base. +------------------------------------------------------------ | VML +------------------------------------------------------------ VML Vector Markup Language +------------------------------------------------------------ | W3C +------------------------------------------------------------ W3C The World Wide Web Consortium (W3C) develops interoperable technologies (specifications, guidelines, software, and tools) to lead the Web to its full potential. W3C is a forum for information, commerce, communication, and collective understanding. +------------------------------------------------------------ | Wavelets +------------------------------------------------------------ Wavelets Wavelets are an alternative basis space. There are infinitely many wavelet bases (Daubechies, Haar, Mexican Hat, "Spline", Zebra, etc), but their primary feature is that they are localized. Fourier basis functions span all space (from negative to positive infinity). Wavelets are basically individual pulses of waves (at various positions and scales). Their value in compression stems from factors like the grouping which generally shows that a good 90% of the data is modelled by the low-pass filters, with the high-pass filters generally showing very small values that are mostly details. (of course, this is not true if the source is noisy in the first place). For images, the greatest value comes from localization of the basis, which means that we can model discontinuities (e.g. edges) VERY well with wavelets. You will NOT get those weird JPEG halos if you use wavelets. +------------------------------------------------------------ | WebDAV +------------------------------------------------------------ WebDAV Web-based Distributed Authoring and Versioning. A set of extensions to the HTTP protocol which allows users to collaboratively edit and managa files on remote web servers. +------------------------------------------------------------ | Widget +------------------------------------------------------------ Widget- objects that make up interfaces, i.e. mouse, menus, textbox, buttons; basic tools and objects. +------------------------------------------------------------ | Windows Media +------------------------------------------------------------ Windows Media Microsoft's proprietary architecture for audio and video on the PC. It's based on a collection of codecs which can be used by the WindowsMedia Player to play files encoded in any supported format. WindowsMedia 7.0 offers a new set of codecs, among them a fully ISO compliant MPEG4 codec (called MS Windows Video V1), an improved MPEG-4 codec called MS Video V7 (although I did not notice any improvement compared with MS Windows Video V3 on which DivX is based), an encoder that supports Deinterlacing and Inverse Telecine. +------------------------------------------------------------ | Win FS +------------------------------------------------------------ Win FS Windows Future Storage file system, planned in Windows Longhorn, the successor of Windows XP. +------------------------------------------------------------ | WYSIWYG +------------------------------------------------------------ WYSIWYG- What You See Is What You Get. Usually to distinguish document authoring tools. Writing a Latex file as a text document is not WYSIWYG, while authoring a word document is. Writing a HTML document with a text editor is not WYSIWYG, while writing it with an authoring tool is. +------------------------------------------------------------ | WORM +------------------------------------------------------------ WORM a program that connects to other machines and replicates itself. Worms have the potential to both damage infected machines and to interfere with networks and services due to congestion caused by the spread of the worm. +------------------------------------------------------------ | WORM +------------------------------------------------------------ XHTML 2 is a general purpose markup language designed for representing documents for a wide range of purposes across the World Wide Web. To this end it does not attempt to be all things to all people, supplying every possible markup idiom, but to supply a generally useful set of elements. Here is an element of a XHTML 2.0 document: Virtual Library

Moved to vlib.org.

+------------------------------------------------------------ | XML +------------------------------------------------------------ XML The Extensible textbased Markup Language is a format for structured documents and data on the Web. It is derived from SGML (ISO 8879). XML is also playing an increasingly important role in the exchange of a wide variety of data on the Web. +------------------------------------------------------------ | XviD +------------------------------------------------------------ XviD XviD is a word play, read it the reverse way and you might find a familiar term. XviD is an open source MPEG-4 codec which depending on whom you're asking yields even better quality than the best DivX codec. +------------------------------------------------------------ | zombie +------------------------------------------------------------ zombie A unix process that has died but has not yet relinquished its process table slot. The parent process hasn't executed a "wait" for it yet). This file is part of the Sofia project sponsored by the Provost's fund for teaching and learning at Harvard university. There are 108 entries in this file. COUNT: 108 NFromContinuedFractionTablek,k,0,100 This file is part of the Sofia project sponsored by the Provost's fund for teaching and learning at Harvard university. There are 0 entries in this file. COUNT: 0 ENTRY CONSTANTS Authors: Oliver Knill: March 2000 - March 2004 Literature: Some from Mario Livio "The golden ratio", www.mathworld.com David Wells: "The Penguin Dictionary of Curious and Interesting Numbers". +------------------------------------------------------------ | Archimedes Constant, pi +------------------------------------------------------------ The Archimedes Constant, pi pi=3.14159 is the length of a half circle with radius 1. It is the area of a disc of radius 1. +------------------------------------------------------------ | Bruns constant +------------------------------------------------------------ Bruns constant is the sum of the reciprocals of all twin primes. Brun has proven that this sum converges evenso it is unknown whether there are infinitely many twin primes. +------------------------------------------------------------ | Catalan constant +------------------------------------------------------------ The Catalan constant is defined as the sum (-1)^n/(2n+1)^2=0.91596. +------------------------------------------------------------ | Champernown's number +------------------------------------------------------------ Champernown's number is 0.12345678910111213... whose digits are those of all natural numbers in succession. +------------------------------------------------------------ | Continued fraction constant +------------------------------------------------------------ Continued fraction constant is the number with continued fraction (0,1,2,3,4,5,6,...) it is about 0.697774658. +------------------------------------------------------------ | Euler Mascheroni constant +------------------------------------------------------------ Euler Mascheroni constant is defined as the limit of (1+1/2+1/3+...+1/n)-log(n) as n goes to infinity. +------------------------------------------------------------ | Aperi constant +------------------------------------------------------------ Aperi constant It is an irrational number zeta(3)=1.20206, the value of the zeta function at 3. % Feigenbaum constant When iterating maps f(x)=ax(1-x) on the unit interval the stable periodic orbits bifurcate when varing a. If a_n are the bifurcation values, then delta=lim (a_n-a_n-1)/(a_n+1-a_n) is a Feigenbaum constant. +------------------------------------------------------------ | golden ratio +------------------------------------------------------------ The golden ratio is tau=(1+sqrt5)/2 = 0.618... If 1,1,2,3,5,8,13,21... are the Fibonnachi numbers (the next number is always the sum of the two previous once), then the ratio of neighboring entries approaches the golden mean. 13/21=0.61904 is already quite close to the golden mean. The Golden ratio has the continued fraction expansion 1,1,1,1... which means that the number can be written as tau = 1+1/(1+1/(1+...)). The golden mean is an example of a Diophantine number, a number which can not be approximated well by rational numbers. Especially, it is irrational. The golden ratio is also called "golden mean" or "divine constant". +------------------------------------------------------------ | golden mean +------------------------------------------------------------ The golden mean see golden ratio. +------------------------------------------------------------ | Khinchin constant +------------------------------------------------------------ The Khinchin constant is defined as the limit (a_1 a_2 ... a_n)^1/n where a_1,a_2,... is the continued fraction of a random number in the sense that the limit is known to exist for almost all real numbers. It is not known for example, if pi is a typical number in the sense that it produces the Khinchin constant. +------------------------------------------------------------ | natural logarithmic base +------------------------------------------------------------ The natural logarithmic base e=2.7182818... can be defined as exp(1)=1+1/1!+1/2!+1/3!+... or lim_n to infinity (1+1/n)^n. +------------------------------------------------------------ | number of the beast +------------------------------------------------------------ The number of the beast is the integer 666. The "beast" is associated with the "antichrist". The origin of the association is the bible: the book of revelations (13:18) reads: "this calls for wisdom: let anyone with understanding calculate the number of the beast, for it is the number of a man. Its number is six hundred and sixty six." +------------------------------------------------------------ | Pythagoras constant +------------------------------------------------------------ The Pythagoras constant is the square root of 2 x=sqrt2=1.41421 .... It is the length of the diagonal of the unit square. It is irrational because x=p/q would imply 2 q^2 = x^2 q^2 = p^2, which is impossible because the prime factorization on the left contains an odd number of 2's, while it contains an even number of 2's on the right. +------------------------------------------------------------ | Smith number +------------------------------------------------------------ Smith number Smith numbers are integers n such that the sum of its digits in the decimal exapansion of n is qual to the sum of the digits of its prime factorization, excluding 1. Smith numbers were defined by A. Wilansky. He called it Smith numbers after his brother in law H. Swmith, whose telephone number 4937775 = 3*5*5*65'837 is a Smith number. Here are the first Smith numbers: 4,22,27,58,85,94,121,166,202,265.... +------------------------------------------------------------ | Wallis constant +------------------------------------------------------------ The Wallis constant is the real solution to the polynomial x^3-2x+5 which is 2.0945514815.... This equation was solved by the English mathematicaian John Wallis 1616-1703 to illustrate Newton's method for the numerical solution of equations. It has since served as a test for many subsequent methods of approximation. +------------------------------------------------------------ | Zero +------------------------------------------------------------ Zero The integer zero 0 is the neutral element in the additive group of integers n+0=n. This file is part of the Sofia project sponsored by the Provost's fund for teaching and learning at Harvard university. There are 16 entries in this file. COUNT: 16 ENTRY CURVES Authors: Oliver Knill, Andrew Chi, 2003 Literature: www.mathworld.com, www.2dcurves.com +------------------------------------------------------------ | astroid +------------------------------------------------------------ An astroid is the curve t mapsto (cos^3(t),a sin^3(t)) with a>0. An asteroid is a 4-cusped hypocycloid. It is sometimes also called a tetracuspid, cubocycloid, or paracycle. +------------------------------------------------------------ | Archimedes spiral +------------------------------------------------------------ An Archimedes spiral is a curve described as the polar graph r(t) = a t where a>0 is a constant. In words: the distance r(t) to the origin grows linearly with the angle. +------------------------------------------------------------ | bowditch curve +------------------------------------------------------------ The bowditch curve is a special Lissajous curve r(t)=( a sin(nt+c), b sin(t)). +------------------------------------------------------------ | brachistochone +------------------------------------------------------------ A brachistochone is a curve along which a particle will slide in the shortest time from one point to an other. It is a cycloid. +------------------------------------------------------------ | Cassini ovals +------------------------------------------------------------ Cassini ovals are curves described by ((x+a) +y^2)((x-a)^2+y^2) = k^4, where k^20, there is a set E_d subset X such that f_n to f uniformly on E setminus E_d and m(E_d)0, there is a set E_d with m(E_d)1 as zeta(s)=1+1/2^s+1/3^s+.... The function can be continued to the entire complex plane except at s=1, where the function has a singularity. The zeta function has zeros at -2,-4,-6 and also zeros on the real line Re(s)=1/2. The famous Riemann hypothesis claims that all the nontrivial zeros are on this line. This conjecture remains unproven until today and is considered one of the most important open problems in mathematics. +------------------------------------------------------------ | Log +------------------------------------------------------------ Log The logarithm is the inverse to the exponential function: log(exp(x)) = x and exp(log(x)) = x. For example: log(1)=0, log(e)=1. The logarithm function satisfies for example the laws log(x y) = log(x) + log(y), log(x/y)=log(x)-log(y), log(x^y)=y log(x). +------------------------------------------------------------ | Sqrt +------------------------------------------------------------ Sqrt The square root of a number x is the number which This file is part of the Sofia project sponsored by the Provost's fund for teaching and learning at Harvard university. There are 24 entries in this file. COUNT: 24 ENTRY GROUP THEORY Authors: started Mark Lezama: October 2003 Literature: "Algebra" by Michael Artin, Mathworld +------------------------------------------------------------ | Group theory +------------------------------------------------------------ Group theory is studies algebraic objects called groups. The German mathematician Karl Friedrich Gauss (1777-1855) developed but did not publish some of the mathematics of group theory. The French mathematician Evariste Galois (1811-1832) is generally credited with being the first to develop the theory, which he did by developing new techniques to study the solubility of equations. Group theory is a powerful method for analyzing abstract and physical systems in which symmetry --the intrinsic property of an object to remain invariant under certain classes of transformations-- is present because the mathematical study of symmetry is systematized and formalized in group theory. Consequently, group theory is an important tool in physics particularly in quantum mechanics. +------------------------------------------------------------ | group +------------------------------------------------------------ A group is an object consisting of a set G and a law of composition (or binary operation) L on G satisfying: L is associative. L has an identity in G. Every element of G has an inverse. The study of groups is known as group theory. If a group G has n elements where n is a positive integer, then G is a finite group with order n. If a group is not finite it is infinite. Examples: Z^+, the integers under addition; R^+ = (R, +), the real numbers under addition; R^x = (R - 0, .), the real numbers without zero under multiplication; GL_n(C), the nx n general linear group under matrix multiplication; S_n, the symmetric group on n objects under composition. +------------------------------------------------------------ | law of composition +------------------------------------------------------------ A law of composition or, binary operation, on a set S is a function from Sx S into S. That is, a law of composition on S prescribes a rule for combining pairs of elements in S to get an element in S. For convenience, functional notation is not used; that is, if a law of composition f sends (a, b) to c, one does not usually write f(a, b) = c. It is customary to instead use notation that resembles that used for multiplication or addition of real numbers, such as ab = c, a. b = c, a circ b = c, a + b = c, and so on. An example of a law of composition is multiplication on the real numbers, R. If m colon R x R to R defines multiplication on R then m(x, y) = x . y. For example m(2, 5) = 2 . 5 = 10. +------------------------------------------------------------ | binary operation +------------------------------------------------------------ A binary operation, or law of composition, on a set S is a function from S x S into S. That is, a binary operation on S prescribes a rule for combining pairs of elements in S to get an element in S. For convenience, functional notation is not used; that is, if a binary operation f sends (a, b) to c, one does not usually write f(a, b) = c. It is customary to instead use notation that resembles that used for multiplication or addition of real numbers, such as ab = c, a. b = c, a circ b = c, a + b = c, and so on. An example of a binary operation is multiplication on the real numbers, R. If mcolon R x R to R defines multiplication on R then m(x, y) = x . y. For example m(2, 5) = 2 . 5 = 10. +------------------------------------------------------------ | associative +------------------------------------------------------------ A law of composition on a set S is associative if for all a, b, c in S, (ab)c = a(bc). The informal intuition behind associativity (the property of being associative) is that if one has an expression in which there are many parentheses and the only operation performed in this expression is that defined by an associative law of composition, then one may ignore the parentheses. For example, if . is an associative law of composition on S and a, b, c, d in S, then ((a.(b. c)). d = ((a. b). c) . d = (a. b) . (c . d) and so on; thus one may write a . b . c . d without being ambiguous. An example of an associative law of composition is addition on the integers, Z. That is, for all a, b, c in Z, (a + b) + c = a + (b + c). +------------------------------------------------------------ | identity +------------------------------------------------------------ An identity for a law of composition on a set S is an element e such that, for all ain S, ea = a and ae = a. Note that a law of composition has at most one identity. The symbols e, 0 and 1 are commonly used to denote the identity element of a group. The number 0 is an identity for addition on the real numbers. +------------------------------------------------------------ | identity +------------------------------------------------------------ Suppose a set S has a law of composition with identity 1. For every element ain S, if there exists an element bin S such that ab = 1 and ba = 1 then b is the inverse of a. When using multiplicative notation for the law of composition, the inverse of a can be written as a^-1. As an example, the inverse of any integer n is -n where the law of composition is addition and the identity is 0. As another example, the inverse of any nonzero real number x is frac1 x, where the law of composition is multiplication and the identity is 1. +------------------------------------------------------------ | general linear group +------------------------------------------------------------ The n x n general linear group GL_n(F) is the set of n x n matrices with entries in the field F and nonzero determinant, under the law of composition of matrix multiplication. Thus GL_n(F) is the group of n x n invertible matrices with entries in F. If F is a finite field of field order q then sometimes the general linear group GL_n(F) is denoted by GL_n(q). The general linear group often appears with respect to the real numbers, R, or the complex numbers, C; that is, the general linear group often appears as GL_n(R) or GL_n(C). The special linear group SL_n(F) is the subgroup of GL_n(F) whose elements have determinant equal to 1. +------------------------------------------------------------ | special linear group +------------------------------------------------------------ The n x n special linear group SL_n(F) is the set of n x n matrices with entries in the field F and determinant equal to 1, under the law of composition of matrix multiplication. If F is a finite field of field order q then sometimes the special linear group SL_n(F) is denoted by SL_n(q). SL_n(F) is a subgroup of the general linear group GL_n(F). +------------------------------------------------------------ | trivial +------------------------------------------------------------ A group is trivial if it contains exactly one element. The one element in the group is the identity element. As all trivial groups are isomorphic, one usually refers to a trivial group as emphthe trivial group. A group that is not trivial is nontrivial. +------------------------------------------------------------ | trivial +------------------------------------------------------------ The group containing exactly one element (the identity) is unique up to isomorphism and is therefore called the trivial group. The trivial group is a normal subgroup of every group. +------------------------------------------------------------ | nontrivial +------------------------------------------------------------ A group is nontrivial if it is not trivial. +------------------------------------------------------------ | abelian +------------------------------------------------------------ A group is abelian if its law of composition is commutative. Examples of abelian groups include the following: R^+ = (R, +), the real numbers under addition; R^x = (R - 0, .), the real numbers without zero under multiplication; any cyclic group. Examples of nonabelian groups, i.e. groups that are not abelian: GL_n(C), the general linear group; The symmetric group on n objects, where n is a positive integer greater than 2. +------------------------------------------------------------ | commutative +------------------------------------------------------------ A law of composition on a set S is commutative if for all a, b in S ab = ba. An example of a commutative law of composition is addition on the real numbers: for example, 3.2 + 4 = 7.2 = 4 + 3.2. +------------------------------------------------------------ | cancellation Law +------------------------------------------------------------ The cancellation Law states that if a, b, and c are elements of a group and if ab = ac then b = c. Similarly, if ba = ca then b = c. The Cancellation Law follows from the fact that every element of a group has an inverse. +------------------------------------------------------------ | permutation +------------------------------------------------------------ If S is a set, then a permutation of S is a bijective map from S into S. The intuition underlying the definition of a permutation is that a permutation determines a reordering of the elements in a list or the rearrangement of objects. For example, the permutation sigma: 1, 2, 3 to 1, 2, 3 defined by sigma(1) = 2, sigma(2) = 1, and sigma(3) = 3 can be thought to represent the reordering of the list 1,2,3 that results in the list 2,1,3. There is an important kind of permutation called a transposition. A transposition of a set S is a permutation sigmacolon S to S satisfying the following: there exist s_1, s_2 in S such that sigma(s_1) = s_2 sigma(s_2) = s_1 and for all sin S, if s neq s_1 and s neq s_2, then sigma(s) = s. Every permutation of a finite set can be written as the composition of a finite number of transpositions of that set. For example, the permutation sigma: 1, 2, 3 to 1, 2, 3 defined by sigma(1) = 2, sigma(2) = 3, and sigma(3) = 1 is equivalent to the composition of two transpositions. Define sigma_1: 1, 2, 3 to 1, 2, 3 by sigma_1(1) = 2, sigma_1(2) = 1, and sigma(3) = 3, and define sigma_2: 1, 2, 3 to 1, 2, 3 by sigma_2(1) = 3, sigma_1(2) = 2, and sigma(3) = 1. Then sigma_1 and sigma_2 are transpositions and sigma = sigma_2 circ sigma_1. The sign of a permutation sigma is (-1)^n where n is a finite positive integer such that there exist n transpositions whose composition equals sigma. If a permutation has sign 1, then it is called an even permutation. If a permutation has sign -1 then it is called an odd permutation. Thus the identity permutation is an even permutation (since it is equal to the composition of any transposition with itself), and any transposition is an odd permutation (since it is equal to one transposition). The sign map encapsulates the notion of the sign of a permutation of a finite set. The set of permutations on a set forms a group where the law of composition is composition of functions. One example of a group of permutations that appears frequently in group theory is the symmetric group on n objects, i.e. the group of permutations of the set 1, 2, ldots, n. +------------------------------------------------------------ | transposition +------------------------------------------------------------ A transposition of a set S is a permutation sigmacolon S to S satisfying the following: there exist s_1, s_2 in S such that sigma(s_1) = s_2 sigma(s_2) = s_1 and for all sin S, if s neq s_1 and s neq s_2, then sigma(s) = s. Every permutation of a finite set can be written as the composition of a finite number of transpositions of that set. For example, the permutation sigma: 1, 2, 3 to 1, 2, 3 defined by sigma(1) = 2, sigma(2) = 3, and sigma(3) = 1 is equivalent to the composition of two transpositions. Define sigma_1: 1, 2, 3 to 1, 2, 3 by sigma_1(1) = 2, sigma_1(2) = 1, and sigma(3) = 3, and define sigma_2: 1, 2, 3 to 1, 2, 3 by sigma_2(1) = 3, sigma_1(2) = 2, and sigma(3) = 1. Then sigma_1 and sigma_2 are transpositions and sigma = sigma_2 circ sigma_1. Every transposition is an odd permutation. +------------------------------------------------------------ | sign +------------------------------------------------------------ The sign of a permutation sigma is (-1)^n where n is a finite positive integer such that there exist n transpositions whose composition equals sigma. If a permutation has sign 1, then it is called an even permutation. If a permutation has sign -1 then it is called an odd permutation. Thus the identity permutation is an even permutation (since it is equal to the composition of any transposition with itself), and any transposition is an odd permutation (since it is equal to one transposition). +------------------------------------------------------------ | even permutation +------------------------------------------------------------ An even permutation is a permutation that has sign 1. That is, an even permutation is the composition of an even number of transpositions. Thus the identity permutation is an even permutation. +------------------------------------------------------------ | odd permutation +------------------------------------------------------------ An odd permutation is a permutation that has sign -1. That is, an odd permutation is the composition of an odd number of transpositions. Thus every tranposition is an odd permutation. +------------------------------------------------------------ | symmetric group +------------------------------------------------------------ The symmetric group on n objects, denoted S_n, is the group of permutations of the set 1, 2, ldots, n; the law of composition is composition of functions. The order of S_n is n! for all positive integers n. For example, S_2 = e, sigma, where e is the identity permutation, and sigma is a transposition. That is, e is the identity element of S_2 and is defined by e(1) = 1 and e(2) = 2; sigma is defined by sigma(1) = 2 and sigma(2) = 1. +------------------------------------------------------------ | sign map +------------------------------------------------------------ The sign map, denoted sign, is a group homomorphism from the symmetric group, S_n, into the group 1, -1 (under multiplication). The sign map is defined by sign(sigma) = (-1)^k where sigma is equal to the composition of k transpositions. The sign map is well-defined because it is a standard result that if sigma is any permutation (of any set), and if sigma is equal to the composition of k transpositions and is also equal to the composition of m transpositions, then (-1)^k = (-1)^m. The kernel of the sign map is the alternating group, A_n; that is, A_n is the group of even permutations on n objects. +------------------------------------------------------------ | alternating group +------------------------------------------------------------ The alternating group is the kernel of the sign map. In other words, the alternating group on n objects, usually denoted A_n, is a normal subgroup of the symmetric group on n objects: A_n = sigma in S_n mid sign(sigma) = 1 . Thus A_n is the group of even permutations in S_n. For n geq 5, A_n is a simple group, i.e. a group that has no proper normal subgroup. +------------------------------------------------------------ | simple group +------------------------------------------------------------ A group G is a simple group if every normal subgroup N of G is not a proper subgroup. That is, G is simple if its only normal subgroups are G and the trivial group. The alternating group A_n is simple for n geq 5. Any cyclic group of prime order is simple. Any cyclic group of prime order is simple. In fact, any simple abelian group is a cyclic group of prime order. +------------------------------------------------------------ | subgroup +------------------------------------------------------------ A subset H of a group G is a subgroup of G if it satisfies the following properties: If ain H and bin H, then ab in H. 1in H, where 1 is the identity element of G. If ain H then the inverse of a, a^-1, is also in H. When it is clear that G is a group, sometimes H subseteq G is used to denote that H is a subgroup of G (as opposed to merely being a subset of G). Every nontrivial group G has at least two subgroups: the whole group G and the subgroup 1 consisting exactly of the identity element of G. If G is trivial then these two subgroups are the same and G has exactly one subgroup. A subgroup is a proper subgroup if it is neither the whole group nor the trivial group. As an example, the integers under addition are a subgroup of the real numbers under addition. By Lagrange's Group Theorem, if H is a subgroup of a finite group G, the order of H divides the order of G. +------------------------------------------------------------ | proper subgroup +------------------------------------------------------------ A proper subgroup of a group G is a nontrivial subgroup of G that is not equal to G. +------------------------------------------------------------ | order +------------------------------------------------------------ A finite group G is said to have order n if G has n elements. More generally, the order of a group G is the cardinality of the set G, both of which are often denoted |G|. For any given element x of a given group, if there exists a positive integer k such that x^k = 1, then x is said to have order m, where m is the least positive integer satisfying x^m = 1. If x^k neq 1 for all positive integers k, then x is said to have infinite order. +------------------------------------------------------------ | cyclic group +------------------------------------------------------------ If G is a group and if x is an element of G, the cyclic group langle x rangle generated by x is the set of all powers of x: langle x rangle = ldots, x^-2, x^-1, 1, x, x^2, ldots. Note that langle x rangle is the smallest subgroup of G which contains x. Further note that any cyclic group is abelian. If x has infinite order then langle x rangle is said to be infinite cyclic. Note that if langle x rangle is infinite cyclic then langle x rangle is isomorphic to Z^+, the integers under addition. As a result, one sometimes refers to any infinite cyclic group as emphthe infinite cyclic group, denoted Z^+. If x has order n, then langle x rangle has order n and is called a cyclic group of order n: langle x rangle = 1, x, x^2, ldots, x^n. If langle x rangle is a cyclic group of order n, then langle x rangle is isomorphic to Z/n, where Z/n is the group satisfying the following properties (we use additive notation as opposed to multiplicative notation for the law of composition of Z/n): 1. Z/n = 0, 1, 2, ldots, n - 1. 2. for any x, y in Z/n, x +_1 y is the unique element in Z/n which is congruent modulo n to x + y, where +_1 denotes the law of composition on Z/n and + denotes conventional integer addition. Normally + is used to denote the law of composition on Z/n, but +_1 is used here to distinguish it from conventional addition. Two integers a and b are congruent modulo n, written aequiv b (mboxmodulo n), if n divides b - a. As a result, one sometimes refers to any cyclic group of order n as emphthe cyclic group of order n, often denoted Z/n. +------------------------------------------------------------ | cyclic group +------------------------------------------------------------ Let G_1 and G_2 be groups. A map varphicolon G_1 to G_2 is a group homomorphism if varphi(ab) = varphi(a)varphi(b) for all a,bin G_1. Here we use the same multiplicative notation for the laws of composition of G_1 and G_2, even though there is no requirement that their laws of composition be the same. Note that varphi(1_G_1)= 1_G_2 and varphi(a^-1) = varphi(a)^-1 for all ain G, where 1_G_i is the identity of G_i. The kernel of varphi, sometimes denoted ker varphi, is the set xin G_1 mid varphi(x) = 1_G_2 . Note that ker varphi is a normal subgroup of G_2. Another subgroup of G_2 determined by varphi is the image of varphi, sometimes denoted im varphi. The image of varphi is im varphi = xin G_2 mid x = varphi(a) for some a in G_1. Sometimes the image of varphi is denoted varphi(G_1). The following are examples of homomorphisms: The inclusion map icolon H to G defined by i(a) = a, where H is a subgroup of G. ker i = 1_G and im i = H. For a fixed a in G, the map varphicolon Z^+ to G defined by varphi(n) = a^n, where Z^+ denotes the integers with addition. ker varphi = n mid a^n = 1_G and im varphi = langle a rangle (the cyclic subgroup generated by a). The determinant map detcolon GL_n(R) to R^x, where GL_n(R) denotes the general linear group and R^x denotes the real numbers without zero under multiplication. ker det = SL_n(R), the special linear group, and im det = R^x. The sign map on permutations signcolon S_n to 1, -1, where S_n denotes the symmetric group on n objects. ker sign = A_n, the alternating group, and im sign = 1, -1. Let R_1 and R_2 be rings. A map varphicolon R_1 to R_2 is a ring homomorphism if varphi(a + b) = varphi(a) + varphi(b), varphi(ab) = varphi(a)varphi(b), and varphi(1_R_1) = 1_R_2, for all a,bin R_1. Here we use the same additive and multiplicative notation for the laws of composition of R_1 and R_2, even though there is no requirement that their laws of composition be the same. +------------------------------------------------------------ | kernel +------------------------------------------------------------ The kernel of a group homomorphism varphicolon G_1 to G_2 is the set ker varphi = x in G_1 mid varphi(x) = 1_G_2 , where 1_G_2 denotes the identity of G_2. The kernel of a homomorphism is an important example of a normal subgroup. There are many results involving the kernel of a homomorphism. +------------------------------------------------------------ | image +------------------------------------------------------------ The image of a map varphicolon G_1 to G_2 is the set xin G_2 mid x = varphi(a) for some a in G_1. In general, the image of varphi is often denoted varphi(G_1). If varphi is a group homomorphism, then the image of varphi is a subgroup of G_2 and is sometimes denoted im varphi. +------------------------------------------------------------ | isomorphism +------------------------------------------------------------ A group isomorphism is a bijective group homomorphism. A ring isomorphism is a bijective ring homomorphism. +------------------------------------------------------------ | isomorphic +------------------------------------------------------------ Two groups G_1 and G_2 are isomorphic if there exists a group isomorphism from G_1 into G_2. Sometimes G_1 cong G_2 is used to denote `G_1 and G_2 are isomorphic.' Note that cong is an equivalence relation on the set of all groups. When one speaks of classifying groups, hat is usually referred to is the classification of isomorphism classes. Thus one might say that there are two groups of order 6 emphup to isomorphism, meaning that there are two isomorphism classes of groups of order 6. One sometimes says that `G_1 is isomorphic to G_2' instead of saying `G_1 and G_2 are isomorphic.' +------------------------------------------------------------ | automorphism +------------------------------------------------------------ An automorphism of a group G is an isomorphism from G into G. The identity map is a simple example of an automorphism. Conjugation by an element of the group is an important example of an automorphism. That is, for a fixed element bin G, conjugation by b is the map varphicolon Gto G defined by varphi(a) = bab^-1. Here we use multiplicative notation for the group law of composition. Note that if G is abelian, then conjugation by any element is the identity map. However, if G is not abelian, then there exists a nontrivial conjugation (i.e. a conjugation not equal to the identity map) of G. +------------------------------------------------------------ | automorphism +------------------------------------------------------------ Let G be a group and let bin G. The map varphicolon Gto G defined by varphi(a) = bab^-1 is conjugation by b. Note that conjugation is an automorphism of G. Further note that if G is abelian, then conjugation by any element is the identity map. However, if G is not abelian, then there exists a nontrivial conjugation (i.e. a conjugation not equal to the identity map) of G. +------------------------------------------------------------ | conjugation +------------------------------------------------------------ Let G be a group and let H be a subgroup of G. H is a normal subgroup of G (sometimes written H triangleleft G) if for all a in H and for all x in G, xax^-1in H. Note that it follows that any subgroup of an abelian group is normal. Normal subgroups appear often in group theory. Every group G has at least one normal subgroup, called the center of G, denoted by Z or Z(G). The center of G is the set of elements that commute with every element of G: Z(G) = zin G mid zx = xz for all xin G. Another important example of a normal subgroup is the kernel of a group homomorphism. +------------------------------------------------------------ | center +------------------------------------------------------------ The center of a group G, denoted by Z or Z(G) is the set of elements that commute with every element of G: Z(G) = zin G mid zx = xz for all xin G. Note that if G is abelian then Z(G) = G. +------------------------------------------------------------ | coset +------------------------------------------------------------ Given a subgroup H of a group G, a coset of H is a subset H' of G such that there exists an a in G such that (1) H' = aH = ah mid h in H, in which case H' is said to be a left coset; or (2) H' = Ha = ha mid h in H, in which case H' is said to be a right coset. Given a in G, aH is not necessarily equal to Ha. However, one can show that the subgroup H of G is a normal subgroup if and only if aH = Ha for every a in G. In what follows, only left cosets will be discussed, though similar statements may be made about right cosets. The left cosets of H are the equivalence classes of the equivalence relation sim defined by a sim b if there exists h in H such that a = bh. Since equivalence classes form a partition, the left cosets of H partition G. The cardinality of the set of left cosets of H is called the index of H in G and is denoted by G:H. Given a in G, h mapsto ah defines a bijective map from H into aH. If G is finite, it follows that |G| = |H|G:H, where |G| denotes the order of G. A very important result follows: if G is finite, then the order of H divides the order of G. Moreover, since the order of any element of G is the order of the cyclic subgroup it generates, if G is finite then the order of an element of G divides the order of G. These results follow from a special case of what is known as Lagrange's Group Theorem: if G is a group, H is a subgroup of G and K is a sugroup of H, then G:K = G:HH:K, where the products are taken as products of cardinals. An important result that follows from Lagrange's Theorem is that if the order of G is a prime number then G = langle a rangle for any a in G such that a is not the identity, where langle a rangle denotes the cyclic group generated by a. Note that if varphi colon G to G' is a group homomorphism, then G: kervarphi = |im varphi|. Thus another result of Langrange's Theorem is that |G| = |ker varphi| . |im varphi|. +------------------------------------------------------------ | left coset +------------------------------------------------------------ Given a subgroup H of a group G, a left coset of H is a subset H' of G such that there exists an a in G such that H' = aH = ah mid h in H. Given a in G, aH is not necessarily equal to Ha. However, one can show that the subgroup H of G is a normal subgroup if and only if aH = Ha for every a in G. In what follows, only left cosets will be discussed, though similar statements may be made about right cosets. The left cosets of H are the equivalence classes of the equivalence relation sim defined by a sim b if there exists h in H such that a = bh. Since equivalence classes form a partition, the left cosets of H partition G. The cardinality of the set of left cosets of H is called the index of H in G and is denoted by G:H. Given a in G, h mapsto ah defines a bijective map from H into aH. If G is finite, it follows that |G| = |H|G:H, where |G| denotes the order of G. A very important result follows: if G is finite, then the order of H divides the order of G. Moreover, since the order of any element of G is the order of the cyclic subgroup it generates, if G is finite then the order of an element of G divides the order of G. These results follow from a special case of what is known as Lagrange's Group Theorem: if G is a group, H is a subgroup of G and K is a sugroup of H, then G:K = G:HH:K, where the products are taken as products of cardinals. An important result that follows from Lagrange's Theorem is that if the order of G is a prime number then G = langle a rangle for any a in G such that a is not the identity, where langle a rangle denotes the cyclic group generated by a. Note that if varphi colon G to G' is a group homomorphism, then G: ker(varphi) = |im(varphi)|. Thus another result of Langrange's Theorem is that |G| = |ker(varphi)| . |im(varphi)|. +------------------------------------------------------------ | right coset +------------------------------------------------------------ Given a subgroup H of a group G, a right coset of H is a subset H' of G such that there exists an a in G such that H' = Ha = ha mid h in H. Given a in G, aH is not necessarily equal to Ha. However, one can show that the subgroup H of G is a normal subgroup if and only if aH = Ha for every a in G. In what follows, only left cosets will be discussed, though similar statements may be made about right cosets. The left cosets of H are the equivalence classes of the equivalence relation sim defined by a sim b if there exists h in H such that a = bh. Since equivalence classes form a partition, the left cosets of H partition G. The cardinality of the set of left cosets of H is called the index of H in G and is denoted by G:H. Given a in G, h mapsto ah defines a bijective map from H into aH. If G is finite, it follows that |G| = |H|G:H, where |G| denotes the order of G. A very important result follows: if G is finite, then the order of H divides the order of G. Moreover, since the order of any element of G is the order of the cyclic subgroup it generates, if G is finite then the order of an element of G divides the order of G. These results follow from a special case of what is known as Lagrange's Group Theorem: if G is a group, H is a subgroup of G and K is a sugroup of H, then G:K = G:HH:K, where the products are taken as products of cardinals. An important result that follows from Lagrange's Theorem is that if the order of G is a prime number then G = langle a rangle for any a in G such that a is not the identity, where langle a rangle denotes the cyclic group generated by a. Note that if varphi colon G to G' is a group homomorphism, then G:ke(varphi) = |im(varphi)|. Thus another result of Langrange's Theorem is that |G| = |ker(varphi)| . |im(varphi)|. +------------------------------------------------------------ | index +------------------------------------------------------------ The index of subgroup H of a group G is the cardinality of the set of left cosets of H in G. The index of H in G is denoted G:H. +------------------------------------------------------------ | quotient group +------------------------------------------------------------ Given a group G and a normal subgroup N of G, the quotient group of N in G, written G / N and read ``G mod(ulo) N'', is the set of cosets of N in G, under the law of composition that is defined as follows: (aN)(bN) = abN, where xN = xn mid n in N. Note that since N is normal, aN = Na for all a in G, so it is not necessary to define this law of composition in terms of left cosets instead of right cosets. The order of G / N is the index G:N of N in G. Quotient groups can be identified by the First Isomorphism Theorem: if varphi colon G to G' is a surjective group homomorphism and if N = ke(varphi) then psicolon G / N to G' is an isomorphism, where psi is defined by psi(aN) = varphi(a). +------------------------------------------------------------ | First Isomorphism Theorem +------------------------------------------------------------ The First Isomorphism Theorem. Suppose varphi colon G to G' is a surjective group homomorphism, and let N denote the kernel of varphi. Then the quotient group G / N is isomorphic to G' by the map psi defined by psi(aN) = varphi(a). The First Isomorphism Theorem is the principle method of identifying quotient groups. As an example, consider the group homomorphism varphi from C^x, the nonzero complex numbers under multiplication, into R^x, the nonzero real numbers under multiplication, defined by varphi(z) = |z|, where |z| denotes the absolute value of z. The kernel of varphi is the unit circle, U, and the image of varphi is the group of positive real numbers. So C^x / U is isomorphic to the multiplicative group of positive real numbers. +------------------------------------------------------------ | operation +------------------------------------------------------------ Given a group G and a set S, an operation of a G on S is a map from G x S into S - often written using multiplicative notation: (g,s) mapsto gs - satisfying: 1 s = s for all s in S, where 1 is the identity of G; and (g g')s = g(g's), for all g, g' in G and for all s in S. There are some terms that are sometimes associated with a group operation: S is often called a G-set; G is sometimes called a transformation group; and the group operation is often also called a group action. Mathworld: "Historically, the first group action studied was the action of the Galois group on the roots of a polynomial. However, there are numerous examples and applications of group actions in many branches of mathematics, including algebra, topology, geometry, number theory, and analysis, as well as the sciences, including chemistry and physics." This file is part of the Sofia project sponsored by the Provost's fund for teaching and learning at Harvard university. There are 44 entries in this file. COUNT: 44 ENTRY HARVARD Authors: Oliver Knill: 2000, Literature: no +------------------------------------------------------------ | Harvard +------------------------------------------------------------ Harvard +------------------------------------------------------------ | Science center +------------------------------------------------------------ Science center The science center is the Polaroid camera shape building near Harvard square. The actual building is close to Memorial Hall. I actually live and think in the science center. You can virtually walk into the science center +------------------------------------------------------------ | president of Harvard +------------------------------------------------------------ president of Harvard The President of Harvard is currently Lawrence H. Summers. To find out more about him, visit the website +------------------------------------------------------------ | chairman of math department +------------------------------------------------------------ chairman of math department The chairman of the Mathematics department is currently Joe Harris. +------------------------------------------------------------ | number people math department +------------------------------------------------------------ number people math department Currently, there are over 190 people at the Math department. +------------------------------------------------------------ | preceptor +------------------------------------------------------------ preceptor Preceptors work alongside other faculty on teaching, developing and supporting sections of entry level courses at the Harvard Mathematics department. +------------------------------------------------------------ | ca +------------------------------------------------------------ ca A course assistant (CA) is an undergraduate student who assists the teaching fellow (TF) with grading, running problem sessions and tutoring in the question center. +------------------------------------------------------------ | tf +------------------------------------------------------------ tf TF stands for teaching fellow. Everybody who is teaching is called a TF. It can be a senior or junior faculty, a visiting fellow or a graduate student. +------------------------------------------------------------ | concentrator +------------------------------------------------------------ concentrator A (math) concentrator is a sophomore or senior undergraduate student. A math concentrator would also be called a math major. There are about one hundred math concentrators. Each year, there are about 30 new concentrators. +------------------------------------------------------------ | head tutor +------------------------------------------------------------ head tutor The head tutor is a professor who is the chief undergraduate advisor. +------------------------------------------------------------ | question center +------------------------------------------------------------ question center The question center QC is a place to work on homework problems or exam preparation. Tutors (both course assistants or teaching fellows) are available for questions. +------------------------------------------------------------ | qc +------------------------------------------------------------ qc see question center +------------------------------------------------------------ | math table +------------------------------------------------------------ math table The math table is a "dinner seminar" which takes place in one of the student houses. A faculty or student presents a half an hour talk just after a dinner. +------------------------------------------------------------ | grade +------------------------------------------------------------ grade Grades are an important issue for most students. Unfortunately, I have no access to your grades. +------------------------------------------------------------ | where harvard +------------------------------------------------------------ where harvard Harvard University is located in Cambridge, Massachusetts USA. We are on the east cost. One can see from here Boston. The Campus is quite close to the Charles River. To look up something specific, it is best to start online with the Harvard search page. +------------------------------------------------------------ | get into Harvard +------------------------------------------------------------ get into Harvard Work hard, have lots of interests. You need of course some luck. +------------------------------------------------------------ | life at Harvard +------------------------------------------------------------ life at Harvard Fun. Beside a great academic environment, there are a lot of things to see. You can spend weeks at Harvard square and still find new things. +------------------------------------------------------------ | grade inflation +------------------------------------------------------------ grade inflation Grade inflation is when most students get A's. One of the problems with grade inflation is that grades start losing their purpose. +------------------------------------------------------------ | computer type +------------------------------------------------------------ computer type At the Mathematics department, people use all kind of operating systems Sun work stations running Solaris Macintoshs running OSX PC's running Linux PC's running Windows. +------------------------------------------------------------ | software +------------------------------------------------------------ software Have a look at http://www.math.harvard.edu/computing. +------------------------------------------------------------ | mathematics +------------------------------------------------------------ mathematics Mathematics is both a science and an art. It is also a language for other sciences. Quite many topics in Mathematics have turned out to be useful. Examples is the theory of operators which provided the framework for Quantum mechanics. An other example is number theory which provides the foundation for many encryption algorithms. +------------------------------------------------------------ | afread of math +------------------------------------------------------------ afread of math You probably had some bad experiences in the past. You should chat more with me! +------------------------------------------------------------ | learn math +------------------------------------------------------------ learn math Just do it! There are hundreds of nice books about Math, many resources on the internet. Take a math class with a good teacher. +------------------------------------------------------------ | physics and math +------------------------------------------------------------ physics and math There are not many differences. Indeed, there are branches of Mathematics like computational number theory where people do a lot of experiments and there are branches of physics, where people do very abstract theory which presumably never can be tested in laboratories. +------------------------------------------------------------ | charles river +------------------------------------------------------------ charles river A great place to row, run bike and relax. +------------------------------------------------------------ | harvard yard +------------------------------------------------------------ harvard yard the center of the Harvard campus +------------------------------------------------------------ | math professors +------------------------------------------------------------ math professors They are all excellent Mathematicians. +------------------------------------------------------------ | pi day +------------------------------------------------------------ pi day +------------------------------------------------------------ | student number +------------------------------------------------------------ student number There are more than 18'000 degree candidates at Harvard. +------------------------------------------------------------ | Harvard founded +------------------------------------------------------------ Harvard founded Harvard College was established in 1636, already 16 years after the arrival of the pilgrims at Plymouth. +------------------------------------------------------------ | people at harvard +------------------------------------------------------------ people at harvard There are over 14'000 people at Harvard including more than 2'000 faculty. There are also 7,000 faculty appointments in affiliated teaching hospitals. +------------------------------------------------------------ | Nobel Laureates +------------------------------------------------------------ Nobel Laureates Harvard produced nearly 40 Nobel Laureates. +------------------------------------------------------------ | US presidents +------------------------------------------------------------ US presidents Harvard produced seven presidents of the United States: John Adams, John Quincy Adams, Theodore and Franklin Delano Roosevelt, Rutherford B. Hayes, John Fitzgerald Kennedy and George W. Bush. +------------------------------------------------------------ | Millenium prize problems +------------------------------------------------------------ Millenium prize problems P versus NP Hodge Conjecture Poincare Conjecture Riemann Hypothesis Yang-Mills Existence and Mass Gap Navier-Stokes Existence and Smoothness Birch-Swinnerton-Dyer Conjecture +------------------------------------------------------------ | Core course +------------------------------------------------------------ Core course In 1978, Harvard adopted a 'core' of courses in fields of inquiry that spanned domains, including historical study, moral reasoning, social analysis, science, music and art, literature and so on. These courses are designed to introduce 'approaches to knowledge' rather than specific information and thus legitimized a trend throughout education toward ways of knowing rather than knowledge. This file is part of the Sofia project sponsored by the Provost's fund for teaching and learning at Harvard university. There are 35 entries in this file. COUNT: 35 ENTRY JOKES Authors: Oliver Knill: 2002 Literature: not yet +------------------------------------------------------------ | JOKE GARAGE SALE +------------------------------------------------------------ JOKE GARAGE SALE Pride is what you feel when your kids net 143 dollars from a garage sale. Panic is what you feel when you realize your car is missing. +------------------------------------------------------------ | JOKE DOORBELL +------------------------------------------------------------ JOKE DOORBELL A priest was walking down the street when he saw a little boy jumping up and down to try to reach a doorbell. So the priest walked over and pressed the button for the youngster. "And now what, my little man?" he asked. "Now." said the boy, "run like hell!" +------------------------------------------------------------ | JOKE FAMOUS LAST WORDS +------------------------------------------------------------ JOKE FAMOUS LAST WORDS postman: "good doggy, nice doggy" butcher: "could you throw me the big knife, please?" computer: "are you sure? (yes/no)" stuntman: "what? reality TV?" doorman: "only over my dead body" detective" "clear case: you are the murderer" muchroom picker: "I never saw this one" boss: "nice present, a lighter which looks like a revolver" submarine crew: "I need some fresh air, open the window" sysadmin: "I recently had a fresh backup" student: "I'm going to eat in the mensa, anybody coming?" bungee jumper: "hurrey!" PC: "loading windows - please wait" +------------------------------------------------------------ | JOKE PAINT JOB +------------------------------------------------------------ JOKE PAINT JOB There was a college student trying to earn some pocket money by going from house to house offering to do odd jobs. He explained this to a man who answered one door. "How much will you charge to paint my porch?" asked the man. "Forty dollars." "Fine" said the man, and gave the student the paint and brushes. Three hours later the paint-splattered lad knocked on the door again. "All done!", he says, and collects his money. "By the way," the student says, "That's not a Porsche, it's a Ferrari." This file is part of the Sofia project sponsored by the Provost's fund for teaching and learning at Harvard university. There are 4 entries in this file. COUNT: 4 ENTRY K12 Authors: Oliver Knill: 2000 Literature: not yet +------------------------------------------------------------ | abacus +------------------------------------------------------------ An abacus is an acient mechanical computing device. It is made of beads arranged on a frame. +------------------------------------------------------------ | absolute value +------------------------------------------------------------ The absolute value |n| of a real number n is the maximum of n and its negative -n. For example, the absolute value of -6 is |-6|=6. The absolute value is the distance from 0. +------------------------------------------------------------ | adjacent angles +------------------------------------------------------------ Two angles that share a ray are called adjacent angles. +------------------------------------------------------------ | affine cipher +------------------------------------------------------------ An affine cipher uses affine functions to scramble the letters in an alphabet of a secret message. For example, with an alphabet of 26 letters, f(x) = bx+a= 5 x + 2 mod(26) produces a new alphabet of the same size if b has no common multiple with 26. The simplest example is the Caesar cipher, where b=2. It rotates the letters in an alphabet x mapsto x+a mod(26). For example, for a=1, we get a b c d e f g h i j k l m n o p q r s t u v w x y z b c d e f g h i j k l m n o p q r s t u v w x y z a This cipher changes the word hello to the word ifmmp. A requently used Caesar cipher is "rot13" defined by f(x)=x+13 mod(13). It has the property that encryption and decryption are the same. For example, applying rot13 on the word "decryption" produces qrpelcgvba and applying rot13 on that word again gives back "decryption. More complicated versions of affine cyphers can be obtained by writing the to encoded text as a sequence of vectors x and then applying Ax+b on each vector. Affine cyphers are very easy to crack. They are only used to illustrate the concept like for educational purposes. +------------------------------------------------------------ | algebra +------------------------------------------------------------ algebra is a branch of elementary mathematics that generalizes arithmetic by using variables. An example of an algebraic identity is x*(y+z)=x*y+x*z. +------------------------------------------------------------ | acute +------------------------------------------------------------ An angle is called acute, if is smaller than 90 degrees. An angle which is 90 degrees is called a right angle. +------------------------------------------------------------ | addition +------------------------------------------------------------ addition is a basic operation for numbers. The result is called the sum of the two numbers. Examples: 5+3=8. 2 3 4 5 + 9 2 3 5 1 1 5 8 0 More generally, a group operation in a commutative group is often called addition. Examples of groups are integers, real numbers, vectors or matrices. +------------------------------------------------------------ | alternate exterior angles +------------------------------------------------------------ alternate exterior angles are angles located outside a set of two parallel lines and on opposite sides of the transversal line. They are equal. +------------------------------------------------------------ | alternate interior angles +------------------------------------------------------------ alternate interior angles are angles located inside a set of parallel lines and on opposite sides of the transversal. +------------------------------------------------------------ | angle bisector +------------------------------------------------------------ A ray that divides an angle into two equal angles is called an angle bisector. The bisector can be constructed with ruler and compass. An angle trisector on the other hand, a ray which splits an angle into three equal parts can not be constructed by ruler and compass. +------------------------------------------------------------ | apex +------------------------------------------------------------ The apex is the highest vertex in a given orientation of a polygon. +------------------------------------------------------------ | Arabic numerals +------------------------------------------------------------ Arabic numerals: symbols 0,1,2,3,4,5,6,7,8,9 that represent successive entries of words representing numbers in the decimal system. For example, 2347 is the number 2000+300+40+7. +------------------------------------------------------------ | area +------------------------------------------------------------ The area of a surface is a measure for the number of square units needed to cover the surface. For example, the sphere of radius 1 has the surface area 4 pi. +------------------------------------------------------------ | arithmetic mean +------------------------------------------------------------ arithmetic mean Given two numbers a,b, the arithmetic mean is defined as (a+b)/2. It is sometimes also called the mean. Other means are the geometric mean sqrta b or the harmonic mean 1/(1/a+1/b). +------------------------------------------------------------ | average +------------------------------------------------------------ The average of a few numbers is the sum of all the numbers divided by the number of numbers. For example, the average of 2,4,6 is (2+4+6)/3=4, the average of the numbers 1,5,8,4 is (1+5+8+5)/3 = 19/3 The average is also called the mean. The average of two numbers is also called the arithmetic mean. +------------------------------------------------------------ | base +------------------------------------------------------------ A base is the number of distinct single-digit numbers in a counting system. Example: the binary system has base 2. The decimal system has base 10. The base is also called radix. Numbers can be represented in any base r>1. Because humans have 10 fingers, the decimal system is the one favoured by this species. Because computers work with circuits which are based on the principle "on" or "off", they like the base 2. The hexadecimal system (base 16) or octal system (base 8) are also used a lot by computers. Modern computers can even work directly with numbers written in base 32 or 64. +------------------------------------------------------------ | bell curve +------------------------------------------------------------ The bell curve is an other term for graph of the normal distribution f(x) = frac1sqrtpi e^-x^2. It is also called the Gauss distribution. The bell curve is often seen in probability distributions. There is a reason for that called the central limit theorem which assures that if we average independent data with some distribution, we approach the normal distribution. +------------------------------------------------------------ | billion +------------------------------------------------------------ A billion is one thousand millions in the American or French system, it is a million millions in the English or German system. In other words One billion in UK,Germany: 10^12 1'000'000'000'000 One billion in US,France: 10^9 1'000'000'000 +------------------------------------------------------------ | binary number +------------------------------------------------------------ A binary number is a number expressed in place-value notation to the base 2. For example: 101101 represents the decimal number 1+0+4+8+0+32 = 45. +------------------------------------------------------------ | cipher +------------------------------------------------------------ cipher Ciphers are codes used to encrypt "secret" messages. +------------------------------------------------------------ | coefficient +------------------------------------------------------------ The word coefficient is used to denote numbers in the front of the variables in an algebraic formula. For example: 4x + 5y =3 has coefficients 4,5. +------------------------------------------------------------ | combinatorics +------------------------------------------------------------ combinatorics The science of counting things. Combinatorics is an important part of probability and statistics. +------------------------------------------------------------ | common factor +------------------------------------------------------------ A common factor of two integers n and m is a number which is a factor of both. A common factor is also called a common divisor. Examples: 3 is a common factor of 18 and 27. Also, 9 is the greatest common factor of 18 and 27: we write 9= gcd(18,27), where gcd stands for the greatest common divisor. +------------------------------------------------------------ | complex numbers +------------------------------------------------------------ complex numbers can be written as a pair of real numbers z=x+iy, where i is a symbol which satisfies i^2=-1. One can add and subtract complex numbers by adding their coefficients x,y. For example 4+5i + 5-7i = 9-2i. +------------------------------------------------------------ | complementary angles +------------------------------------------------------------ Two angles whose sum is 90 degrees form complementary angles. For example, the two non-right angles in a right triangle form complementary angles. +------------------------------------------------------------ | concave up +------------------------------------------------------------ A graph of a function f is concave up if f has the property f((x+y)/2) leq (f(x) + f(y))/2. If f is concave up, then -f is concave down. For example, the graph of the function f(x)=x^2 is concave up, the graph of the function f(x)=-x^4 is concave down. +------------------------------------------------------------ | conditional probability +------------------------------------------------------------ The conditional probability is the probability that an event A happens provided a second event B occurs. One writes PA|B. It satisfies PA|B = PA cap B/PB, where PB is the probability of the event B and PA cap B probability of the intersection of A and B. For example, if we throw 2 coins and we know one of the coins is head H, then the probability that the there is also a coin with tail is 2/3. Proof: The probability space is X=HT,TH,TT,HH. The event that one of the coins is head is A=HT,TH,HH. The event B that one of the coins shows tail is B=HT,TH,TT. The intersection of B and A is TH,HT. We have PB | A = PB cap A/PA = (1/2)/(3/4)=2/3. +------------------------------------------------------------ | congruent +------------------------------------------------------------ Two figures are called congruent if one can move one to an other by translation and rotation. +------------------------------------------------------------ | constant +------------------------------------------------------------ A quantity that does not change in an equation is called a constant. +------------------------------------------------------------ | constant function +------------------------------------------------------------ A constant function is a function which takes the same value whatever input we enter to it. +------------------------------------------------------------ | coordinate +------------------------------------------------------------ A coordinate is an entry in a collection of numbers identifiing the point in coordinate space. +------------------------------------------------------------ | continuous graph +------------------------------------------------------------ The graph of a continuous function is called a continuous graph. Roughly speaking, a graph of a function defined on some interval a,b is a continuous graph if one can draw the graph using a pencil without having to the lift the pencil. Examples: 1/x is not continuous on -1,1. x^2+1 is continuous on -1,1. 1/x^2 is not continuous on -1,1. sin(1/x) is not continuous on -1,1. x sin(1/x) is continuous on -1,1. f'(x) is not continuous on -1,1 if f(x)=|x|. +------------------------------------------------------------ | corresponding angles +------------------------------------------------------------ corresponding angles are two angles in the same relative position on two straight lines when those lines are intersected by a a transversal straight line. +------------------------------------------------------------ | decimal number +------------------------------------------------------------ decimal number is a fraction, where the denominator is a power of 10. It can be expressed using a decimal point. For example: 0.872 is the decimal equivalent of 872/1000. +------------------------------------------------------------ | degrees +------------------------------------------------------------ An angle is often measured in degrees. The entire circle has 360 degrees, a half a circle is 180 degrees, a quarter circle is a right angle and has 90 degrees. A more natural unit is the length unit where the entire circle has angle 2pi and the right angle is the angle pi/2. +------------------------------------------------------------ | denominator +------------------------------------------------------------ The denominator is the integer q below the fraction in a rational number p/q. The other number p is called the nominator. +------------------------------------------------------------ | discontinuous graph +------------------------------------------------------------ A discontinuous graph is the graph of a function which is not continuous. Discontinuities can occur in different ways. The function can jump from one value to an other. The function can also be infinite at some point or the function can oscillate infinitely much at some point. Examples: The graph of the function f(x)=1/x on -1,1. The graph of the function f(x)=sin(1/x) on -1,1. The graph of the function f(x)=sign(x), which is 1 for x>0 equal to 0 for x=0 and -1 for x=-1. +------------------------------------------------------------ | disjoint events +------------------------------------------------------------ Two events are called disjoint events if they have no common elements. +------------------------------------------------------------ | division +------------------------------------------------------------ The inverse operation of multiplication is called division. +------------------------------------------------------------ | domain +------------------------------------------------------------ The domain of a function f is the set of numbers x for which f(x) is defined. For example, the domain of the function f(x)=1/x is the entire real line except the point 0. +------------------------------------------------------------ | element +------------------------------------------------------------ An element of a set is is a member of that set. For example table is an element of the set table, chair, floor . +------------------------------------------------------------ | empty set +------------------------------------------------------------ The empty set emptyset is the set which does not contain any elements. +------------------------------------------------------------ | equally likely +------------------------------------------------------------ If two events have the same probability they are called equally likely. For example, the event of throwing an even number with one dice is equally likely than throwing an odd number. +------------------------------------------------------------ | event +------------------------------------------------------------ An event is a subset of the entire probability space. For example, if X=HH,HT,TH,TT is the probability space of all throwing of two coins, then A=HH,HT is the event that in the second throw one had a head. +------------------------------------------------------------ | exponent +------------------------------------------------------------ The exponent of an expression a^x is part x. One can get the exponent of y=a^x by the formula x=log(y)/log(a). +------------------------------------------------------------ | Fibonacci numbers +------------------------------------------------------------ Fibonacci numbers are numbers obtained in the Fibonacci sequence defined by starting the numbers 0,1 and defining the next element as the sum of the two previous ones: 0,1,1,2,3,5,8,13,21,34,.... The sequence is named after Leonardo of Pisa, who called himself Fibonacci, short for Filius Bonacci (= Son of Bonacci). The original problem he investigated in 1202 A.D. was the growth of rabbits. Explicit expressions for the n'th term of the sequence can be obtained using linear algebra. More generally, one can find explicit formulas for the n'th term in a linear recursion of the form a_n+1=sum_j=0^k c_j a_n-j. +------------------------------------------------------------ | fractal +------------------------------------------------------------ A fractal is a set which has non-integer dimension. The term was coined by Benoit Mandelbrot in 1975. Many objects in nature appear to be fractals, like coast lines, trees, mountains. One can mathematically define fractals using iterative constructions. Examples are the Koch curve, the snow flake, the Menger Sponge, the Shirpinsky carpet, the Cantor set. +------------------------------------------------------------ | fraction +------------------------------------------------------------ A fraction is a rational number written in the form a/b, where a is called the numerator and b is called the denominator. +------------------------------------------------------------ | function +------------------------------------------------------------ A function f of a variable x is a rule that assigns to each number x in the function's domain a single number f(x). For example f(x)=x^2 is a function which assigns to each number its square like f(4)=16. +------------------------------------------------------------ | geometric sequence +------------------------------------------------------------ The geometric sequence is a sequence where each element is a multiple of the previous element. For example: 1,2,4,8,16,32,64,... is a geometric sequence. +------------------------------------------------------------ | graph of the function +------------------------------------------------------------ The graph of the function is the set of all points (x,f(x)) in the plane, where x in the domain of f. +------------------------------------------------------------ | greatest common factor +------------------------------------------------------------ The greatest common factor of two numbers n,m is the larest common factor of both. One denotes the greatest common factor with "gcd". Examples: 6 is the greatest common factor of 12 and 18 6=gcd(12,18) 8 is the greatest common factor of 8 and 80. 8=gcd(8,80) 1 is the greatest common factor of 7 and 11 1=gcd(7,11) +------------------------------------------------------------ | greatest common divisor +------------------------------------------------------------ greatest common divisor see greatest common factor. +------------------------------------------------------------ | histogram +------------------------------------------------------------ A histogram is a bar graph in which area over each range of values is proportional to the relative frequency of the data in this interval. +------------------------------------------------------------ | hypotenuse +------------------------------------------------------------ The hypotenuse of a right triangle is the opposite side to the right angle. +------------------------------------------------------------ | independent events +------------------------------------------------------------ Two events A and B are called independent events if the probability that both happen is the product of the probabilities that each occurs alone: PA cap B = P(A) P(B). Using conditional probability one can write this as PA|B=PA. Knowing A under the condition B is the same as knowing A without knowing B. +------------------------------------------------------------ | infinity +------------------------------------------------------------ infinity is a "number" which is larger than any other number. One writes infinity. One should rather treat of it as a symbol evenso some computations can be extended to the real numbers including infinity like infinity + x = infinity, infinity + infinity=infinity, x * infinity=infinity for x>0, x * infinity=-infinity for x<0 or infinity * infinity=infinity, (-infinity) * infinity=-infinity. One can not define infinity - infinity in a consistent way nor can one do that with 0 * infinity. Also the expression 1/0 = infinity is ill defined because 1/x takes near x=0 arbitrary large and arbitrary small values. +------------------------------------------------------------ | integer +------------------------------------------------------------ An integer is a number of the form n or -n, where n is a natural number. Examples of integers are ...-3,2,1,0,1,2,3,4.... The fraction 2/5 is not an integer. +------------------------------------------------------------ | intersection +------------------------------------------------------------ The intersection of two or more sets is the set of elements which are in both sets. One writes A cap B for the intersection of A and B. +------------------------------------------------------------ | isosceles triangle +------------------------------------------------------------ An isosceles triangle is a triangle which has at least two congruent sides. A special case is the isocline triangle in which all sides are congruent. +------------------------------------------------------------ | least common multiple +------------------------------------------------------------ The least common multiple of two numbers n,m is the least common multiple of both. One denotes the least common multiple with "lcm". Examples: 18 is the least common multiple of 9 and 6 18=lcm(9,6) 77 is the least common multiple of 7 and 11 77=gcd(7,11) +------------------------------------------------------------ | limit +------------------------------------------------------------ The limit of a sequence of numbers is the limiting value the sequence converges to. It needs not to exist. For example, the sequence a_n = 1/n converges to 0. One sais that 0 is the limit of that sequence. The sequence a_n=n has no finite limit. One could assign infinity as a limit. The sequence 1,-1,1,-1,1,-1,... has no limit. +------------------------------------------------------------ | logarithm +------------------------------------------------------------ The logarithm of b is the exponent to which one has to rize a base number to get b. For example, 2 is the logarithm of 100 to the base 10 or 10 is the logarithm of 1024 to the base 2. +------------------------------------------------------------ | mean +------------------------------------------------------------ The mean of a list of numbers is their sum divided by the total number of members in the list. It is also called arithmetic mean. +------------------------------------------------------------ | median +------------------------------------------------------------ The median is the "middle value" of a list. If the list has an odd number 2m+1 elements, the median is the number in the list such that m scores are smaller or equal and m scores are bigger or equal. If the list has an even number of elements, one usually takes the algebraic average between the middle to elements Examples: med(1,1,2,2)=3/2, med(1,2,3,4,7)=3, med(1,2,3,4,5,6)=7/2. +------------------------------------------------------------ | multiplication table +------------------------------------------------------------ multiplication table A table of products of numbers which has to be memorized. l|llllllllll 1 2 3 4 5 6 7 8 9 10 1 1 2 3 4 5 6 7 8 9 10 2 2 4 6 8 10 12 14 16 18 20 3 3 6 9 12 15 18 21 24 27 30 4 4 8 12 16 20 24 28 32 36 40 5 5 10 15 20 25 30 35 40 45 50 6 6 12 18 24 30 36 42 48 54 60 7 7 14 21 28 35 42 49 56 63 70 8 8 16 24 32 40 48 56 64 72 80 9 9 18 27 36 45 54 63 72 81 90 10 10 20 30 40 50 60 70 80 90 100 The diagonal contains squares. All numbers between 11 and 99 which do not appear in this table are prime numbers, numbers only divisible by 1 and itself. +------------------------------------------------------------ | obtuse angle +------------------------------------------------------------ An angle whose measure is greater than 90 degrees is called an obtuse angle. +------------------------------------------------------------ | optical illusion +------------------------------------------------------------ An optical illusion is a drawing of an object that makes certain things appear which it does not have. +------------------------------------------------------------ | palindrome +------------------------------------------------------------ A palindrome is a word or number that is the same when read backwards. Examples: "otto", "anna", "racecar", "78777787". +------------------------------------------------------------ | paradox +------------------------------------------------------------ A paradox is a statement that appears to contradict itself. For example, the statement "I always lie" is a paradox. If I tell the truth, then I lie, if I lie, then I tell the truth. +------------------------------------------------------------ | parallel +------------------------------------------------------------ Two lines which do not intersect are called parallel. +------------------------------------------------------------ | parallelogram +------------------------------------------------------------ A parallelogram is a quadrilateral that contains two pairs of parallel sides. +------------------------------------------------------------ | pattern +------------------------------------------------------------ A pattern is a characteristic observed in one item that may be repeated in other items. For example, the sequence 3,4,5,4,3,4,5,4,3,4,5,4,3,... has a pattern which is also visible in a similar way in the sequence 1,3,4,3,1,3,4,3,1,3,4,3,1,.... +------------------------------------------------------------ | percent +------------------------------------------------------------ A percent is one hunderth. The symbol for percent is %. For example 0.1 is 10 percent. 2 is two hundred percent. +------------------------------------------------------------ | perimeter +------------------------------------------------------------ The perimeter of a polygon is the sum of the lengths of all the sides of the polygon. +------------------------------------------------------------ | permutation +------------------------------------------------------------ A permutation is a rearrangement of objects in a set. There are for example 6 permutations of the set A=(a,b,c). They are (a,b,c),(a,c,b),(b,a,c),(b,c,a),(c,a,b),(c,b,a). +------------------------------------------------------------ | polygon +------------------------------------------------------------ A polygon is a closed plane figure formed by connecting a finite set of points in such a way that they do not cross each other. +------------------------------------------------------------ | polyhedra +------------------------------------------------------------ polyhedra A solid figure for which the outer surface is composed of polygons. +------------------------------------------------------------ | prime number +------------------------------------------------------------ A prime number is a number which is divisible only by 1 and itself. The first prime numbers are 2,3,5,7,11,13,17,19,23,29,31,37.... The number 33 for example is no prime number because it is divisible by 3. +------------------------------------------------------------ | quadrant +------------------------------------------------------------ A quadrant is one of the regions in the plane obtained when cutting the plane along the coordinate axes. The first quadrant contains all the points with positive x and positive y coordinates. The second quadrant contains all the points with negative x and positive y coordinates. The third quadrant contains all the points with negative x and negative y coordinates. The fourth quadrant contains all the points with positive x and negative y coordinates. +------------------------------------------------------------ | quadratic function +------------------------------------------------------------ A function of the form f(x)=ax^2+bx+c is called a quadratic function. For example f(x)=x^2+2 is a quadratic function. The graph of a quadratic function is a parabola if a is not zero. If a is zero it is a linear function which has as the graph a line. +------------------------------------------------------------ | quotient +------------------------------------------------------------ The quotient of two numbers n and m is the largest integer smaller or equal to n/m. for example the quotient of 11 and 4 is 2 with areminder of 3. +------------------------------------------------------------ | smallest common multiple +------------------------------------------------------------ smallest common multiple see least common multiple. +------------------------------------------------------------ | subtraction +------------------------------------------------------------ subtraction is a basic operation for numbers. Examples: 5-3=2. 2 3 5 2 - 1 4 5 6 8 9 6 More generally in any commutative group, the addition of the inverse -b of an element b to an element a, denoted by a-b is called a subtraction. +------------------------------------------------------------ | multiplication +------------------------------------------------------------ multiplication is a basic operation for numbers. Example: 4*5=20. 1 2 4 x 1 2 3 3 4 + 4 9 3 3 6 + 2 4 6 6 8 + 1 2 3 3 4 1 5 2 9 4 1 6 More generally, in any group the group operation is called multiplication. Example: The composition of two transformations in the plane is a multiplication, matrix multiplication is a multiplication. +------------------------------------------------------------ | fraction +------------------------------------------------------------ fraction: The representation of a rational number r as p/q, where p and q neq 0 are integers is called a fraction. Fractions can be added, subtracted, multiplied and one can divide by nonzero fractions. Addition of fractions: 5/3+7/5=25/15+21/15=46/15 Subtraction of fractions: 5/3-7/5=25/15-21/15=4/15 Multiplication of fractions: 5/3 x 7/5 = 35/15 Division of fractions: 5/3 : 7/5 = 5/3 x 5/7 = 25/21 Not all numbers are rational numbers. Non rational numbers are called irrational. For example, sqr(2) can not be written as p/q because sqrt(2)=p/q would imply 2=p^2/q^2 or 2 q^2=p^2. But by the uniqueness of prime factorisation of integers the left hand side has an odd number of factors 2, the right hand side an even number. sqrt(2) is an example of an irrational number. +------------------------------------------------------------ | hexagon +------------------------------------------------------------ A hexagon is a polygon with six sides. +------------------------------------------------------------ | parallelogram +------------------------------------------------------------ A parallelogram is a quadrilateral whose opposite sides are parallel and congruent. +------------------------------------------------------------ | polygon +------------------------------------------------------------ A polygon is a closed curve in the plane formed by three or more line segments. One usually assumes that the segments don't intersect. Examples: 3 sides: triangle 4 sides: quadrilateral, (i.e. rectangle, rhombus, rhombus) 5 sides: pentagon 6 sides: hexagon 7 sides: septagon 8 sides: octogon +------------------------------------------------------------ | quadrilateral +------------------------------------------------------------ A quadrilateral is a polygon with four sides. +------------------------------------------------------------ | parallel +------------------------------------------------------------ Two lines in the plane are called parallel if they do not intersect. Two parallel lines can be translated into each other. Two lines in space are called parallel if they can be translated into each other. Unlike in the plane, two lines in space which are not parallel do not need to intersect. +------------------------------------------------------------ | triangle +------------------------------------------------------------ A triangle is a polygon defined by three points in the plane. The three points form the edges of the triangles, the three connections of the points form the sides of the triangle. +------------------------------------------------------------ | random number generator +------------------------------------------------------------ A random number generator is a device used to produce random numbers. In daily life like for gambling, one often uses dice or coin tossing to find random numbers. Computers often use pseudo random number generators, which are deterministic sequences which look random. Computers can also access hardware internal staes of the computer to improve randomness. +------------------------------------------------------------ | range of a function +------------------------------------------------------------ The range of a function is the set of all values f(x), where x is in the domain of f. +------------------------------------------------------------ | ratio +------------------------------------------------------------ ratio A rational number of the form a/b where a is called the numerator and b is called the denominator. +------------------------------------------------------------ | rectangle +------------------------------------------------------------ A rectangle is a parallelogram with four right angles. It is a quadrilateral, a polygon with four points in the plane. All angles have to be right angles. In a rectangle, opposite sides are parallel. A rectangle is therefore a special parallelogram. +------------------------------------------------------------ | regular polygon +------------------------------------------------------------ A regular polygon is a polygon which has sides of equal length and equal angles. Squares, equilateral triangles or regular hexagons are examples of regular polygons. +------------------------------------------------------------ | remainder +------------------------------------------------------------ The remainder of a division p/q is the amount left after subtracting the maximal integer multiple of q from p. For example 7/3 has the reminder 1 because 7-2*3=1. -11/5 has the reminder -1. +------------------------------------------------------------ | rhombus +------------------------------------------------------------ A rhombus is a parallelogram with four congruent sides. A special case is the square. +------------------------------------------------------------ | right angle +------------------------------------------------------------ An angle of 90 degrees is called a right angle. +------------------------------------------------------------ | right triangle +------------------------------------------------------------ A triangle which has a right angle is also called a right triangle. +------------------------------------------------------------ | sequence +------------------------------------------------------------ An ordered list of elements is called a sequence For example, (1,3,2,1) is a list of elements which form a finite sequence. The list (1,2,3,4,5,6,....) forms an infinite sequence. +------------------------------------------------------------ | set +------------------------------------------------------------ A set is a collection of things, without regard to their order. +------------------------------------------------------------ | slope +------------------------------------------------------------ The slope of a line y = mx + b is the number m. One often measures it in percentages. m=1 means 100 percent. If a street has a slope of 10 percent, one climbes for every 10 meters going forwards 1 meter up. +------------------------------------------------------------ | square +------------------------------------------------------------ A square is a polygonal shape in the plane with four sides where each side has the same length and all sides are perpendicular on each other. A square is also a number of the form n*n like 65=8*8. A square with integer side length has a square number as the area. +------------------------------------------------------------ | subset +------------------------------------------------------------ A subset of a set is a set of elements which are all contained in that set. For example the set A=1,2,3,8,4 has B=2,3 as a subset. +------------------------------------------------------------ | subtraction +------------------------------------------------------------ subtraction is the operation of taking the difference between two numbers. For example 7-2=5. +------------------------------------------------------------ | tessellation +------------------------------------------------------------ A tessellation is a cover of the plane using a finite set of polygons without leaving gaps or overlaps. Examples are regular tessellation into triangles or squares or regular hexagons. Semiregular tesselations allow to cover the plane with different types of shapes. Tesselations are also called tilings and can be defined also in higher dimensions. +------------------------------------------------------------ | trapezoid +------------------------------------------------------------ A trapezoid is a quadrilateral with one pair of parallel sides. +------------------------------------------------------------ | union of sets +------------------------------------------------------------ The union of sets is the set which contains the elements of all sets. One writes A cup B. For example, if A=1,2,3,4 and B=0,2,4,6, then A cup B = 0,1,2,3,4,6. +------------------------------------------------------------ | Venn Diagram +------------------------------------------------------------ In a Venn Diagram sets are represented as simple geometric shapes. It visualizes intersections and unions of sets. For example if A is the set of all even numbers between 0 and 10 and B is the set of all numbers divisible by 3 between 0 and 10 one can visualize this with two circles, one of which contains 2,5,8,6, the other 3,6,9. The circles intersect in a region which has the single element 6. +------------------------------------------------------------ | whole number +------------------------------------------------------------ A whole number is one of the numbers 0,1,2,3,4,... A whole number is also called natural number or nonnegative integer. This file is part of the Sofia project sponsored by the Provost's fund for teaching and learning at Harvard university. There are 112 entries in this file. COUNT: 112 you sound tired I worked all night +------------------------------------------------------------ | This is great +------------------------------------------------------------ This is great this is fantastic +------------------------------------------------------------ | ok +------------------------------------------------------------ ok not ok +------------------------------------------------------------ | switzerland +------------------------------------------------------------ switzerland country center europe +------------------------------------------------------------ | Germany +------------------------------------------------------------ Germany country in Europe +------------------------------------------------------------ | Austria +------------------------------------------------------------ Austria country in Europe This file is part of the Sofia project sponsored by the Provost's fund for teaching and learning at Harvard university. There are 5 entries in this file. COUNT: 5 Famous Literature List from http://www.literaturepage.com +------------------------------------------------------------ | A Christmas Carol +------------------------------------------------------------ A Christmas Carol Charles Dickens +------------------------------------------------------------ | A Connecticut Yankee in King Arthur's Court +------------------------------------------------------------ A Connecticut Yankee in King Arthur's Court Mark Twain +------------------------------------------------------------ | A Dream Within a Dream +------------------------------------------------------------ A Dream Within a Dream Edgar Allan Poe +------------------------------------------------------------ | A Midsummer Night's Dream +------------------------------------------------------------ A Midsummer Night's Dream William Shakespeare +------------------------------------------------------------ | A Princess of Mars +------------------------------------------------------------ A Princess of Mars Edgar Rice Burroughs +------------------------------------------------------------ | A Room With a View +------------------------------------------------------------ A Room With a View E. M. Forster +------------------------------------------------------------ | A Study in Scarlet +------------------------------------------------------------ A Study in Scarlet Sir Arthur Conan Doyle +------------------------------------------------------------ | A Tale of Two Cities +------------------------------------------------------------ A Tale of Two Cities Charles Dickens +------------------------------------------------------------ | A Thief in the Night +------------------------------------------------------------ A Thief in the Night E. W. Hornung +------------------------------------------------------------ | A Treatise on Government +------------------------------------------------------------ A Treatise on Government Aristotle +------------------------------------------------------------ | A Woman of No Importance +------------------------------------------------------------ A Woman of No Importance Oscar Wilde +------------------------------------------------------------ | A Woman of Thirty +------------------------------------------------------------ A Woman of Thirty Honore de Balzac +------------------------------------------------------------ | Adventures of Sherlock Holmes +------------------------------------------------------------ Adventures of Sherlock Holmes Sir Arthur Conan Doyle +------------------------------------------------------------ | Aesop's Fables +------------------------------------------------------------ Aesop's Fables Aesop +------------------------------------------------------------ | Agnes Grey +------------------------------------------------------------ Agnes Grey Anne Bronte +------------------------------------------------------------ | Alice's Adventures in Wonderland +------------------------------------------------------------ Alice's Adventures in Wonderland Lewis Carroll +------------------------------------------------------------ | All's Well That Ends Well +------------------------------------------------------------ All's Well That Ends Well William Shakespeare +------------------------------------------------------------ | Allan Quatermain +------------------------------------------------------------ Allan Quatermain H. Rider Haggard +------------------------------------------------------------ | Allan's Wife +------------------------------------------------------------ Allan's Wife H. Rider Haggard +------------------------------------------------------------ | An Ideal Husband +------------------------------------------------------------ An Ideal Husband Oscar Wilde +------------------------------------------------------------ | An Occurrence at Owl Creek Bridge +------------------------------------------------------------ An Occurrence at Owl Creek Bridge Ambrose Bierce +------------------------------------------------------------ | Andersen's Fairy Tales +------------------------------------------------------------ Andersen's Fairy Tales Hans Christian Andersen +------------------------------------------------------------ | Anna Karenina +------------------------------------------------------------ Anna Karenina Leo Tolstoy +------------------------------------------------------------ | Annabel Lee +------------------------------------------------------------ Annabel Lee Edgar Allan Poe +------------------------------------------------------------ | Anne of Green Gables +------------------------------------------------------------ Anne of Green Gables L. M. Montgomery +------------------------------------------------------------ | Antony and Cleopatra +------------------------------------------------------------ Antony and Cleopatra William Shakespeare +------------------------------------------------------------ | Apology +------------------------------------------------------------ Apology Plato +------------------------------------------------------------ | Around the World in Eighty Days +------------------------------------------------------------ Around the World in Eighty Days Jules Verne +------------------------------------------------------------ | As You Like It +------------------------------------------------------------ As You Like It William Shakespeare +------------------------------------------------------------ | At the Earth's Core +------------------------------------------------------------ At the Earth's Core Edgar Rice Burroughs +------------------------------------------------------------ | Barnaby Rudge +------------------------------------------------------------ Barnaby Rudge Charles Dickens +------------------------------------------------------------ | Berenice +------------------------------------------------------------ Berenice Edgar Allan Poe +------------------------------------------------------------ | Black Beauty +------------------------------------------------------------ Black Beauty Anna Sewell +------------------------------------------------------------ | Bleak House +------------------------------------------------------------ Bleak House Charles Dickens +------------------------------------------------------------ | Buttered Side Down +------------------------------------------------------------ Buttered Side Down Edna Ferber +------------------------------------------------------------ | Cousin Betty +------------------------------------------------------------ Cousin Betty Honore de Balzac +------------------------------------------------------------ | Crime and Punishment +------------------------------------------------------------ Crime and Punishment Fyodor Dostoevsky +------------------------------------------------------------ | Daisy Miller +------------------------------------------------------------ Daisy Miller Henry James +------------------------------------------------------------ | David Copperfield +------------------------------------------------------------ David Copperfield Charles Dickens +------------------------------------------------------------ | Dead Men Tell No Tales +------------------------------------------------------------ Dead Men Tell No Tales E. W. Hornung +------------------------------------------------------------ | Discourse on the Method +------------------------------------------------------------ Discourse on the Method Rene Descartes +------------------------------------------------------------ | Dorothy and the Wizard in Oz +------------------------------------------------------------ Dorothy and the Wizard in Oz L. Frank Baum +------------------------------------------------------------ | Dracula +------------------------------------------------------------ Dracula Bram Stoker +------------------------------------------------------------ | Eight Cousins +------------------------------------------------------------ Eight Cousins Louisa May Alcott +------------------------------------------------------------ | Emma +------------------------------------------------------------ Emma Jane Austen +------------------------------------------------------------ | Essays and Lectures +------------------------------------------------------------ Essays and Lectures Oscar Wilde +------------------------------------------------------------ | Essays of Francis Bacon +------------------------------------------------------------ Essays of Francis Bacon Sir Francis Bacon +------------------------------------------------------------ | Essays, First Series +------------------------------------------------------------ Essays, First Series Ralph Waldo Emerson +------------------------------------------------------------ | Essays, Second Series +------------------------------------------------------------ Essays, Second Series Ralph Waldo Emerson +------------------------------------------------------------ | Ethan Frome +------------------------------------------------------------ Ethan Frome Edith Wharton +------------------------------------------------------------ | Cousin Betty +------------------------------------------------------------ Cousin Betty Honore de Balzac +------------------------------------------------------------ | Lady Windermere's Fan +------------------------------------------------------------ Lady Windermere's Fan Oscar Wilde +------------------------------------------------------------ | Lincoln's First Inaugural Address +------------------------------------------------------------ Lincoln's First Inaugural Address Abraham Lincoln +------------------------------------------------------------ | Lincoln's Second Inaugural Address +------------------------------------------------------------ Lincoln's Second Inaugural Address Abraham Lincoln +------------------------------------------------------------ | Little Men +------------------------------------------------------------ Little Men Louisa May Alcott +------------------------------------------------------------ | Little Women +------------------------------------------------------------ Little Women Louisa May Alcott +------------------------------------------------------------ | Macbeth +------------------------------------------------------------ Macbeth William Shakespeare +------------------------------------------------------------ | Main Street +------------------------------------------------------------ Main Street Sinclair Lewis +------------------------------------------------------------ | Mansfield Park +------------------------------------------------------------ Mansfield Park Jane Austen +------------------------------------------------------------ | Memoirs of Sherlock Holmes +------------------------------------------------------------ Memoirs of Sherlock Holmes Sir Arthur Conan Doyle +------------------------------------------------------------ | Middlemarch +------------------------------------------------------------ Middlemarch George Eliot +------------------------------------------------------------ | Moby Dick +------------------------------------------------------------ Moby Dick Herman Melville +------------------------------------------------------------ | Moll Flanders +------------------------------------------------------------ Moll Flanders Daniel Defoe +------------------------------------------------------------ | Much Ado About Nothing +------------------------------------------------------------ Much Ado About Nothing William Shakespeare +------------------------------------------------------------ | Night and Day +------------------------------------------------------------ Night and Day Virginia Woolf +------------------------------------------------------------ | Northanger Abbey +------------------------------------------------------------ Northanger Abbey Jane Austen +------------------------------------------------------------ | Nostromo +------------------------------------------------------------ Nostromo Joseph Conrad +------------------------------------------------------------ | O Pioneers! +------------------------------------------------------------ O Pioneers! Willa Cather +------------------------------------------------------------ | Of Human Bondage +------------------------------------------------------------ Of Human Bondage W. Somerset Maugham +------------------------------------------------------------ | Oliver Twist +------------------------------------------------------------ Oliver Twist Charles Dickens +------------------------------------------------------------ | On the Decay of the Art of Lying +------------------------------------------------------------ On the Decay of the Art of Lying Mark Twain +------------------------------------------------------------ | On the Duty of Civil Disobedience +------------------------------------------------------------ On the Duty of Civil Disobedience Henry David Thoreau +------------------------------------------------------------ | Othello, Moor of Venice +------------------------------------------------------------ Othello, Moor of Venice William Shakespeare +------------------------------------------------------------ | Ozma of Oz +------------------------------------------------------------ Ozma of Oz L. Frank Baum +------------------------------------------------------------ | Pandora +------------------------------------------------------------ Pandora Henry James +------------------------------------------------------------ | Paradise Lost +------------------------------------------------------------ Paradise Lost John Milton +------------------------------------------------------------ | Persuasion +------------------------------------------------------------ Persuasion Jane Austen +------------------------------------------------------------ | Poems of Edgar Allan Poe +------------------------------------------------------------ Poems of Edgar Allan Poe Edgar Allan Poe +------------------------------------------------------------ | Pollyanna +------------------------------------------------------------ Pollyanna Eleanor H. Porter +------------------------------------------------------------ | Pride and Prejudice +------------------------------------------------------------ Pride and Prejudice Jane Austen +------------------------------------------------------------ | Pygmalion +------------------------------------------------------------ Pygmalion George Bernard Shaw +------------------------------------------------------------ | Raffles: Further Adventures of the Amateur Cracksman +------------------------------------------------------------ Raffles: Further Adventures of the Amateur Cracksman E. W. Hornung +------------------------------------------------------------ | Rebecca Of Sunnybrook Farm +------------------------------------------------------------ Rebecca Of Sunnybrook Farm Kate Douglas Wiggin +------------------------------------------------------------ | Robinson Crusoe +------------------------------------------------------------ Robinson Crusoe Daniel Defoe +------------------------------------------------------------ | Romeo and Juliet +------------------------------------------------------------ Romeo and Juliet William Shakespeare +------------------------------------------------------------ | Rose in Bloom +------------------------------------------------------------ Rose in Bloom Louisa May Alcott +------------------------------------------------------------ | Sense and Sensibility +------------------------------------------------------------ Sense and Sensibility Jane Austen +------------------------------------------------------------ | Silas Marner +------------------------------------------------------------ Silas Marner George Eliot +------------------------------------------------------------ | Tarzan of the Apes +------------------------------------------------------------ Tarzan of the Apes Edgar Rice Burroughs +------------------------------------------------------------ | Tess of the d'Urbervilles +------------------------------------------------------------ Tess of the d'Urbervilles Thomas Hardy +------------------------------------------------------------ | The Adventures of Huckleberry Finn +------------------------------------------------------------ The Adventures of Huckleberry Finn Mark Twain +------------------------------------------------------------ | The Adventures of Pinocchio +------------------------------------------------------------ The Adventures of Pinocchio Carlo Collodi +------------------------------------------------------------ | The Adventures of Tom Sawyer +------------------------------------------------------------ The Adventures of Tom Sawyer Mark Twain +------------------------------------------------------------ | The Aeneid +------------------------------------------------------------ The Aeneid Virgil +------------------------------------------------------------ | The Age of Innocence +------------------------------------------------------------ The Age of Innocence Edith Wharton +------------------------------------------------------------ | The Amateur Cracksman +------------------------------------------------------------ The Amateur Cracksman E. W. Hornung +------------------------------------------------------------ | The Analysis of Mind +------------------------------------------------------------ The Analysis of Mind Bertrand Russell +------------------------------------------------------------ | The Ballad of Reading Gaol +------------------------------------------------------------ The Ballad of Reading Gaol Oscar Wilde +------------------------------------------------------------ | The Bells +------------------------------------------------------------ The Bells Edgar Allan Poe +------------------------------------------------------------ | The Call of the Wild +------------------------------------------------------------ The Call of the Wild Jack London +------------------------------------------------------------ | The Cask of Amontillado +------------------------------------------------------------ The Cask of Amontillado Edgar Allan Poe +------------------------------------------------------------ | The Categories +------------------------------------------------------------ The Categories Aristotle +------------------------------------------------------------ | The Chessmen of Mars +------------------------------------------------------------ The Chessmen of Mars Edgar Rice Burroughs +------------------------------------------------------------ | The Comedy of Errors +------------------------------------------------------------ The Comedy of Errors William Shakespeare +------------------------------------------------------------ | The Count of Monte Cristo +------------------------------------------------------------ The Count of Monte Cristo Alexandre Dumas +------------------------------------------------------------ | The Emerald City of Oz +------------------------------------------------------------ The Emerald City of Oz L. Frank Baum +------------------------------------------------------------ | The Fall of the House of Usher +------------------------------------------------------------ The Fall of the House of Usher Edgar Allan Poe +------------------------------------------------------------ | The Four Million +------------------------------------------------------------ The Four Million O. Henry +------------------------------------------------------------ | The Gambler +------------------------------------------------------------ The Gambler Fyodor Dostoevsky +------------------------------------------------------------ | The Gettysburg Address +------------------------------------------------------------ The Gettysburg Address Abraham Lincoln +------------------------------------------------------------ | The Gods of Mars +------------------------------------------------------------ The Gods of Mars Edgar Rice Burroughs +------------------------------------------------------------ | The Happy Prince and Other Tales +------------------------------------------------------------ The Happy Prince and Other Tales Oscar Wilde +------------------------------------------------------------ | The History of Tom Jones, a foundling +------------------------------------------------------------ The History of Tom Jones, a foundling Henry Fielding +------------------------------------------------------------ | The History of Troilus and Cressida +------------------------------------------------------------ The History of Troilus and Cressida William Shakespeare +------------------------------------------------------------ | The Hound of the Baskervilles +------------------------------------------------------------ The Hound of the Baskervilles Sir Arthur Conan Doyle +------------------------------------------------------------ | The Hunchback of Notre Dame +------------------------------------------------------------ The Hunchback of Notre Dame Victor Hugo +------------------------------------------------------------ | The Hunting of the Snark +------------------------------------------------------------ The Hunting of the Snark Lewis Carroll +------------------------------------------------------------ | The Idiot +------------------------------------------------------------ The Idiot Fyodor Dostoevsky +------------------------------------------------------------ | The Iliad +------------------------------------------------------------ The Iliad Homer +------------------------------------------------------------ | The Importance of Being Earnest +------------------------------------------------------------ The Importance of Being Earnest Oscar Wilde +------------------------------------------------------------ | The Innocence of Father Brown +------------------------------------------------------------ The Innocence of Father Brown G. K. Chesterton +------------------------------------------------------------ | The Innocents Abroad +------------------------------------------------------------ The Innocents Abroad Mark Twain +------------------------------------------------------------ | The Invisible Man +------------------------------------------------------------ The Invisible Man H. G. Wells +------------------------------------------------------------ | The Island of Doctor Moreau +------------------------------------------------------------ The Island of Doctor Moreau H. G. Wells +------------------------------------------------------------ | The Jungle Book +------------------------------------------------------------ The Jungle Book Rudyard Kipling +------------------------------------------------------------ | The Last Days of Pompeii +------------------------------------------------------------ The Last Days of Pompeii Edward Bulwer-Lytton +------------------------------------------------------------ | The Last of the Mohicans +------------------------------------------------------------ The Last of the Mohicans James Fenimore Cooper +------------------------------------------------------------ | The Legend of Sleepy Hollow +------------------------------------------------------------ The Legend of Sleepy Hollow Washington Irving +------------------------------------------------------------ | The Life and Adventures of Nicholas Nickleby +------------------------------------------------------------ The Life and Adventures of Nicholas Nickleby Charles Dickens +------------------------------------------------------------ | The Life and Death of King Richard III +------------------------------------------------------------ The Life and Death of King Richard III William Shakespeare +------------------------------------------------------------ | The Life of King Henry V +------------------------------------------------------------ The Life of King Henry V William Shakespeare +------------------------------------------------------------ | The Lost Continent +------------------------------------------------------------ The Lost Continent Edgar Rice Burroughs +------------------------------------------------------------ | The Lost World +------------------------------------------------------------ The Lost World Sir Arthur Conan Doyle +------------------------------------------------------------ | The Man in the Iron Mask +------------------------------------------------------------ The Man in the Iron Mask Alexandre Dumas +------------------------------------------------------------ | The Man Upstairs and Other Stories +------------------------------------------------------------ The Man Upstairs and Other Stories P. G. Wodehouse +------------------------------------------------------------ | The Man Who Knew Too Much +------------------------------------------------------------ The Man Who Knew Too Much G. K. Chesterton +------------------------------------------------------------ | The Man with Two Left Feet +------------------------------------------------------------ The Man with Two Left Feet P. G. Wodehouse +------------------------------------------------------------ | The Masque of the Red Death +------------------------------------------------------------ The Masque of the Red Death Edgar Allan Poe +------------------------------------------------------------ | The Merchant of Venice +------------------------------------------------------------ The Merchant of Venice William Shakespeare +------------------------------------------------------------ | The Merry Adventures of Robin Hood +------------------------------------------------------------ The Merry Adventures of Robin Hood Howard Pyle +------------------------------------------------------------ | The Merry Wives of Windsor +------------------------------------------------------------ The Merry Wives of Windsor William Shakespeare +------------------------------------------------------------ | The Moon and Sixpence +------------------------------------------------------------ The Moon and Sixpence W. Somerset Maugham +------------------------------------------------------------ | The Moonstone +------------------------------------------------------------ The Moonstone Wilkie Collins +------------------------------------------------------------ | The Murders in the Rue Morgue +------------------------------------------------------------ The Murders in the Rue Morgue Edgar Allan Poe +------------------------------------------------------------ | The Mysterious Affair at Styles +------------------------------------------------------------ The Mysterious Affair at Styles Agatha Christie +------------------------------------------------------------ | The Mystery of Edwin Drood +------------------------------------------------------------ The Mystery of Edwin Drood Charles Dickens +------------------------------------------------------------ | The Mystery of the Yellow Room +------------------------------------------------------------ The Mystery of the Yellow Room Gaston Leroux +------------------------------------------------------------ | The Odyssey +------------------------------------------------------------ The Odyssey Homer +------------------------------------------------------------ | The Origin of Species means of Natural Selection +------------------------------------------------------------ The Origin of Species means of Natural Selection Charles Darwin +------------------------------------------------------------ | The Phantom of the Opera +------------------------------------------------------------ The Phantom of the Opera Gaston Leroux +------------------------------------------------------------ | The Picture of Dorian Gray +------------------------------------------------------------ The Picture of Dorian Gray Oscar Wilde +------------------------------------------------------------ | The Pit and the Pendulum +------------------------------------------------------------ The Pit and the Pendulum Edgar Allan Poe +------------------------------------------------------------ | The Portrait of a Lady +------------------------------------------------------------ The Portrait of a Lady Henry James +------------------------------------------------------------ | The Purloined Letter +------------------------------------------------------------ The Purloined Letter Edgar Allan Poe +------------------------------------------------------------ | The Raven +------------------------------------------------------------ The Raven Edgar Allan Poe +------------------------------------------------------------ | The Red Badge of Courage +------------------------------------------------------------ The Red Badge of Courage Stephen Crane +------------------------------------------------------------ | The Republic +------------------------------------------------------------ The Republic Plato +------------------------------------------------------------ | The Return of Sherlock Holmes +------------------------------------------------------------ The Return of Sherlock Holmes Sir Arthur Conan Doyle +------------------------------------------------------------ | The Rime of the Ancient Mariner +------------------------------------------------------------ The Rime of the Ancient Mariner Samuel Taylor Coleridge +------------------------------------------------------------ | The Scarecrow of Oz +------------------------------------------------------------ The Scarecrow of Oz L. Frank Baum +------------------------------------------------------------ | The Scarlet Letter +------------------------------------------------------------ The Scarlet Letter Nathaniel Hawthorne +------------------------------------------------------------ | The Sonnets +------------------------------------------------------------ The Sonnets William Shakespeare +------------------------------------------------------------ | The Strange Case of Dr. Jekyll and Mr. Hyde +------------------------------------------------------------ The Strange Case of Dr. Jekyll and Mr. Hyde Robert Louis Stevenson +------------------------------------------------------------ | The Taming of the Shrew +------------------------------------------------------------ The Taming of the Shrew William Shakespeare +------------------------------------------------------------ | The Tell-Tale Heart +------------------------------------------------------------ The Tell-Tale Heart Edgar Allan Poe +------------------------------------------------------------ | The Tempest +------------------------------------------------------------ The Tempest William Shakespeare +------------------------------------------------------------ | The Tenant of Wildfell Hall +------------------------------------------------------------ The Tenant of Wildfell Hall Anne Bronte +------------------------------------------------------------ | The Three Musketeers +------------------------------------------------------------ The Three Musketeers Alexandre Dumas +------------------------------------------------------------ | The Time Machine +------------------------------------------------------------ The Time Machine H. G. Wells +------------------------------------------------------------ | The Tin Woodman of Oz +------------------------------------------------------------ The Tin Woodman of Oz L. Frank Baum +------------------------------------------------------------ | The Tragedy of Coriolanus +------------------------------------------------------------ The Tragedy of Coriolanus William Shakespeare +------------------------------------------------------------ | The Tragedy of King Lear +------------------------------------------------------------ The Tragedy of King Lear William Shakespeare +------------------------------------------------------------ | The Tragedy of King Richard the Second +------------------------------------------------------------ The Tragedy of King Richard the Second William Shakespeare +------------------------------------------------------------ | The Voyage Out +------------------------------------------------------------ The Voyage Out Virginia Woolf +------------------------------------------------------------ | The War in the Air +------------------------------------------------------------ The War in the Air H. G. Wells +------------------------------------------------------------ | The War of the Worlds +------------------------------------------------------------ The War of the Worlds H. G. Wells +------------------------------------------------------------ | The Wind in the Willows +------------------------------------------------------------ The Wind in the Willows Kenneth Grahame +------------------------------------------------------------ | The Wisdom of Father Brown +------------------------------------------------------------ The Wisdom of Father Brown G. K. Chesterton +------------------------------------------------------------ | The Wonderful Wizard of Oz +------------------------------------------------------------ The Wonderful Wizard of Oz L. Frank Baum +------------------------------------------------------------ | Three Men in a Boat +------------------------------------------------------------ Three Men in a Boat Jerome K. Jerome +------------------------------------------------------------ | Through the Looking Glass +------------------------------------------------------------ Through the Looking Glass Lewis Carroll +------------------------------------------------------------ | Thus Spake Zarathustra +------------------------------------------------------------ Thus Spake Zarathustra Friedrich Nietzsche +------------------------------------------------------------ | Thuvia, Maid of Mars +------------------------------------------------------------ Thuvia, Maid of Mars Edgar Rice Burroughs +------------------------------------------------------------ | To Helen +------------------------------------------------------------ To Helen Edgar Allan Poe +------------------------------------------------------------ | Treasure Island +------------------------------------------------------------ Treasure Island Robert Louis Stevenson +------------------------------------------------------------ | Twelfth Night +------------------------------------------------------------ Twelfth Night William Shakespeare +------------------------------------------------------------ | Twenty Thousand Leagues Under the Seas +------------------------------------------------------------ Twenty Thousand Leagues Under the Seas Jules Verne +------------------------------------------------------------ | Twenty Years After +------------------------------------------------------------ Twenty Years After Alexandre Dumas +------------------------------------------------------------ | Typee +------------------------------------------------------------ Typee Herman Melville +------------------------------------------------------------ | Ulalume +------------------------------------------------------------ Ulalume Edgar Allan Poe +------------------------------------------------------------ | Uneasy Money +------------------------------------------------------------ Uneasy Money P. G. Wodehouse +------------------------------------------------------------ | Up From Slavery: An Autobiography +------------------------------------------------------------ Up From Slavery: An Autobiography Booker T. Washington +------------------------------------------------------------ | Vanity Fair +------------------------------------------------------------ Vanity Fair William Makepeace Thackeray +------------------------------------------------------------ | Walden +------------------------------------------------------------ Walden Henry David Thoreau +------------------------------------------------------------ | War and Peace +------------------------------------------------------------ War and Peace Leo Tolstoy +------------------------------------------------------------ | Warlord of Mars +------------------------------------------------------------ Warlord of Mars Edgar Rice Burroughs +------------------------------------------------------------ | White Fang +------------------------------------------------------------ White Fang Jack London +------------------------------------------------------------ | Wuthering Heights +------------------------------------------------------------ This file is part of the Sofia project sponsored by the Provost's fund for teaching and learning at Harvard university. There are 198 entries in this file. COUNT: 198 ENTRY SINGLE VARIABLE CALCULUS I Authors: Oliver Knill: 2001 Literature: not yet +------------------------------------------------------------ | Abel's partial summation formula +------------------------------------------------------------ Abel's partial summation formula is a discrete version of the partial integration formula: with A_n = sum_k=1^n a_k one has sum_k=m^n a_k b_k = sum_k=m^n A_k(b_k-b_k+1) + A_n b_n+1 - A_m-1 b_m. +------------------------------------------------------------ | Abel's test +------------------------------------------------------------ Abel's test: if a_n is a bounded monotonic sequence and b_n is a convergent series, then the sum sum_n a_n b_n converges. +------------------------------------------------------------ | absolute value +------------------------------------------------------------ The absolute value of a real number x is denoted by |x| and defined as the maximum of x and -x. We can also write |x|=+sqrtx^2. The absolute value of a complex number z=x+iy is defined as sqrtx^2+y^2. +------------------------------------------------------------ | accumulation point +------------------------------------------------------------ An accumulation point of a sequence a_n of real numbers is a point a which the limit of a subsequence a_n_k of a_n. A sequence a_n converges if and only if there is exactly one accumulation point. Example: The sequence a_n = sin( pi n) has two accumulation points, a=1 and a=-1. The sequence a_n = sin( pi n)/n has only the accumulation point a=0. It converges. +------------------------------------------------------------ | Achilles paradox +------------------------------------------------------------ The Achilles paradox is one of Zenos paradoxon. It argues that motion can not exist: "set up a race between Achilles A and tortoise T. At the initial time t_0=0, A is at the spot s=0 while T is at position s_1=1. Lets assume A runs twice as fast. The reace starts. When A reaches s_1 at time t_1=1, its opponent T has already advanced to a point s_2=1+1/2. Whenever A reaches a point s_k at time t_k, where T has been at time t_k-1, T has already advanced further to location s_k+1. Because an infinite number of timesteps is necessary for A to reach T, it is impossible that A overcomes T." The paradox exploits a misunderstanding of the concept of summation of infinite series. At the finite time t = sum_n=1^infinity (t_n-t_n-1)=2, both A and T will be at the same spot s=lim_n to infinity s_n=2. +------------------------------------------------------------ | addition formulas +------------------------------------------------------------ The addition formulas for trigonometric functions are begineqnarray* cos(a+b) = sin(a) cos(b) + cos(a) sin(b) sin(a+b) = cos(a) cos(b) - sin(a) sin(b) endeqnarray* +------------------------------------------------------------ | alternating series +------------------------------------------------------------ An alternating series is a series in which terms are alternatively positive and negative. An example is sum_n=1^infinity a_n = sum_n=1^infinity (-1)^n/n=-1+1/2-1/3+1/4-.... An alternating series with a_n to 0 converges by the alternating series test. +------------------------------------------------------------ | alternating series test +------------------------------------------------------------ Leibniz's alternating series test assures that an alternating series sum_n a_n with |a_n| to 0 is a convergent series. +------------------------------------------------------------ | acute +------------------------------------------------------------ An angle is acute, if it is smaller than a right angle. For example alpha=pi/3=60^circ is an acute angle. The angle alpha=2pi/3=120^circ is not an acute angle. The right angle alpha=pi/2 = 90^circ does not count as an acute angle. The angle alpha=-pi/6=-30^circ is an acute angle. +------------------------------------------------------------ | antiderivative +------------------------------------------------------------ The antiderivative of a function f is a function F(x) such that the derivative of F is f that is if d/dx F(x) = f(x). The antiderivative is not unique. For example, every function F(x) = cos(x) +C is the antiderivative of f(x)=sin(x). Every function F(x)=x^n+1/(n+1) + C is the antiderivative of f(x)=x^n. +------------------------------------------------------------ | Arithmetic progression +------------------------------------------------------------ Arithmetic progression A sequence of numbers a_n for which b_n=a_n+1-a_n is constant, is called an arithmetic progression. For example, 3,7,11,15,19,... is an arithmetic progression. The sequence 0,1,2,4,5,6,7 is not an arithmetic progression. +------------------------------------------------------------ | arrow paradox +------------------------------------------------------------ The arrow paradox is a classical Zeno paradox with conclusion that motion can not exist: "an object occupies at each time a space equal to itself, but something which occupies a space equal to itself can not move. Therefore, the arrow is always at rest." +------------------------------------------------------------ | asymptotic +------------------------------------------------------------ Two real functions are called asymptotic at a point a if lim_x to a f(x)/g(x) =1. For example, f(x) = sin(x) and g(x) = x are asymptotic at a=0. The point a can also be infinite: for example, f(x)=x and g(x)=sqrtx^2+1 are asymptotic at a=infinity. +------------------------------------------------------------ | Bernstein polynomials +------------------------------------------------------------ The Bernstein polynomials of a continuous function f on the unit interval 0leq x leq 1 are defined as B_n(x) = sum_k=1^n f(k/n) x^k (1-x)^n-k n!/(k!(n-k!). +------------------------------------------------------------ | Binomial coefficients +------------------------------------------------------------ Binomial coefficients The coefficients B(n,k) of the polynomial (x+1)^n for integer n are called Binomial coefficients. Explicitly one has B(n,k) = n!/(k! (n-k)!), where k!=k (k-1)!, 0!=1 is the factorial of k. The function B(n,k) can be defined for any real numbers n,k by writing n!=Gamma(n+1), where Gamma is the Gamma function. If p is a positive real number and k is an integer, one has one has B(p,k)=p (p-1)...(p-k+1)/k!. For example, B(1/2,0)=B(1/2,1)=1/2,B(1/2,2)=-1/8. Indeed, (1+x)^1/2=1+x/2-x^2/8+.... +------------------------------------------------------------ | Binominal theorem +------------------------------------------------------------ The Binominal theorem tells that for a real number |z|<1 and real number p, one has (1+z)^p=sum_k=0^infinity B(p,k) z^k, where B(p,k) is called the Binomial coefficient. If p is a positive integer, then (1+z)^p is a polynomial. For example: (1+z)^4 = 1+4z+6z^2+4z^3+z^4 . If p is a noninteger or negative, then (1+z)^p is an infinite sum. For example (1+z)^-1/2 = 1-x/2+3x^2/8-5x^3/16+... +------------------------------------------------------------ | bisector +------------------------------------------------------------ A bisector is a straight line that bisects a given angle or a given line segment. For example, the y-axis x=0 in the plane bisects the line segment connecting (-1,0) with (1,0). The line x=y bisects the angle angle(CAB) where C=(0,1), A=(0,0),B=(1,0) at the point A. +------------------------------------------------------------ | Bolzano's theorem +------------------------------------------------------------ Bolzano's theorem also called intermediate value theorem says that a continuous function on an interval (a,b) takes each value between f(a) and f(b). For example, the function f(x) = sin(x) takes any value between -1 and 1 because f is continuous and f(-pi/2)=-1 and f(pi/2)=1. +------------------------------------------------------------ | Fermat principle +------------------------------------------------------------ The Fermat principle tells that if f is a function which is differentiable at z and f(x)>f(z) for all points in an interval (z-a,z+a) with a>0, then f'(z)=0. +------------------------------------------------------------ | fundamental theorem of calculus +------------------------------------------------------------ The fundamental theorem of calculus: if f is a differentiable function on a leq x leq b where a0, we can find n, such that for all k>n,m>n one has |a_m-a_k|0, and the integer n closest to e^2c+1 one has for k>n and m>n |a_m-a_k| leq c. +------------------------------------------------------------ | Cauchy's convergence test +------------------------------------------------------------ Cauchy's convergence test. Given a series sum_k=1^infinity a_k with positive summands a_k. If r=lim_n to infinity a_n^1/n<1 then the series is a convergent series. If r>1, then the series diverges. +------------------------------------------------------------ | continuous +------------------------------------------------------------ A function f is called continuous at a point x if for every open interval V around f(x) there exists an open interval U around x such that f(U) is a subset of V. A function f is continuous in a set Y if it is continous at every point in Y. This definition is equivalent to: for every sequence x_n to x, the sequence f(x_n) converges to f(x). Examples: Any polynomial like x^5+5x^3+3x is continuous on the entire line. The sum and product of continuous functions is continuous. the composition of two continuous functions is continuous. Discontinuities can happen in different ways: the function can become infinite like f(x)=1/x at 0 or tan(x) at x=pi/2, the function can jump like f(x)=sign(x) which is 1 if x>0, -1 if x<0 and 0 if x=0. A function can also become too oscillatory at a point like f(x)=sin(1/x) at x=0. Note that f(x)=x sin(1/x) is continuous on the entire real line. There are functions which are discontinuous at every point. An example is f(x)=1 if x is rational and f(x)=-1 if x is irrational. Note that be restricting the domain of a function, one an make it continuous. For example: f(x)=1/x is continuous on the positive real axes. +------------------------------------------------------------ | converges +------------------------------------------------------------ A function f(x) converges to a value z at x if the function g which agrees with f away from x and satisfies g(x)=z is continouous at x. The value z is called the limit of f at x. For example the function f(x)=(1-x^2)/(1-x) has the limit z=2 at x=1. The function g(x) which is defined bo be f(x) for x neq 1 and g(1)=2 is indeed continuous. One writes z = lim_y to x f(y). One has lim_x to z (f(x) + g(x)) = lim_x to z f(x) + lim_x to z g(x). lim_x to z (f(x) g(x)) = lim_x to z f(x) + g(x). lim_x to z f(g(x)) = f(lim_x to z g(x)). A series a_n is a convergent series, if the partial sum sequence b_n = sum_k=1^n a_k converges to a finite limit a. +------------------------------------------------------------ | absolutely convergent series +------------------------------------------------------------ A series sum_n a_n is called an absolutely convergent series if sum_n |a_n| is a convergent series. +------------------------------------------------------------ | series +------------------------------------------------------------ Summing up a sequence is called a series. An important example is the geometric series 1+1/2+1/4+1/8+1/16+..., which sums up to 2. An other example is the harmonic series 1+1/2+1/3+1/4+1/5+... which has no finite limit. +------------------------------------------------------------ | change of variables +------------------------------------------------------------ The change of variables in integration theory is the formula int f(x) dx = int f(g(u)) g'(u) du if x=g(u). For example, int sqrt1-x^2 dx becomes with x=g(u) = sin(u) and dx = g'(u) du = cos(u) du the integral int cos2(u) du. +------------------------------------------------------------ | differentiable +------------------------------------------------------------ A function f is called differentiable at z if there exists a function g which is continuous at z such that f(x)=f(z)+(x-z) g(x). The derivative of f at z is g(z) and also denoted f'(x). By solving for g(x) and letting x to z one can write g(z) = lim_x to z (f(x)-f(z))/(x-z). The quotient is called the differential quotient. The sum of two at z differentiable functions is differentiable at z and (f+g)'(z)=f'(z)+g'(z). This is called the sum rule. The prduct of two at z differentiable functions is differentiable at z and (f g)'=f'g + f g'. This is called the product rule. The composition of two differentiable functions is differentiable and (f circ g)' = (f' circ g) g'. This is called the chain rule. Functions can be continuous without being differentiable. For example f(x)=|x| is continuous at 0 but not differentiable at 0. There are functions which are continuous everywhere but not differentiable at most points. An example is the Weierstrass function f(x) = sum_k=1^infinity cos(k^2 x)/k^2. +------------------------------------------------------------ | Extended mean value theorem +------------------------------------------------------------ Extended mean value theorem. If f(x) and g(x) are differentiable on the interval (a,b) and are continuous on the closed interval I = a leq x leq b then there exists a point x in I for which f'(x)/g'(x)=(f(b)-f(a))/(g(b)-g(a)) . Proof. Otherwise one would have one of the following two possibilies: f'(x) (g(b)-g(a)) < g'(x) (f(b)-f(a)) for all x in (a,b) or f'(x) (g(b)-g(a)) < g'(x) (f(b)-f(a)) for all x in (a,b). Integration of these expressions using the fundamental theorem of calculus gives (f(b)-f(a)) (g(b)-g(a)) < (g(b)-g(a)) (f(b)-f(a)) or (f(b)-f(a)) (g(b)-g(a)) > (g(b)-g(a)) (f(b)-f(a)) which both are not possible. The special case g(x)=x is called the mean value theorem. +------------------------------------------------------------ | factorial +------------------------------------------------------------ The factorial of a positive integer n is defined recursively by n! = n (n-1)! and 0!=1. For example, 5!=120. The factorial function can be extended to the real line and is then called the Gamma function: n! = Gamma(n+1), where Gamma(z) = int_0^infinity t^z-1 e^-t dt . which is finite everywhere except at z=0,-1,-2,.... +------------------------------------------------------------ | limit +------------------------------------------------------------ The limit of a sequence of numbers a_n is a number a such that a_n converges to a in the following sense: for every c>0 there exists an integer m such that |a_n -a|m. Limits can be defined in any metric space and more generally in any topological space. +------------------------------------------------------------ | maximum-value theorem +------------------------------------------------------------ The maximum-value theorem assures that a continuous function on an interval aC on (a,b). Integration gives using the fundamental theorem of calculus f(x)-f(a)=int_a^x f'(t) dt < C (x-a) or f(x)-f(a)=int_a^x f'(t) dt > C (x-a) especially f(x)-f(a)=int_a^b f'(t) dt < C (b-a) or f(x)-f(a)=int_a^b f'(t) dt > C (b-a) which is a contradiction The mean value theorem is a special case of the extended mean value theorem. +------------------------------------------------------------ | Rolle's theorem +------------------------------------------------------------ Rolle's theorem If f(x) is a continuous function on the interval I = a leq x leq b which is differentiable on the open interval (a,b) and f(a)=f(b), then there exists a point x in (a,b), for which f'(x)=0. Proof. f takes both its maximum and minimum on I. If the maximum is equal to the minimum, then f(x) is constant on I, otherwise, either the minium or the maximum is a point x in (a,b). At that point f'(x)=0. qed. Rolle's theorem is a special case of the mean value theorem +------------------------------------------------------------ | rule of three +------------------------------------------------------------ The rule of three ia a rough rule of thumb when solving calculus problems or teaching calculus: Look at a calculus problem graphically, numerically and analytically. In other words, one should try to understand a calculus problem geometrically, algebraically and computationally. For example, the notion of the derivative of a function of one variable can be understood geometrically as a slope, can be understood through algebraic manipulations like (x^n)' = n x^n-1 or computationally by plugging in numbers or doing things on a computer. +------------------------------------------------------------ | Weierstrass function +------------------------------------------------------------ A Weierstrass function is an example of a function which is continuous but almost nowhere differentiable. An example is f(x) = sum_k=1^infinity cos(k^2 x)/k^2. This file is part of the Sofia project sponsored by the Provost's fund for teaching and learning at Harvard university. There are 42 entries in this file. COUNT: 42 ENTRY MULTIVARIABLE CALCULUS Author: Oliver Knill: March 2000 -March 2004 Literature: Standard glossary of multivariable calculus course as taught at the Harvard mathematics department. +------------------------------------------------------------ | acceleration +------------------------------------------------------------ The acceleration of a parametrized curve r(t) = (x(t),y(t),z(t)) is defined as the vector r''(t). It is the rate of change of the velocity r'(t). It is significant, because Newtons law relates the acceleration r''(t) of a mass point of mass m with the force F acting on it: m r''(t) = F(r(t)) . This ordinary differential equation determines completely the motion of the particle. +------------------------------------------------------------ | advection equation +------------------------------------------------------------ The advection equation u_t= c u_x is a linear partial differential equation. Its general solution is u(t,x)=f(x+ct), where f(x)=u(0,x). The advection equation is also called transport equation. In higher dimensions, it generalizes to the gradient flow u_t = c grad(u). +------------------------------------------------------------ | Archimedes spiral +------------------------------------------------------------ The Archimedes spiral is the plane curve defined in polar coordinates as r(t) = c t, where c is a constant. In Euclidean coordinates, it is given by the parametrization r(t) = (c t cos(t), c t sin(t) ). +------------------------------------------------------------ | axis of rotation +------------------------------------------------------------ The axis of rotation of a rotation in Euclidean space is the set of fixed points of that rotation. +------------------------------------------------------------ | Antipodes +------------------------------------------------------------ Two points on the sphere of radius r are called Antipodes (=anti-podal points) if their Euclidean distance is maximal 2r. If the sphere is centered at the origin, the antipodal point to (x,y,z) is the point (-x,-y,-z). +------------------------------------------------------------ | boundary +------------------------------------------------------------ The boundary of a geometric object. Examples: The boundary of an interval I= a leq x leq b is the set with two points a,b. For example, 0 leq x leq 1 the boundary 0,1. The boundary of a region G in the plane is the union of curves which bound the region. The unit disc has as a boundary the unit circle. The entire plane has an empty boundary. The boundary of a surface S in space is the union of curves which bound the surface. For example: A semisphere has as the boundary the equator. The entire sphere has an empty boundary. The boundary of a region G in space is the union of surfaces which bound the region. For example, the unit ball has the unit sphere as a boundary. A cube has as a boundary the union of 6 faces. The boundary of a curve r(t), t in a,b consists of the two points r(a),r(b). The boundary can be defined also in higher dimensions where surfaces are also called manifolds. The dimension of the boundary is always one less then the dimension of the object itself. In cases like the half cone, the tip of the cone is not considered a part of the boundary. It is a singular point which belongs to the surface. While the boundary can be defined for far more general objects in a mathematical field called "topology", the boundaries of objects occuring in multivariable calculus are assumed to be of dimension one less than the object itself. +------------------------------------------------------------ | Burger's equation +------------------------------------------------------------ The Burger's equation u_t = u u_t is a nonlinear partial differential equation in one dimension. It is a simple model for the formation of shocks. +------------------------------------------------------------ | Cartesian coordinates +------------------------------------------------------------ Cartesian coordinates in three-dimensional space describe a point P with coordinates x, y and z. Other possible coordinate systems are cylindrical coordinates and spherical coordinates. Going from one coordinate system to an other is called a coordinate change. +------------------------------------------------------------ | Cavalieri principle +------------------------------------------------------------ Cavalieri principle tells that if two solids have equal heights and their sections at equal distances have have areas with a given ratio, then the volumes of the solids have the same ratio. +------------------------------------------------------------ | change of variables +------------------------------------------------------------ A change of variables is defined by a coordinate transformation. Examples are changes between cylindrical coordinates, spherical coordinates or Cartesian coordinates. Often one uses also rotations, allowing to use a convenient coordinate system, like for example, when one puts a coordinate system so that a surface of revolution has as the symmetry axes the z-axes. +------------------------------------------------------------ | circle +------------------------------------------------------------ A circle is a curve in the plane whose distance from a given point is constant. The fixed point is called the center of the circle. The distance is the radius of the circle. One can parametrize a circle by r(t) = (cos(t),sin(t) or given as an implicit equation g(x,y)=x^2+y^2 = 1. The circle is an example of a conic section, the intersection of a cone with a plane to which ellipses, hyperbola and parabolas belong to. +------------------------------------------------------------ | cone +------------------------------------------------------------ A cone in space is the set of points x^2+y^2=z^2 in space. Also translates, scaled and rotated versions of this set are still called a cone. For example 2x^2+3 y^2 = 7 z^2 is an elliptical cone. +------------------------------------------------------------ | conic section +------------------------------------------------------------ A conic section is the intersection of a cone with a plane. Hyperbola, ellipses and parabola lines and pairs of intersecting lines are examples of conic sections. +------------------------------------------------------------ | continuity equation +------------------------------------------------------------ The continuity equation is the partial differential equation rho_t + div(rho v)=0, where rho is the density of the fluid and v is the velocity of the fluid. The continuity equation is the consequence of the fact that the negative change of mass in a small ball is equal to the amount of mass which leaves the ball. The later is the flux of the current j=v rho through the surface and by the divergence theorem the integral of div(j). +------------------------------------------------------------ | cos theorem +------------------------------------------------------------ The cos theorem relates the length of the edges a,b,c in a triangle ABC with one of the angles alpha: a^2 = b^2+c^2-2bccos(alpha) Especially, if alpha=pi/2, it becomes the theorem of Pythagoras. +------------------------------------------------------------ | critical point +------------------------------------------------------------ A critical point of a function f(x,y) is a point (x_0,y_0), where the gradient grad f(x_0,y_0) vanishes. Critical points are also called stationary points. For functions of two variables f(x,y), critical points are typically maxima, minima or saddle points realized by f(x,y) = -x^2-y^2, f(x,y)=x^2+y^2 or f(x,y)=x^2-y^2. +------------------------------------------------------------ | chain rule +------------------------------------------------------------ The chain rule expresses the derivative of the composition of two functions in terms of the derivatives of the functions. It is (f g)'(x) = f'(g(x)) g'(x). For example, if r(t) is a curve in space and F a function in three variables, then (d/dt) f(r(t)) = grad(f) . r'(t). Example. If T and S are maps on the plane, then (T S)' = T'(S) S', where T' is the Jacobean of T and S' is the Jacobean of S. +------------------------------------------------------------ | change of variables +------------------------------------------------------------ A change of variables on a region R in Euclidean space is given by an invertible map T: R to T(R). The change of variables formula int_T(R) f(x) dx = int_R f(Tx) det(T'(x)) dx allows to evaluate integrals of a function f of several variables on a complicated region by integrating on a simple region R. In one dimensions, the change of variable formula is the formula for substitution. Example: (2D polar coordinates) T(r,phi) = (xcos(theta), y sin(theta). with det(T')=r maps the rectangle 0,s x 0,2 pi into the disc. Example of 3D spherical coordinates are T(r,theta,phi) = ( rcos(theta) sin(phi), r sin(theta) sin(phi), rcos(phi)), det(T')=r^2 sin(phi) maps the rectangular region (0,s) x (0,2pi0 x (0,pi) onto a sphere of radius s. +------------------------------------------------------------ | curl +------------------------------------------------------------ The curl of a vector field F=(P,Q,R) in space is the vector field (R_y-Q_z,P_z-R_x,Q_x-P_y). It measures the amount of circulation = vorticity of the vector field. The curl of a vector field F=(P,Q) in the plane is the scalar field (Q_x-P_y). It measures the vorticity of the vector field in the plane. +------------------------------------------------------------ | curvature +------------------------------------------------------------ The curvature of a parametrized curve r(t) = (x(t),y(t),z(t)) is defined as k(t) = |r'(t) x r''(t)|/|r'(t)|^3. Examples: The curvature of a line is zero. The curvature of a circle of radius r is 1/r. +------------------------------------------------------------ | curve +------------------------------------------------------------ A curve in space is the image of a map X: t -> r(t)=(x(t),y(t),z(t)), where x(t),y(t),z(t) are three piecewise smooth functions. For general continuous maps x(t),y(t),z(t), the length or the velocity of the curve would no more be defined. +------------------------------------------------------------ | cross product +------------------------------------------------------------ The cross product of two vectors v=(v_1,v_2,v_3) and w=(w_1,w_2,w_3) is the vector (v_2 w_3-w_2 v_3,v_3 w_1-w_3 v_1,v_1 w_2-w_2 v_1). +------------------------------------------------------------ | curve +------------------------------------------------------------ A curve in three-dimensional space is the image of a map r(t)=(x(t),y(t),z(t)), where x(t),y(t),z(t) are three continuous functions. A curve in two-dimensional space is the image of a map r(t)=(x(t),y(t)). +------------------------------------------------------------ | cylinder +------------------------------------------------------------ A cylinder is a surface in three dimensional space such that its defining equation f(x,y,z)=0 does not involve one of the variables. For example, z=2 sin(y) defines a cylinder. A cylinder usually means the surface x^2+y^2=r or a translated rotated version of this surface. +------------------------------------------------------------ | derivative +------------------------------------------------------------ The derivative of a function f(x) of one variable at a point x is the rate of change of the function at this point. Formally, it is defined as lim_dx to 0 (f(x+dx)-f(x))/dx. One writes f'(x) for the derivative of f. The derivative measures the slope of the graph of f(x) at the point. If the derivative exists for all x, the function is called differentiable. Functions like sin(x) or cos(x) are differentiable. One has for example f'(x)=cos(x) if f(x)=sin(x). An example of a function which is not differentiable everywhere is f(x)=|x|. The derivative at 0 is not defined. +------------------------------------------------------------ | cylindrical coordinates +------------------------------------------------------------ cylindrical coordinates in three dimensional space describe a point P by the coordinates r=(x^2+y^2+z^2)^1/2, phi=arctan(y/x),z, where P=(x,y,z) are the Cartesian coordinates of P. Other coordinate systems are Cartesian coordinates or spherical coordinates. +------------------------------------------------------------ | determinant +------------------------------------------------------------ The determinant of a matrix A=( a b c d ) is ad-bc. The determinant of a matrix A=( a b c d e f g h i ) is a e i+ b f g + c d h - c e g - f h a - i b d. The determinant is relevant when changing variables in integration. +------------------------------------------------------------ | directional derivative +------------------------------------------------------------ The directional derivative of f(x,y,z) in the direction v is the dot product of the gradient of f with v. It measures the rate of change of f at a point P when moving trough the point (x,y,z) with velocity v. +------------------------------------------------------------ | distance +------------------------------------------------------------ The distance of two points P=(a,b,c) and Q=(u,v,w) in three dimensional Euclidean space is the square root of (a-u)^2+(b-v)^2+(c-w)^2. The distance of two points P=(a,b) and Q=(u,v) in the plane is the square root of (a-u)^2+(b-v)^2. +------------------------------------------------------------ | distance +------------------------------------------------------------ The distance between two nonparallel lines in three dimensional Euclidean space is given by the forumla d = | (v x w) . u |/| (v x w)|, where v and w are arbitrary nonzero vectors in each line and u is an arbitrary vector connecting a point on the first line to a point of the second line. +------------------------------------------------------------ | divergence +------------------------------------------------------------ The divergence of a vector field F=(P,Q,R) is the scalar field div(F) = P_x+Q_y+R_z. The value div(F)(x,y,z) measures the amount of expansion of the vector field at the point (x,y,z). +------------------------------------------------------------ | dot product +------------------------------------------------------------ The dot product of two vectors v=(v_1,v_2,v_3) and w=(w_1,w_2,w_3) is the scalar v_1 w_1+v_2 w_2+v_3 w_3. +------------------------------------------------------------ | ellipse +------------------------------------------------------------ An ellipse is the set of points in the plane which satisfy an equation (x-a)^2/A^2+(y-b)^2/B^2=1. It is inscribed in a rectangle of length A and width B centered at (a,b). Ellipses can also be defined as the set of points in the plance whose sum of the distances to two points is constants. The two fixed points are called the foci of the ellipse. The line through the foci of a noncircular ellipse is called the focal line, the points where focal axes and a noncircular ellipse cross, are called vertices of the ellipse. The major axis of the ellipse is the line segment connecting the two vertices, the minor axis is the symmetry line of the ellipse which mirrors the two focal points or the two vertices. Ellipses are examples of conic sections, the intersection of a cone with a plane. +------------------------------------------------------------ | ellipsoid +------------------------------------------------------------ An ellipsoid is the set of points in three dimensional Euclidean space, which satisfy an equation (x-a)^2/A^2+(y-b)^2/B^2+(z-c)^2/C^2=1. It is inscribed in a box of length, width and height A,B,C centered at (a,b,c). +------------------------------------------------------------ | equation of motion +------------------------------------------------------------ The equation of motion of a fluid is the partial differential equation rho Dv/dt = - grad(p) + F, where F are external forces like gravity rho g, or magnetic force j x B and Dv/dt is the total time derivative Dv/dt = v_t+v grad(v). The term - grad(p) is the pressure force. Together with an incompressibility assumption div(v)=0, these equations of motion are called Navier Stokes equations. +------------------------------------------------------------ | flux integral +------------------------------------------------------------ The flux integral of a vector field F through a surface S=X(R) is defined as the double integral of X(F).n over R, where n=X_u x X_v is the normal vector of the surface as defined through the parameterization X(u,v). +------------------------------------------------------------ | Fubinis theorem +------------------------------------------------------------ Fubinis theorem tells that int_a^b int_c^d f(x,y) dx dy = int_c^d int_a^b f(x,y) dy dx. +------------------------------------------------------------ | gradient +------------------------------------------------------------ The gradient of a function f at a point P=(x,y,z) is the vector (f_x(x,y,z),f_y(x,y,z),f_z(x,y,z)) where f_x denotes the partial derivative of f with respect to x. +------------------------------------------------------------ | Hamilton equations +------------------------------------------------------------ The Hamilton equations to a function f(x,y) is the system of ordinary differential equations x'(t) = f_y(x,y), y'(t) = - f_x(x,y). which is called Hamilton system. Solution curves of this system are located on level curves f(x,y)=c because by the chain rule one has d/dt f(x(t),y(t)) = f_x x' + f_y y' = f_x f_y - f_x f_y = 0. The preservation of f is in physics called energy conservation. +------------------------------------------------------------ | heat equation +------------------------------------------------------------ The heat equation is the Partial differential equation u_t = mu Delta(u), where mu is a constant, and Delta u is the Laplacian of u. The heat equation is also called the diffusion equation. +------------------------------------------------------------ | Hessian +------------------------------------------------------------ The Hessian is the determinant of the Hessian matrix. +------------------------------------------------------------ | Hessian matrix +------------------------------------------------------------ The Hessian matrix of a function f(x,y,z) at a point (u,v,w) is the 3x3 matrix f''(u,v,w)=H(u,v,w) = ( f_xx f_xy f_xz f_yx f_yy f_yz f_zx f_zy f_zz ). The Hessian matrix of a function f(x,y) at a point (u,v) is the 2x2 matrix f'' = H(u,v,w) = ( f_xx f_xy f_yx f_yy ). The Hessian matrix is useful to classify critical points of f(x,y) using the second derivative test. +------------------------------------------------------------ | hyperbola +------------------------------------------------------------ A hyperbola is a plane curve which can be defined as the level curve g(x,y)=x^2/a^2+y^2/b^2=1 or given as a parametrized curve r(t)=(a cosh(t),b sinh(t)). A hyperbola can geometrically also be defined as the set of points whose distances from two fixed points in the plane is constant. The two fixed points are called the focal points of the hyperbola. The line through the focal points of a hyperbola is called the focal axis. The points, where the focal axis and the hyperbola cross are called vertices. A hyperbola is an example of a conic section, the intersection of a cone with a plane. +------------------------------------------------------------ | hyperboloid +------------------------------------------------------------ A hyperboloid is the set of points in three dimensional Euclidean space, which satisfy an equation (x-u)^2/a^2-(y-v)^2/b^2-(z-w)^2/c^2= I, where I=1 or I=-1. For a=b=c=1, the hyperboloid is obtained by rotating a hyperbola x^2-y^2=1 around the x-axes. It is two-sided for I=-1 (the intersection of the plane z=c with the hyperboloid is then empty) and one-sided for I=1. +------------------------------------------------------------ | incompressible +------------------------------------------------------------ A vector field F is called incompressible if its divergence is zero div(F)=0. The notation has its origins from fluid dynamics, where velocity fields F of fluids, gases or plasma often are assumed to be incompressible. If a vector field is incompressible and is a velocity field, then the corresponding flow preserves the volume. +------------------------------------------------------------ | continuity equation +------------------------------------------------------------ The continuity equation rho_t + div(i) = 0 links density rho and velocity field i. It is an infinitesimal description which is equivalent to the preservation of mass by the theorem of Gauss. The change of mass M(t) int int int_R rho dV inside a region R in space is the minus the flux of mass through the boundary S of R. +------------------------------------------------------------ | interval +------------------------------------------------------------ An interval is a subset of the real line defined by two points a,b. One can write I = a leq x leq b for a closed interval, I = a < x < b for an open interval and I = a leq x < b , I = a < x leq b for half open intervals. If a=-infinity and b=infinity, then the interval is the entire real line. If a=0,b-infinity, then I = (a,b) is the set of positive real numbers. Intervals can be characterized as the connected sets in the real line. +------------------------------------------------------------ | integral +------------------------------------------------------------ An integral of f(x) over an interval I on the line is the limit (1/n) sum_i=1^n f(i/n) for n to infinity over the integers and the sum is taken over all i such that i/n is in I. An integral of f(x,y) over a region R in the plane is the limit (1/n^2) sum_(i/n,j/n) in R f(i/n,j/n) for n to infinity. Such an integral is also called double integral. Often, double integrals can be evaluated by iterating two one-dimensional integrals. An integral of f(x,y,z) over a domain R in space is the limit (1/n^3) sum_(i/n,j/n,k/n) in R f(i/n,j/n,k/n) for n to infinity. Such an integral is also called a triple integral. Often, triple integrals can be evaluated by iterating three one-dimensional integrals. +------------------------------------------------------------ | intercept +------------------------------------------------------------ An intercept is the intersection of a surface with a coordinate axes. Like traces, intercepts are useful for drawing surfaces by hand. For example, the two sheeted hyperboloid x^2+y^2-z^2=-1 has the intercepts x^2-z^2=-1 and y^2-z^2=-1 (hyperbola) and an empty intercept with the z axes. +------------------------------------------------------------ | jerk +------------------------------------------------------------ The jerk of a parametrized curve r(t)=(x(t),y(t),z(t)) is defined as r'''(t). It is the rate of change of the acceleration. By Newtons law, the jerk measures the rate of change of the force acting on the body. +------------------------------------------------------------ | Lagrange multiplier +------------------------------------------------------------ A Lagrange multiplier is an additional variable introduced for solving extremal problems under constraints. To extremize f(x,y,z) on a surface g(x,y,z)=0 then an extremum satisfies the equations f'= L g',g=0, where L is the Lagrange multiplier. These are four equations for four unknowns x,y,z,l. Additionally, one has to check for solutions of g'(x,y,z)=0. Example. If we want to extremize F(x,y,z)= -x log(x)-y log(y)-z log(z) under the constraint G(x,y,z)= x+y+z=1, we solve the equations -1-log(x)=L 1 -1-log(y)=L 1 -1-log(z)=L 1 x+y+z = 1, the solution of which is x=y=z=1/3. +------------------------------------------------------------ | Lagrange method +------------------------------------------------------------ The Lagrange method to solve extremal problems under constraints: 1) in order that a function f of several variables is extremal on a constraint set g=c, we either have grad g=0 or the point is a solution to the Lagrange equations grad f = L grad g, g=c. 2) in order to extremize a function f of several variables under the contraint set g=c,h=d, we have to solve the Lagrange equations grad f = L grad g + mu grad h, g=c,h=d or solve grad g = grad h = 0. +------------------------------------------------------------ | Laplacian +------------------------------------------------------------ The Laplacian of a function f(x,y,z) is defined as Delta(f)=f_xx+f_yy+f_zz. One can write it as Delta= div grad(f). Functions for which the Laplacian vanish are called harmonic. Laplacian appear often in PDE's Examples: the Laplace equation Delta(f)=0, the Poisson equation Delta(f)=rho, the Heat equation f_t=mu Delta(f) or the wave equation f_tt = c^2 Delta f. % The length of a curve r(t)=(x(t),y(t),z(t)) from t=a to t=b is the integral of the speed |r'(t)| over the interval a,b. Example. the length of the curve r(t)=(cos(t),sin(t)) from t=0 to t=pi is pi because the speed |r'(t)| is 1. +------------------------------------------------------------ | length +------------------------------------------------------------ The length of a vector v=(a,b,c) is the square root of v . v=a^2+b^2+c^2. An other word for length is norm. If a vector has length 1, it is called normalized or a unit vector. +------------------------------------------------------------ | level curve +------------------------------------------------------------ A level curve of a function f(x,y) of two variables is the set of points which satisfy the equation f(x,y)=c. For example, if f(x,y)=x^2-y^2, then its level curves are hyperbola. Level curves are orthogonal to the gradient vector field grad(f). +------------------------------------------------------------ | level surface +------------------------------------------------------------ A level surface of a scalar function f(x,y,z) is the set of points which satisfy f(x,y,z)=c. For example, if f(x,y,z)=x^2+y^2+3 z^2, then its level surfaces are ellipsoids. Level surfaces are orthogonal to the gradient field grad(f). +------------------------------------------------------------ | linear approximation +------------------------------------------------------------ The linear approximation of a function f(x,y,z) at a point (u,v,w) is the linear function L(x,y,z) = f(u,v,w) + grad f(u,v,w) . (x-u,y-v,z-w). +------------------------------------------------------------ | line +------------------------------------------------------------ A line in three-dimensional space is a curve in space given by r(t) = P + t v, where P is a point in space and v is a vector in space. The representation r(t)=P+tv is called a parameterization of the line. Algebraically, a line can also be given as the intersection of two planes: a x + b y + c z=d, u x + v y + w z = q. The corresponding vector v in the line is the cross product of (a,b,c) and (u,v,w). A point P=(x,y,z) on the line can be obtained by fixing one of the coordinates, say z=0 and solving the system a x + b y=d, u x + v y = q for the unknowns x and y. +------------------------------------------------------------ | line integral +------------------------------------------------------------ The line integral of a vector field F(x,y) along a curve C: r(t)=(x(t),y(t)), t in a,b in the plane is defined as int_C F . ds = int_a^b F(r(t)) . r'(t) dt , where r'(t)= (x'(t),y'(t)) is the velocity. The definition is similar in three dimensions where F(x,y,z) is a vector field and C: r(t)=(x(t),y(t),z(t)), t in a,b is a curve in space. +------------------------------------------------------------ | Maxwell equations +------------------------------------------------------------ The Maxwell equations are a set of partial differential equations which determine the electric field E and magnetic field B, when the charge density rho and the current density j are given. There are 4 equations: div(B) = 0 no magnetic monopoles curl(E) = -B_t/c Faradays law, change of magnetic flux produces voltage curl(B) = E_t/c+ (4pi/c) j Ampere's law, current or E change produce magnetism div(E) = 4 pi rho Gauss law, electric charges produce an electric field +------------------------------------------------------------ | nabla +------------------------------------------------------------ nabla is a mathematical symbol used when writing the gradient grad f of a function f(x,y,z). Nabla looks like an upside down Delta. Etymologically, the name has the meaning of an Egyption harp. +------------------------------------------------------------ | nabla calculus +------------------------------------------------------------ The nabla calculus introduces the vector grad=(partial_x,partial_y,partial_z). It satisfies grad(f) = grad(f), grad x F = curl(F), grad . F = div(F). Using basic vector operation rules and differentiation rules like grad (f g) = (grad f) g + f (grad g) one can verify identities: like for example div ( curl) F = 0, curl ( grad) f = 0, curl ( curl F) = grad (div F) - Delta (F), div(E x F) = F . curl(E) - E . curl(F). +------------------------------------------------------------ | nonparallel +------------------------------------------------------------ Two vectors v and w are called nonparallel if they are not parallel. Two vectors in space are parallel if and only if their cross product v x w is nonzero. +------------------------------------------------------------ | normal vector +------------------------------------------------------------ A normal vector to a parametrized surface X(u,v)=(x(u,v),y(u,v),z(u,v)) at a point P=(x,y,z) is the vector X_u x X_v. It is orthogonal to the tangent plane spanned by the two tangent vectors X_u and X_v. +------------------------------------------------------------ | normalized +------------------------------------------------------------ A vector is called normalized if its length is equal to 1. For example, the vector (3/5,4/5) is normalized. The vector (2,1) is not normalized. +------------------------------------------------------------ | octant +------------------------------------------------------------ An octant is one of the 8 regions when dividing three dimensional space with coordinate planes. It is the analogue of quadrant in two dimensions. +------------------------------------------------------------ | open set +------------------------------------------------------------ An open set R in the plane or in space is a set for which every point P is contained in a small disc U which is still contained in R. The disc x^2+y^2<1 is an example of an open set. The set x^2+y^2 leq 1 is not open because the point (1,0) for example has no neighborhood disc contained in R. +------------------------------------------------------------ | open +------------------------------------------------------------ A set is called open, if it is an open set. It means that every point in the set is contained in a neighborhood which still is in the set. The complement of open sets are called closed. +------------------------------------------------------------ | ordinary differential equation +------------------------------------------------------------ An ordinary differential equation (ODE) is an equation for a function or curve f(t) which relates derivatives f,f',f''.... of f. An example is f'=c f which has the solution f(t) = C e^(ct), where C is a constant. Only derivatives with respect to one variable may appear in an ODE. In most cases, the variable t is associated with time. Examples: f'= c f population model c>0. f'= - c f radioactive decay c>0 f'= c f (1-f) logistic equation f''=-c f harmonic oscillator f''= F(f) general form of Newton equations By increasing the dimension of the phase space, every ordinary differential equation can be written as a first order autonomous system x'=F(x). For example, f''=-f can be written with the vector x=(x_1,x_2)=(f,f') as (x_1',x_2')=(f',f'')=(f',f)=(x_2',-x_2'). There is a 2 x 2 matrix such that x'=A x. +------------------------------------------------------------ | orthogonal +------------------------------------------------------------ Two vectors v and w are called orthogonal if v . w=0. An other word for orthogonal is perpendicular. The zero vector 0 is orthogonal to any other vector. +------------------------------------------------------------ | parabola +------------------------------------------------------------ A parabola is a plane curve. It can be defined as the set of points which have the same distance to a line and a point. The line is called the directrix, the point is called the focus of the parabola. One can parametrize a parabola as r(t)=(t,t^2). It is also possible to give a parabola as a level curve g(x,y)=y-x^2=0 of a function of two variables. A parabola is an example of a conic section, to which also circles, ellipses and hyperbola belong. +------------------------------------------------------------ | parallelogram +------------------------------------------------------------ A parallelogram E can be defined as the image of the unit square under a map T(s,t) = s v + t w, where u and v are vectors in the plane. One says, E is spanned by the vectors v and w. The area of a parallelogram is |v x w|. +------------------------------------------------------------ | parallelepiped +------------------------------------------------------------ A parallelepiped E can be defined as the image of the unit cube under a linear map T(r,s,t) = r u + s v + t v, where u,v,w are vectors in space. One says, E is spanned by the vectors u,v and w. The volume of a parallelepiped is |u . (v x w)|. +------------------------------------------------------------ | perpendicular +------------------------------------------------------------ Two vectors v and w are called perpendicular if v . w=0. An other word for perpendicular is orthogonal. The zero vector v=0 is perpendicular to any other vector. +------------------------------------------------------------ | quadratic approximation +------------------------------------------------------------ The quadratic approximation of a function f(x,y,z) at a point (u,v,w) is the quadratic function Q(x,y,z) = L(x,y,z) + H(u,v,w) (x-u,y-v,z-w) . (x-u,y-v,z-w)/2, where H(u,v,w) is the Hessian matrix of f at (u,v,w) and where L(x,y,z) is the linear approximation of f(x,y,z) at (u,v,w). For example, the function f(x,y) = 3+sin(x+y)+cos(x + 2 y) has the linear approximation L(x,y) = 4+x+y and the quadratic approximation Q(x,y) = 4+x+y + (x+2y)^2/2. +------------------------------------------------------------ | quadrant +------------------------------------------------------------ A quadrant is one of the 4 regions when dividing the two dimensional space using coordinate axes. It is the analogue of octant in three dimensions. For example, the set x>0,y>0 is the open upper right quadrant. The set x geq 0, y geq 0 is the closed upper right quadrant. +------------------------------------------------------------ | parallel +------------------------------------------------------------ Two vectors v and w are called parallel if there exists a real number L such that v = L w. Two vectors in space are parallel if and only if their cross product v x w is zero. +------------------------------------------------------------ | parametrized surface +------------------------------------------------------------ A parametrized surface is defined by a map X(u,v) = (x(u,v),y(u,v),z(u,v)) from a region R in the uv-plane to xyz-space. Examples Sphere: X(u,v) = (rcos(u) sin(v), r sin(u) sin(v), rcos(v)) R = 0,2 pi) x 0,pi, u and v are called Euler angles. Plane X(u,v) = P + u U + v V, where P is a point, U,V are vectors and R is the entire plane. Surface of revolution is parametrized by X(u,v) = (f(v) cos(u), f(v) sin(u), v) where u is an angle measuring the rotation round the z axes and f(v) is a nonnegative function giving the distance to the z-axes at the height v. A graph of a function f(x,y) is parametrized by X(u,v) = (u,v,f(u,v)). A torus is parametrized by X(u,v) = (a+bcos(v))cos(u), (a+bcos(v)) sin(u),sin(v)) on R=0,2 pi) x 0,2 pi). +------------------------------------------------------------ | parametrized curve +------------------------------------------------------------ A parametrized curve in space is defined by a map r(t) = (x(t),y(t),z(t)) from an interval I to space. Examples are Circle in the xy-plane r(t) = (cos(t), sin(t), 0) with t in 0,2pi. Helix r(t) = (cos(t), sin(t), t) with t in a,b. Line r(t) = P + t V, where V is a vector and P is a point and -infinity0 and and f_xx(x,y)<0 then (x,y) is a local maximum. If det(f''(x,y))<0 and f_xx(x,y)>0 then (x,y) is a local minimum. +------------------------------------------------------------ | Space +------------------------------------------------------------ Space is usually used as an abbreviation for three dimensional Euclidean space. In a wider sense, it can mean linear space a vector space in which on can add and scale. +------------------------------------------------------------ | speed +------------------------------------------------------------ The speed of a curve r(t)=(x(t),y(t),z(t)) at time t is the length of the velocity vector r'(t)=(x'(t),y'(t),z'(t)). +------------------------------------------------------------ | sphere +------------------------------------------------------------ A sphere is the set of points in space, which have a given distance r from a point P=(a,b,c). It is the set (x-a)^2+(y-b)^2+(z-c)^2=r^2. For a=b=c=0,r=1 one obtains the unit sphere: x^2+y^2+z^2=1. Spheres can be define in any dimenesions. A sphere in two dimensions is a circle. A sphere in 1 dimension is the union of two points. The unit sphere in 4 dimensions is the set of points (x,y,z,w) in R^4 which satisfy x^2+y^2+z^2+w^2=1 Spheres can be defined in any space equiped with a distance like d((x,y),(u,v))=|x-u|+|y-v| in the plane. +------------------------------------------------------------ | superformula +------------------------------------------------------------ The superformula describes a class of curves with a few parameters m,n_1,n_2,n_3,a,b. It is the polar graph r(t) = (|cos( m t/4 )|^n_1/a + |sin( m t/4 )|^n_2/b)^-1/n_3 . It had been proposed by the Belgian Biologist Johan Gielis in 1997. +------------------------------------------------------------ | superposition +------------------------------------------------------------ The principle of superposition tells that the sum of two solutions of a linear partial differential equation (PDE) is again a solution of the PDE. For example, f(x,y) = sin(x-t) and g(x,y) = e^x-t are both solutions to the transport equation f_t(t,x) + f_x(t,x) = 0. Therefore also the sum sin(x-t)+e^x-t is a solution. For nonlinear partial differential equations the superposition principle is no more true which is one of the reasons for the difficulty with dealing with nonlinear systems. +------------------------------------------------------------ | surface +------------------------------------------------------------ A surface can either be described as a parametrized surface or implicitely as a level surface g(x,y,z) = 0. In the first case, the surface is given as the image of a map X:(u,v) mapsto (x(u,v),y(u,v),z(u,v)) where u,v ranges over a parameter domain R in the plane. In the second case, the surface is determined by a function of three variables. Sometimes, one can describe a surface in both ways like in the following examples: Sphere: X(u,v) = (rcos(u) sin(v),r sin(u) sin(v),rcos(v)), g(x,y,z) = x^2+y^2+z^2=r^2 Graphs: X(u,v) = (u,v,f(u,v)), g(x,y,z) = z-f(x,y) = 0 Planes: X(u,v) = P + u U + v V, g(x,y,z) = a x + b y + c z = d, (a,b,c)= U x V. Surface of revolution: X(u,v) = (f(v)cos(u), f(v) sin(u), v), g(x,y,z) = f( (x^2+y^2)^(1/2) ) - z = 0 +------------------------------------------------------------ | surface of revolution +------------------------------------------------------------ A surface of revolution is a surface which is obtained by rotating a curve around a fixed line. If that line is the z-axes, the surface can be given in cylindrical coordinates as r = f(z). A parametrization is X(t,z) = (f(z)cos(t), f(z) sin(t), z). +------------------------------------------------------------ | surface area +------------------------------------------------------------ The surface area of surface S=X(R) is defined as the integral of int int_R |X_u x X_v(u,v)| dudv. For example, for X(u,v) = ( r cos(u) sin(v), r sin(u) sin(v), r cos(v)) on R = 0 leq u leq 2 pi, 0 leq v leq pi , where S=X(R) is the sphere of radius r, one has X_u x X_v = r sin(v) X and |X_u x X_v| = sin(v) r^2. The surface area is int_0^2pi int_0^pi r^2 sin(v) du dv = 4 pi r^2. +------------------------------------------------------------ | surface integral +------------------------------------------------------------ A surface integral of a function f(x,y,z) over a surface S=X(R) is defined as the integral of f(X(u,v)) |X_u x X_v(u,v)| over R. In the special case when f(x,y,z)=1, the surface integral is the surface area of the surface S. +------------------------------------------------------------ | tangent plane +------------------------------------------------------------ The tangent plane to an implicitely defined surface g(x,y,z)=c at the point (x_0,y_0,z_0) is the plane a x + b y + c z = d, where (a,b,c) = grad f(x_0,y_0,z_0) is the gradient of g at (x_0,y_0,z_0) and d = a x_0 + b y_0 + c z_0. +------------------------------------------------------------ | tangent line +------------------------------------------------------------ The tangent line to an implicitely defined curve g(x,y)=c at the point (x_0,y_0) is the line a x + b y = d, where (a,b) is the gradient of g(x,y) at the point (x_0,y_0) and d = a x_0 + b y_0. +------------------------------------------------------------ | theorem of Clairot +------------------------------------------------------------ The theorem of Clairot assures that one can interchange the order of differentiation when taking partial derivatives. More precicely, if f(x,y) is a function of two variables for which both f_xy = f_yx are continuous, then f_xy = f_yx. +------------------------------------------------------------ | theorem of Gauss +------------------------------------------------------------ The theorem of Gauss states that the flux of a vector field F through the boundary S of a solid R in three-dimensional space is the integral of the divergence div(F) of F over the region R: int int int_R div(F) dV = int int_S F . dS . +------------------------------------------------------------ | theorem of Green +------------------------------------------------------------ The theorem of Green states that the integral of the curl(F)=Q_x-P_y of a vector field F=(P,Q) over a region R in the plane is the same as the line integral of F along the boundary C of R. int int_R curl(F) dA= int_C F ds . The boundary C is traced in such a way that the region is to the left. The boundary has to be piecewise smooth. The theorem of Green can be derived from the theorem of Stokes. +------------------------------------------------------------ | Green's theorem +------------------------------------------------------------ Green's theorem see theorem of Green. +------------------------------------------------------------ | Green's theorem +------------------------------------------------------------ The determinant of the Jacobean matrix is often called Jacobean or Jacobean determinant. +------------------------------------------------------------ | Jacobean matrix +------------------------------------------------------------ Jacobean matrix If T(u,v) = (f(u,v),g(u,v)) is a transformation from a region R to a region S in the plane, the Jacobean matrix dT is defined as ( f_u(u,v) f_v(u,v) g_u(u,v) g_v(u,v) ). It is the linearization of T near (u,v). Its determinant called the Jacobean determiant measures the area change of a small area element dA=dudv when maped by T. For example, if T(r,theta) = (r cos(theta), r sin(theta))=(x,y) is the coordinate transformation which maps R = r geq 0, theta in 0,2pi) to the plane, then dT is the matrix ( cos(theta) sin(theta) -r sin(theta) r cos(theta) ) which has determinant r. +------------------------------------------------------------ | theorem of Stokes +------------------------------------------------------------ The theorem of Stokes states that the flux of a vector field F in space through a surface S is equal to the line integral of F along the boundary C of S: int int_S curl(F) . dS = int_C F ds . +------------------------------------------------------------ | three dimensional space +------------------------------------------------------------ The three dimensional space consists of all points (x,y,z) where x,y,z ranges over the set of real numbers. To distinguish it from other three-dimensional spaces, one calls it also Euclidean space. +------------------------------------------------------------ | torus +------------------------------------------------------------ A torus is a surface in space defined as the set of points which have a fixed distance from a circle. It can be parametrized by X(u,v) = (a+bcos(v))cos(u), (a+bcos(v)) sin(u),sin(v)) on R=0,2 pi) x 0,2 pi), where a,b are positive constants. +------------------------------------------------------------ | trace +------------------------------------------------------------ The trace of a surface in three dimensional space is the intersection of the surface with one of the coordinate planes x=0 or y=0 or z=0. Traces help to draw a surface when given the task to do so by hand. Other marking points are intercepts, the intersection of the surface with the coordinate axes. +------------------------------------------------------------ | triangle +------------------------------------------------------------ A triangle in the plane or in space is defined by three points P,Q,R. If v=PQ,w=PR, then |v x w|/2 is the area of the triangle. +------------------------------------------------------------ | triple product +------------------------------------------------------------ The triple product between three vectors u,v,w in space is defined as the scalar u . (v x w). The absolute value |u . (v x w)| is the volume of the paralelepiped spanned by u,v and w. +------------------------------------------------------------ | triple dot product +------------------------------------------------------------ triple dot product (see triple product). +------------------------------------------------------------ | unit sphere +------------------------------------------------------------ The unit sphere is the sphere x^2+y^2+z^2=1. It is an example of a two-dimensional surface in three dimensional space. +------------------------------------------------------------ | unit tangent vector +------------------------------------------------------------ The unit tangent vector to a parametrized curve r(t)=(x(t),y(t),z(t)) is the normalized velocity vector T(t)=r'(t)/|r'(t)|. Together with the normal vector N(t) = T'(t)/|T'(t)| and the binormal vector B(t)=T(t) x N(t), it forms a triple of mutually orthogonal vectors. +------------------------------------------------------------ | vector +------------------------------------------------------------ A vector in the plane is defined by two points P,Q. It is the line segment v pointing from P to Q. If P=(a,b) and Q=(c,d) then the coordinates of the vector are v=(c-a,d-b). Points P in the plane can be identified by vectors pointing from 0 to P. A vector in space is defined by two points P,Q in space. If P=(a,b,c) and Q=(d,e,f), then the coordinates of the vector are v=(d-a,e-b,f-c). Points P in space can be identified by vectors pointing from 0 to P. Two vectors which can be translated into each other are considered equal. Remarks. One could define vectors more precisely as affine vectors and introduce an equivalence relation among them: two vectors are equivalent if they can be translated into each other. The equivalence classes are the vectors one deals with in calculus. Since the concept of equivalence relation would unnessesarily confuse students, the more fuzzy definition above is prefered. One should avoid definitions like "Vectors are objects which have length and direction" given in some Encyclopedias. The zero vector (0,0,0) is an example of an object which has length but no direction. It nevertheless is a vector. +------------------------------------------------------------ | vector field +------------------------------------------------------------ A vector field in the plane is a map F(x,y)=(P(x,y),Q(x,y)) which assigns to each point (x,y) in the plane a vector F(x,y). An example of a vector field in the plane is F(x,y) = (-y,x). An other example is the gradient field F(x,y) = grad f(x,y) where f(x,y) is a function. A vector field in space is a map F(x,y,z)=(P(x,y,z),Q(x,y,z),R(x,y,z)) which assigns to each point (x,y,z) in space a vector F(x,y,z). An example is the vector field F(x,y,z) = (x^2,y z,x-y). An other example is the gradient field F(x,y,z) = grad f(x,y,z) of a function f(x,y,z). +------------------------------------------------------------ | velocity +------------------------------------------------------------ The velocity of a parametrized curve r(t)=(x(t),y(t),z(t)) at time t is the vector r'(t) = (x'(t),y'(t),z'(t)). It is tangent to the curve at the point r(t). +------------------------------------------------------------ | volume +------------------------------------------------------------ The volume of a body G is defined as the integral of the constant function f(x,y,z)=1 over the body G. +------------------------------------------------------------ | wave equation +------------------------------------------------------------ The wave equation is the partial differential equation u_tt=c^2 Delta(u), where Delta(u) is the Laplacian of u. Light in vacuum satisfies the wave equation. This can be derived from the Maxwell equations: the identity Delta(B) = grad( div(B)- curl( curl(B)) gives together with div(B)=0 and curl(B)=E_t/c the relation Delta(B)=-d/dt curl(E)/c which leads with the Maxwell equation B_t=-c curl(E) to the wave equation Delta B = B_tt/c^2. The equation E_tt= c^2 Delta E is derived in the same way. +------------------------------------------------------------ | zero vector +------------------------------------------------------------ The zero vector is the vector for which all components are zero. In the plane it is v=(0,0), in space it is v=(0,0,0). The zero vector is a vector. It has length 0 and no direction. Definitions like "a vector is a quantity which has both length and direction" are misleading. This file is part of the Sofia project sponsored by the Provost's fund for teaching and learning at Harvard university. There are 119 entries in this file. COUNT: 119 ENTRY LINEAR ALGEBRA Author: Oliver Knill: Spring 2002-Spring 2004 Literature: Standard glossary of multivariable calculus course as taught at the Harvard mathematics department. +------------------------------------------------------------ | adjacency matrix +------------------------------------------------------------ The adjacency matrix of a graph is a matrix A_ij, where A_ij=1 whenever there is an edge from node i to node j in the graph. Otherwise, A_ij=0. Example: the graph with three nodes with the shape of a V has the adjacency matrix A=( 0 1 0 1 0 1 0 1 0 ), where node 2 is conneced to both node 1 and 3 and node 1 and 3 are not connected to each other. +------------------------------------------------------------ | affine transformation +------------------------------------------------------------ An affine transformation is the composition of a linear transformation with a shift like for example: T(x,y) =(2x+y,3x+4 y) + (2,3). +------------------------------------------------------------ | Algebra +------------------------------------------------------------ Algebra was originally the art of solving equations and systems of equations. The word comves from the Arabic "al-jabr" meaning "restauration". The term was used by Mohammed al-Khowarizmi, who worked in Bhagdad. +------------------------------------------------------------ | algebraic multiplicity +------------------------------------------------------------ The algebraic multiplicity of a root y of a polynomial p is the maximal integer k for which p(x)=(x-y)^k q(x). The algebraic multiplicity is bigger or equal than the geometric multiplicity. +------------------------------------------------------------ | angle +------------------------------------------------------------ The angle between two vectors v and w is arccos( (x . y)/(||x|| ||y||), where x dot y is the dot product between x and y and ||x||=sqrtx . x is the length of x. The inverse of cos gives two angles in 0,2 pi. One usually choses the smaller angle. +------------------------------------------------------------ | argument +------------------------------------------------------------ The argument of a complex number z=x+iy is phi if z=r e^i phi. The argument is determined only up to addition of 2 pi. It can be determined as phi = arctan(y/x) + A, where A=0 if x>0 or x=0,y>0 and A=pi if x<0 or x=0 and y<0. For example, arg(i)=pi/2 and arg(-i)=3 pi/2. The argument is the imaginary part of log(z) because log(r e^i phi) = log(r) + i phi. +------------------------------------------------------------ | associative law +------------------------------------------------------------ The associative law is (A B) C = A (B C). It is an identity which some mathematical operations satisfy. For example, the matrix multiplication satisfies the associative law. One says also, that the operation is associative. An example of a product which is not associative is the cross product v x w: if i,j,k are the standard basis vectors, then i x (i x j) = i x k = -j and (i x i) x j = 0 x j = 0. +------------------------------------------------------------ | augmented matrix +------------------------------------------------------------ The augmented matrix of a linear equation Ax=b is the n x (n+1) matrix ( A b ). One considers the augmented matrix when solving a linear system Ax=b. The reduced row echelon form rref ( A b ) contains the solution vector x in the last column, if a solution exists. More generally, a matrix which contains a given matrix as a submatrix is called an augmented matrix. +------------------------------------------------------------ | basis +------------------------------------------------------------ A basis of a linear space is a finite set of vectors v_1, ...,v_n, which are linearly independent and which span the linear space. If the basis contains n vectors, the vector space has dimension n. +------------------------------------------------------------ | basis theorem +------------------------------------------------------------ The basis theorem states that d linearly independent vectors in a vector space of dimension d forms a basis. +------------------------------------------------------------ | block matrix +------------------------------------------------------------ A block matrix is a matrix A, where the only non-zero elements are contained in a sequence of smaller square matrices arranged along the main diagonal of A. Such matrices are also called block diagonal matrices. The matrix A=( 1 2 0 0 0 3 2 0 0 0 0 0 5 0 0 0 0 0 6 7 0 0 0 8 9 ). is an example of a block-diagonal matrix containing a two 2x2, and a 1x1 matrix in its diagonal. +------------------------------------------------------------ | Cauchy-Schwarz inequality +------------------------------------------------------------ The Cauchy-Schwarz inequality tells that |x . y| is smaller or equal to ||x|| ||y||. Equality holds if and only if x and y are parallel vectors. +------------------------------------------------------------ | Cayley-Hamilton theorem +------------------------------------------------------------ The Cayley-Hamilton theorem assures that every square matrix A satisfies p(A)=0, where p(x) = det(A-x) is the characteristic polynomial of A and the right hand side 0 is the zero matrix. +------------------------------------------------------------ | change of basis +------------------------------------------------------------ A change of basis from an old basis v_j to a new basis w_j is described by an invertible matrix S which relates the coordinates (a_1,...,a_n) of a vector a = sum_i a_i v_i in the old v-basis with the coordinates (b_1,...,b_n) of the same vector b=sum_i b_i w_i in the new w-basis. The relation of the coordinates is b = S a In that case, one has v_j = sum_j S^T_ij w_j, where S^T is the transpose of S. For example if v_1 = (1,0), v_2 = (0,1), w_1 = (3,4) w_2 = (2,3), then a=(a_1,a_2)=(5,7) in the v-basis has the coordinates b=(b_1,b_2)=(1,1) in the w-basis. With S = ( 3 2 4 3 ) and S^T=( 3 4 2 3 ) we have b = S a and w_1 = 3 v_1 + 4 v_2, w_2 = 2 v_1 + 3 v_2. +------------------------------------------------------------ | characteristic matrix +------------------------------------------------------------ The characteristic matrix of a square matrix A is the matrix A(x) = (x I-A) , where I is the identity matrix. The characteristic matrix is a function of the free variable x. +------------------------------------------------------------ | characteristic polynomial +------------------------------------------------------------ The characteristic polynomial of a matrix A is the polynomial p(x) = det(x I-A), where I is the identity matrix. It has the form p(x) = x^n - tr(A) x^(n-1) + ... + (-1)^n det(A) where tr(A) is the trace of A and det(A) is the determinant of the matrix A. The eigenvalues of A are the roots of the characteristic polynomial of A. +------------------------------------------------------------ | Cholesky factoriztion +------------------------------------------------------------ The Cholesky factoriztion of a symmetric and positive definite matrix A is A = R^T R, where R is upper triangular with positive diagonal entries. +------------------------------------------------------------ | circulant matrix +------------------------------------------------------------ A circulant matrix is a square matrix, where the entries in each diagonal are constant. If S is the shift matrix which has 1 in the side diagonal and 0 everywhere else like in the 3x3 case: S = ( 0 1 0 0 0 1 1 0 0 ), then a circular matrix can be written as A=a_0 + a_1 S + ... + a_n-1 S^n-1. A general 3x3 circulant matrix has the form A = a + b S + c S^2 which is S = ( a b c c a b b c a ). +------------------------------------------------------------ | classical adjoint +------------------------------------------------------------ The classical adjoint adj(A) of a n x n matrix A is the n x n matrix whose entry a_ij is a_ij = (-1)^(i+j) det(A_ji), where A_ji is a minor of A. The classical adjoint plays a role in Cramer's rule A^(-1) = adj(A)/ det(A). The name "adjoint" comes from the fact that we have a change indices like in the adjoint. However, the classical adjoint has nothing to do with the adjoint. +------------------------------------------------------------ | codomain +------------------------------------------------------------ The codomain of a linear transformation T: X to Y is the target space Y. The name has its origin from naming X the domain of A. +------------------------------------------------------------ | cofactor +------------------------------------------------------------ A cofactor C_ij of a n x n matrix A is the determinant of the (n-1) x (n-1) matrix obtained by removing row i and column j from A and multiplying the result with (-1)^i+j. +------------------------------------------------------------ | coefficient +------------------------------------------------------------ A coefficient of a matrix A is an entry A_ij in the i'th row and the j'th column. For a real matrix, all entries are real numbers, for a complex matrix, the entries can be complex numbers. +------------------------------------------------------------ | column +------------------------------------------------------------ A column of a matrix is one of the vectors ( A_11 A_21 A_31 dots A_m1 ), ( A_1n A_2n A_3n dots A_mn ) of a m x n matrix A. Column vectors are in the image of the transformation x mapsto A(x). +------------------------------------------------------------ | column +------------------------------------------------------------ The matrix A defining a linear equation Ax=b or begineqnarray* A_11 x_1 + dots A_1n x_n = b_1 dots = dots A_m1 x_1 + dots A_mn x_n = b_m endeqnarray* is called the coefficient matrix of the system. The augmented matrix is the m x (n+1) matrix ( A b ), where b forms an additional column. +------------------------------------------------------------ | column picture +------------------------------------------------------------ The column picture of a linear equation Ax = b is that the vector b becomes a linear combination of the columns of A. The linear equation is solvable if the vector b is in the column space of A. +------------------------------------------------------------ | column space +------------------------------------------------------------ The column space of a matrix A is the linear space spanned by the columns of A. +------------------------------------------------------------ | commuting matrices +------------------------------------------------------------ Two commuting matrices A,B satisfy A B = B A. In that case, if A is diagonalizable, then also B is diagonalizable and both A and B share the same n eigenvectors. +------------------------------------------------------------ | commutative law +------------------------------------------------------------ The commutative law A*B = B*A for some operation * is an identity which holds for certain operations like the addition of matrices. Other operations like the multiplication of matrices does not satisfy the commutative law. One says: matrix multiplication is not commutative. +------------------------------------------------------------ | complex conjugate +------------------------------------------------------------ The complex conjugate of a complex number z=x+iy is the complex number x-iy. It has the same length |z| as z. +------------------------------------------------------------ | Complex numbers +------------------------------------------------------------ Complex numbers form an extension of the real numbers. They are obtained by introducing i = (-1)^1/2 and extending the rules of addition (a + ib) + (c + id) = (a + c) + i(b + d) and multiplication (a + ib)(c + id) = (ac - bd) + i(ad + bc). The absolute value r=|x+iy| is the length of the vector (a,b). The argument of z, phi= arg(z) is defined as the angle in 0,2pi) between the x axes and the vector (a,b). Using these polar coordinates one can see the Euler identity z = r exp(i phi) = r cos(phi) + i r sin(phi). +------------------------------------------------------------ | consistent +------------------------------------------------------------ A system of linear equations Ax=b is called consistent, if there exists for every vector b a solution vector x to the equation Ax=b. If the system has no solution, the system is called inconsistent. +------------------------------------------------------------ | continuous dynamical system +------------------------------------------------------------ A continuous dynamical system is defined by an ordinary differential equation d/dt u = f(u) where u=u(t) is a vector valued function and f(u) is a vector field. If f(u) is linear, the equation has the form d/dt u = A u. The name "continuous" comes from the fact that the time variable t is taken in the continuum. This distinguishes the system from discrete dynamical systems u(t+1)=f(u(t)) determined by a map f and where t is an integer. For linear continuous dynamical systems, the origin 0 is invariant. The origin is called asymptotically stable if x(t) to 0 for all initial conditions x(0). For continuous dynamical systems u_t = A u, this is equivalent with the requirement that all eigenvalues of A have a negative real part. In two dimensions, where the trace and the determinant determine the eigenvalues, linear stability is characterized by det(A)>0, tr(A)<0 (stability quadrant). +------------------------------------------------------------ | covariance matrix +------------------------------------------------------------ A covariance matrix A of two finite dimensional random variables x,y with expectation Ex=Ey=0 is defined as A_ij = E x_i y_j , where Ex= (x_1+...+x_n)/n is the mean or expectation of x. The covariance matrix is always symmetric. If the covariance matrix is diagonal, the random variables x,y are called uncorrelated. +------------------------------------------------------------ | Cramer's rule +------------------------------------------------------------ Cramer's rule tells that a solution x of a linear equation Ax=b can be obtained as x_i = det(A_i)/ det(A), where A_i is the matrix obtained by replacing the column i of A with the vector b. +------------------------------------------------------------ | de Moivre formula +------------------------------------------------------------ The de Moivre formula is (cos(x)+i sin(x))^n = cos(nx) + i sin(n x). It is useful to derive trigonometric identities like cos(x)^3-3 sin(x)^2 cos(x)=cos(3x). +------------------------------------------------------------ | determinant +------------------------------------------------------------ The determinant of a n x n square matrix A is the sum over all products A1,pi(1) ... An,pi(n) (-1)^pi, where pi runs over all permutations of 1,2,...,n and (-1)^pi is the sign of the permutation pi. Example: for a 2 x 2 matrix A= ( a b c d ) the determinant is det(A)=ad-bc. Example: For a 3x3 matrix A = ( a b c d e f g h i ), the determinant is det(A)=a e i + b f g + c d h - c e g - b f g - c d h. Properties of the determinant are det(A B)= det(A) det(B), det(A^T)= det(A), det(A^-1)=1/ det(A). +------------------------------------------------------------ | differential equation +------------------------------------------------------------ A differential equation is an equation for a function f in one or several variables which involves derivatives with respect to these variables. An ordinary differential equation is a differential equation, where derivatives appear only with respect to one variable. By adding new variables if necessary (for example for t, or derivatives u_t,u_tt etc, one can write an ordinary differential equation always in the form x_t = f(x). +------------------------------------------------------------ | dilation +------------------------------------------------------------ A dilation is a linear transformation x to b x. Dilations scale each vector v by the factor b but leave the direction of v invariant. +------------------------------------------------------------ | dimension +------------------------------------------------------------ The dimension of a vector space X is the number of basis vectors in a basis of X. +------------------------------------------------------------ | distributive law +------------------------------------------------------------ The distributive law is A*(B+C) = A*B + A*C. The set of matrices with matrix multiplication * and addition + is an example where the distributive law applies. +------------------------------------------------------------ | dot product +------------------------------------------------------------ The dot product v . w of two vectors v and w is the sum of the products v_i w_i of their components v_i, w_i. For complex vectors, the dot product is defined as sum_i overlinev_i w_i. Examples: (3,2,1) . (1,2,-1) = 6. if v . w=0, then the vectors are orthogonal. the length of the vector |v| is the square root of v . v. v . w = |v| |w| cos(alpha), where alpha is the angle between v and w. if A,B are two n x n matrices, then (AB)_ij is the dot product of the i'th row of A with the j'th column of B +------------------------------------------------------------ | echelon matrix +------------------------------------------------------------ The echelon matrix of a matrix A is a matrix rref(A), where the pivot in each row comes after the pivot in the previous row. The pivot is the first nonzero entry in each row. The echelon matrix is also called a matrix in reduced row echelon form. +------------------------------------------------------------ | eigenbasis +------------------------------------------------------------ An eigenbasis to a matrix A is a basis which consists of eigenvectors of A. +------------------------------------------------------------ | eigenvalue +------------------------------------------------------------ An eigenvalue L of a matrix A is a number for which there exists a vector v such that A v= L v. Example: A = ( 3 2 0 4 ) has the eigenvector v = (0,1) with eigenvalue x=4. +------------------------------------------------------------ | eigenvector +------------------------------------------------------------ An eigenvector v of a matrix A is a nonzero vector v for which Av=L v with some number L (called eigenvalue). Example: ( -1 1 1 1 ) has the eigenvector v = (1,1) with eigenvalue L=0. +------------------------------------------------------------ | Elimination +------------------------------------------------------------ Elimination is a process which reduces a matrix A to its echelon matrix rref(A). See row reduced echelon form. +------------------------------------------------------------ | ellipsoid +------------------------------------------------------------ An ellipsoid can be written as the set of points x which satisfy x dot Ax =1, where A is a positive definit matrix. The axes v_i of the ellipse are the eigenvectors of A have the length 1/x_i , where x_i are the eigenvalues of A. +------------------------------------------------------------ | entry +------------------------------------------------------------ An entry or coefficient of a matrix is the number or the variable A(i,j) of a matrix. +------------------------------------------------------------ | expansion factor +------------------------------------------------------------ The expansion factor of a linear map is the absolute value of the determinant of A. It is the volume of the parallelepiped obtained as the image of the unit cube under A. +------------------------------------------------------------ | exponential +------------------------------------------------------------ The exponential exp(A) of a matrix A is defined as the sum exp(A) = 1 + A + A^2/2! + A^3/3! + .... The linear system of differential equations x'=Ax for x(t) has the solution x(t) = exp(A t) x(0). +------------------------------------------------------------ | factorization +------------------------------------------------------------ The factorization of a polynomial p(x) is the representation p(x)= a (L_1-x) ... (L_n-x), where lambda_i are the n roots of the polynomials whose existence is assured by the fundamental theorem of algebra. +------------------------------------------------------------ | Fourier coefficients +------------------------------------------------------------ The Fourier coefficients of a 2pi periodic function f(x) on -pi,pi is c_n = (1/2 pi) int_-pi^pi f(x) exp(-i n x). One has f(x) = sum_n c_n exp(i n x). By writing f(x)=g(x)+h(x), where g(x)=f(x)+f(-x)/2 is even and h(x)=f(x)-f(-x)/2 is odd one can obtain real versions: the even function can be written as a cos-series g(x) = sum_n=0^infinity a_n cos(n x), where a_n=(2/pi) int_0^pi g(x) cos(n x) dx for n>0 and a_0=(1/pi) int_0^pi g(x) dx. The odd function can be written as the sin-series h(x)= sum_n=1^infinity b_n sin(n x), where b_n=(2/pi) int_0^pi h(x) sin(n x) dx. The complex Fourier coefficients c_n are coordinates of f(x) with respect to the orthonormal basis exp(i n x). The real Fourier coefficients are the coordinates of f(x) with respect to orthogonal basis 1, cos(n x), sin(n x), n>0. % The Fourier series of a function f is f(x) = sum_n=-infinity^infinity c_n exp(i n x) or f(x) = sum_n a_n cos(n x) for even functions or f(x)=sum_n=1^infinity b_n sin(n x) for odd functions. +------------------------------------------------------------ | fundamental theorem of algebra +------------------------------------------------------------ The fundamental theorem of algebra states that a polynomial p(x) =x^n+... a_1 x + a_0 of degree n has exactly n roots. +------------------------------------------------------------ | Gauss Jordan elimination +------------------------------------------------------------ The Gauss Jordan elimination is a method for solving linear equations. It was already known by the Chinese 2000 years ago. Gauss called it "eliminatio vulgaris". The method does linear combinations of the rows of a n x (n+1) matrix until the system is solved. Example: 2 x + 4y = 2 3 x + y = 12 x + 2y = 1 3 x + y = 13 x + 2y = 1 - 5y = 10 x + 2y = 1 y = -2 x = 5 y = -2 First, the top equation was scaled, then three times the first equation was subtracted from the second equation. Then the the second equation was scaled. Finally, twice the the second equation was subtracted from the first. +------------------------------------------------------------ | geometric multiplicity +------------------------------------------------------------ The geometric multiplicity of an eigenvalue L is the dimension of ker(L-A). The geometric multiplicity is smaller or equal to the algebraic multiplicity. +------------------------------------------------------------ | Gibbs phenomenon +------------------------------------------------------------ The Gibbs phenomenon describes the error when doing a Fourier approximation of the discontinuous Heavyside function f(x)= -1 if -pi leq x leq 0 1 if 0 < x 0 a_n sin(n x) e^-mu n^2 t, where a_n=(2/pi) int_0^pi f(x) sin(n x) dx are the Fourier coefficients. +------------------------------------------------------------ | Hermitian matrix +------------------------------------------------------------ A Hermitian matrix satisfies A^* = overlineA^T = A, where A^T is the transpose of A and overlineA is the complex conjugate matrix, where all entries are replaced by their complex conjugates. +------------------------------------------------------------ | Hessenberg matrix +------------------------------------------------------------ A Hessenberg matrix A is an upper triangular matrix with only one extra nonzero adjacent diagonal below the diagonal. Example: a general 3 x 3 Hessenberg matrix is A = ( a b c d e f 0 g h ). +------------------------------------------------------------ | Hilbert matrix +------------------------------------------------------------ A Hilbert matrix is a symmetric square matrix, where A_ij = 1/(i+j-1). It is an example of a Hankel matrix and positive definit. Hilbert matrices are examples of matrices which are difficult to invert, because their determinant is small. For example, for n=3, the Hilbert matrix A = ( 1 1/2 1/3 1/2 1/3 1/4 1/3 1/4 1/5 ) has determinant 1/2160. % A hyperplane in n-dimensional space V is a (n-1)-dimensional linear subspace of V. +------------------------------------------------------------ | identity matrix +------------------------------------------------------------ The identity matrix is the matrix I which has 1 in the diagonal and zero everywhere else. The identity matrix I satisfies I A = A for any matrix A. The 3 x 3 identity matrix is A = ( 1 0 0 0 1 0 0 0 1 ). +------------------------------------------------------------ | incidence matrix +------------------------------------------------------------ The incidence matrix of a directed graph with n nodes and m edges is a m x n matrix which has a row for each edge connecting nodes i and j with entries -1 and 1 in columns i, j. Example. The directed graph 1 Rightarrow 2 Leftarrow 3, 1 arrow 4 with 3 edges and 4 nodes has the 3 x 4 incidence matrix A = ( -1 1 0 0 0 1 -1 0 -1 0 0 1 ). +------------------------------------------------------------ | inconsistent +------------------------------------------------------------ A system of linear equations Ax=b is called inconsistent if the system has no solutions. +------------------------------------------------------------ | indefinite matrix +------------------------------------------------------------ An indefinite matrix is a matrix, whith eigenvalues of different sign. A positive definite matrix is an example of a matrix which is not indefinite. +------------------------------------------------------------ | Independent vectors +------------------------------------------------------------ Independent vectors. If no linear combination a_1 v_1 + ... + a_n v_n is zero unless all a_i are zero, then the vectors v_1,...,v_n are called independent. If A is the matrix which contains the vectors v_i as columns, then the kernel of A is trivial. A basis consists of independent vectors. +------------------------------------------------------------ | Independent vectors +------------------------------------------------------------ The linear transformation corresponding to the identity matrix is called the identity transformation. +------------------------------------------------------------ | image +------------------------------------------------------------ The image of a linear transformation T: X to Y, T(x)=Ax is the subset of all vectors y=Ax, x in X in Y. The image is denoted by im(T) or im(A) and is a subset of the codomain Y of T. The image is also called the range. The dimension of the image of T is equal to the rank of A and the dimension satisfies dim( ker(A)) + dim( im(A)) = n, where n is the dimension of the linear space X. +------------------------------------------------------------ | index +------------------------------------------------------------ The index of a linear map T is defined as ind(A) = dim ( ker A) - dim ( coker A), where coker(A) is the orthogonal complement of the image of A. Examples are: The index of a n x n matrix A is dim ( ker A) - dim ( coker A) = 0. The index of the differential operator Df = f' acting on smooth functions on the real line is 1-0 = 1 because D has a one dimensional kernel (the constant functions) and a zero dimensional cokernel (all functions can be obtained as the image of D). The index of D^n is n. The index of the differential operator Df=f' acting on smooth functions on the circle is 1-1=0 because D has a one dimensional kernel (the constant functions) and a one-dimensional cokernel (the constant functions, one can not find a periodic function g such that g'=1). The Atiyah-Singer index theorem relates topological properties of a surface M with the index of a "Dirac operator" T on it. The previous two examples exemply that. T=D is a Dirac operator and the topology of the circle or the line are different. +------------------------------------------------------------ | inverse +------------------------------------------------------------ The inverse of a square matrix A is a matrix B satisfying A B=I and B A=I where I is the identity matrix. For example, the inverse of the transformation A= ( a b c d ) is the transformation B = ( d -b -c a ) / (ad-bc). +------------------------------------------------------------ | invertible +------------------------------------------------------------ A square matrix A is invertible if there exists a matrix B such that A B=I. A matrix A is invertible if and only if the determinant of A is different from zero. +------------------------------------------------------------ | Jordan normal form +------------------------------------------------------------ The Jordan normal form J = S^-1 A S of a square matrix A is a block matrix J = diag(J_1, ..., J_k), where each block is of the form J_k = x_k I_k + N_k, where x_k is an eigenvalue of A, I_k is an identity matrix and N_k is a matrix with 1 in the first sidediagonal. If all eigenvalues of A are different, then the Jordan normal form is a diagonal matrix. +------------------------------------------------------------ | kernel +------------------------------------------------------------ The kernel of a linear transformation T: X to Y, T(x)=Ax is the linear space x in X such that Ax=0 . The kernel is denoted by ker(T) or ker(A) and is a subset of the domain X of T. The dimension of the kernel dim( ker(A)) and the dimension of the image dim( im(A)) are related by dim( ker(A)) + dim( im(A)) = n, where n is the dimension of the linear space X. The kernel of a transformation is computed by building rref(A), the reduced row echelon form of A. Echelon = "series of steps". Every vector in the kernel of rref(A) is also in the kernel of A. +------------------------------------------------------------ | Leontief +------------------------------------------------------------ Wassily Leontief. A Russian-born US economist who was working also at Harvard University. Leontief was a winner of the 1973 Economics Nobel prize for the development of the input-output method and for its application to important economic problems. Linear algebra students find the following problem in textbooks: two industries A and B produce output with value x and y (in millions of dollars). Assume that the consumer demand is a for the product of A and b for the product of B. Assume also an industry demand: p x is transfered from A to B, and q y is transfered from B to A. For which x and y are both the industry and conumer demand satisfied? The problem is equivalent to solving the linear system begineqnarray* x - q y = a -p x + y = b endeqnarray* +------------------------------------------------------------ | Laplace equation +------------------------------------------------------------ The Laplace equation in a region G is the linear partial differential equation u_xx + u_yy = 0. A solution is determined if u(x,y) is prescribed on the boundary of G. On the square 0,pi x 0,pi with boundary conditions 0 except at the side y=pi, where one has u(x,pi)=f(x), one can find a solution via Fourier series: u(x,y) = sum_n>0 a_n sin(n x) sinh(n y)/sinh(n pi), where a_n=(2/pi) int_0^pi f(x) sin(n x) dx. The case with general boundary conditions can be solved by adding corresponding solutions u(x,y),u(y,x),u(x,pi-y),u(pi-y,x) for the other 3 sides of the square. +------------------------------------------------------------ | Laplace expansion +------------------------------------------------------------ The Laplace expansion is a formula for the determinant of A: det(A) = (-1)^i+1 a_i1 det(A_i1)+ ... +(-1)^i+n a_in det(A_in). +------------------------------------------------------------ | leading one +------------------------------------------------------------ A leading one is an entry of a matrix in reduced row echelon form which is contained in a row with this element as the first nonzero entry. +------------------------------------------------------------ | leading variable +------------------------------------------------------------ A leading variable is a variable which corresponds to a leading one in rref(A). +------------------------------------------------------------ | least-squares solution +------------------------------------------------------------ A vector x in R^n is called a least-squares solution of the system Ax=b where A is a m x n matrix, if ||b-Ay|| is less or equal then ||b-Ax|| for all y in R^n. If x is the least-squares solution of Ax=b then Ax is the orthogonal projection of b onto the image im(A). The explicit formula is x = (A^T A)^(-1) A^T b and derived from that A^T (Ax-b) =0 which itself just means that Ax-b is orthogonal to the image of A. +------------------------------------------------------------ | length +------------------------------------------------------------ The length of a vector v is ||v||=(v . v)^1/2 = (v_1^2+ ... + v_n^2)^(1/2). The length of a complex number x+iy is the length of (x,y). The length of a vector depends on the basis, usually it is understood with respect to the standard basis. +------------------------------------------------------------ | linear combination +------------------------------------------------------------ A linear combination of n vectors v_1, ..., v_n is a vector a_1 v_1 + ... + a_n v_n. +------------------------------------------------------------ | Linearly dependent vectors +------------------------------------------------------------ Linearly dependent vectors. If there exist a_1,...,a_n which are not all zero such that a_1 v_1 + ... + a_n v_n =0, then the vectors v_1,...,v_n are called linearly dependent. +------------------------------------------------------------ | Linearity +------------------------------------------------------------ Linearity is a property of maps between linear spaces: it means that lines are maped into lines and the image of the sum of two vectors is the same as sum of the images. For example: T(x,y,z) = (2x+z,y-x) is linear. T(x,y) = (x^2-y,x) is nonlinear. +------------------------------------------------------------ | linear dynamical system +------------------------------------------------------------ A linear dynamical system is defined by a linear map x mapsto Ax. The orbits of the dynamical system are x, Ax, A^2x, ... . +------------------------------------------------------------ | linear space +------------------------------------------------------------ A linear space is the same as a vector space. It is a set which is closed under addition and multiplication with real numbers. +------------------------------------------------------------ | linear combination +------------------------------------------------------------ A sum a_1 v_1 + ... + a_n v_n is called a linear combination of the vectors v_1, ... , v_n. +------------------------------------------------------------ | linear subspace +------------------------------------------------------------ A linear subspace of a vector space V is a subset of V which is also a vector space. In particular, it is closed under addition, scalar multiplication and contains a neutral element. +------------------------------------------------------------ | linear system of equations +------------------------------------------------------------ A linear system of equations is an equation of the form Ax=b, where A is a m x n matrix, x is a n-vector and b is a m-vector. There are three possibilities: consistent with one solution: no row vector (0 ... 0 | 1 ) in rref(A | b). There is exactly one solution if there is a leading one in each column of rref(A). consistent with infinitely many solutions: there are columns with no leading one. Inconsistent with no solutions: there is a row (0 ... 0 | 1 ) in rref(A | b). +------------------------------------------------------------ | logarithm +------------------------------------------------------------ The logarithm log(z) of a complex number z=x+iy neq 0 is defined as log|z|+i arg(z), where arg(z) is the argument of z. The imaginary part of the logarithm is only defined up to a multiple of 2 pi. +------------------------------------------------------------ | Markov matrix +------------------------------------------------------------ A Markov matrix is a square matrix, where all entries are nonnegative and the sum of each column is 1. One of the eigenvalues of a Markov matrix is 1 because A^T has the eigenvector (1,1,...,1). If all entries of a Markov matrix are positive, then A^k v converges to the eigenvector v with eigenvalue 1. This vector is called the "steady state" vector. +------------------------------------------------------------ | matrix +------------------------------------------------------------ A matrix is a rectangular array of numbers. The following 3 x 4 matrix for example consists of three rows and four columns: A = ( 2 8 4 2 1 2 1 3 2 -1 1 2 ). The first index addresses the row, the second the column of the matrix. A n x m matrix maps the m-dimensional space to the n-dimensional space. +------------------------------------------------------------ | Matrix multiplication +------------------------------------------------------------ Matrix multiplication is an operation defined between a (n x m) matrix A and a (m x p) matrix B. (A B)_ij is the dot product between the i'th row of A with the j'th column of B. Example: (n=2, m=3, p=4) ( 2 1 0 1 0 1 ) ( 2 1 0 0 0 0 1 2 1 3 1 1 ) =( 4 2 1 2 3 4 1 1 ). +------------------------------------------------------------ | minor +------------------------------------------------------------ A minor of a matrix A is a matrix A(i,j) which is obtained from A by deleting row i and column j. +------------------------------------------------------------ | nilpotent matrix +------------------------------------------------------------ A nilpotent matrix is a matrix A for which some power A^k is the zero matrix. A nilpotent matrix has only zero eigenvalues. he matrix A = ( 0 1 1 0 0 2 0 0 0 ) for example satisfies A^3=0 and is therefore nilpotent. +------------------------------------------------------------ | non-leading coefficient +------------------------------------------------------------ A non-leading coefficient is an entry in the row reduced echelon form of a matrix A which is nonzero and which comes after the leading 1. The relevance of this definition comes from the fact that the number of columns with non-leading coefficients is the dimension of the kernel of the map. +------------------------------------------------------------ | normal equation +------------------------------------------------------------ The normal equation to the linear equation Ax=b is the consistent system A^T A x = A^T b. +------------------------------------------------------------ | normal matrix +------------------------------------------------------------ A matrix A is called a normal matrix, if A A^T = A^T A. A normal matrix has orthonormal possibly complex eigenvectors. +------------------------------------------------------------ | null space +------------------------------------------------------------ The null space of a matrix A is the same as the kernal of A. It is spanned by the solutions A v = 0. The dimension of the null space is n-r, where n is the number of columns of A and r is the rank of A. +------------------------------------------------------------ | ordinary differential equation +------------------------------------------------------------ An ordinary differential equation is a differential equation, where derivatives appear only with respect to one variable. By adding new variables if necessary (for example for t, or derivatives u_t,u_tt etc. one can write such an equation always in the form x_t = f(x). An ordinary differential equation defines a continuous dynamical system. The initial condition x(0) determines the trajectories x(t). An ordinary differential equation of the form u_t = A u, where A is a matrix is called a linear ordinary differential equation. +------------------------------------------------------------ | orthogonal +------------------------------------------------------------ Two vectors v,w are orthogonal if their dot product v . w vanishes. +------------------------------------------------------------ | orthogonal basis +------------------------------------------------------------ An orthogonal basis is a basis such that all vectors in the basis are orthogonal. +------------------------------------------------------------ | orthonormal basis +------------------------------------------------------------ An orthonormal basis is a basis such that all vectors are orthogonal and normed. +------------------------------------------------------------ | orthonormal complement +------------------------------------------------------------ The orthonormal complement of a linear subspace V in R^n is the set of vectors which are orthogonal to V. +------------------------------------------------------------ | orthogonal projection +------------------------------------------------------------ The orthogonal projection onto a linear space V is proj_V(x) = (x . v_1) v_1 + ... + (x . v_n) v_n, where the v_j form an orthonormal basis in V. Despite the name, an orthogonal projection is not an orthogonal transformation. It has a kernel. In an eigenbasis, a projection has the form (x,y) to (x,0). +------------------------------------------------------------ | Euclidean space +------------------------------------------------------------ Euclidean space is the linear space of all vectors = 1 x n matrices. R^0 is the space 0. The space R^1 is the real linear space of all real numbers, the R^2 is the plane, the R^3 the Euclidean three dimensional space. +------------------------------------------------------------ | parallel +------------------------------------------------------------ Two vectors v and w are called parallel if v both are nonzero and one is a multiple of the other. +------------------------------------------------------------ | parallelepiped +------------------------------------------------------------ A set in R^n is a parallelepiped E if it is the linear image A(Q) of the unit cube Q. The volume of a n-dimensional parallelepiped E in R^n satisfies vol(E)=| det(A)|, in general, vol(E)=( det(A^T A) )^1/2. +------------------------------------------------------------ | partial differential equation +------------------------------------------------------------ A partial differential equation (PDE) is an equation for a multi-variable function which involves partial derivatives. It is called linear if (u+v) and r v are solutions whenever u and v are solutions. Examples of linear PDEs: u_t = c u_x transport equation u_t = b u_xx heat equation u_tt = a u_xx wave equation u_xx + u_yy = 0 Laplace equation u_xx + u_yy = f(x,y) Poisson equation i h u_t = - u_xx + V(x) Schroedinger equation Examples of nonlinear PDE: u_t + u u_x = a u_xx Burger equation u_t + u u_x = - u_xxx Korteweg de Vries equation u_tt-u_xx= sin(x) Sine Gordon equation u_tt-u_xx = f(x) Nonlinear wave equation i h u_t = -u_xx-|x|^2 x Nonlinear Schroedinger equation u_t + u_x(x,t)^2/2 + V(x)=0 Hamilton Jacobi equation +------------------------------------------------------------ | permutation matrix +------------------------------------------------------------ A permutation matrix A is a square matrix with entries A_ij = I_i pi(j) where pi is a permutation of 1,...,n and where I is the identiy matrix. There are n! permutation matrices. Example: for n=3 the permutation pi(1,2,3) = (2,1,3) defines the permutation matrix A = ( 0 1 0 1 0 0 0 0 1 ). +------------------------------------------------------------ | pivot +------------------------------------------------------------ A pivot d is the first nonzero diagonal entry when a row is used in Gaussian elimination. +------------------------------------------------------------ | pivot column +------------------------------------------------------------ A column of a matrix is called a pivot column if the corresponding column of rref(A) contains a leading one. The pivot columns are important because they form a basis for the image of A. +------------------------------------------------------------ | polar decomposition +------------------------------------------------------------ The polar decomposition of a matrix A is A = O B, where O is orthogonal and where B is positive semidefinit. +------------------------------------------------------------ | positive definite matrix +------------------------------------------------------------ A positive definite matrix is a symmetric matrix which satisfies v . A v > 0, for every nonzero vector v. +------------------------------------------------------------ | power +------------------------------------------------------------ The n'th power of a matrix A is defined as A^n=A A^(n-1) = A A .... A. The eigenvalues of A^n are L_i^n, where L_i are the eigenvalues of A. +------------------------------------------------------------ | QR decomposition +------------------------------------------------------------ The QR decomposition of a matrix A is obtained during the Gramm-Schmitt orthogonalization process. It is A=QR, where Q is an orthogonal matrix and where R is an upper triangular matrix. +------------------------------------------------------------ | rank +------------------------------------------------------------ The rank of a linear matrix A is the set of leading 1's in the matrix rref(A). +------------------------------------------------------------ | orientation +------------------------------------------------------------ The orientation of n vectors v_1, ... ,v_n in the n-dimensional Euclidean space is defined as the sign of det(A), where A is the matrix with v_i in the columns. +------------------------------------------------------------ | orthogonal +------------------------------------------------------------ A square matrix A is orthogonal if it preserves length: ||Av|| = ||v|| for all vectors v. +------------------------------------------------------------ | perpendicular +------------------------------------------------------------ Two vectors v and w are called perpendicular if their dot product vanishes: v . w=0. A synonym of perpendicular is orthogonal. +------------------------------------------------------------ | projection matrix +------------------------------------------------------------ A projection matrix is a matrix P which satisfies P^2=P=P^T. It has eigenvalues 1 or 0. The image is a linear subspace S. The vectors in S are eigenvectors to the eigenvalues 1. The vectors in the orthogonal complement of S are eigenvectors to the eigenvalue 0. If A is the matrix which contains the basis of S as the columns, then P = A (A^T A)^-1 A^T is the projection onto S. +------------------------------------------------------------ | pseudoinverse +------------------------------------------------------------ The pseudoinverse of a (m x n) matrix A is the (n x m) matrix A^+ that maps the image of A to the image of A^T. The kernel of A^+ is the kernel of A^T and the rank of A^+ is equal to the rank of A. A^+ A is the projection on the image of A^T and A A^+ is the projection on the image of A. Especially, if A is an invertible (n x n) matrix, then A^+ is the inverse of A. The pseudoinverse is also called Moore-Penrose inverse. +------------------------------------------------------------ | rank +------------------------------------------------------------ The rank of a matrix A is the dimension of the image of A. +------------------------------------------------------------ | Rayleigh quotient +------------------------------------------------------------ The Rayleigh quotient of a symmetric matrix A is defined as the function q(v) = (v . Av)/(v . v). The maximal value of q(v) is the maximal eigenvalue of A and the minimal value of q(v) is the minimal eigenvalue of A. +------------------------------------------------------------ | rotation +------------------------------------------------------------ A rotation is a linear transformation which preserves the angle between two vectors as well as their lengths. A rotation in three dimensional space is determined by the axis of rotation as well as the rotation angle. +------------------------------------------------------------ | rotation matrix +------------------------------------------------------------ A rotation matrix is the matrix belonging to a rotation. A rotation matrix is an example of an orthogonal matrix. Example: In two dimensions, a rotation matrix has the form A = ( cos(t) sin(t) -sin(t) cos(t) ). +------------------------------------------------------------ | rotation-dilation matrix +------------------------------------------------------------ A rotation-dilation matrix is a 2x2 matrix of the form A = ( p -q q p ). It has the eigenvalues p pm i q. The action of A represents the complex multiplication with the complex number p+iq in the complex plane. +------------------------------------------------------------ | row +------------------------------------------------------------ A row of a matrix is formed by horizontal lines A_1j, j=1, .. n of a m x n matrix A. +------------------------------------------------------------ | shear +------------------------------------------------------------ A shear is a linear transformation in the plane which has in a suitable basis the form T(x,y) = (x,y+a x). More generally, in n dimensions, one can define as shear along a m-dimensional plane. If a basis is chosen so that the plane has the from (x,0) then a shear is T(x,y) = (x,y+a x). Shears have determinant 1 and preserve therefore volume. +------------------------------------------------------------ | singular +------------------------------------------------------------ A square matrix A is called singular if it has no inverse. A matrix A is singular if and only if det(A)=0. +------------------------------------------------------------ | singular value decomposition +------------------------------------------------------------ The singular value decomposition (SVD) of a matrix writes a matrix A in the form A = U D V^T, where U,V are orthogonal and D is diagonal. The first r columns of U form an orthonormal basis of the image of A and the first r columns of V form an orthonormal basis of the image of A^T. The last columns of U form an orthonormal basis of the kernel of A^T and the last columns of V form a basis of the kernel of A. +------------------------------------------------------------ | skew symmetric +------------------------------------------------------------ A matrix A is skew symmetric if it is minus its transpose that is if A^T=-A. The eigenvalues of a skew-symmetric matrix are purely imaginary. The eigenvectors are orthogonal. If A is skew symmetric, then B=exp(A t) is an orthogonal matrix, because B^T B = exp(-A t) exp(A t) = 1. For example A = ( 0 1 -1 0 ) gives the rotation matrix exp(A t). +------------------------------------------------------------ | span +------------------------------------------------------------ The span of a set of vectors v_1, ... v_n is the set of all linear combinations of v_1, ... ,v_n. +------------------------------------------------------------ | spectral theorem +------------------------------------------------------------ The spectral theorem tells that a real symmetric matrix A can be diagonalized A = U D U^T, where U is an orthonormal matrix containing an orthonormal eigenbasis in the columns and where D is a diagonal matrix D = diag(x_1,...,x_n), where x_i are the eigenvalues of A. +------------------------------------------------------------ | square matrix +------------------------------------------------------------ A square matrix is a matrix which has the same number of rows than columns. +------------------------------------------------------------ | asymptotically stable +------------------------------------------------------------ A linear dynamical system is called asymptotically stable if A^n x to 0 for all initial values x, where A^n is the n'th power of the matrix A. This is equivalent to the fact that all eigenvalues L of A satisfy |L|<1. +------------------------------------------------------------ | Stability triangle +------------------------------------------------------------ Stability triangle. A discrete dynamical system in the plane is asymptotically stable if and only if the trace and determinant are in the stability triangle | tr(A) |-1 < det(A)<1. A rotation-dilation A is asymptotically stable if and only if det(A)<1. +------------------------------------------------------------ | reduced row echelon form +------------------------------------------------------------ The reduced row echelon form rref(A) of a m x n matrix A is the end product of Gauss-Jordan elimination. The matrix rref(A) has the following properties: if a row has nonzero entries, then the first nonzero entry is 1, called leading 1. if a columns contains a leading 1, then all other entries in that column are 0. if a row contains a leading 1, then every row above contains a leading 1 further left. The algorithm to produce rref(A) from A is obtained by putting the cursor to the upper left corner and repeating the following steps until nothing changes anymore beginenumerate if the cursor entry is zero swap the cursor row with the first row below that has a nonzero entry in that column divide the cursor row by the cursor entry to make the cursor entry = 1 eliminate all other entries in cursor column by subtracting suitable multiples of the cursor row from the other row move the cursor down one row and and to the right one column. If the cursor entry is zero and all entries below are zero, move the cursor to the next column. repeat 4 if as long as necessary and move then to 1 endenumerate +------------------------------------------------------------ | reflection +------------------------------------------------------------ A reflection is a linear transformation T different from the identity transformation which satisfies T^2=1. The eigenvalues of T are -1 or 1. In an eigenbasis, the reflection has the form T(x,y)=(x,-y). The determinant of a reflection is 1 if and only if the dimension of the eigenspace to -1 is even. For example, a reflection at a line in the plane has the matrix A = ( cos(2 x) sin(2 x) sin(2 x) -cos(2 x) ) which has determinant -1. A reflection at the origin in the plane is -I with determinant 1. +------------------------------------------------------------ | root +------------------------------------------------------------ A root of a polynomial p(x) is a complex value z such that p(z)=0. According to the fundamental theorem of algebra, a polynomial of degree n has exactly n roots. +------------------------------------------------------------ | symmetric +------------------------------------------------------------ A matrix A is symmetric if it is equal to its transpose. The spectral theorem for symmetric matrices tells that they have real eigenvalues and symmetric matrices can always be diagonalized with an orthogonal matrix S. +------------------------------------------------------------ | span +------------------------------------------------------------ The span of a finite set of vectors v_1, ... ,v_n is the set of all possible linear combinations c_1 v_1 + c_2 v_2 + ... + c_n v_n where c_i are real numbers. For example, if v_1=(1,0,0) and v_2=(0,1,0), then the span of v_1,v_2 in three dimensional space is the xy-plane. The span is a linear space. +------------------------------------------------------------ | spectral theorem +------------------------------------------------------------ The spectral theorem for a symmetric matrix A assures that A can be diagonalized: there exists an orthogonal matrix S such that A^(-1) A S is diagonal and contains the eigenvalues of A in the diagonal. +------------------------------------------------------------ | standard basis +------------------------------------------------------------ The standard basis of the n-dimensional Euclidean space consists of the columns of the identity matrix I. +------------------------------------------------------------ | symmetric +------------------------------------------------------------ A matrix A is called symmetric if A^T=A. A symmetric matrix has to be a square matrix. Real symmetric matrices can be diagonalized. +------------------------------------------------------------ | Toepitz matrix +------------------------------------------------------------ A Toepitz matrix is a square matrix A, where the entries are constant along the diagonal. In other ways A_ij depends only on j-i. Example: a 3 x 3 Toeplitz matrix is of the form A = ( c d e b c d a b c ). +------------------------------------------------------------ | trace +------------------------------------------------------------ The trace of a matrix A is the sum of the diagonal entries of A. The trace is independent of the basis and is equal to the sum of the eigenvalues of A. +------------------------------------------------------------ | transpose +------------------------------------------------------------ The transpose A^T of a matrix A is the matrix with entries A_ij if A has the entries A_ji. The rank of A^T is equal to the rank of A. For square matrices, the eigenvalues of A^T and A agree because A and A^T have the same eigenvalues. Transposition satisfies (A^T)^T=A, (A B)^T = B^T A^T and (A^(-1))^T = (A^T)^(-1). +------------------------------------------------------------ | triangle inequality +------------------------------------------------------------ The triangle inequality tells that in a linear space, ||v+w|| leq ||v||+||w||. One has equality if and only if the vectors v and w are orthogonal. +------------------------------------------------------------ | eigenvalues of a two times two matrix +------------------------------------------------------------ The eigenvalues of a two times two matrix A = ( a b c d ) are L_1= tr(A)/2 + (( tr(A)/2)^2- det(A))^1/2 and L_2= tr(A)/2 - (( tr(A)/2)^2- det(A))^1/2. The eigenvectors are v_i = ( L_i-d c ) if c neq 0 or v_i = ( b L_i-a ) if b neq 0. (If b=0,c=0, then the standard vectors are eigenvector.) +------------------------------------------------------------ | unit vector +------------------------------------------------------------ A unit vector is a vector of length 1. A given nonzero vector can be made a unit vector by scaling: v/||v|| is a unit vector. +------------------------------------------------------------ | Vandermonde Matrix +------------------------------------------------------------ A Vandermonde Matrix is a square matrix with entries A_ij = x_i^j-1, where x_1,...,x_n are some real numbers. The determinant of a Vandermonde Matrix is prod_j>i (x_j-x_i). Example: x_1=2,x_2=3,x_3=-1 defines the Vandermonde Matrix A = ( 1 2 4 1 3 9 1 -1 1 ) which has the determinant det(A) = (3-2) (-1-2) (-1-3) = 12. +------------------------------------------------------------ | vector +------------------------------------------------------------ A vector is a matrix with one column. The entries of the vector are called coefficients. +------------------------------------------------------------ | vector space +------------------------------------------------------------ A vector space X is a set equipped with addition and scalar multiplication. A vector space is also called a linear space. The addition operation is a group: f+g = g+f Commutativity (f+g)+h = f+(g+h) Associativity f+0 = 0 Existence of a neutral element f+x = 0 Existence of a unique inverse The scalar multiplication satisfies: r (f+g) = r f + r g Distributivity (r+s) f = r f + s f Distributivity r (s f) = (r s) f Associativity 1 f = f One element +------------------------------------------------------------ | wave equation +------------------------------------------------------------ The wave equation is the linear partial differential equation u_tt = c^2 u_xx where c is a constant. The wave equation on a finite interval 00 a_n sin(n x) cos(n c t) + b_n sin(n x) sin(n c t) where a_n=(2/pi) int_0^pi f(x) sin(n x) dx, and b_n=(2/pi) int_0^pi g(x) sin(n x) dx/(c n) are Fourier coefficients. +------------------------------------------------------------ | zero matrix +------------------------------------------------------------ This file is part of the Sofia project sponsored by the Provost's fund for teaching and learning at Harvard university. There are 157 entries in this file. COUNT: 157 MATHEMATICIANS Authors: Oliver Knill: 2000 Literature: Started from a list of names with birthdates grabbed from mactutor in 2000. +------------------------------------------------------------ | Abbe +------------------------------------------------------------ Abbe Abbe Ernst (1840-1909) +------------------------------------------------------------ | Abel +------------------------------------------------------------ Abel Abel Niels Henrik (1802-1829) Norwegian mathematician. Significant contributions to algebra and analysis, in particular the study of groups and series. Famous for proving the insolubility of the quintic equation at the age of 19. +------------------------------------------------------------ | AbrahamMax +------------------------------------------------------------ AbrahamMax Abraham Max (1875-1922) +------------------------------------------------------------ | Ackermann +------------------------------------------------------------ Ackermann Ackermann Wilhelm (1896-1962) +------------------------------------------------------------ | AdamsFrank +------------------------------------------------------------ AdamsFrank Adams J Frank (1930-1989) +------------------------------------------------------------ | Adams +------------------------------------------------------------ Adams Adams John Couch (1819-1892) +------------------------------------------------------------ | Adelard +------------------------------------------------------------ Adelard Adelard of Bath (1075-1160) +------------------------------------------------------------ | Adler +------------------------------------------------------------ Adler Adler August (1863-1923) +------------------------------------------------------------ | Adrain +------------------------------------------------------------ Adrain Adrain Robert (1775-1843) +------------------------------------------------------------ | Aepinus +------------------------------------------------------------ Aepinus Aepinus Franz (1724-1802) +------------------------------------------------------------ | Agnesi +------------------------------------------------------------ Agnesi Agnesi Maria (1718-1799) +------------------------------------------------------------ | Ahlfors +------------------------------------------------------------ Ahlfors Ahlfors Lars (1907-1996) Finnish mathematician working in complex analysis, was also professor at Harvard from 1946, retiring in 1977. Ahlfors won both the Fields medal in 1936 and the Wolf prize in 1981. +------------------------------------------------------------ | Ahmes +------------------------------------------------------------ Ahmes Ahmes (1680BC-1620BC) +------------------------------------------------------------ | Aida +------------------------------------------------------------ Aida Aida Yasuaki (1747-1817) +------------------------------------------------------------ | Aiken +------------------------------------------------------------ Aiken Aiken Howard (1900-1973) +------------------------------------------------------------ | Airy +------------------------------------------------------------ Airy Airy George (1801-1892) +------------------------------------------------------------ | Aitken +------------------------------------------------------------ Aitken Aitken Alec (1895-1967) +------------------------------------------------------------ | Ajima +------------------------------------------------------------ Ajima Ajima Naonobu (1732-1798) +------------------------------------------------------------ | Akhiezer +------------------------------------------------------------ Akhiezer Akhiezer Naum Ilich (1901-1980) +------------------------------------------------------------ | Albanese +------------------------------------------------------------ Albanese Albanese Giacomo (1890-1948) +------------------------------------------------------------ | Albert +------------------------------------------------------------ Albert Albert of Saxony (1316-1390) +------------------------------------------------------------ | AlbertAbraham +------------------------------------------------------------ AlbertAbraham Albert A Adrian (1905-1972) +------------------------------------------------------------ | Alberti +------------------------------------------------------------ Alberti Alberti Leone (1404-1472) +------------------------------------------------------------ | Albertus +------------------------------------------------------------ Albertus Albertus Magnus Saint (1200-1280) +------------------------------------------------------------ | Alcuin +------------------------------------------------------------ Alcuin Alcuin of York (735-804) +------------------------------------------------------------ | Alexander +------------------------------------------------------------ Alexander Alexander James (1888-1971) +------------------------------------------------------------ | AlexanderArchie +------------------------------------------------------------ AlexanderArchie Alexander Archie (1888-1958) +------------------------------------------------------------ | Aleksandrov +------------------------------------------------------------ Aleksandrov Alexandroff Pave (1896-1982) +------------------------------------------------------------ | AleksandrovAleksandr +------------------------------------------------------------ AleksandrovAleksandr Alexandroff Alexander +------------------------------------------------------------ | Ampere +------------------------------------------------------------ Ampere Amp`ere Andr'e-Marie (1775-1836) +------------------------------------------------------------ | Amsler +------------------------------------------------------------ Amsler Amsler Jacob (1823-1912) +------------------------------------------------------------ | Anaxagoras +------------------------------------------------------------ Anaxagoras Anaxagoras of Clazomenae (499BC-428BC) +------------------------------------------------------------ | Anderson +------------------------------------------------------------ Anderson Anderson Oskar (1887-1960) +------------------------------------------------------------ | Andreev +------------------------------------------------------------ Andreev Andreev Konstantin (1848-1921) +------------------------------------------------------------ | Angeli +------------------------------------------------------------ Angeli Angeli Stephano degli (1623-1697) +------------------------------------------------------------ | Anstice +------------------------------------------------------------ Anstice Anstice Robert (1813-1853) +------------------------------------------------------------ | Anthemius +------------------------------------------------------------ Anthemius Anthemius of Tralles (474-534) +------------------------------------------------------------ | Antiphon +------------------------------------------------------------ Antiphon Antiphon the Sophist (480BC-411BC) +------------------------------------------------------------ | Apastamba +------------------------------------------------------------ Apastamba Apastamba (600BC-540BC) +------------------------------------------------------------ | Apollonius +------------------------------------------------------------ Apollonius Apollonius of Perga (262BC-190BC) +------------------------------------------------------------ | Appell +------------------------------------------------------------ Appell Appell Paul (1855-1930) +------------------------------------------------------------ | Arago +------------------------------------------------------------ Arago Arago Francois (1786-1853) +------------------------------------------------------------ | Arbogast +------------------------------------------------------------ Arbogast Arbogast Louis (1759-1803) +------------------------------------------------------------ | Arbuthnot +------------------------------------------------------------ Arbuthnot Arbuthnot John (1667-1735) +------------------------------------------------------------ | Archimedes +------------------------------------------------------------ Archimedes Archimedes of Syracuse (287BC-212BC) +------------------------------------------------------------ | Archytas +------------------------------------------------------------ Archytas Archytas of Tarentum (428BC-350BC) +------------------------------------------------------------ | Arf +------------------------------------------------------------ Arf Arf Cahit (1910-1997) +------------------------------------------------------------ | Argand +------------------------------------------------------------ Argand Argand Jean (1768-1822) +------------------------------------------------------------ | Aristaeus +------------------------------------------------------------ Aristaeus Aristaeus the Elder (360BC-300BC) +------------------------------------------------------------ | Aristarchus +------------------------------------------------------------ Aristarchus Aristarchus of Samos (310BC-230BC) +------------------------------------------------------------ | Aristotle +------------------------------------------------------------ Aristotle Aristotle (384BC-322BC) +------------------------------------------------------------ | Arnauld +------------------------------------------------------------ Arnauld Arnauld Antoine (1612-1694) +------------------------------------------------------------ | Aronhold +------------------------------------------------------------ Aronhold Aronhold Siegfried (1819-1884) +------------------------------------------------------------ | Artin +------------------------------------------------------------ Artin Artin Emil (1898-1962) +------------------------------------------------------------ | AryabhataII +------------------------------------------------------------ AryabhataII Aryabhata II +------------------------------------------------------------ | AryabhataI +------------------------------------------------------------ AryabhataI Aryabhata I (476-550) +------------------------------------------------------------ | Atiyah +------------------------------------------------------------ Atiyah Atiyah Michael +------------------------------------------------------------ | Atwood +------------------------------------------------------------ Atwood Atwood George (1745-1807) +------------------------------------------------------------ | Auslander +------------------------------------------------------------ Auslander Auslander Maurice (1926-1994) +------------------------------------------------------------ | Autolycus +------------------------------------------------------------ Autolycus Autolycus of Pitane (360BC-290BC) +------------------------------------------------------------ | Bezout +------------------------------------------------------------ Bezout B'ezout Etienne (1730-1783) French geometer and analyst. +------------------------------------------------------------ | Bocher +------------------------------------------------------------ Bocher Bocher Maxime (1867-1918) +------------------------------------------------------------ | Burgi +------------------------------------------------------------ Burgi B"urgi Joost (1552-1632) +------------------------------------------------------------ | Babbage +------------------------------------------------------------ Babbage Babbage Charles (1791-1871) +------------------------------------------------------------ | Bachet +------------------------------------------------------------ Bachet Bachet Claude (1581-1638) +------------------------------------------------------------ | Bachmann +------------------------------------------------------------ Bachmann Bachmann Paul (1837-1920) +------------------------------------------------------------ | Backus +------------------------------------------------------------ Backus Backus John +------------------------------------------------------------ | Bacon +------------------------------------------------------------ Bacon Bacon Roger (1219-1292) +------------------------------------------------------------ | Baer +------------------------------------------------------------ Baer Baer Reinhold (1902-1979) +------------------------------------------------------------ | Baire +------------------------------------------------------------ Baire Baire Ren'e-Louis (1874-1932) +------------------------------------------------------------ | BakerAlan +------------------------------------------------------------ BakerAlan Baker Alan +------------------------------------------------------------ | Baker +------------------------------------------------------------ Baker Baker Henry (1866-1956) +------------------------------------------------------------ | Ball +------------------------------------------------------------ Ball Ball Walter W Rouse (1850-1925) +------------------------------------------------------------ | Balmer +------------------------------------------------------------ Balmer Balmer Johann (1825-1898) +------------------------------------------------------------ | Banach +------------------------------------------------------------ Banach, Stefan, (1892-1945) Polish mathematician who founded functional analysis. +------------------------------------------------------------ | Banneker +------------------------------------------------------------ Banneker Banneker Benjamin (1731-1806) +------------------------------------------------------------ | BanuMusaMuhammad +------------------------------------------------------------ BanuMusaMuhammad Banu Musa Jafar (810-873) +------------------------------------------------------------ | BanuMusa +------------------------------------------------------------ BanuMusa Banu Musa brothers +------------------------------------------------------------ | BanuMusaal-Hasan +------------------------------------------------------------ BanuMusaal-Hasan Banu Musa al-Hasan (810-873) +------------------------------------------------------------ | BanuMusaAhmad +------------------------------------------------------------ BanuMusaAhmad Banu Musa Ahmad (810-873) +------------------------------------------------------------ | Barbier +------------------------------------------------------------ Barbier Barbier Joseph Emile (1839-1889) +------------------------------------------------------------ | Bari +------------------------------------------------------------ Bari Bari Nina (1901-1961) +------------------------------------------------------------ | Barlow +------------------------------------------------------------ Barlow Barlow Peter (1776-1862) +------------------------------------------------------------ | Barnes +------------------------------------------------------------ Barnes Barnes Ernest (1874-1953) +------------------------------------------------------------ | Barocius +------------------------------------------------------------ Barocius Barozzi Francesco (1537-1604) +------------------------------------------------------------ | Barrow +------------------------------------------------------------ Barrow Barrow Isaac (1630-1677) +------------------------------------------------------------ | Bartholin +------------------------------------------------------------ Bartholin Bartholin Erasmus (1625-1698) +------------------------------------------------------------ | Batchelor +------------------------------------------------------------ Batchelor Batchelor George (1920-2000) +------------------------------------------------------------ | Bateman +------------------------------------------------------------ Bateman Bateman Harry (1882-1946) +------------------------------------------------------------ | Battaglini +------------------------------------------------------------ Battaglini Battaglini Guiseppe (1826-1894) +------------------------------------------------------------ | Al-Battani +------------------------------------------------------------ Al-Battani Battani Abu al- (850-929) +------------------------------------------------------------ | Baudhayana +------------------------------------------------------------ Baudhayana Baudhayana (800BC-740BC) +------------------------------------------------------------ | Bayes +------------------------------------------------------------ Bayes Thomas, (1702-1761). English probability theorist and theologian. +------------------------------------------------------------ | Beaugrand +------------------------------------------------------------ Beaugrand Beaugrand Jean (1595-1640) +------------------------------------------------------------ | Bell +------------------------------------------------------------ Bell Bell Eric Temple (1883-1960) +------------------------------------------------------------ | Bellavitis +------------------------------------------------------------ Bellavitis Bellavitis Giusto (1803-1880) +------------------------------------------------------------ | Beltrami +------------------------------------------------------------ Beltrami Beltrami Eugenio (1835-1899) +------------------------------------------------------------ | Bendixson +------------------------------------------------------------ Bendixson Bendixson Ivar Otto (1861-1935) +------------------------------------------------------------ | Benedetti +------------------------------------------------------------ Benedetti Benedetti Giovanni (1530-1590) +------------------------------------------------------------ | Bergman +------------------------------------------------------------ Bergman Bergman Stefan (1895-1977) +------------------------------------------------------------ | Berkeley +------------------------------------------------------------ Berkeley Berkeley George (1685-1753) +------------------------------------------------------------ | Bernays +------------------------------------------------------------ Bernays Bernays Paul Isaac (1888-1977) +------------------------------------------------------------ | BernoulliDaniel +------------------------------------------------------------ BernoulliDaniel Bernoulli Daniel (1700-1782) +------------------------------------------------------------ | BernoulliJohann +------------------------------------------------------------ BernoulliJohann Bernoulli Johann (1667-1748) +------------------------------------------------------------ | BernoulliNicolaus +------------------------------------------------------------ BernoulliNicolaus Bernoulli Nicolaus +------------------------------------------------------------ | BernoulliJacob +------------------------------------------------------------ BernoulliJacob Bernoulli Jakob (1654-1705) Swiss analyst, probability theorist and physicist. +------------------------------------------------------------ | Bernstein +------------------------------------------------------------ Bernstein Sergei Natanovich (1880-1968). Russian analyst. +------------------------------------------------------------ | BernsteinFelix +------------------------------------------------------------ BernsteinFelix Bernstein Felix (1878-1956) +------------------------------------------------------------ | Bers +------------------------------------------------------------ Bers Bers Lipa (1914-1993) +------------------------------------------------------------ | Bertini +------------------------------------------------------------ Bertini Bertini Eugenio (1846-1933) +------------------------------------------------------------ | Bertrand +------------------------------------------------------------ Bertrand Bertrand Joseph (1822-1900) +------------------------------------------------------------ | Berwald +------------------------------------------------------------ Berwald Berwald Lugwig (1883-1942) +------------------------------------------------------------ | Berwick +------------------------------------------------------------ Berwick Berwick William (1888-1944) +------------------------------------------------------------ | Besicovitch +------------------------------------------------------------ Besicovitch Besicovitch Abram (1891-1970) +------------------------------------------------------------ | Bessel +------------------------------------------------------------ Bessel Friedrich Wilhelm, (1784-1846) calculated orbit of Halley's orbit as 20 year old. Made accurate measurements of stellar positions. Professor of Astronomy at Koenigsberg. +------------------------------------------------------------ | Betti +------------------------------------------------------------ Betti Betti Enrico (1823-1892) +------------------------------------------------------------ | Beurling +------------------------------------------------------------ Beurling Beurling Arne (1905-1986) +------------------------------------------------------------ | BhaskaraI +------------------------------------------------------------ BhaskaraI Bhaskara I (600-680) +------------------------------------------------------------ | BhaskaraII +------------------------------------------------------------ BhaskaraII Bhaskaracharya (1114-1185) +------------------------------------------------------------ | Bianchi +------------------------------------------------------------ Bianchi Bianchi Luigi (1856-1928) +------------------------------------------------------------ | Bieberbach +------------------------------------------------------------ Bieberbach Bieberbach Ludwig (1886-1982) +------------------------------------------------------------ | Bienayme +------------------------------------------------------------ Bienayme Bienaym'e Ir'ene'e-Jules (1796-1878) +------------------------------------------------------------ | Binet +------------------------------------------------------------ Binet Binet Jacques (1786-1856) +------------------------------------------------------------ | Bing +------------------------------------------------------------ Bing Bing R (1940-1986) +------------------------------------------------------------ | Biot +------------------------------------------------------------ Biot Biot Jean-Baptiste (1774-1862) +------------------------------------------------------------ | Birkhoff +------------------------------------------------------------ Birkhoff Birkhoff George (1884-1944) +------------------------------------------------------------ | BirkhoffGarrett +------------------------------------------------------------ BirkhoffGarrett Birkhoff Garrett (1911-1996) +------------------------------------------------------------ | Al-Biruni +------------------------------------------------------------ Al-Biruni Biruni Abu al +------------------------------------------------------------ | BjerknesVilhelm +------------------------------------------------------------ BjerknesVilhelm Bjerknes Vilhelm (1862-1951) +------------------------------------------------------------ | BjerknesCarl +------------------------------------------------------------ BjerknesCarl Bjerknes Carl (1825-1903) +------------------------------------------------------------ | Black +------------------------------------------------------------ Black Black Max (1909-1988) +------------------------------------------------------------ | Blaschke +------------------------------------------------------------ Blaschke Blaschke Wilhelm (1885-1962) +------------------------------------------------------------ | Blichfeldt +------------------------------------------------------------ Blichfeldt Blichfeldt Hans (1873-1945) +------------------------------------------------------------ | Bliss +------------------------------------------------------------ Bliss Bliss Gilbert (1876-1951) +------------------------------------------------------------ | Bloch +------------------------------------------------------------ Bloch Bloch Andr'e (1893-1948) +------------------------------------------------------------ | Bobillier +------------------------------------------------------------ Bobillier Bobillier Etienne (1798-1840) +------------------------------------------------------------ | Bochner +------------------------------------------------------------ Bochner Bochner Salomon (1899-1982) +------------------------------------------------------------ | Boethius +------------------------------------------------------------ Boethius Boethius Anicus (475-524) +------------------------------------------------------------ | Boggio +------------------------------------------------------------ Boggio Boggio Tommaso (1877-1963) +------------------------------------------------------------ | Bohl +------------------------------------------------------------ Bohl Bohl Piers (1865-1921) +------------------------------------------------------------ | BohrHarald +------------------------------------------------------------ BohrHarald Bohr Harald (1887-1951) +------------------------------------------------------------ | BohrNiels +------------------------------------------------------------ BohrNiels Bohr Niels (1885-1962) +------------------------------------------------------------ | Boltzmann +------------------------------------------------------------ Boltzmann Boltzmann Ludwig (1844-1906) +------------------------------------------------------------ | BolyaiFarkas +------------------------------------------------------------ BolyaiFarkas Bolyai Farkas (1775-1856) +------------------------------------------------------------ | Bolyai +------------------------------------------------------------ Bolyai Bolyai J"anos (1802-1860) +------------------------------------------------------------ | Bolza +------------------------------------------------------------ Bolza, Oscar (1857-1943), German-born American analyst. +------------------------------------------------------------ | Bolzano +------------------------------------------------------------ Bolzano Bolzano Bernhard (1781-1848) +------------------------------------------------------------ | Bombelli +------------------------------------------------------------ Bombelli Bombelli Rafael (1526-1573) +------------------------------------------------------------ | Bombieri +------------------------------------------------------------ Bombieri Bombieri Enrico +------------------------------------------------------------ | Bonferroni +------------------------------------------------------------ Bonferroni Bonferroni Carlo (1892-1960) +------------------------------------------------------------ | Bonnet +------------------------------------------------------------ Bonnet Bonnet Pierre (1819-1892) +------------------------------------------------------------ | Boole +------------------------------------------------------------ Boole, George (1815-1864), English Logician, who made also contributions to analysis and probability theory. +------------------------------------------------------------ | Boone +------------------------------------------------------------ Boone Boone Bill (1920-1983) +------------------------------------------------------------ | Borchardt +------------------------------------------------------------ Borchardt Borchardt Carl (1817-1880) +------------------------------------------------------------ | Borda +------------------------------------------------------------ Borda Borda Jean (1733-1799) +------------------------------------------------------------ | Borel +------------------------------------------------------------ Borel, Felix Edouard Justin Emile, (1871-1956) French measure theorist and probability theorist. +------------------------------------------------------------ | Borgi +------------------------------------------------------------ Borgi Borgi Piero (1424-1484) +------------------------------------------------------------ | Born +------------------------------------------------------------ Born Born Max (1882-1970) +------------------------------------------------------------ | Borsuk +------------------------------------------------------------ Borsuk Borsuk Karol (1905-1982) +------------------------------------------------------------ | Bortkiewicz +------------------------------------------------------------ Bortkiewicz Bortkiewicz Ladislaus (1868-1931) +------------------------------------------------------------ | Bortolotti +------------------------------------------------------------ Bortolotti Bortolotti Ettore (1866-1947) +------------------------------------------------------------ | Bosanquet +------------------------------------------------------------ Bosanquet Bosanquet Stephen (1903-1984) +------------------------------------------------------------ | Boscovich +------------------------------------------------------------ Boscovich Boscovich Ruggero (1711-1787) +------------------------------------------------------------ | Bose +------------------------------------------------------------ Bose Bose Satyendranath (1894-1974) +------------------------------------------------------------ | Bossut +------------------------------------------------------------ Bossut Bossut Charles (1730-1814) +------------------------------------------------------------ | Bouguer +------------------------------------------------------------ Bouguer Bouguer Pierre (1698-1758) +------------------------------------------------------------ | Boulliau +------------------------------------------------------------ Boulliau Boulliau Ismael (1605-1694) +------------------------------------------------------------ | Bouquet +------------------------------------------------------------ Bouquet Bouquet Jean Claude (1819-1885) +------------------------------------------------------------ | Bour +------------------------------------------------------------ Bour Bour Edmond (1832-1866) +------------------------------------------------------------ | Bourbaki +------------------------------------------------------------ Bourbaki, Nicolas (1939- ) Collective pseudonym of a group of mostly French mathematicians. +------------------------------------------------------------ | Bourgain +------------------------------------------------------------ Bourgain Bourgain Jean +------------------------------------------------------------ | Boutroux +------------------------------------------------------------ Boutroux Boutroux Pierre L'eon (1880-1922) +------------------------------------------------------------ | Bowditch +------------------------------------------------------------ Bowditch Bowditch Nathaniel (1773-1838) +------------------------------------------------------------ | Bowen +------------------------------------------------------------ Bowen Bowen Rufus (1947-1978) +------------------------------------------------------------ | Boyle +------------------------------------------------------------ Boyle Boyle Robert (1627-1691) +------------------------------------------------------------ | Boys +------------------------------------------------------------ Boys Boys Charles (1855-1944) +------------------------------------------------------------ | Bradwardine +------------------------------------------------------------ Bradwardine Bradwardine Thomas (1290-1349) +------------------------------------------------------------ | Brahe +------------------------------------------------------------ Brahe Brahe Tycho (1546-1601) +------------------------------------------------------------ | Brahmadeva +------------------------------------------------------------ Brahmadeva Brahmadeva (1060-1130) +------------------------------------------------------------ | Brahmagupta +------------------------------------------------------------ Brahmagupta Brahmagupta (598-670) +------------------------------------------------------------ | Braikenridge +------------------------------------------------------------ Braikenridge Braikenridge William (1700-1762) +------------------------------------------------------------ | Bramer +------------------------------------------------------------ Bramer Bramer Benjamin (1588-1652) +------------------------------------------------------------ | Brashman +------------------------------------------------------------ Brashman Brashman Nikolai (1796-1866) +------------------------------------------------------------ | BrauerAlfred +------------------------------------------------------------ BrauerAlfred Brauer Alfred (1894-1985) +------------------------------------------------------------ | Brauer +------------------------------------------------------------ Brauer Brauer Richard (1901-1977) +------------------------------------------------------------ | Brianchon +------------------------------------------------------------ Brianchon Brianchon Charles (1783-1864) +------------------------------------------------------------ | Briggs +------------------------------------------------------------ Briggs Briggs Henry (1561-1630) English mathematician producing tables of common logarithms up to 15 digits. +------------------------------------------------------------ | Brillouin +------------------------------------------------------------ Brillouin Brillouin Marcel (1854-1948) +------------------------------------------------------------ | Bring +------------------------------------------------------------ Bring Bring Erland (1736-1798) +------------------------------------------------------------ | Brioschi +------------------------------------------------------------ Brioschi Brioschi Francesco (1824-1897) +------------------------------------------------------------ | Briot +------------------------------------------------------------ Briot Briot Charlese (1817-1882) +------------------------------------------------------------ | Brisson +------------------------------------------------------------ Brisson Brisson Barnab'e (1777-1828) +------------------------------------------------------------ | Britton +------------------------------------------------------------ Britton Britton John (1927-1994) +------------------------------------------------------------ | Brocard +------------------------------------------------------------ Brocard Brocard Henri (1845-1922) +------------------------------------------------------------ | Brodetsky +------------------------------------------------------------ Brodetsky Brodetsky Selig +------------------------------------------------------------ | Bromwich +------------------------------------------------------------ Bromwich Bromwich Thomas (1875-1929) +------------------------------------------------------------ | Bronowski +------------------------------------------------------------ Bronowski Bronowski Jacob (1908-1974) +------------------------------------------------------------ | Brouncker +------------------------------------------------------------ Brouncker Brouncker William (1620-1684) +------------------------------------------------------------ | Brouwer +------------------------------------------------------------ Brouwer Brouwer Luitzen Egbertus Jan (1881-1966) Dutch matematician and philosopher. +------------------------------------------------------------ | Brown +------------------------------------------------------------ Brown Brown Ernest (1866-1938) +------------------------------------------------------------ | Browne +------------------------------------------------------------ Browne Browne Marjorie (1914-1979) +------------------------------------------------------------ | Bruno +------------------------------------------------------------ Bruno Bruno Giuseppe (1828-1893) +------------------------------------------------------------ | Bruns +------------------------------------------------------------ Bruns Bruns Heinrich (1848-1919) +------------------------------------------------------------ | Bryson +------------------------------------------------------------ Bryson Bryson of Heraclea (450BC-390BC) +------------------------------------------------------------ | Buffon +------------------------------------------------------------ Buffon Buffon Georges Comte de (1707-1788) +------------------------------------------------------------ | Bugaev +------------------------------------------------------------ Bugaev Bugaev Nicolay (1837-1903) +------------------------------------------------------------ | Bukreev +------------------------------------------------------------ Bukreev Bukreev Boris (1859-1962) +------------------------------------------------------------ | Bunyakovsky +------------------------------------------------------------ Bunyakovsky Bunyakovsky Viktor (1804-1889) +------------------------------------------------------------ | Burchnall +------------------------------------------------------------ Burchnall Burchnall Joseph (1892-1975) +------------------------------------------------------------ | Burkhardt +------------------------------------------------------------ Burkhardt Burkhardt Heinrich (1861-1914) +------------------------------------------------------------ | Burkill +------------------------------------------------------------ Burkill Burkill John (1900-1993) +------------------------------------------------------------ | Burnside +------------------------------------------------------------ Burnside Burnside William (1852-1927) +------------------------------------------------------------ | Caccioppoli +------------------------------------------------------------ Caccioppoli Caccioppoli Renato (1904-1959) +------------------------------------------------------------ | Cajori +------------------------------------------------------------ Cajori Cajori Florian (1859-1930) +------------------------------------------------------------ | Calderon +------------------------------------------------------------ Calderon Calder'on Alberto (1920-1998) +------------------------------------------------------------ | Callippus +------------------------------------------------------------ Callippus Callippus (370BC-310BC) +------------------------------------------------------------ | Campanus +------------------------------------------------------------ Campanus Campanus of Novara (1220-1296) +------------------------------------------------------------ | Campbell +------------------------------------------------------------ Campbell Campbell John (1862-1924) +------------------------------------------------------------ | Camus +------------------------------------------------------------ Camus Camus Charles (1699-1768) +------------------------------------------------------------ | Cannell +------------------------------------------------------------ Cannell Cannell Doris (1913-2000) +------------------------------------------------------------ | Cantelli +------------------------------------------------------------ Cantelli Cantelli Francesco (1875-1966) +------------------------------------------------------------ | CantorMoritz +------------------------------------------------------------ CantorMoritz Cantor Moritz (1829-1920) +------------------------------------------------------------ | Cantor +------------------------------------------------------------ Cantor Cantor Georg (1845-1918) Cantor Georg, (1845-1918) German mathematician. Precise definition of infinite set. +------------------------------------------------------------ | Caramuel +------------------------------------------------------------ Caramuel Caramuel Juan (1606-1682) +------------------------------------------------------------ | Caratheodory +------------------------------------------------------------ Caratheodory Carath'eodory Constantin (1873-1950) +------------------------------------------------------------ | Cardan +------------------------------------------------------------ Cardan Cardano Girolamo (1501-1576) Italian mathematicaian, physician and astrologer. First publication for the solution of the general cubic equation (solution found by Tartaglia). +------------------------------------------------------------ | Carlitz +------------------------------------------------------------ Carlitz Carlitz Leonard (1907-1999) +------------------------------------------------------------ | Carlyle +------------------------------------------------------------ Carlyle Carlyle Thomas (1795-1881) +------------------------------------------------------------ | Carnot +------------------------------------------------------------ Carnot Carnot Lazare (1753-1823) French mathematician and politican best known for his work on the foundations of calculus and modern geometry. +------------------------------------------------------------ | CarnotSadi +------------------------------------------------------------ CarnotSadi Carnot Sadi (1796-1832) French mathematical physisist working on the foundations of thermodynamics. Carnot's work led directly to the discovery of the second law of thermodynamics. +------------------------------------------------------------ | Carslaw +------------------------------------------------------------ Carslaw Carslaw Horatio (1870-1954) +------------------------------------------------------------ | Cartan +------------------------------------------------------------ Cartan Cartan Elie (1869-1951) +------------------------------------------------------------ | CartanHenri +------------------------------------------------------------ CartanHenri Cartan Henri +------------------------------------------------------------ | Cartwright +------------------------------------------------------------ Cartwright Cartwright Dame Mary (1900-1998) +------------------------------------------------------------ | Casorati +------------------------------------------------------------ Casorati Casorati Felice (1835-1890) +------------------------------------------------------------ | Cassels +------------------------------------------------------------ Cassels Cassels John +------------------------------------------------------------ | Cassini +------------------------------------------------------------ Cassini Cassini Giovanni (1625-1712) +------------------------------------------------------------ | Castel +------------------------------------------------------------ Castel Castel Louis (1688-1757) +------------------------------------------------------------ | Castelnuovo +------------------------------------------------------------ Castelnuovo Castelnuovo Guido (1865-1952) +------------------------------------------------------------ | Castigliano +------------------------------------------------------------ Castigliano Castigliano Alberto (1847-1884) +------------------------------------------------------------ | Castillon +------------------------------------------------------------ Castillon Castillon Johann (1704-1791) +------------------------------------------------------------ | Catalan +------------------------------------------------------------ Catalan Catalan Eug`ene (1814-1894) +------------------------------------------------------------ | Cataldi +------------------------------------------------------------ Cataldi Cataldi Pietro (1548-1626) +------------------------------------------------------------ | Cauchy +------------------------------------------------------------ Cauchy Cauchy Augustin-Louis (1789-1857) French mathematician who introduced modern notions of continuity limit, convergence and differentiability, proved Cauchy's theorem in group theory, contributed to the calculus of variations, probability theory and the study of differential equations. +------------------------------------------------------------ | Cavalieri +------------------------------------------------------------ Cavalieri Cavalieri Bonaventura (1598-1647) Italian mathematician. Introduced method of indivisibles, a forerunner of integral calculus to determine the area enclosed by certain curves. +------------------------------------------------------------ | Cayley +------------------------------------------------------------ Cayley Cayley Arthur (1821-1895) English mathematician working in the theory of matrices, abstract groups and algebraic geometry. +------------------------------------------------------------ | Cech +------------------------------------------------------------ Cech Cech Eduard (1893-1960) +------------------------------------------------------------ | Cesaro +------------------------------------------------------------ Cesaro Ces`aro Ernesto (1859-1906) +------------------------------------------------------------ | CevaGiovanni +------------------------------------------------------------ CevaGiovanni Ceva Giovanni (1647-1734) +------------------------------------------------------------ | CevaTommaso +------------------------------------------------------------ CevaTommaso Ceva Tommaso (1648-1737) +------------------------------------------------------------ | Chatelet +------------------------------------------------------------ Chatelet Chatelet Gabrielle du (1706-1749) +------------------------------------------------------------ | Chandrasekhar +------------------------------------------------------------ Chandrasekhar Chandrasekhar Subrah. (1910-1995) +------------------------------------------------------------ | Chang +------------------------------------------------------------ Chang Chang Sun-Yung Alice +------------------------------------------------------------ | Chaplygin +------------------------------------------------------------ Chaplygin Chaplygin Sergi (1869-1942) +------------------------------------------------------------ | Chapman +------------------------------------------------------------ Chapman Chapman Sydney (1888-1970) +------------------------------------------------------------ | Chasles +------------------------------------------------------------ Chasles Chasles Michel (1793-1880) +------------------------------------------------------------ | Chebotaryov +------------------------------------------------------------ Chebotaryov Chebotaryov Nikolai (1894-1947) +------------------------------------------------------------ | Chebyshev +------------------------------------------------------------ Chebyshev Chebyshev Pafnuty (1821-1894) +------------------------------------------------------------ | Chern +------------------------------------------------------------ Chern Chern Shiing-shen +------------------------------------------------------------ | Chernikov +------------------------------------------------------------ Chernikov Chernikov Sergei (1912-1987) +------------------------------------------------------------ | Chevalley +------------------------------------------------------------ Chevalley Chevalley Claude (1909-1984) +------------------------------------------------------------ | Ch'in +------------------------------------------------------------ Ch'in Chiu-Shao Ch'in (1202-1261) +------------------------------------------------------------ | Chowla +------------------------------------------------------------ Chowla Chowla Sarvadaman (1907-1995) +------------------------------------------------------------ | Christoffel +------------------------------------------------------------ Christoffel Christoffel Elwin (1829-1900) +------------------------------------------------------------ | Chrysippus +------------------------------------------------------------ Chrysippus Chrysippus (280BC-206BC) +------------------------------------------------------------ | Chrystal +------------------------------------------------------------ Chrystal Chrystal George (1851-1911) +------------------------------------------------------------ | Chuquet +------------------------------------------------------------ Chuquet Chuquet Nicolas (1445-1500) +------------------------------------------------------------ | Church +------------------------------------------------------------ Church Church Alonzo (1903-1995) American mathematical logician. +------------------------------------------------------------ | Clairaut +------------------------------------------------------------ Clairaut Clairaut Alexis (1713-1765) French mathematician and physicist who worked on the problem of geodesic flows, celestial mechanics and cubic curves. +------------------------------------------------------------ | Clapeyron +------------------------------------------------------------ Clapeyron Clapeyron Emile (1799-1864) +------------------------------------------------------------ | Clarke +------------------------------------------------------------ Clarke Clarke Samuel (1675-1729) +------------------------------------------------------------ | Clausen +------------------------------------------------------------ Clausen Clausen Thomas (1801-1885) +------------------------------------------------------------ | Clausius +------------------------------------------------------------ Clausius Clausius Rudolf (1822-1888) +------------------------------------------------------------ | Clavius +------------------------------------------------------------ Clavius Clavius Christopher (1537-1612) +------------------------------------------------------------ | Clebsch +------------------------------------------------------------ Clebsch Clebsch Alfred (1833-1872) +------------------------------------------------------------ | Cleomedes +------------------------------------------------------------ Cleomedes Cleomedes Cleomedes (10AD-70) +------------------------------------------------------------ | Clifford +------------------------------------------------------------ Clifford Clifford William (1845-1879) +------------------------------------------------------------ | Coates +------------------------------------------------------------ Coates Coates John +------------------------------------------------------------ | Coble +------------------------------------------------------------ Coble Coble Arthur (1878-1966) +------------------------------------------------------------ | Cochran +------------------------------------------------------------ Cochran Cochran William (1909-1980) +------------------------------------------------------------ | Cocker +------------------------------------------------------------ Cocker Cocker Edward (1631-1675) +------------------------------------------------------------ | Codazzi +------------------------------------------------------------ Codazzi Codazzi Delfino (1824-1873) +------------------------------------------------------------ | Cohen +------------------------------------------------------------ Cohen Paul Joseph (1934-) American mathematician who resolved the status of the continuum hypothesis. +------------------------------------------------------------ | Cole +------------------------------------------------------------ Cole Cole Frank (1861-1926) +------------------------------------------------------------ | Collingwood +------------------------------------------------------------ Collingwood Collingwood Edward (1900-1970) +------------------------------------------------------------ | Collins +------------------------------------------------------------ Collins Collins John (1625-1683) +------------------------------------------------------------ | Condorcet +------------------------------------------------------------ Condorcet Condorcet Marie Jean (1743-1794) +------------------------------------------------------------ | Connes +------------------------------------------------------------ Connes Connes Alain +------------------------------------------------------------ | Conon +------------------------------------------------------------ Conon Conon of Samos (280BC-220BC) +------------------------------------------------------------ | ConwayArthur +------------------------------------------------------------ ConwayArthur Conway Arthur (1875-1950) +------------------------------------------------------------ | Conway +------------------------------------------------------------ Conway Conway John +------------------------------------------------------------ | Coolidge +------------------------------------------------------------ Coolidge Coolidge Julian (1873-1954) +------------------------------------------------------------ | Cooper +------------------------------------------------------------ Cooper Cooper Lionel (1915-1977) +------------------------------------------------------------ | Copernicus +------------------------------------------------------------ Copernicus Copernicus Nicolaus (1473-1543) +------------------------------------------------------------ | Copson +------------------------------------------------------------ Copson Copson Edward (1901-1980) +------------------------------------------------------------ | Cosserat +------------------------------------------------------------ Cosserat Cosserat Eug`ene (1866-1931) +------------------------------------------------------------ | Cotes +------------------------------------------------------------ Cotes Cotes Roger (1682-1716) +------------------------------------------------------------ | Courant +------------------------------------------------------------ Courant Courant Richard (1888-1972) +------------------------------------------------------------ | Cournot +------------------------------------------------------------ Cournot Cournot Antoine (1801-1877) +------------------------------------------------------------ | Couturat +------------------------------------------------------------ Couturat Couturat Louis (1868-1914) +------------------------------------------------------------ | Cox +------------------------------------------------------------ Cox Cox Gertrude (1900-1978) +------------------------------------------------------------ | Coxeter +------------------------------------------------------------ Coxeter Coxeter Donald +------------------------------------------------------------ | Craig +------------------------------------------------------------ Craig Craig John (1663-1731) +------------------------------------------------------------ | CramerHarald +------------------------------------------------------------ CramerHarald Cram'er Harald (1893-1985) +------------------------------------------------------------ | Cramer +------------------------------------------------------------ Cramer Cramer Gabriel (1704-1752) +------------------------------------------------------------ | Crank +------------------------------------------------------------ Crank Crank John +------------------------------------------------------------ | Crelle +------------------------------------------------------------ Crelle Crelle August (1780-1855) +------------------------------------------------------------ | Cremona +------------------------------------------------------------ Cremona Cremona Luigi (1830-1903) +------------------------------------------------------------ | Crighton +------------------------------------------------------------ Crighton Crighton David (1942-2000) +------------------------------------------------------------ | Cunha +------------------------------------------------------------ Cunha Cunha Anast"acio da (1744-1787) +------------------------------------------------------------ | Cunningham +------------------------------------------------------------ Cunningham Cunningham Ebenezer (1881-1977) +------------------------------------------------------------ | Curry +------------------------------------------------------------ Curry Curry Haskell (1900-1982) +------------------------------------------------------------ | Cusa +------------------------------------------------------------ Cusa Cusa Nicholas of (1401-1464) +------------------------------------------------------------ | Durer +------------------------------------------------------------ Durer D"urer Albrecht (1471-1528) +------------------------------------------------------------ | Dandelin +------------------------------------------------------------ Dandelin Dandelin Germinal (1794-1847) +------------------------------------------------------------ | Danti +------------------------------------------------------------ Danti Danti Egnatio (1536-1586) +------------------------------------------------------------ | DantzigGeorge +------------------------------------------------------------ DantzigGeorge Dantzig George +------------------------------------------------------------ | Darboux +------------------------------------------------------------ Darboux Darboux Gaston (1842-1917) +------------------------------------------------------------ | Darwin +------------------------------------------------------------ Darwin Darwin George (1845-1912) +------------------------------------------------------------ | Dase +------------------------------------------------------------ Dase Dase Zacharias (1824-1861) +------------------------------------------------------------ | Davenport +------------------------------------------------------------ Davenport Davenport Harold (1907-1969) +------------------------------------------------------------ | Davidov +------------------------------------------------------------ Davidov Davidov August (1823-1885) +------------------------------------------------------------ | Davies +------------------------------------------------------------ Davies Davies Evan Tom (1904-1973) +------------------------------------------------------------ | Dechales +------------------------------------------------------------ Dechales Dechales Claude (1621-1678) +------------------------------------------------------------ | Dedekind +------------------------------------------------------------ Dedekind Dedekind Richard (1831-1916) +------------------------------------------------------------ | Dee +------------------------------------------------------------ Dee Dee John (1527-1608) +------------------------------------------------------------ | Dehn +------------------------------------------------------------ Dehn Dehn Max (1878-1952) +------------------------------------------------------------ | Delamain +------------------------------------------------------------ Delamain Delamain Richard (1600-1644) +------------------------------------------------------------ | Delambre +------------------------------------------------------------ Delambre Delambre Jean Baptiste (1749-1822) +------------------------------------------------------------ | Delaunay +------------------------------------------------------------ Delaunay Delaunay Charles (1816-1872) +------------------------------------------------------------ | Deligne +------------------------------------------------------------ Deligne Deligne Pierre +------------------------------------------------------------ | Delone +------------------------------------------------------------ Delone Delone Boris (1890-1973) +------------------------------------------------------------ | Delsarte +------------------------------------------------------------ Delsarte Delsarte Jean (1903-1968) +------------------------------------------------------------ | Democritus +------------------------------------------------------------ Democritus Democritus of Abdera (460BC-370BC) +------------------------------------------------------------ | Denjoy +------------------------------------------------------------ Denjoy Denjoy Arnaud (1884-1974) +------------------------------------------------------------ | Deparcieux +------------------------------------------------------------ Deparcieux Deparcieux Antoine (1703-1768) +------------------------------------------------------------ | Desargues +------------------------------------------------------------ Desargues Desargues Girard (1591-1661) +------------------------------------------------------------ | Descartes +------------------------------------------------------------ Descartes Descartes Ren'e (1596-1650) +------------------------------------------------------------ | Dickson +------------------------------------------------------------ Dickson Dickson Leonard (1874-1954) +------------------------------------------------------------ | Dickstein +------------------------------------------------------------ Dickstein Dickstein Samuel (1851-1939) +------------------------------------------------------------ | Dieudonne +------------------------------------------------------------ Dieudonne Dieudonn'e Jean (1906-1992) +------------------------------------------------------------ | Digges +------------------------------------------------------------ Digges Digges Thomas (1546-1595) +------------------------------------------------------------ | Dilworth +------------------------------------------------------------ Dilworth Dilworth Robert +------------------------------------------------------------ | Dinghas +------------------------------------------------------------ Dinghas Dinghas Alexander (1908-1974) +------------------------------------------------------------ | Dini +------------------------------------------------------------ Dini Dini Ulisse (1845-1918) +------------------------------------------------------------ | Dinostratus +------------------------------------------------------------ Dinostratus Dinostratus (390BC-320BC) +------------------------------------------------------------ | Diocles +------------------------------------------------------------ Diocles Diocles (240BC-180BC) +------------------------------------------------------------ | Dionis +------------------------------------------------------------ Dionis Dionis du S'ejour A (1734-1794) +------------------------------------------------------------ | Dionysodorus +------------------------------------------------------------ Dionysodorus Dionysodorus (250BC-190BC) +------------------------------------------------------------ | Diophantus +------------------------------------------------------------ Diophantus Diophantus of Alexandria (200-284) +------------------------------------------------------------ | Dirac +------------------------------------------------------------ Dirac Dirac Paul (1902-1984) +------------------------------------------------------------ | Dirichlet +------------------------------------------------------------ Dirichlet Dirichlet Lejeune (1805-1859) +------------------------------------------------------------ | DixonArthur +------------------------------------------------------------ DixonArthur Dixon Arthur Lee (1867-1955) +------------------------------------------------------------ | Dixon +------------------------------------------------------------ Dixon Dixon Alfred (1865-1936) +------------------------------------------------------------ | Dodgson +------------------------------------------------------------ Dodgson Dodgson Charles (1832-1898) +------------------------------------------------------------ | Doeblin +------------------------------------------------------------ Doeblin Doeblin Wolfang (1915-1940) +------------------------------------------------------------ | Domninus +------------------------------------------------------------ Domninus Domninus of Larissa (420-480) +------------------------------------------------------------ | Donaldson +------------------------------------------------------------ Donaldson Donaldson Simon +------------------------------------------------------------ | Doob +------------------------------------------------------------ Doob Doob Joseph +------------------------------------------------------------ | Doppelmayr +------------------------------------------------------------ Doppelmayr Doppelmayr Johann (1671-1750) +------------------------------------------------------------ | Doppler +------------------------------------------------------------ Doppler Doppler Christian (1803-1853) +------------------------------------------------------------ | Douglas +------------------------------------------------------------ Douglas Douglas Jesse (1897-1965) +------------------------------------------------------------ | Dowker +------------------------------------------------------------ Dowker Dowker Clifford (1912-1982) +------------------------------------------------------------ | Drach +------------------------------------------------------------ Drach Drach Jules (1871-1941) +------------------------------------------------------------ | Drinfeld +------------------------------------------------------------ Drinfeld Drinfeld Vladimir +------------------------------------------------------------ | Dubreil +------------------------------------------------------------ Dubreil Dubreil Paul (1904-1994) +------------------------------------------------------------ | Dudeney +------------------------------------------------------------ Dudeney Dudeney Henry (1857-1931) +------------------------------------------------------------ | Duhamel +------------------------------------------------------------ Duhamel Duhamel Jean-Marie (1797-1872) +------------------------------------------------------------ | Duhem +------------------------------------------------------------ Duhem Duhem Pierre (1861-1916) +------------------------------------------------------------ | Dupin +------------------------------------------------------------ Dupin Dupin Pierre (1784-1873) +------------------------------------------------------------ | Dupre +------------------------------------------------------------ Dupre Dupr'e Athanase (1808-1869) +------------------------------------------------------------ | Dynkin +------------------------------------------------------------ Dynkin Dynkin Evgenii +------------------------------------------------------------ | EckertJohn +------------------------------------------------------------ EckertJohn Eckert J Presper (1919-1995) +------------------------------------------------------------ | EckertWallace +------------------------------------------------------------ EckertWallace Eckert Wallace J (1902-1971) +------------------------------------------------------------ | Eckmann +------------------------------------------------------------ Eckmann Eckmann Beno +------------------------------------------------------------ | Eddington +------------------------------------------------------------ Eddington Eddington Arthur (1882-1944) +------------------------------------------------------------ | Edge +------------------------------------------------------------ Edge Edge William (1904-1997) +------------------------------------------------------------ | Edgeworth +------------------------------------------------------------ Edgeworth Edgeworth Francis (1845-1926) +------------------------------------------------------------ | Egorov +------------------------------------------------------------ Egorov Egorov Dimitri (1869-1931) +------------------------------------------------------------ | Ehrenfest +------------------------------------------------------------ Ehrenfest Ehrenfest Paul (1880-1933) +------------------------------------------------------------ | Ehresmann +------------------------------------------------------------ Ehresmann Ehresmann Charles (1905-1979) +------------------------------------------------------------ | Eilenberg +------------------------------------------------------------ Eilenberg Eilenberg Samuel (1913-1998) +------------------------------------------------------------ | Einstein +------------------------------------------------------------ Einstein Einstein Albert (1879-1955) +------------------------------------------------------------ | Eisenhart +------------------------------------------------------------ Eisenhart Eisenhart Luther (1876-1965) +------------------------------------------------------------ | Eisenstein +------------------------------------------------------------ Eisenstein Eisenstein Gotthold (1823-1852) +------------------------------------------------------------ | Elliott +------------------------------------------------------------ Elliott Elliott Edwin (1851-1937) +------------------------------------------------------------ | Empedocles +------------------------------------------------------------ Empedocles Empedocles (492BC-432BC) +------------------------------------------------------------ | Engel +------------------------------------------------------------ Engel Engel Friedrich (1861-1941) +------------------------------------------------------------ | Enriques +------------------------------------------------------------ Enriques Enriques Federigo (1871-1946) +------------------------------------------------------------ | Enskog +------------------------------------------------------------ Enskog Enskog David (1884-1947) +------------------------------------------------------------ | Epstein +------------------------------------------------------------ Epstein Epstein Paul (1871-1939) +------------------------------------------------------------ | Eratosthenes +------------------------------------------------------------ Eratosthenes Eratosthenes of Cyrene (276BC-197BC) +------------------------------------------------------------ | Erdelyi +------------------------------------------------------------ Erdelyi Erd'elyi Arthur (1908-1977) +------------------------------------------------------------ | Erdos +------------------------------------------------------------ Erdos Erd"os Paul (1913-1996) +------------------------------------------------------------ | Erlang +------------------------------------------------------------ Erlang Erlang Agner (1878-1929) +------------------------------------------------------------ | Escher +------------------------------------------------------------ Escher Escher Maurits (1898-1972) +------------------------------------------------------------ | Esclangon +------------------------------------------------------------ Esclangon Esclangon Ernest (1876-1954) +------------------------------------------------------------ | Euclid +------------------------------------------------------------ Euclid Euclid of Alexandria (325BC-265BC) +------------------------------------------------------------ | Eudemus +------------------------------------------------------------ Eudemus Eudemus of Rhodes (350BC-290BC) +------------------------------------------------------------ | Eudoxus +------------------------------------------------------------ Eudoxus Eudoxus of Cnidus (408BC-355BC) +------------------------------------------------------------ | Euler +------------------------------------------------------------ Euler Euler Leonhard (1707-1783) Swiss mathematician, worked on practially all fields of mathematics. +------------------------------------------------------------ | Eutocius +------------------------------------------------------------ Eutocius Eutocius of Ascalon (480-540) +------------------------------------------------------------ | Evans +------------------------------------------------------------ Evans Evans Griffith (1887-1973) +------------------------------------------------------------ | Ezra +------------------------------------------------------------ Ezra Ezra Rabbi Ben (1092-1167) +------------------------------------------------------------ | FaadiBruno +------------------------------------------------------------ FaadiBruno Fa`a di Bruno Francesco (1825-1907) +------------------------------------------------------------ | Faber +------------------------------------------------------------ Faber Faber Georg +------------------------------------------------------------ | Fabri +------------------------------------------------------------ Fabri Fabri Honor'e (1607-1688) +------------------------------------------------------------ | FagnanoGiovanni +------------------------------------------------------------ FagnanoGiovanni Fagnano Giovanni (1715-1797) +------------------------------------------------------------ | FagnanoGiulio +------------------------------------------------------------ FagnanoGiulio Fagnano Giulio (1682-1766) +------------------------------------------------------------ | Faltings +------------------------------------------------------------ Faltings Faltings Gerd +------------------------------------------------------------ | Fano +------------------------------------------------------------ Fano Fano Gino (1871-1952) +------------------------------------------------------------ | Faraday +------------------------------------------------------------ Faraday Faraday Michael (1791-1867) +------------------------------------------------------------ | Farey +------------------------------------------------------------ Farey Farey John (1766-1826) +------------------------------------------------------------ | Fatou +------------------------------------------------------------ Fatou Fatou Pierre (1878-1929) +------------------------------------------------------------ | Faulhaber +------------------------------------------------------------ Faulhaber Faulhaber Johann (1580-1635) +------------------------------------------------------------ | Fefferman +------------------------------------------------------------ Fefferman Fefferman Charles +------------------------------------------------------------ | Feigenbaum +------------------------------------------------------------ Feigenbaum Feigenbaum Mitchell +------------------------------------------------------------ | Feigl +------------------------------------------------------------ Feigl Feigl Georg (1890-1945) +------------------------------------------------------------ | Fejer +------------------------------------------------------------ Fejer Fej'er Lip'ot (1880-1959) +------------------------------------------------------------ | Feller +------------------------------------------------------------ Feller Feller William (1906-1970) +------------------------------------------------------------ | Fermat +------------------------------------------------------------ Fermat Fermat Pierre de (1601-1665) +------------------------------------------------------------ | Ferrar +------------------------------------------------------------ Ferrar Ferrar Bill (1893-1990) +------------------------------------------------------------ | Ferrari +------------------------------------------------------------ Ferrari Ferrari Lodovico (1522-1565) +------------------------------------------------------------ | Ferrel +------------------------------------------------------------ Ferrel Ferrel William (1817-1891) +------------------------------------------------------------ | Ferro +------------------------------------------------------------ Ferro Ferro Scipione del (1465-1526) +------------------------------------------------------------ | Feuerbach +------------------------------------------------------------ Feuerbach Feuerbach Karl (1800-1834) +------------------------------------------------------------ | Feynman +------------------------------------------------------------ Feynman Feynman Richard (1918-1988) +------------------------------------------------------------ | Fields +------------------------------------------------------------ Fields Fields John (1863-1932) +------------------------------------------------------------ | Finck +------------------------------------------------------------ Finck Finck Pierre-Joseph (1797-1870) +------------------------------------------------------------ | Fincke +------------------------------------------------------------ Fincke Fincke Thomas (1561-1656) +------------------------------------------------------------ | FineHenry +------------------------------------------------------------ FineHenry Fine Henry (1858-1928) +------------------------------------------------------------ | Fine +------------------------------------------------------------ Fine Fine Oronce (1494-1555) +------------------------------------------------------------ | Finsler +------------------------------------------------------------ Finsler Finsler Paul (1894-1970) +------------------------------------------------------------ | Fischer +------------------------------------------------------------ Fischer Fischer Ernst (1875-1959) +------------------------------------------------------------ | Fisher +------------------------------------------------------------ Fisher Fisher Sir Ronald (1890-1962) +------------------------------------------------------------ | Fiske +------------------------------------------------------------ Fiske Fiske Thomas (1865-1944) +------------------------------------------------------------ | FitzGerald +------------------------------------------------------------ FitzGerald FitzGerald George (1851-1901) +------------------------------------------------------------ | Flugge-Lotz +------------------------------------------------------------ Flugge-Lotz Fl"ugge-Lotz Irmgard (1903-1974) +------------------------------------------------------------ | Flamsteed +------------------------------------------------------------ Flamsteed Flamsteed John +------------------------------------------------------------ | Fomin +------------------------------------------------------------ Fomin Fomin Sergei (1917-1975) +------------------------------------------------------------ | FontainedesBertins +------------------------------------------------------------ FontainedesBertins Fontaine des Bertins A (1704-1771) +------------------------------------------------------------ | Fontenelle +------------------------------------------------------------ Fontenelle Fontenelle Bernard de (1657-1757) +------------------------------------------------------------ | Forsyth +------------------------------------------------------------ Forsyth Forsyth Andrew (1858-1942) +------------------------------------------------------------ | Burali-Forti +------------------------------------------------------------ Burali-Forti Forti Cesare Burali- (1861-1931) +------------------------------------------------------------ | Fourier +------------------------------------------------------------ Fourier Fourier Joseph (1768-1830) +------------------------------------------------------------ | Fowler +------------------------------------------------------------ Fowler Fowler Ralph (1889-1944) +------------------------------------------------------------ | Fox +------------------------------------------------------------ Fox Fox Charles (1897-1977) +------------------------------------------------------------ | Frechet +------------------------------------------------------------ Frechet Fr'echet Maurice (1878-1973) +------------------------------------------------------------ | Fraenkel +------------------------------------------------------------ Fraenkel Fraenkel Adolf (1891-1965) +------------------------------------------------------------ | FrancaisJacques +------------------------------------------------------------ FrancaisJacques Francais Jacques (1775-1833) +------------------------------------------------------------ | FrancaisFrancois +------------------------------------------------------------ FrancaisFrancois Francais Francois (1768-1810) +------------------------------------------------------------ | Francoeur +------------------------------------------------------------ Francoeur Francoeur Louis (1773-1849) +------------------------------------------------------------ | Frank +------------------------------------------------------------ Frank Frank Philipp (1884-1966) +------------------------------------------------------------ | Franklin +------------------------------------------------------------ Franklin Franklin Philip (1898-1965) +------------------------------------------------------------ | FranklinBenjamin +------------------------------------------------------------ FranklinBenjamin Franklin Benjamin (1706-1790) +------------------------------------------------------------ | Frattini +------------------------------------------------------------ Frattini Frattini Giovanni (1852-1925) +------------------------------------------------------------ | Fredholm +------------------------------------------------------------ Fredholm Fredholm Ivar (1866-1927) +------------------------------------------------------------ | Freedman +------------------------------------------------------------ Freedman Freedman Michael +------------------------------------------------------------ | Frege +------------------------------------------------------------ Frege Frege Gottlob (1848-1925) +------------------------------------------------------------ | Freitag +------------------------------------------------------------ Freitag Freitag Herta (1908-2000) +------------------------------------------------------------ | Frenet +------------------------------------------------------------ Frenet Frenet Jean (1816-1900) +------------------------------------------------------------ | FrenicledeBessy +------------------------------------------------------------ FrenicledeBessy Frenicle de Bessy B (1605-1675) +------------------------------------------------------------ | Frenkel +------------------------------------------------------------ Frenkel Frenkel Jacov (1894-1952) +------------------------------------------------------------ | Fresnel +------------------------------------------------------------ Fresnel Fresnel Augustin (1788-1827) +------------------------------------------------------------ | Freudenthal +------------------------------------------------------------ Freudenthal Freudenthal Hans (1905-1990) +------------------------------------------------------------ | Freundlich +------------------------------------------------------------ Freundlich Freundlich Finlay (1885-1964) +------------------------------------------------------------ | Friedmann +------------------------------------------------------------ Friedmann Friedmann Alexander (1888-1925) +------------------------------------------------------------ | Friedrichs +------------------------------------------------------------ Friedrichs Friedrichs Kurt (1901-1982) +------------------------------------------------------------ | Frisi +------------------------------------------------------------ Frisi Frisi Paolo (1728-1784) +------------------------------------------------------------ | Frobenius +------------------------------------------------------------ Frobenius Frobenius Georg (1849-1917) +------------------------------------------------------------ | Fubini +------------------------------------------------------------ Fubini Fubini Guido (1879-1943) +------------------------------------------------------------ | Fuchs +------------------------------------------------------------ Fuchs Fuchs Lazarus (1833-1902) +------------------------------------------------------------ | Fueter +------------------------------------------------------------ Fueter Fueter Rudolph (1880-1950) +------------------------------------------------------------ | Fuller +------------------------------------------------------------ Fuller Fuller R Buckminster (1895-1983) +------------------------------------------------------------ | Fuss +------------------------------------------------------------ Fuss Fuss Nicolai (1755-1826) +------------------------------------------------------------ | Godel +------------------------------------------------------------ Godel G"odel Kurt (1906-1978) +------------------------------------------------------------ | Gopel +------------------------------------------------------------ Gopel G"opel Adolph (1812-1847) +------------------------------------------------------------ | Galerkin +------------------------------------------------------------ Galerkin Galerkin Boris (1871-1945) +------------------------------------------------------------ | Galileo +------------------------------------------------------------ Galileo Galileo Galilei (1564-1642) +------------------------------------------------------------ | Gallarati +------------------------------------------------------------ Gallarati Gallarati Dionisio +------------------------------------------------------------ | Galois +------------------------------------------------------------ Galois Galois Evariste (1811-1832) +------------------------------------------------------------ | Galton +------------------------------------------------------------ Galton Galton Francis (1822-1911) +------------------------------------------------------------ | Gassendi +------------------------------------------------------------ Gassendi Gassendi Pierre (1592-1655) +------------------------------------------------------------ | Gauss +------------------------------------------------------------ Gauss Gauss Carl Friedrich (1777-1855) +------------------------------------------------------------ | Gegenbauer +------------------------------------------------------------ Gegenbauer Gegenbauer Leopold (1849-1903) +------------------------------------------------------------ | Geiser +------------------------------------------------------------ Geiser Geiser Karl (1843-1934) +------------------------------------------------------------ | Gelfand +------------------------------------------------------------ Gelfand Gelfand Israil +------------------------------------------------------------ | Gelfond +------------------------------------------------------------ Gelfond Gelfond Aleksandr (1906-1968) +------------------------------------------------------------ | Gellibrand +------------------------------------------------------------ Gellibrand Gellibrand Henry (1597-1636) +------------------------------------------------------------ | Geminus +------------------------------------------------------------ Geminus Geminus (10BC-60AD) +------------------------------------------------------------ | GemmaFrisius +------------------------------------------------------------ GemmaFrisius Gemma Frisius Regnier (1508-1555) +------------------------------------------------------------ | Genocchi +------------------------------------------------------------ Genocchi Genocchi Angelo (1817-1889) +------------------------------------------------------------ | Gentzen +------------------------------------------------------------ Gentzen Gentzen Gerhard (1909-1945) +------------------------------------------------------------ | Gergonne +------------------------------------------------------------ Gergonne Gergonne Joseph (1771-1859) +------------------------------------------------------------ | Germain +------------------------------------------------------------ Germain Germain Sophie (1776-1831) +------------------------------------------------------------ | Gherard +------------------------------------------------------------ Gherard Gherard of Cremona (1114-1187) +------------------------------------------------------------ | Ghetaldi +------------------------------------------------------------ Ghetaldi Ghetaldi Marino (1566-1626) +------------------------------------------------------------ | Gibbs +------------------------------------------------------------ Gibbs Gibbs J Willard (1839-1903) +------------------------------------------------------------ | GirardAlbert +------------------------------------------------------------ GirardAlbert Girard Albert (1595-1632) +------------------------------------------------------------ | GirardPierre +------------------------------------------------------------ GirardPierre Girard Pierre Simon (1765-1836) +------------------------------------------------------------ | Glaisher +------------------------------------------------------------ Glaisher Glaisher James (1848-1928) +------------------------------------------------------------ | Glenie +------------------------------------------------------------ Glenie Glenie James (1750-1817) +------------------------------------------------------------ | Gohberg +------------------------------------------------------------ Gohberg Gohberg Israel +------------------------------------------------------------ | Goldbach +------------------------------------------------------------ Goldbach Goldbach Christian (1690-1764) +------------------------------------------------------------ | Goldstein +------------------------------------------------------------ Goldstein Goldstein Sydney (1903-1989) +------------------------------------------------------------ | Gompertz +------------------------------------------------------------ Gompertz Gompertz Benjamin (1779-1865) +------------------------------------------------------------ | Goodstein +------------------------------------------------------------ Goodstein Goodstein Reuben (1912-1985) +------------------------------------------------------------ | Gordan +------------------------------------------------------------ Gordan Gordan Paul (1837-1912) +------------------------------------------------------------ | Gorenstein +------------------------------------------------------------ Gorenstein Gorenstein Daniel (1923-1992) +------------------------------------------------------------ | Gosset +------------------------------------------------------------ Gosset Gosset William (1876-1937) +------------------------------------------------------------ | Goursat +------------------------------------------------------------ Goursat Goursat Edouard (1858-1936) +------------------------------------------------------------ | Govindasvami +------------------------------------------------------------ Govindasvami Govindasvami (800-860) +------------------------------------------------------------ | Graffe +------------------------------------------------------------ Graffe Gr"affe Karl (1799-1873) +------------------------------------------------------------ | Gram +------------------------------------------------------------ Gram Gram Jorgen (1850-1916) +------------------------------------------------------------ | Grandi +------------------------------------------------------------ Grandi Grandi Guido (1671-1742) +------------------------------------------------------------ | Granville +------------------------------------------------------------ Granville Granville Evelyn +------------------------------------------------------------ | Grassmann +------------------------------------------------------------ Grassmann Grassmann Hermann (1808-1877) +------------------------------------------------------------ | Grave +------------------------------------------------------------ Grave Grave Dmitry (1863-1939) +------------------------------------------------------------ | Green +------------------------------------------------------------ Green Green George (1793-1841) +------------------------------------------------------------ | Greenhill +------------------------------------------------------------ Greenhill Greenhill Alfred (1847-1927) +------------------------------------------------------------ | Gregory +------------------------------------------------------------ Gregory Gregory James (1638-1675) +------------------------------------------------------------ | GregoryDuncan +------------------------------------------------------------ GregoryDuncan Gregory Duncan (1813-1844) +------------------------------------------------------------ | GregoryDavid +------------------------------------------------------------ GregoryDavid Gregory David (1659-1708) +------------------------------------------------------------ | DeGroot +------------------------------------------------------------ DeGroot Groot Johannes de (1914-1972) +------------------------------------------------------------ | Grosseteste +------------------------------------------------------------ Grosseteste Grosseteste Robert (1168-1253) +------------------------------------------------------------ | Grossmann +------------------------------------------------------------ Grossmann Grossmann Marcel (1878-1936) +------------------------------------------------------------ | Grothendieck +------------------------------------------------------------ Grothendieck Grothendieck Alexander +------------------------------------------------------------ | Grunsky +------------------------------------------------------------ Grunsky Grunsky Helmut (1904-1986) +------------------------------------------------------------ | Guarini +------------------------------------------------------------ Guarini Guarini Guarino (1624-1683) +------------------------------------------------------------ | Guccia +------------------------------------------------------------ Guccia Guccia Giovanni (1855-1914) +------------------------------------------------------------ | Gudermann +------------------------------------------------------------ Gudermann Gudermann Christoph (1798-1852) +------------------------------------------------------------ | Guenther +------------------------------------------------------------ Guenther Guenther Adam (1848-1923) +------------------------------------------------------------ | Guinand +------------------------------------------------------------ Guinand Guinand Andy (1912-1987) +------------------------------------------------------------ | Guldin +------------------------------------------------------------ Guldin Guldin Paul (1577-1643) +------------------------------------------------------------ | Gunter +------------------------------------------------------------ Gunter Gunter Edmund (1581-1626) +------------------------------------------------------------ | Hajek +------------------------------------------------------------ Hajek H"ajek Jaroslav (1926-1974) +------------------------------------------------------------ | Herigone +------------------------------------------------------------ Herigone H'erigone Pierre (1580-1643) +------------------------------------------------------------ | Holder +------------------------------------------------------------ Holder H"older Otto (1859-1937) +------------------------------------------------------------ | Hormander +------------------------------------------------------------ Hormander H"ormander Lars +------------------------------------------------------------ | Haar +------------------------------------------------------------ Haar Haar Alfr'ed (1885-1933) +------------------------------------------------------------ | Hachette +------------------------------------------------------------ Hachette Hachette Jean (1769-1834) +------------------------------------------------------------ | Hadamard +------------------------------------------------------------ Hadamard Hadamard Jacques (1865-1963) +------------------------------------------------------------ | Hadley +------------------------------------------------------------ Hadley Hadley John (1682-1744) +------------------------------------------------------------ | Hahn +------------------------------------------------------------ Hahn Hahn Hans (1879-1934) +------------------------------------------------------------ | Hall +------------------------------------------------------------ Hall Hall Philip (1904-1982) +------------------------------------------------------------ | HallMarshall +------------------------------------------------------------ HallMarshall Hall Marshall Jr. (1910-1990) +------------------------------------------------------------ | Halley +------------------------------------------------------------ Halley Halley Edmond (1656-1742) +------------------------------------------------------------ | Halmos +------------------------------------------------------------ Halmos Halmos Paul +------------------------------------------------------------ | Halphen +------------------------------------------------------------ Halphen Halphen George (1844-1889) +------------------------------------------------------------ | Halsted +------------------------------------------------------------ Halsted Halsted George (1853-1922) +------------------------------------------------------------ | Hamill +------------------------------------------------------------ Hamill Hamill Christine +------------------------------------------------------------ | Hamilton +------------------------------------------------------------ Hamilton Hamilton William R (1805-1865) +------------------------------------------------------------ | HamiltonWilliam +------------------------------------------------------------ HamiltonWilliam Hamilton William (1788-1856) +------------------------------------------------------------ | Hamming +------------------------------------------------------------ Hamming Hamming Richard W (1915-1998) +------------------------------------------------------------ | Hankel +------------------------------------------------------------ Hankel Hankel Hermann (1839-1873) +------------------------------------------------------------ | HardyClaude +------------------------------------------------------------ HardyClaude Hardy Claude (1598-1678) +------------------------------------------------------------ | Hardy +------------------------------------------------------------ Hardy Hardy G H (1877-1947) +------------------------------------------------------------ | Harish-Chandra +------------------------------------------------------------ Harish-Chandra Harish-Chandra (1923-1983) +------------------------------------------------------------ | Harriot +------------------------------------------------------------ Harriot Harriot Thomas (1560-1621) +------------------------------------------------------------ | Hartley +------------------------------------------------------------ Hartley Hartley Brian (1939-1994) +------------------------------------------------------------ | Hartree +------------------------------------------------------------ Hartree Hartree Douglas (1897-1958) +------------------------------------------------------------ | Hasse +------------------------------------------------------------ Hasse Hasse Helmut (1898-1979) +------------------------------------------------------------ | Hausdorff +------------------------------------------------------------ Hausdorff Hausdorff Felix (1869-1942) +------------------------------------------------------------ | Hawking +------------------------------------------------------------ Hawking Hawking Stephen +------------------------------------------------------------ | Al-Haytham +------------------------------------------------------------ Al-Haytham Haytham Abu Ali al +------------------------------------------------------------ | Heath +------------------------------------------------------------ Heath Heath Thomas (1861-1940) +------------------------------------------------------------ | Heaviside +------------------------------------------------------------ Heaviside Heaviside Oliver (1850-1925) +------------------------------------------------------------ | Heawood +------------------------------------------------------------ Heawood Heawood Percy (1861-1955) +------------------------------------------------------------ | Hecht +------------------------------------------------------------ Hecht Hecht Daniel (1777-1833) +------------------------------------------------------------ | Hecke +------------------------------------------------------------ Hecke Hecke Erich (1887-1947) +------------------------------------------------------------ | Hedrick +------------------------------------------------------------ Hedrick Hedrick Earle (1876-1943) +------------------------------------------------------------ | Heegaard +------------------------------------------------------------ Heegaard Heegaard Poul (1871-1948) +------------------------------------------------------------ | Heilbronn +------------------------------------------------------------ Heilbronn Heilbronn Hans (1908-1975) +------------------------------------------------------------ | Heine +------------------------------------------------------------ Heine Heine Eduard (1821-1881) +------------------------------------------------------------ | Heisenberg +------------------------------------------------------------ Heisenberg Heisenberg Werner (1901-1976) +------------------------------------------------------------ | Hellinger +------------------------------------------------------------ Hellinger Hellinger Ernst (1883-1950) +------------------------------------------------------------ | Helly +------------------------------------------------------------ Helly Helly Eduard (1884-1943) +------------------------------------------------------------ | Heng +------------------------------------------------------------ Heng Heng Zhang (78AD-139) +------------------------------------------------------------ | Henrici +------------------------------------------------------------ Henrici Henrici Olaus (1840-1918) +------------------------------------------------------------ | Hensel +------------------------------------------------------------ Hensel Hensel Kurt (1861-1941) +------------------------------------------------------------ | Heraclides +------------------------------------------------------------ Heraclides Heraclides of Pontus (387BC-312BC) +------------------------------------------------------------ | Herbrand +------------------------------------------------------------ Herbrand Herbrand Jacques (1908-1931) +------------------------------------------------------------ | Hermann +------------------------------------------------------------ Hermann Hermann Jakob (1678-1733) +------------------------------------------------------------ | Hermite +------------------------------------------------------------ Hermite Hermite Charles (1822-1901) +------------------------------------------------------------ | Heron +------------------------------------------------------------ Heron Heron of Alexandria (10AD-75) +------------------------------------------------------------ | HerschelCaroline +------------------------------------------------------------ HerschelCaroline Herschel Caroline (1750-1848) +------------------------------------------------------------ | Herschel +------------------------------------------------------------ Herschel Herschel John (1792-1871) +------------------------------------------------------------ | Herstein +------------------------------------------------------------ Herstein Herstein Yitz (1923-1988) +------------------------------------------------------------ | Hesse +------------------------------------------------------------ Hesse Hesse Otto (1811-1874) +------------------------------------------------------------ | Heyting +------------------------------------------------------------ Heyting Heyting Arend (1898-1980) +------------------------------------------------------------ | Higman +------------------------------------------------------------ Higman Higman Graham +------------------------------------------------------------ | Hilbert +------------------------------------------------------------ Hilbert Hilbert David (1862-1943) +------------------------------------------------------------ | Hill +------------------------------------------------------------ Hill Hill George (1838-1914) +------------------------------------------------------------ | Hille +------------------------------------------------------------ Hille Hille Einar (1894-1980) +------------------------------------------------------------ | Hindenburg +------------------------------------------------------------ Hindenburg Hindenburg Carl (1741-1808) +------------------------------------------------------------ | Hipparchus +------------------------------------------------------------ Hipparchus Hipparchus of Rhodes (190BC-120BC) +------------------------------------------------------------ | Hippias +------------------------------------------------------------ Hippias Hippias of Elis (460BC-400BC) +------------------------------------------------------------ | Hippocrates +------------------------------------------------------------ Hippocrates Hippocrates of Chios (470BC-410BC) +------------------------------------------------------------ | Hironaka +------------------------------------------------------------ Hironaka Hironaka Heisuke +------------------------------------------------------------ | Hirsch +------------------------------------------------------------ Hirsch Hirsch Kurt (1906-1986) +------------------------------------------------------------ | Hirst +------------------------------------------------------------ Hirst Hirst Thomas (1830-1891) +------------------------------------------------------------ | Gnedenko +------------------------------------------------------------ Gnedenko Hniedenko Boris (1912-1995) +------------------------------------------------------------ | Houel +------------------------------------------------------------ Houel Ho"uel Jules (1823-1886) +------------------------------------------------------------ | Hobbes +------------------------------------------------------------ Hobbes Hobbes Thomas (1588-1679) +------------------------------------------------------------ | Hobson +------------------------------------------------------------ Hobson Hobson Ernest (1856-1933) +------------------------------------------------------------ | Hodge +------------------------------------------------------------ Hodge Hodge William (1903-1975) +------------------------------------------------------------ | Hollerith +------------------------------------------------------------ Hollerith Hollerith Herman (1860-1929) +------------------------------------------------------------ | Holmboe +------------------------------------------------------------ Holmboe Holmboe Bernt (1795-1850) +------------------------------------------------------------ | Honda +------------------------------------------------------------ Honda Honda Taira (1932-1975) +------------------------------------------------------------ | Hooke +------------------------------------------------------------ Hooke Hooke Robert (1635-1703) +------------------------------------------------------------ | HopfEberhard +------------------------------------------------------------ HopfEberhard Hopf Eberhard (1902-1983) +------------------------------------------------------------ | Hopf +------------------------------------------------------------ Hopf Hopf Heinz (1894-1971) +------------------------------------------------------------ | Hopkins +------------------------------------------------------------ Hopkins Hopkins William (1793-1866) +------------------------------------------------------------ | Hopkinson +------------------------------------------------------------ Hopkinson Hopkinson John (1849-1898) +------------------------------------------------------------ | Hopper +------------------------------------------------------------ Hopper Hopper Grace (1906-1992) +------------------------------------------------------------ | Horner +------------------------------------------------------------ Horner Horner William (1786-1837) +------------------------------------------------------------ | Householder +------------------------------------------------------------ Householder Householder Alston (1904-1993) +------------------------------------------------------------ | Hsu +------------------------------------------------------------ Hsu Hsu Pao-Lu (1910-1970) +------------------------------------------------------------ | Hubble +------------------------------------------------------------ Hubble Hubble Edwin (1889-1953) +------------------------------------------------------------ | Hudde +------------------------------------------------------------ Hudde Hudde Johann (1628-1704) +------------------------------------------------------------ | HumbertPierre +------------------------------------------------------------ HumbertPierre Humbert Pierre (1891-1953) +------------------------------------------------------------ | HumbertGeorges +------------------------------------------------------------ HumbertGeorges Humbert Georges (1859-1921) +------------------------------------------------------------ | Huntington +------------------------------------------------------------ Huntington Huntington Edward (1874-1952) +------------------------------------------------------------ | Hurewicz +------------------------------------------------------------ Hurewicz Hurewicz Witold (1904-1956) +------------------------------------------------------------ | Hurwitz +------------------------------------------------------------ Hurwitz Hurwitz Adolf (1859-1919) +------------------------------------------------------------ | Hutton +------------------------------------------------------------ Hutton Hutton Charles (1737-1823) +------------------------------------------------------------ | Huygens +------------------------------------------------------------ Huygens Huygens Christiaan (1629-1695) +------------------------------------------------------------ | Hypatia +------------------------------------------------------------ Hypatia Hypatia of Alexandria (370-415) +------------------------------------------------------------ | Hypsicles +------------------------------------------------------------ Hypsicles Hypsicles of Alexandria (190BC-120BC) +------------------------------------------------------------ | Ibrahim +------------------------------------------------------------ Ibrahim Ibrahim ibn Sinan (908-946) +------------------------------------------------------------ | Ingham +------------------------------------------------------------ Ingham Ingham Albert (1900-1967) +------------------------------------------------------------ | Ito +------------------------------------------------------------ Ito Ito Kiyosi +------------------------------------------------------------ | Ivory +------------------------------------------------------------ Ivory Ivory James (1765-1842) +------------------------------------------------------------ | Iwasawa +------------------------------------------------------------ Iwasawa Iwasawa Kenkichi (1917-1998) +------------------------------------------------------------ | Iyanaga +------------------------------------------------------------ Iyanaga Iyanaga Skokichi +------------------------------------------------------------ | JabiribnAflah +------------------------------------------------------------ JabiribnAflah Jabir ibn Aflah (1100-1160) +------------------------------------------------------------ | Jacobi +------------------------------------------------------------ Jacobi Jacobi Carl (1804-1851) +------------------------------------------------------------ | Jacobson +------------------------------------------------------------ Jacobson Jacobson Nathan (1910-1999) +------------------------------------------------------------ | Jagannatha +------------------------------------------------------------ Jagannatha Jagannatha Samrat +------------------------------------------------------------ | James +------------------------------------------------------------ James James Ioan +------------------------------------------------------------ | Janiszewski +------------------------------------------------------------ Janiszewski Janiszewski Zygmunt (1888-1920) +------------------------------------------------------------ | Janovskaja +------------------------------------------------------------ Janovskaja Janovskaja Sof'ja (1896-1966) +------------------------------------------------------------ | Jarnik +------------------------------------------------------------ Jarnik Jarnik Vojtech (1897-1970) +------------------------------------------------------------ | Al-Jawhari +------------------------------------------------------------ Al-Jawhari Jawhari al-Abbas al (800-860) +------------------------------------------------------------ | Al-Jayyani +------------------------------------------------------------ Al-Jayyani Jayyani Abu al +------------------------------------------------------------ | Jeans +------------------------------------------------------------ Jeans Jeans Sir James (1877-1946) +------------------------------------------------------------ | Jeffrey +------------------------------------------------------------ Jeffrey Jeffrey George (1891-1957) +------------------------------------------------------------ | Jeffreys +------------------------------------------------------------ Jeffreys Jeffreys Sir Harold (1891-1989) +------------------------------------------------------------ | Jensen +------------------------------------------------------------ Jensen Jensen Johan (1859-1925) +------------------------------------------------------------ | Jerrard +------------------------------------------------------------ Jerrard Jerrard George (1804-1863) +------------------------------------------------------------ | Jevons +------------------------------------------------------------ Jevons Jevons William (1835-1882) +------------------------------------------------------------ | Joachimsthal +------------------------------------------------------------ Joachimsthal Joachimsthal Ferdinand (1818-1861) +------------------------------------------------------------ | John +------------------------------------------------------------ John John Fritz +------------------------------------------------------------ | JohnsonBarry +------------------------------------------------------------ JohnsonBarry Johnson Barry +------------------------------------------------------------ | Johnson +------------------------------------------------------------ Johnson Johnson William (1858-1931) +------------------------------------------------------------ | JonesBurton +------------------------------------------------------------ JonesBurton Jones F (1910-1999) +------------------------------------------------------------ | Jones +------------------------------------------------------------ Jones Jones William (1675-1749) +------------------------------------------------------------ | JonesVaughan +------------------------------------------------------------ JonesVaughan Jones Vaughan +------------------------------------------------------------ | Jonquieres +------------------------------------------------------------ Jonquieres Jonqui`eres Ernest de (1820-1901) +------------------------------------------------------------ | Jordan +------------------------------------------------------------ Jordan Jordan Camille (1838-1922) +------------------------------------------------------------ | Jordanus +------------------------------------------------------------ Jordanus Jordanus Nemorarius (1225-1260) +------------------------------------------------------------ | Jourdain +------------------------------------------------------------ Jourdain Jourdain Philip (1879-1921) +------------------------------------------------------------ | Juel +------------------------------------------------------------ Juel Juel Christian (1855-1935) +------------------------------------------------------------ | Julia +------------------------------------------------------------ Julia Julia Gaston (1893-1978) +------------------------------------------------------------ | Jungius +------------------------------------------------------------ Jungius Jungius Joachim (1587-1657) +------------------------------------------------------------ | Jyesthadeva +------------------------------------------------------------ Jyesthadeva Jyesthadeva (1500-1575) +------------------------------------------------------------ | KonigJulius +------------------------------------------------------------ KonigJulius K"onig Julius (1849-1913) +------------------------------------------------------------ | KonigSamuel +------------------------------------------------------------ KonigSamuel K"onig Samuel (1712-1757) +------------------------------------------------------------ | Konigsberger +------------------------------------------------------------ Konigsberger K"onigsberger Leo (1837-1921) +------------------------------------------------------------ | Kurschak +------------------------------------------------------------ Kurschak K"ursch"ak J'ozsef (1864-1933) +------------------------------------------------------------ | Kac +------------------------------------------------------------ Kac Kac Mark (1914-1984) +------------------------------------------------------------ | Kaestner +------------------------------------------------------------ Kaestner Kaestner Abraham (1719-1800) +------------------------------------------------------------ | Kagan +------------------------------------------------------------ Kagan Kagan Benjamin (1869-1953) +------------------------------------------------------------ | Kakutani +------------------------------------------------------------ Kakutani Kakutani Shizuo +------------------------------------------------------------ | Kalmar +------------------------------------------------------------ Kalmar Kalm"ar L"aszl'o (1905-1976) +------------------------------------------------------------ | Kaluza +------------------------------------------------------------ Kaluza Kaluza Theodor (1885-1945) +------------------------------------------------------------ | Kaluznin +------------------------------------------------------------ Kaluznin Kaluznin Lev (1914-1990) +------------------------------------------------------------ | Al-Farisi +------------------------------------------------------------ Al-Farisi Kamal al-Farisi (1260-1320) +------------------------------------------------------------ | Kamalakara +------------------------------------------------------------ Kamalakara Kamalakara (1616-1700) +------------------------------------------------------------ | AbuKamil +------------------------------------------------------------ AbuKamil Kamil Abu Shuja (850-930) +------------------------------------------------------------ | Kantorovich +------------------------------------------------------------ Kantorovich Kantorovich Leonid (1912-1986) +------------------------------------------------------------ | Kaplansky +------------------------------------------------------------ Kaplansky Kaplansky Irving +------------------------------------------------------------ | Al-Karaji +------------------------------------------------------------ Al-Karaji Karkhi al +------------------------------------------------------------ | Karp +------------------------------------------------------------ Karp Karp Carol (1926-1972) +------------------------------------------------------------ | Al-Kashi +------------------------------------------------------------ Al-Kashi Kashi Ghiyath al (1390-1450) +------------------------------------------------------------ | Katyayana +------------------------------------------------------------ Katyayana Katyayana (200BC-140BC) +------------------------------------------------------------ | Keill +------------------------------------------------------------ Keill Keill John (1671-1721) +------------------------------------------------------------ | Kelland +------------------------------------------------------------ Kelland Kelland Philip (1808-1879) +------------------------------------------------------------ | Kellogg +------------------------------------------------------------ Kellogg Kellogg Oliver (1878-1957) +------------------------------------------------------------ | Kemeny +------------------------------------------------------------ Kemeny Kemeny John (1926-1992) +------------------------------------------------------------ | Kempe +------------------------------------------------------------ Kempe Kempe Alfred (1849-1922) +------------------------------------------------------------ | KendallMaurice +------------------------------------------------------------ KendallMaurice Kendall Maurice (1907-1983) +------------------------------------------------------------ | Kendall +------------------------------------------------------------ Kendall Kendall David +------------------------------------------------------------ | Kepler +------------------------------------------------------------ Kepler Kepler Johannes (1571-1630) +------------------------------------------------------------ | Kerekjarto +------------------------------------------------------------ Kerekjarto Ker'ekj"art'o B'ela (1898-1946) +------------------------------------------------------------ | Keynes +------------------------------------------------------------ Keynes Keynes John Maynard (1883-1946) +------------------------------------------------------------ | Al-Khalili +------------------------------------------------------------ Al-Khalili Khalili Shams al (1320-1380) +------------------------------------------------------------ | Al-Khazin +------------------------------------------------------------ Al-Khazin Khazin Abu Jafar al (900-971) +------------------------------------------------------------ | Khinchin +------------------------------------------------------------ Khinchin Khinchin Aleksandr (1894-1959) +------------------------------------------------------------ | Al-Khujandi +------------------------------------------------------------ Al-Khujandi Khujandi Abu al +------------------------------------------------------------ | Al-Khwarizmi +------------------------------------------------------------ Al-Khwarizmi Khwarizmi Abu al- (790-850) +------------------------------------------------------------ | Killing +------------------------------------------------------------ Killing Killing Wilhelm (1847-1923) +------------------------------------------------------------ | Al-Kindi +------------------------------------------------------------ Al-Kindi Kindi Abu al (805-873) +------------------------------------------------------------ | Kingman +------------------------------------------------------------ Kingman Kingman John +------------------------------------------------------------ | Kirchhoff +------------------------------------------------------------ Kirchhoff Kirchhoff Gustav (1824-1887) +------------------------------------------------------------ | Kirkman +------------------------------------------------------------ Kirkman Kirkman Thomas (1806-1895) +------------------------------------------------------------ | Klugel +------------------------------------------------------------ Klugel Kl"ugel Georg (1739-1812) +------------------------------------------------------------ | Kleene +------------------------------------------------------------ Kleene Kleene Stephen (1909-1994) +------------------------------------------------------------ | KleinOskar +------------------------------------------------------------ KleinOskar Klein Oskar (1894-1977) +------------------------------------------------------------ | Klein +------------------------------------------------------------ Klein Klein Felix (1849-1925) +------------------------------------------------------------ | Klingenberg +------------------------------------------------------------ Klingenberg Klingenberg Wilhelm +------------------------------------------------------------ | Kloosterman +------------------------------------------------------------ Kloosterman Kloosterman Hendrik (1900-1968) +------------------------------------------------------------ | Kneser +------------------------------------------------------------ Kneser Kneser Adolf (1862-1930) +------------------------------------------------------------ | KneserHellmuth +------------------------------------------------------------ KneserHellmuth Kneser Hellmuth (1898-1973) +------------------------------------------------------------ | Knopp +------------------------------------------------------------ Knopp Knopp Konrad (1882-1957) +------------------------------------------------------------ | Kober +------------------------------------------------------------ Kober Kober Hermann (1888-1973) +------------------------------------------------------------ | Kochin +------------------------------------------------------------ Kochin Kochin Nikolai (1901-1944) +------------------------------------------------------------ | Kodaira +------------------------------------------------------------ Kodaira Kodaira Kunihiko (1915-1997) +------------------------------------------------------------ | Koebe +------------------------------------------------------------ Koebe Koebe Paul (1882-1945) +------------------------------------------------------------ | Koenigs +------------------------------------------------------------ Koenigs Koenigs Gabriel (1858-1931) +------------------------------------------------------------ | Kolmogorov +------------------------------------------------------------ Kolmogorov Kolmogorov Andrey (1903-1987) Russian probabilist who established in 1933 the mathematical foundation of probability theory and did important work also in other fields like Hamiltonian dynamics (KAM theorem) or turbulence Kolmogorov scaling. +------------------------------------------------------------ | Kolosov +------------------------------------------------------------ Kolosov Kolosov Gury (1867-1936) +------------------------------------------------------------ | KonigDenes +------------------------------------------------------------ KonigDenes Konig Denes (1884-1944) +------------------------------------------------------------ | Korteweg +------------------------------------------------------------ Korteweg Korteweg Diederik (1848-1941) +------------------------------------------------------------ | Kotelnikov +------------------------------------------------------------ Kotelnikov Kotelnikov Aleksandr (1865-1944) +------------------------------------------------------------ | Kovalevskaya +------------------------------------------------------------ Kovalevskaya Kovalevskaya Sofia (1850-1891) +------------------------------------------------------------ | Kramp +------------------------------------------------------------ Kramp Kramp Christian (1760-1826) +------------------------------------------------------------ | Krawtchouk +------------------------------------------------------------ Krawtchouk Krawtchouk Mikhail (1892-1942) +------------------------------------------------------------ | Krein +------------------------------------------------------------ Krein Krein Mark (1907-1989) +------------------------------------------------------------ | Kreisel +------------------------------------------------------------ Kreisel Kreisel Georg +------------------------------------------------------------ | Kronecker +------------------------------------------------------------ Kronecker Kronecker Leopold (1823-1891) +------------------------------------------------------------ | Krull +------------------------------------------------------------ Krull Krull Wolfgang (1899-1971) +------------------------------------------------------------ | KrylovAleksei +------------------------------------------------------------ KrylovAleksei Krylov Aleksei (1863-1945) +------------------------------------------------------------ | KrylovNikolai +------------------------------------------------------------ KrylovNikolai Krylov Nikolai (1879-1955) +------------------------------------------------------------ | Kulik +------------------------------------------------------------ Kulik Kulik Yakov (1783-1863) +------------------------------------------------------------ | Kumano-Go +------------------------------------------------------------ Kumano-Go Kumano-Go Hitoshi (1935-1982) +------------------------------------------------------------ | Kummer +------------------------------------------------------------ Kummer Kummer Eduard (1810-1893) +------------------------------------------------------------ | Kuratowski +------------------------------------------------------------ Kuratowski Kuratowski Kazimierz (1896-1980) +------------------------------------------------------------ | Kurosh +------------------------------------------------------------ Kurosh Kurosh Aleksandr (1908-1971) +------------------------------------------------------------ | Kutta +------------------------------------------------------------ Kutta Kutta Martin (1867-1944) +------------------------------------------------------------ | Kuttner +------------------------------------------------------------ Kuttner Kuttner Brian (1908-1992) +------------------------------------------------------------ | Leger +------------------------------------------------------------ Leger L'eger Emile (1795-1838) +------------------------------------------------------------ | LevyPaul +------------------------------------------------------------ LevyPaul L'evy Paul (1886-1971) +------------------------------------------------------------ | Lowenheim +------------------------------------------------------------ Lowenheim L"owenheim Leopold (1878-1957) +------------------------------------------------------------ | Loewner +------------------------------------------------------------ Loewner L"owner Karl (1893-1968) +------------------------------------------------------------ | DeL'Hopital +------------------------------------------------------------ DeL'Hopital L'Hopital Guillaume de (1661-1704) +------------------------------------------------------------ | LaHire +------------------------------------------------------------ LaHire La Hire Philippe de (1640-1718) +------------------------------------------------------------ | LaFaille +------------------------------------------------------------ LaFaille La Faille Charles de (1597-1652) +------------------------------------------------------------ | LaCondamine +------------------------------------------------------------ LaCondamine La Condamine Charles de (1701-1774) +------------------------------------------------------------ | Lacroix +------------------------------------------------------------ Lacroix Lacroix Sylvestre (1765-1843) +------------------------------------------------------------ | Lagny +------------------------------------------------------------ Lagny Lagny Thomas de (1660-1734) +------------------------------------------------------------ | Lagrange +------------------------------------------------------------ Lagrange Lagrange Joseph-Louis (1736-1813) +------------------------------------------------------------ | Laguerre +------------------------------------------------------------ Laguerre Laguerre Edmond (1834-1886) +------------------------------------------------------------ | Lakatos +------------------------------------------------------------ Lakatos Lakatos Imre (1922-1974) +------------------------------------------------------------ | Lalla +------------------------------------------------------------ Lalla Lalla (720-790) +------------------------------------------------------------ | Lame +------------------------------------------------------------ Lame Lam'e Gabriel (1795-1870) +------------------------------------------------------------ | Lamb +------------------------------------------------------------ Lamb Lamb Horace (1849-1934) +------------------------------------------------------------ | Lambert +------------------------------------------------------------ Lambert Lambert Johann (1728-1777) +------------------------------------------------------------ | HermannofReichenau +------------------------------------------------------------ HermannofReichenau Lame Hermann the (1013-1054) +------------------------------------------------------------ | Lamy +------------------------------------------------------------ Lamy Lamy Bernard (1640-1715) +------------------------------------------------------------ | Lanczos +------------------------------------------------------------ Lanczos Lanczos Cornelius (1893-1974) +------------------------------------------------------------ | Landau +------------------------------------------------------------ Landau Landau Edmund (1877-1938) +------------------------------------------------------------ | LandauLev +------------------------------------------------------------ LandauLev Landau Lev (1908-1968) +------------------------------------------------------------ | Landen +------------------------------------------------------------ Landen Landen John (1719-1790) +------------------------------------------------------------ | Landsberg +------------------------------------------------------------ Landsberg Landsberg Georg (1865-1912) +------------------------------------------------------------ | Langlands +------------------------------------------------------------ Langlands Langlands Robert +------------------------------------------------------------ | Laplace +------------------------------------------------------------ Laplace Laplace Pierre-Simon (1749-1827) +------------------------------------------------------------ | Larmor +------------------------------------------------------------ Larmor Larmor Sir Joseph (1857-1942) +------------------------------------------------------------ | Lasker +------------------------------------------------------------ Lasker Lasker Emanuel (1868-1941) +------------------------------------------------------------ | Kramer +------------------------------------------------------------ Kramer Lassar Edna Kramer (1902-1984) +------------------------------------------------------------ | LaurentHermann +------------------------------------------------------------ LaurentHermann Laurent Hermann (1841-1908) +------------------------------------------------------------ | LaurentPierre +------------------------------------------------------------ LaurentPierre Laurent Pierre (1813-1854) +------------------------------------------------------------ | Lavanha +------------------------------------------------------------ Lavanha Lavanha Joao Baptista (1550-1624) +------------------------------------------------------------ | Lavrentev +------------------------------------------------------------ Lavrentev Lavrentev Mikhail (1900-1980) +------------------------------------------------------------ | Lax +------------------------------------------------------------ Lax Lax Gaspar (1487-1560) +------------------------------------------------------------ | LeFevre +------------------------------------------------------------ LeFevre Le F`evre Jean (1652-1706) +------------------------------------------------------------ | Lebesgue +------------------------------------------------------------ Lebesgue Lebesgue Henri (1875-1941) +------------------------------------------------------------ | Ledermann +------------------------------------------------------------ Ledermann Ledermann Walter +------------------------------------------------------------ | Leech +------------------------------------------------------------ Leech Leech John (1926-1992) +------------------------------------------------------------ | Lefschetz +------------------------------------------------------------ Lefschetz Lefschetz Solomon (1884-1972) +------------------------------------------------------------ | Legendre +------------------------------------------------------------ Legendre Legendre Adrien-Marie (1752-1833) +------------------------------------------------------------ | Lemoine +------------------------------------------------------------ Lemoine Lemoine Emile (1840-1912) +------------------------------------------------------------ | Leray +------------------------------------------------------------ Leray Leray Jean (1906-1998) +------------------------------------------------------------ | Lerch +------------------------------------------------------------ Lerch Lerch Mathias (1860-1922) +------------------------------------------------------------ | Leshniewski +------------------------------------------------------------ Leshniewski Leshniewski Stanislaw (1886-1939) +------------------------------------------------------------ | Leslie +------------------------------------------------------------ Leslie Leslie John (1766-1832) +------------------------------------------------------------ | Leucippus +------------------------------------------------------------ Leucippus Leucippus (480BC-420BC) +------------------------------------------------------------ | Levi +------------------------------------------------------------ Levi Levi ben Gerson (1288-1344) +------------------------------------------------------------ | Levi-Civita +------------------------------------------------------------ Levi-Civita Levi-Civita Tullio (1873-1941) +------------------------------------------------------------ | Levinson +------------------------------------------------------------ Levinson Levinson Norman (1912-1975) +------------------------------------------------------------ | LevyHyman +------------------------------------------------------------ LevyHyman Levy Hyman (1889-1975) +------------------------------------------------------------ | Levytsky +------------------------------------------------------------ Levytsky Levytsky Volodymyr (1872-1956) +------------------------------------------------------------ | Lexell +------------------------------------------------------------ Lexell Lexell Anders (1740-1784) +------------------------------------------------------------ | Lexis +------------------------------------------------------------ Lexis Lexis Wilhelm (1837-1914) +------------------------------------------------------------ | Lhuilier +------------------------------------------------------------ Lhuilier Lhuilier Simon (1750-1840) +------------------------------------------------------------ | Libri +------------------------------------------------------------ Libri Libri Guglielmo (1803-1869) +------------------------------------------------------------ | Lie +------------------------------------------------------------ Lie Lie Sophus (1842-1899) +------------------------------------------------------------ | Lifshitz +------------------------------------------------------------ Lifshitz Lifshitz Evgenii (1915-1985) +------------------------------------------------------------ | Lighthill +------------------------------------------------------------ Lighthill Lighthill Sir James (1924-1998) +------------------------------------------------------------ | Lindelof +------------------------------------------------------------ Lindelof Lindel"of Ernst (1870-1946) +------------------------------------------------------------ | Linnik +------------------------------------------------------------ Linnik Linnik Yuri (1915-1972) +------------------------------------------------------------ | Lions +------------------------------------------------------------ Lions Lions Pierre-Louis +------------------------------------------------------------ | Liouville +------------------------------------------------------------ Liouville Liouville Joseph (1809-1882) +------------------------------------------------------------ | Lipschitz +------------------------------------------------------------ Lipschitz Lipschitz Rudolf (1832-1903) +------------------------------------------------------------ | Lissajous +------------------------------------------------------------ Lissajous Lissajous Jules (1822-1880) +------------------------------------------------------------ | Listing +------------------------------------------------------------ Listing Listing Johann (1808-1882) +------------------------------------------------------------ | Littlewood +------------------------------------------------------------ Littlewood Littlewood John E (1885-1977) +------------------------------------------------------------ | LittlewoodDudley +------------------------------------------------------------ LittlewoodDudley Littlewood Dudley (1903-1979) +------------------------------------------------------------ | Livsic +------------------------------------------------------------ Livsic Livsic Moshe +------------------------------------------------------------ | Llull +------------------------------------------------------------ Llull Llull Ramon (1235-1316) +------------------------------------------------------------ | Lobachevsky +------------------------------------------------------------ Lobachevsky Lobachevsky Nikolai (1792-1856) +------------------------------------------------------------ | Loewy +------------------------------------------------------------ Loewy Loewy Alfred (1873-1935) +------------------------------------------------------------ | Lopatynsky +------------------------------------------------------------ Lopatynsky Lopatynsky Yaroslav (1906-1981) +------------------------------------------------------------ | Lorentz +------------------------------------------------------------ Lorentz Lorentz Hendrik (1853-1928) +------------------------------------------------------------ | Love +------------------------------------------------------------ Love Love Augustus (1863-1940) +------------------------------------------------------------ | Lovelace +------------------------------------------------------------ Lovelace Lovelace Augusta Ada (1815-1852) +------------------------------------------------------------ | Loyd +------------------------------------------------------------ Loyd Loyd Samuel (1841-1911) +------------------------------------------------------------ | Lucas +------------------------------------------------------------ Lucas Lucas F Edouard (1842-1891) +------------------------------------------------------------ | Lueroth +------------------------------------------------------------ Lueroth Lueroth Jacob (1844-1910) +------------------------------------------------------------ | Lukacs +------------------------------------------------------------ Lukacs Lukacs Eugene (1906-1987) +------------------------------------------------------------ | Lukasiewicz +------------------------------------------------------------ Lukasiewicz Lukasiewicz Jan (1878-1956) +------------------------------------------------------------ | Luke +------------------------------------------------------------ Luke Luke Yudell (1918-1983) +------------------------------------------------------------ | Luzin +------------------------------------------------------------ Luzin Luzin Nikolai (1883-1950) +------------------------------------------------------------ | Lyapunov +------------------------------------------------------------ Lyapunov Lyapunov Aleksandr (1857-1918) +------------------------------------------------------------ | Lyndon +------------------------------------------------------------ Lyndon Lyndon Roger (1917-1988) +------------------------------------------------------------ | Meray +------------------------------------------------------------ Meray M'eray Charles (1835-1911) +------------------------------------------------------------ | Mobius +------------------------------------------------------------ Mobius M"obius August (1790-1868) +------------------------------------------------------------ | MacCullagh +------------------------------------------------------------ MacCullagh MacCullagh James (1809-1896) +------------------------------------------------------------ | MacLane +------------------------------------------------------------ MacLane MacLane Saunders +------------------------------------------------------------ | MacMahon +------------------------------------------------------------ MacMahon MacMahon Percy (1854-1929) +------------------------------------------------------------ | Macaulay +------------------------------------------------------------ Macaulay Macaulay Francis (1862-1937) +------------------------------------------------------------ | Macdonald +------------------------------------------------------------ Macdonald Macdonald Hector (1865-1935) +------------------------------------------------------------ | Maclaurin +------------------------------------------------------------ Maclaurin Maclaurin Colin (1698-1746) +------------------------------------------------------------ | Madhava +------------------------------------------------------------ Madhava Madhava Sangamagramma (1350-1425) +------------------------------------------------------------ | Al-Maghribi +------------------------------------------------------------ Al-Maghribi Maghribi Muhyi al (1220-1280) +------------------------------------------------------------ | Magnitsky +------------------------------------------------------------ Magnitsky Magnitsky Leonty (1669-1739) +------------------------------------------------------------ | Magnus +------------------------------------------------------------ Magnus Magnus Wilhelm (1907-1990) +------------------------------------------------------------ | Al-Mahani +------------------------------------------------------------ Al-Mahani Mahani Abu al (820-880) +------------------------------------------------------------ | Mahavira +------------------------------------------------------------ Mahavira Mahavira Mahavira (800-870) +------------------------------------------------------------ | MahendraSuri +------------------------------------------------------------ MahendraSuri Mahendra Suri (1340-1410) +------------------------------------------------------------ | Mahler +------------------------------------------------------------ Mahler Mahler Kurt (1903-1988) +------------------------------------------------------------ | Maior +------------------------------------------------------------ Maior Maior John (1469-1550) +------------------------------------------------------------ | Malcev +------------------------------------------------------------ Malcev Malcev Anatoly (1909-1967) +------------------------------------------------------------ | Malebranche +------------------------------------------------------------ Malebranche Malebranche Nicolas (1638-1715) +------------------------------------------------------------ | Malfatti +------------------------------------------------------------ Malfatti Malfatti Francesco (1731-1807) +------------------------------------------------------------ | Malus +------------------------------------------------------------ Malus Malus Etienne Louis (1775-1812) +------------------------------------------------------------ | Manava +------------------------------------------------------------ Manava Manava (750BC-690BC) +------------------------------------------------------------ | Mandelbrot +------------------------------------------------------------ Mandelbrot Mandelbrot Benoit +------------------------------------------------------------ | Mannheim +------------------------------------------------------------ Mannheim Mannheim Am'ed'ee (1831-1906) +------------------------------------------------------------ | Mansion +------------------------------------------------------------ Mansion Mansion Paul (1844-1919) +------------------------------------------------------------ | Mansur +------------------------------------------------------------ Mansur Mansur ibn Ali Abu +------------------------------------------------------------ | Marchenko +------------------------------------------------------------ Marchenko Marchenko Vladimir +------------------------------------------------------------ | Marcinkiewicz +------------------------------------------------------------ Marcinkiewicz Marcinkiewicz Jozef (1910-1940) +------------------------------------------------------------ | Marczewski +------------------------------------------------------------ Marczewski Marczewski Edward (1907-1976) +------------------------------------------------------------ | Margulis +------------------------------------------------------------ Margulis Margulis Gregori +------------------------------------------------------------ | Marinus +------------------------------------------------------------ Marinus Marinus of Neapolis (450-500) +------------------------------------------------------------ | Markov +------------------------------------------------------------ Markov Markov Andrei (1856-1922) +------------------------------------------------------------ | Al-Banna +------------------------------------------------------------ Al-Banna Marrakushi al (1256-1321) +------------------------------------------------------------ | Mascheroni +------------------------------------------------------------ Mascheroni Mascheroni Lorenzo (1750-1800) +------------------------------------------------------------ | Maschke +------------------------------------------------------------ Maschke Maschke Heinrich (1853-1908) +------------------------------------------------------------ | Maseres +------------------------------------------------------------ Maseres Maseres Francis (1731-1824) +------------------------------------------------------------ | Maskelyne +------------------------------------------------------------ Maskelyne Maskelyne Nevil (1732-1811) +------------------------------------------------------------ | Mason +------------------------------------------------------------ Mason Mason Max (1877-1961) +------------------------------------------------------------ | Mathews +------------------------------------------------------------ Mathews Mathews George (1861-1922) +------------------------------------------------------------ | MathieuClaude +------------------------------------------------------------ MathieuClaude Mathieu Claude-Louis (1783-1875) +------------------------------------------------------------ | MathieuEmile +------------------------------------------------------------ MathieuEmile Mathieu Emile (1835-1890) +------------------------------------------------------------ | Matsushima +------------------------------------------------------------ Matsushima Matsushima Yozo (1921-1983) +------------------------------------------------------------ | Mauchly +------------------------------------------------------------ Mauchly Mauchly John (1907-1980) +------------------------------------------------------------ | Maupertuis +------------------------------------------------------------ Maupertuis Maupertuis Pierre de (1698-1759) +------------------------------------------------------------ | Maurolico +------------------------------------------------------------ Maurolico Maurolico Francisco (1494-1575) +------------------------------------------------------------ | Maxwell +------------------------------------------------------------ Maxwell Maxwell James Clerk (1831-1879) +------------------------------------------------------------ | MayerAdolph +------------------------------------------------------------ MayerAdolph Mayer Adolph (1839-1903) +------------------------------------------------------------ | MayerTobias +------------------------------------------------------------ MayerTobias Mayer Tobias (1723-1762) +------------------------------------------------------------ | Mazur +------------------------------------------------------------ Mazur Mazur Stanislaw (1905-1981) +------------------------------------------------------------ | Mazurkiewicz +------------------------------------------------------------ Mazurkiewicz Mazurkiewicz Stefan (1888-1945) +------------------------------------------------------------ | McClintock +------------------------------------------------------------ McClintock McClintock John (1840-1916) +------------------------------------------------------------ | McDuff +------------------------------------------------------------ McDuff McDuff Margaret +------------------------------------------------------------ | McShane +------------------------------------------------------------ McShane McShane Edward (1904-1989) +------------------------------------------------------------ | Meissel +------------------------------------------------------------ Meissel Meissel Ernst (1826-1895) +------------------------------------------------------------ | Mellin +------------------------------------------------------------ Mellin Mellin Hjalmar (1854-1933) +------------------------------------------------------------ | Menabrea +------------------------------------------------------------ Menabrea Menabrea Luigi (1809-1896) +------------------------------------------------------------ | Menaechmus +------------------------------------------------------------ Menaechmus Menaechmus (380BC-320BC) +------------------------------------------------------------ | Menelaus +------------------------------------------------------------ Menelaus Menelaus of Alexandria (70AD-130) +------------------------------------------------------------ | Menger +------------------------------------------------------------ Menger Menger Karl +------------------------------------------------------------ | Mengoli +------------------------------------------------------------ Mengoli Mengoli Pietro (1626-1686) +------------------------------------------------------------ | Menshov +------------------------------------------------------------ Menshov Menshov Dmitrii (1892-1988) +------------------------------------------------------------ | MercatorGerardus +------------------------------------------------------------ MercatorGerardus Mercator Gerardus (1512-1592) +------------------------------------------------------------ | MercatorNicolaus +------------------------------------------------------------ MercatorNicolaus Mercator Nicolaus (1620-1687) +------------------------------------------------------------ | Mercer +------------------------------------------------------------ Mercer Mercer James (1883-1932) +------------------------------------------------------------ | Merrifield +------------------------------------------------------------ Merrifield Merrifield Charles (1827-1884) +------------------------------------------------------------ | Merrill +------------------------------------------------------------ Merrill Merrill Winifred (1862-1951) +------------------------------------------------------------ | Mersenne +------------------------------------------------------------ Mersenne Mersenne Marin (1588-1648) +------------------------------------------------------------ | Mertens +------------------------------------------------------------ Mertens Mertens Franz (1840-1927) +------------------------------------------------------------ | Meshchersky +------------------------------------------------------------ Meshchersky Meshchersky Ivan (1859-1935) +------------------------------------------------------------ | Meyer +------------------------------------------------------------ Meyer Meyer Wilhelm (1856-1934) +------------------------------------------------------------ | Miller +------------------------------------------------------------ Miller Miller George (1863-1951) +------------------------------------------------------------ | Milne +------------------------------------------------------------ Milne Milne Edward (1896-1950) +------------------------------------------------------------ | Milnor +------------------------------------------------------------ Milnor Milnor John +------------------------------------------------------------ | Minding +------------------------------------------------------------ Minding Minding Ferdinand (1806-1885) +------------------------------------------------------------ | Mineur +------------------------------------------------------------ Mineur Mineur Henri (1899-1954) +------------------------------------------------------------ | Minkowski +------------------------------------------------------------ Minkowski Minkowski Hermann (1864-1909) +------------------------------------------------------------ | Mirsky +------------------------------------------------------------ Mirsky Mirsky Leon (1918-1983) +------------------------------------------------------------ | Mittag-Leffler +------------------------------------------------------------ Mittag-Leffler Mittag-Leffler G"osta (1846-1927) +------------------------------------------------------------ | Mohr +------------------------------------------------------------ Mohr Mohr Georg (1640-1697) +------------------------------------------------------------ | DeMoivre +------------------------------------------------------------ DeMoivre Moivre Abraham de (1667-1754) +------------------------------------------------------------ | Molin +------------------------------------------------------------ Molin Molin Fedor (1861-1941) +------------------------------------------------------------ | Monge +------------------------------------------------------------ Monge Monge Gaspard (1746-1818) +------------------------------------------------------------ | Monte +------------------------------------------------------------ Monte Monte Guidobaldo del (1545-1607) +------------------------------------------------------------ | Montel +------------------------------------------------------------ Montel Montel Paul (1876-1975) +------------------------------------------------------------ | Montmort +------------------------------------------------------------ Montmort Montmort Pierre R'emond de (1678-1719) +------------------------------------------------------------ | Montucla +------------------------------------------------------------ Montucla Montucla Jean (1725-1799) +------------------------------------------------------------ | MooreJonas +------------------------------------------------------------ MooreJonas Moore Jonas (1627-1679) +------------------------------------------------------------ | MooreRobert +------------------------------------------------------------ MooreRobert Moore Robert (1882-1974) +------------------------------------------------------------ | MooreEliakim +------------------------------------------------------------ MooreEliakim Moore Eliakim (1862-1932) +------------------------------------------------------------ | Morawetz +------------------------------------------------------------ Morawetz Morawetz Cathleen +------------------------------------------------------------ | Mordell +------------------------------------------------------------ Mordell Mordell Louis (1888-1972) +------------------------------------------------------------ | DeMorgan +------------------------------------------------------------ DeMorgan Morgan Augustus De (1806-1871) +------------------------------------------------------------ | Mori +------------------------------------------------------------ Mori Mori Shigefumi +------------------------------------------------------------ | Morin +------------------------------------------------------------ Morin Morin Arthur (1795-1880) +------------------------------------------------------------ | MorinJean-Baptiste +------------------------------------------------------------ MorinJean-Baptiste Morin Jean-Baptiste +------------------------------------------------------------ | Morley +------------------------------------------------------------ Morley Morley Frank (1860-1937) +------------------------------------------------------------ | Morse +------------------------------------------------------------ Morse Morse Harald Marston (1892-1977) +------------------------------------------------------------ | Mostowski +------------------------------------------------------------ Mostowski Mostowski Andrzej (1913-1975) +------------------------------------------------------------ | Motzkin +------------------------------------------------------------ Motzkin Motzkin Theodore (1908-1970) +------------------------------------------------------------ | Moufang +------------------------------------------------------------ Moufang Moufang Ruth (1905-1977) +------------------------------------------------------------ | Mouton +------------------------------------------------------------ Mouton Mouton Gabriel (1618-1694) +------------------------------------------------------------ | Muir +------------------------------------------------------------ Muir Muir Thomas (1844-1934) +------------------------------------------------------------ | Mumford +------------------------------------------------------------ Mumford Mumford David +------------------------------------------------------------ | Mydorge +------------------------------------------------------------ Mydorge Mydorge Claude (1585-1647) +------------------------------------------------------------ | Mytropolshy +------------------------------------------------------------ Mytropolshy Mytropolshy Yurii +------------------------------------------------------------ | Naimark +------------------------------------------------------------ Naimark Naimark Mark (1909-1978) +------------------------------------------------------------ | Napier +------------------------------------------------------------ Napier Napier John (1550-1617) +------------------------------------------------------------ | Narayana +------------------------------------------------------------ Narayana Narayana Pandit (1340-1400) +------------------------------------------------------------ | Al-Nasawi +------------------------------------------------------------ Al-Nasawi Nasawi Abu al (1010-1075) +------------------------------------------------------------ | Nash +------------------------------------------------------------ Nash Nash John +------------------------------------------------------------ | Navier +------------------------------------------------------------ Navier Navier Claude (1785-1836) +------------------------------------------------------------ | Al-Nayrizi +------------------------------------------------------------ Al-Nayrizi Nayrizi Abu'l al (875-940) +------------------------------------------------------------ | Neile +------------------------------------------------------------ Neile Neile William (1637-1670) +------------------------------------------------------------ | Nekrasov +------------------------------------------------------------ Nekrasov Nekrasov Aleksandr (1883-1957) +------------------------------------------------------------ | Netto +------------------------------------------------------------ Netto Netto Eugen (1848-1919) +------------------------------------------------------------ | Neuberg +------------------------------------------------------------ Neuberg Neuberg Joseph (1840-1926) +------------------------------------------------------------ | Neugebauer +------------------------------------------------------------ Neugebauer Neugebauer Otto (1899-1990) +------------------------------------------------------------ | NeumannHanna +------------------------------------------------------------ NeumannHanna Neumann Hanna (1914-1971) +------------------------------------------------------------ | NeumannCarl +------------------------------------------------------------ NeumannCarl Neumann Carl Gottfried (1832-1925) +------------------------------------------------------------ | NeumannFranz +------------------------------------------------------------ NeumannFranz Neumann Franz Ernst (1798-1895) +------------------------------------------------------------ | NeumannBernhard +------------------------------------------------------------ NeumannBernhard Neumann Bernhard +------------------------------------------------------------ | Nevanlinna +------------------------------------------------------------ Nevanlinna Nevanlinna Rolf (1895-1980) +------------------------------------------------------------ | Newcomb +------------------------------------------------------------ Newcomb Newcomb Simon (1835-1909) +------------------------------------------------------------ | Newman +------------------------------------------------------------ Newman Newman Maxwell (1897-1984) +------------------------------------------------------------ | Newton +------------------------------------------------------------ Newton Newton Sir Isaac (1643-1727) +------------------------------------------------------------ | Neyman +------------------------------------------------------------ Neyman Neyman Jerzy (1894-1981) +------------------------------------------------------------ | Nicolson +------------------------------------------------------------ Nicolson Nicolson Phyllis (1917-1968) +------------------------------------------------------------ | Nicomachus +------------------------------------------------------------ Nicomachus Nicomachus of Gerasa (60AD-120) +------------------------------------------------------------ | Nicomedes +------------------------------------------------------------ Nicomedes Nicomedes (280BC-210BC) +------------------------------------------------------------ | Nielsen +------------------------------------------------------------ Nielsen Nielsen Niels (1865-1931) +------------------------------------------------------------ | NielsenJakob +------------------------------------------------------------ NielsenJakob Nielsen Jacob +------------------------------------------------------------ | Nightingale +------------------------------------------------------------ Nightingale Nightingale Florence (1820-1910) +------------------------------------------------------------ | Nilakantha +------------------------------------------------------------ Nilakantha Nilakantha Somayaji (1444-1544) +------------------------------------------------------------ | Niven +------------------------------------------------------------ Niven Niven William (1843-1917) +------------------------------------------------------------ | NoetherMax +------------------------------------------------------------ NoetherMax Noether Max (1844-1921) +------------------------------------------------------------ | NoetherEmmy +------------------------------------------------------------ NoetherEmmy Noether Emmy (1882-1935) +------------------------------------------------------------ | Novikov +------------------------------------------------------------ Novikov Novikov Petr (1901-1975) +------------------------------------------------------------ | NovikovSergi +------------------------------------------------------------ NovikovSergi Novikov Sergi +------------------------------------------------------------ | Oenopides +------------------------------------------------------------ Oenopides Oenopides of Chios (490BC-420BC) +------------------------------------------------------------ | Ohm +------------------------------------------------------------ Ohm Ohm Georg Simon (1789-1854) +------------------------------------------------------------ | Oka +------------------------------------------------------------ Oka Oka Kiyoshi (1901-1978) +------------------------------------------------------------ | Olivier +------------------------------------------------------------ Olivier Olivier Th'eodore (1793-1853) +------------------------------------------------------------ | Khayyam +------------------------------------------------------------ Khayyam Omar Khayyam (1048-1122) +------------------------------------------------------------ | Oresme +------------------------------------------------------------ Oresme Oresme Nicole d' (1323-1382) +------------------------------------------------------------ | Orlicz +------------------------------------------------------------ Orlicz Orlicz Wladyslaw (1903-1990) +------------------------------------------------------------ | Ortega +------------------------------------------------------------ Ortega Ortega Juan de (1480-1568) +------------------------------------------------------------ | Osgood +------------------------------------------------------------ Osgood Osgood William (1864-1943) +------------------------------------------------------------ | Osipovsky +------------------------------------------------------------ Osipovsky Osipovsky Timofei (1765-1832) +------------------------------------------------------------ | Ostrogradski +------------------------------------------------------------ Ostrogradski Ostrogradski Mikhail (1801-1862) +------------------------------------------------------------ | Ostrowski +------------------------------------------------------------ Ostrowski Ostrowski Alexander (1893-1986) +------------------------------------------------------------ | Oughtred +------------------------------------------------------------ Oughtred Oughtred William (1574-1660) +------------------------------------------------------------ | D'Ovidio +------------------------------------------------------------ D'Ovidio Ovidio Enrico D' (1842-1933) +------------------------------------------------------------ | Ozanam +------------------------------------------------------------ Ozanam Ozanam Jacques (1640-1717) +------------------------------------------------------------ | Peres +------------------------------------------------------------ Peres P'er`es Joseph (1890-1962) +------------------------------------------------------------ | Peter +------------------------------------------------------------ Peter P'eter R'ozsa (1905-1977) +------------------------------------------------------------ | Polya +------------------------------------------------------------ Polya P'olya George (1887-1985) +------------------------------------------------------------ | Pacioli +------------------------------------------------------------ Pacioli Pacioli Luca (1445-1517) +------------------------------------------------------------ | Pade +------------------------------------------------------------ Pade Pad'e Henri (1863-1953) +------------------------------------------------------------ | Padoa +------------------------------------------------------------ Padoa Padoa Alessandro (1868-1937) +------------------------------------------------------------ | LePaige +------------------------------------------------------------ LePaige Paige Constantin Le (1852-1929) +------------------------------------------------------------ | Painleve +------------------------------------------------------------ Painleve Painlev'e Paul (1863-1933) +------------------------------------------------------------ | Paley +------------------------------------------------------------ Paley Paley Raymond (1907-1933) +------------------------------------------------------------ | Paman +------------------------------------------------------------ Paman Paman Roger (1710-1748) +------------------------------------------------------------ | Panini +------------------------------------------------------------ Panini Panini (520BC-460BC) +------------------------------------------------------------ | Papin +------------------------------------------------------------ Papin Papin Denis (1647-1712) +------------------------------------------------------------ | Pappus +------------------------------------------------------------ Pappus Pappus of Alexandria (290-350) +------------------------------------------------------------ | Pars +------------------------------------------------------------ Pars Pars Leopold (1896-1985) +------------------------------------------------------------ | Parseval +------------------------------------------------------------ Parseval Parseval des Chees M-A (1755-1836) +------------------------------------------------------------ | Pascal +------------------------------------------------------------ Pascal Pascal Blaise (1623-1662) +------------------------------------------------------------ | PascalEtienne +------------------------------------------------------------ PascalEtienne Pascal Etienne (1588-1640) +------------------------------------------------------------ | Pasch +------------------------------------------------------------ Pasch Pasch Moritz (1843-1930) +------------------------------------------------------------ | Patodi +------------------------------------------------------------ Patodi Patodi Vijay (1945-1976) +------------------------------------------------------------ | Pauli +------------------------------------------------------------ Pauli Pauli Wolfgang (1900-1958) +------------------------------------------------------------ | Peacock +------------------------------------------------------------ Peacock Peacock George (1791-1858) +------------------------------------------------------------ | Peano +------------------------------------------------------------ Peano Peano Giuseppe (1858-1932) +------------------------------------------------------------ | Pearson +------------------------------------------------------------ Pearson Pearson Karl (1857-1936) +------------------------------------------------------------ | PearsonEgon +------------------------------------------------------------ PearsonEgon Pearson Egon (1895-1980) +------------------------------------------------------------ | PeirceBenjamin +------------------------------------------------------------ PeirceBenjamin Peirce Benjamin (1809-1880) +------------------------------------------------------------ | PeirceCharles +------------------------------------------------------------ PeirceCharles Peirce Charles (1839-1914) +------------------------------------------------------------ | Pell +------------------------------------------------------------ Pell Pell John (1611-1685) +------------------------------------------------------------ | Penney +------------------------------------------------------------ Penney Penney Bill (1909-1991) +------------------------------------------------------------ | Perron +------------------------------------------------------------ Perron Perron Oskar (1880-1975) +------------------------------------------------------------ | Perseus +------------------------------------------------------------ Perseus Perseus (180BC-120BC) +------------------------------------------------------------ | Petersen +------------------------------------------------------------ Petersen Petersen Julius (1839-1910) +------------------------------------------------------------ | Peterson +------------------------------------------------------------ Peterson Peterson Karl (1828-1881) +------------------------------------------------------------ | Petit +------------------------------------------------------------ Petit Petit Al'exis (1791-1820) +------------------------------------------------------------ | Petrovsky +------------------------------------------------------------ Petrovsky Petrovsky Ivan (1901-1973) +------------------------------------------------------------ | Petryshyn +------------------------------------------------------------ Petryshyn Petryshyn Volodymyr +------------------------------------------------------------ | Petzval +------------------------------------------------------------ Petzval Petzval J'ozeph (1807-1891) +------------------------------------------------------------ | Peurbach +------------------------------------------------------------ Peurbach Peurbach Georg (1423-1461) +------------------------------------------------------------ | Pfaff +------------------------------------------------------------ Pfaff Pfaff Johann (1765-1825) +------------------------------------------------------------ | Pfeiffer +------------------------------------------------------------ Pfeiffer Pfeiffer Georgii (1872-1946) +------------------------------------------------------------ | Philon +------------------------------------------------------------ Philon Philon of Byzantium (280BC-220BC) +------------------------------------------------------------ | PicardEmile +------------------------------------------------------------ PicardEmile Picard Emile (1856-1941) +------------------------------------------------------------ | PicardJean +------------------------------------------------------------ PicardJean Picard Jean (1620-1682) +------------------------------------------------------------ | Pieri +------------------------------------------------------------ Pieri Pieri Mario (1860-1913) +------------------------------------------------------------ | Francesca +------------------------------------------------------------ Francesca Piero della Francesca (1412-1492) +------------------------------------------------------------ | Pillai +------------------------------------------------------------ Pillai Pillai K C Sreedharan (1920-1980) +------------------------------------------------------------ | Pincherle +------------------------------------------------------------ Pincherle Pincherle Salvatore (1853-1936) +------------------------------------------------------------ | Fibonacci +------------------------------------------------------------ Fibonacci Pisano Leonardo Fibonacci (1170-1250) +------------------------------------------------------------ | Pitiscus +------------------------------------------------------------ Pitiscus Pitiscus Bartholomeo (1561-1613) +------------------------------------------------------------ | Plucker +------------------------------------------------------------ Plucker Pl"ucker Julius (1801-1868) +------------------------------------------------------------ | Plana +------------------------------------------------------------ Plana Plana Giovanni (1781-1864) +------------------------------------------------------------ | Planck +------------------------------------------------------------ Planck Planck Max (1858-1947) +------------------------------------------------------------ | Plateau +------------------------------------------------------------ Plateau Plateau Joseph (1801-1883) +------------------------------------------------------------ | Plato +------------------------------------------------------------ Plato Plato (428BC-347BC) +------------------------------------------------------------ | Playfair +------------------------------------------------------------ Playfair Playfair John (1748-1819) +------------------------------------------------------------ | Plessner +------------------------------------------------------------ Plessner Plessner Abraham +------------------------------------------------------------ | Poincare +------------------------------------------------------------ Poincare Poincar'e J Henri (1854-1912) +------------------------------------------------------------ | Poinsot +------------------------------------------------------------ Poinsot Poinsot Louis (1777-1859) +------------------------------------------------------------ | Poisson +------------------------------------------------------------ Poisson Poisson Sim'eon (1781-1840) +------------------------------------------------------------ | Poleni +------------------------------------------------------------ Poleni Poleni Giovanni (1683-1761) +------------------------------------------------------------ | Polozii +------------------------------------------------------------ Polozii Polozii Georgii (1914-1968) +------------------------------------------------------------ | Poncelet +------------------------------------------------------------ Poncelet Poncelet Jean-Victor (1788-1867) +------------------------------------------------------------ | Pontryagin +------------------------------------------------------------ Pontryagin Pontryagin Lev (1908-1988) +------------------------------------------------------------ | Poretsky +------------------------------------------------------------ Poretsky Poretsky Platon (1846-1907) +------------------------------------------------------------ | Porphyry +------------------------------------------------------------ Porphyry Porphyry of Malchus (233-309) +------------------------------------------------------------ | Porta +------------------------------------------------------------ Porta Porta Giambattista Della (1535-1615) +------------------------------------------------------------ | Posidonius +------------------------------------------------------------ Posidonius Posidonius of Rhodes (135BC-51BC) +------------------------------------------------------------ | Post +------------------------------------------------------------ Post Post Emil (1897-1954) +------------------------------------------------------------ | Potapov +------------------------------------------------------------ Potapov Potapov Vladimir (1914-1980) +------------------------------------------------------------ | Prufer +------------------------------------------------------------ Prufer Pr"ufer Heinz (1896-1934) +------------------------------------------------------------ | Pratt +------------------------------------------------------------ Pratt Pratt John (1809-1871) +------------------------------------------------------------ | Pringsheim +------------------------------------------------------------ Pringsheim Pringsheim Alfred (1850-1941) +------------------------------------------------------------ | Privalov +------------------------------------------------------------ Privalov Privalov Ivan (1891-1941) +------------------------------------------------------------ | PrivatdeMolieres +------------------------------------------------------------ PrivatdeMolieres Privat de Moli`eres Joseph (1677-1742) +------------------------------------------------------------ | Proclus +------------------------------------------------------------ Proclus Proclus Diadochus (411-485) +------------------------------------------------------------ | DeProny +------------------------------------------------------------ DeProny Prony Gaspard de (1755-1839) +------------------------------------------------------------ | Prthudakasvami +------------------------------------------------------------ Prthudakasvami Prthudakasvami (830-890) +------------------------------------------------------------ | Ptolemy +------------------------------------------------------------ Ptolemy Ptolemy (85AD-165) +------------------------------------------------------------ | Puiseux +------------------------------------------------------------ Puiseux Puiseux Victor (1820-1883) +------------------------------------------------------------ | Puissant +------------------------------------------------------------ Puissant Puissant Louis (1769-1943) +------------------------------------------------------------ | Pythagoras +------------------------------------------------------------ Pythagoras Pythagoras of Samos (580BC-520BC) +------------------------------------------------------------ | Al-Qalasadi +------------------------------------------------------------ Al-Qalasadi Qalasadi Abu'l al (1412-1486) +------------------------------------------------------------ | Quetelet +------------------------------------------------------------ Quetelet Quetelet Adolphe (1796-1874) +------------------------------------------------------------ | Al-Quhi +------------------------------------------------------------ Al-Quhi Quhi Abu al +------------------------------------------------------------ | Quillen +------------------------------------------------------------ Quillen Quillen Daniel +------------------------------------------------------------ | Quine +------------------------------------------------------------ Quine Quine Willard Van (1908-2000) +------------------------------------------------------------ | Renyi +------------------------------------------------------------ Renyi R'enyi Alfr'ed (1921-1970) +------------------------------------------------------------ | Rado +------------------------------------------------------------ Rado Rad'o Tibor (1895-1965) +------------------------------------------------------------ | Rademacher +------------------------------------------------------------ Rademacher Rademacher Hans (1892-1969) +------------------------------------------------------------ | RadoRichard +------------------------------------------------------------ RadoRichard Rado Richard (1906-1989) +------------------------------------------------------------ | Radon +------------------------------------------------------------ Radon Radon Johann (1887-1956) +------------------------------------------------------------ | Rahn +------------------------------------------------------------ Rahn Rahn Johann (1622-1676) +------------------------------------------------------------ | Rajagopal +------------------------------------------------------------ Rajagopal Rajagopal Cadambathur (1903-1978) +------------------------------------------------------------ | Ramanujam +------------------------------------------------------------ Ramanujam Ramanujam Chidambaram (1938-1974) +------------------------------------------------------------ | Ramanujan +------------------------------------------------------------ Ramanujan Ramanujan Srinivasa (1887-1920) +------------------------------------------------------------ | Ramsden +------------------------------------------------------------ Ramsden Ramsden Jesse (1735-1800) +------------------------------------------------------------ | Ramsey +------------------------------------------------------------ Ramsey Ramsey Frank (1903-1930) +------------------------------------------------------------ | Ramus +------------------------------------------------------------ Ramus Ramus Peter (1515-1572) +------------------------------------------------------------ | Rankin +------------------------------------------------------------ Rankin Rankin Robert (1915-2001) +------------------------------------------------------------ | Rankine +------------------------------------------------------------ Rankine Rankine William (1820-1872) +------------------------------------------------------------ | Raphson +------------------------------------------------------------ Raphson Raphson Joseph (1648-1715) +------------------------------------------------------------ | Rasiowa +------------------------------------------------------------ Rasiowa Rasiowa Helena (1917-1994) +------------------------------------------------------------ | Razmadze +------------------------------------------------------------ Razmadze Razmadze Andrei (1889-1929) +------------------------------------------------------------ | Recorde +------------------------------------------------------------ Recorde Recorde Robert (1510-1558) +------------------------------------------------------------ | Rees +------------------------------------------------------------ Rees Rees Mina (1902-1997) +------------------------------------------------------------ | Regiomontanus +------------------------------------------------------------ Regiomontanus Regiomontanus Johann (1436-1476) +------------------------------------------------------------ | Reichenbach +------------------------------------------------------------ Reichenbach Reichenbach Hans (1891-1953) +------------------------------------------------------------ | Reidemeister +------------------------------------------------------------ Reidemeister Reidemeister Kurt (1893-1971) +------------------------------------------------------------ | Reiner +------------------------------------------------------------ Reiner Reiner Irving (1924-1986) +------------------------------------------------------------ | Remak +------------------------------------------------------------ Remak Remak Robert (1888-1942) +------------------------------------------------------------ | Remez +------------------------------------------------------------ Remez Remez Evgeny (1896-1975) +------------------------------------------------------------ | ReyPastor +------------------------------------------------------------ ReyPastor Rey Pastor Julio (1888-1962) +------------------------------------------------------------ | Reye +------------------------------------------------------------ Reye Reye Theodor (1838-1919) +------------------------------------------------------------ | DuBois-Reymond +------------------------------------------------------------ DuBois-Reymond Reymond Paul du Bois- (1831-1889) +------------------------------------------------------------ | Reynaud +------------------------------------------------------------ Reynaud Reynaud Antoine-Andr'e (1771-1844) +------------------------------------------------------------ | Reyneau +------------------------------------------------------------ Reyneau Reyneau Charles (1656-1728) +------------------------------------------------------------ | Reynolds +------------------------------------------------------------ Reynolds Reynolds Osborne (1842-1912) +------------------------------------------------------------ | DeRham +------------------------------------------------------------ DeRham Rham Georges de (1903-1990) +------------------------------------------------------------ | Rheticus +------------------------------------------------------------ Rheticus Rheticus Georg Joachim (1514-1574) +------------------------------------------------------------ | Riccati +------------------------------------------------------------ Riccati Riccati Jacopo (1676-1754) +------------------------------------------------------------ | RiccatiVincenzo +------------------------------------------------------------ RiccatiVincenzo Riccati Vincenzo (1707-1775) +------------------------------------------------------------ | RicciMatteo +------------------------------------------------------------ RicciMatteo Ricci Matteo (1552-1610) +------------------------------------------------------------ | Ricci +------------------------------------------------------------ Ricci Ricci Michelangelo (1619-1682) +------------------------------------------------------------ | Ricci-Curbastro +------------------------------------------------------------ Ricci-Curbastro Ricci-Curbastro Georgorio (1853-1925) +------------------------------------------------------------ | RichardLouis +------------------------------------------------------------ RichardLouis Richard Louis (1795-1849) +------------------------------------------------------------ | RichardJules +------------------------------------------------------------ RichardJules Richard Jules (1862-1956) +------------------------------------------------------------ | Richardson +------------------------------------------------------------ Richardson Richardson Lewis (1881-1953) +------------------------------------------------------------ | Richer +------------------------------------------------------------ Richer Richer Jean (1630-1696) +------------------------------------------------------------ | Richmond +------------------------------------------------------------ Richmond Richmond Herbert (1863-1948) +------------------------------------------------------------ | Riemann +------------------------------------------------------------ Riemann Riemann G F Bernhard (1826-1866) +------------------------------------------------------------ | Ries +------------------------------------------------------------ Ries Ries Adam (1492-1559) +------------------------------------------------------------ | RieszMarcel +------------------------------------------------------------ RieszMarcel Riesz Marcel (1886-1969) +------------------------------------------------------------ | Riesz +------------------------------------------------------------ Riesz Riesz Frigyes (1880-1956) +------------------------------------------------------------ | Ringrose +------------------------------------------------------------ Ringrose Ringrose John +------------------------------------------------------------ | Roberts +------------------------------------------------------------ Roberts Roberts Samuel (1827-1913) +------------------------------------------------------------ | Roberval +------------------------------------------------------------ Roberval Roberval Gilles de (1602-1675) +------------------------------------------------------------ | Robins +------------------------------------------------------------ Robins Robins Benjamin (1707-1751) +------------------------------------------------------------ | RobinsonJulia +------------------------------------------------------------ RobinsonJulia Robinson Julia Bowman (1919-1985) +------------------------------------------------------------ | Robinson +------------------------------------------------------------ Robinson Robinson Abraham (1918-1974) +------------------------------------------------------------ | Rocard +------------------------------------------------------------ Rocard Rocard Yves-Andr'e (1903-1992) +------------------------------------------------------------ | LaRoche +------------------------------------------------------------ LaRoche Roche Estienne de La (1470-1530) +------------------------------------------------------------ | Rogers +------------------------------------------------------------ Rogers Rogers Ambrose +------------------------------------------------------------ | Rohn +------------------------------------------------------------ Rohn Rohn Karl (1855-1920) +------------------------------------------------------------ | Rolle +------------------------------------------------------------ Rolle Rolle Michel (1652-1719) +------------------------------------------------------------ | Rosanes +------------------------------------------------------------ Rosanes Rosanes Jakob (1842-1922) +------------------------------------------------------------ | Rosenhain +------------------------------------------------------------ Rosenhain Rosenhain Johann (1816-1887) +------------------------------------------------------------ | Rota +------------------------------------------------------------ Rota Rota Gian-Carlo (1932-1999) +------------------------------------------------------------ | Roth +------------------------------------------------------------ Roth Roth Leonard (1904-1968) +------------------------------------------------------------ | RothKlaus +------------------------------------------------------------ RothKlaus Roth Klaus +------------------------------------------------------------ | Routh +------------------------------------------------------------ Routh Routh Edward (1831-1907) +------------------------------------------------------------ | Rudio +------------------------------------------------------------ Rudio Rudio Ferdinand (1856-1929) +------------------------------------------------------------ | Rudolff +------------------------------------------------------------ Rudolff Rudolff Christoff (1499-1545) +------------------------------------------------------------ | Ruffini +------------------------------------------------------------ Ruffini Ruffini Paolo (1765-1822) +------------------------------------------------------------ | Runge +------------------------------------------------------------ Runge Runge Carle (1856-1927) +------------------------------------------------------------ | RussellScott +------------------------------------------------------------ RussellScott Russell John (1808-1882) +------------------------------------------------------------ | Russell +------------------------------------------------------------ Russell Russell Bertrand (1872-1970) +------------------------------------------------------------ | Rutherford +------------------------------------------------------------ Rutherford Rutherford Daniel E (1906-1966) +------------------------------------------------------------ | Rydberg +------------------------------------------------------------ Rydberg Rydberg Johannes (1854-1919) +------------------------------------------------------------ | Saccheri +------------------------------------------------------------ Saccheri Saccheri Giovanni (1667-1733) +------------------------------------------------------------ | Sacrobosco +------------------------------------------------------------ Sacrobosco Sacrobosco Johannes de (1195-1256) +------------------------------------------------------------ | Saks +------------------------------------------------------------ Saks Saks Stanislaw (1897-1942) +------------------------------------------------------------ | Nunez +------------------------------------------------------------ Nunez Salaciense Pedro Nunez (1502-1587) +------------------------------------------------------------ | Salem +------------------------------------------------------------ Salem Salem Rapha"el (1898-1963) +------------------------------------------------------------ | Salmon +------------------------------------------------------------ Salmon Salmon George (1819-1904) +------------------------------------------------------------ | Al-Samarqandi +------------------------------------------------------------ Al-Samarqandi Samarqandi Shams al (1250-1310) +------------------------------------------------------------ | Al-Samawal +------------------------------------------------------------ Al-Samawal Samawal Ibn al (1130-1180) +------------------------------------------------------------ | Samoilenko +------------------------------------------------------------ Samoilenko Samoilenko Anatoly +------------------------------------------------------------ | Sang +------------------------------------------------------------ Sang Sang Edward (1805-1890) +------------------------------------------------------------ | Sankara +------------------------------------------------------------ Sankara Sankara Narayana (840-900) +------------------------------------------------------------ | Sasaki +------------------------------------------------------------ Sasaki Sasaki Shigeo +------------------------------------------------------------ | Saurin +------------------------------------------------------------ Saurin Saurin Joseph (1659-1737) +------------------------------------------------------------ | Savage +------------------------------------------------------------ Savage Savage Leonard (1917-1971) +------------------------------------------------------------ | Savart +------------------------------------------------------------ Savart Savart Felix (1791-1841) +------------------------------------------------------------ | Savary +------------------------------------------------------------ Savary Savary F'elix (1797-1841) +------------------------------------------------------------ | Abraham +------------------------------------------------------------ Abraham Savasorda (1070-1130) +------------------------------------------------------------ | Savile +------------------------------------------------------------ Savile Savile Sir Henry (1549-1622) +------------------------------------------------------------ | Schonflies +------------------------------------------------------------ Schonflies Sch"onflies Arthur (1853-1928) +------------------------------------------------------------ | Schatten +------------------------------------------------------------ Schatten Schatten Robert (1911-1977) +------------------------------------------------------------ | Schauder +------------------------------------------------------------ Schauder Schauder Juliusz (1899-1943) +------------------------------------------------------------ | Scheffe +------------------------------------------------------------ Scheffe Scheff'e Henry (1907-1977) +------------------------------------------------------------ | Scheffers +------------------------------------------------------------ Scheffers Scheffers Georg (1866-1945) +------------------------------------------------------------ | Schickard +------------------------------------------------------------ Schickard Schickard Wilhelm (1592-1635) +------------------------------------------------------------ | Schlafli +------------------------------------------------------------ Schlafli Schl"afli Ludwig (1814-1895) +------------------------------------------------------------ | Schlomilch +------------------------------------------------------------ Schlomilch Schl"omilch Oscar (1823-1901) +------------------------------------------------------------ | Schmidt +------------------------------------------------------------ Schmidt Schmidt Erhard (1876-1959) +------------------------------------------------------------ | Schoenberg +------------------------------------------------------------ Schoenberg Schoenberg Isaac (1903-1990) +------------------------------------------------------------ | Schottky +------------------------------------------------------------ Schottky Schottky Friedrich (1851-1935) +------------------------------------------------------------ | Schoute +------------------------------------------------------------ Schoute Schoute Pieter (1846-1923) +------------------------------------------------------------ | Schouten +------------------------------------------------------------ Schouten Schouten Jan (1883-1971) +------------------------------------------------------------ | Schroder +------------------------------------------------------------ Schroder Schr"oder Ernst (1841-1902) +------------------------------------------------------------ | Schrodinger +------------------------------------------------------------ Schrodinger Schr"odinger Erwin (1887-1961) +------------------------------------------------------------ | Schreier +------------------------------------------------------------ Schreier Schreier Otto (1901-1929) +------------------------------------------------------------ | Schroeter +------------------------------------------------------------ Schroeter Schroeter Heinrich (1829-1892) +------------------------------------------------------------ | Schubert +------------------------------------------------------------ Schubert Schubert Hermann (1848-1911) +------------------------------------------------------------ | Schur +------------------------------------------------------------ Schur Schur Issai (1875-1941) +------------------------------------------------------------ | Schwartz +------------------------------------------------------------ Schwartz Schwartz Laurent +------------------------------------------------------------ | SchwarzStefan +------------------------------------------------------------ SchwarzStefan Schwarz Stefan (1914-1996) +------------------------------------------------------------ | Schwarz +------------------------------------------------------------ Schwarz Schwarz Herman (1843-1921) +------------------------------------------------------------ | Schwarzschild +------------------------------------------------------------ Schwarzschild Schwarzschild Karl (1873-1916) +------------------------------------------------------------ | Schwinger +------------------------------------------------------------ Schwinger Schwinger Julian (1918-1994) +------------------------------------------------------------ | Scott +------------------------------------------------------------ Scott Scott Charlotte (1858-1931) +------------------------------------------------------------ | Macintyre +------------------------------------------------------------ Macintyre Scott Sheila (1910-1960) +------------------------------------------------------------ | SegreBeniamino +------------------------------------------------------------ SegreBeniamino Segre Beniamino (1903-1977) +------------------------------------------------------------ | SegreCorrado +------------------------------------------------------------ SegreCorrado Segre Corrado (1863-1924) +------------------------------------------------------------ | Seifert +------------------------------------------------------------ Seifert Seifert Karl (1907-1996) +------------------------------------------------------------ | Selberg +------------------------------------------------------------ Selberg Selberg Atle +------------------------------------------------------------ | Selten +------------------------------------------------------------ Selten Selten Reinhard +------------------------------------------------------------ | Semple +------------------------------------------------------------ Semple Semple Jack (1904-1985) +------------------------------------------------------------ | Serenus +------------------------------------------------------------ Serenus Serenus (300-360) +------------------------------------------------------------ | Serre +------------------------------------------------------------ Serre Serre Jean-Pierre +------------------------------------------------------------ | Serret +------------------------------------------------------------ Serret Serret Joseph (1819-1885) +------------------------------------------------------------ | Servois +------------------------------------------------------------ Servois Servois Francois (1768-1847) +------------------------------------------------------------ | Severi +------------------------------------------------------------ Severi Severi Francesco (1879-1961) +------------------------------------------------------------ | Shanks +------------------------------------------------------------ Shanks Shanks William (1812-1882) +------------------------------------------------------------ | Shannon +------------------------------------------------------------ Shannon Shannon Claude (1916-2001) +------------------------------------------------------------ | Sharkovsky +------------------------------------------------------------ Sharkovsky Sharkovsky Oleksandr +------------------------------------------------------------ | Shatunovsky +------------------------------------------------------------ Shatunovsky Shatunovsky Samuil (1859-1929) +------------------------------------------------------------ | Shen +------------------------------------------------------------ Shen Shen Kua (1031-1095) +------------------------------------------------------------ | Shewhart +------------------------------------------------------------ Shewhart Shewhart Walter (1891-1967) +------------------------------------------------------------ | Shields +------------------------------------------------------------ Shields Shields Allen (1927-1989) +------------------------------------------------------------ | Shnirelman +------------------------------------------------------------ Shnirelman Shnirelman Lev (1905-1938) +------------------------------------------------------------ | Shoda +------------------------------------------------------------ Shoda Shoda Kenjiro (1902-1977) +------------------------------------------------------------ | Shtokalo +------------------------------------------------------------ Shtokalo Shtokalo Josif (1897-1987) +------------------------------------------------------------ | Siacci +------------------------------------------------------------ Siacci Siacci Francesco (1839-1907) +------------------------------------------------------------ | Siegel +------------------------------------------------------------ Siegel Siegel Carl (1896-1981) +------------------------------------------------------------ | Sierpinski +------------------------------------------------------------ Sierpinski Sierpinski Waclaw (1882-1969) +------------------------------------------------------------ | Siguenza +------------------------------------------------------------ Siguenza Siguenza y Gongora (1645-1700) +------------------------------------------------------------ | Al-Sijzi +------------------------------------------------------------ Al-Sijzi Sijzi Abu al +------------------------------------------------------------ | Simplicius +------------------------------------------------------------ Simplicius Simplicius Simplicius (490-560) +------------------------------------------------------------ | Simpson +------------------------------------------------------------ Simpson Simpson Thomas (1710-1761) +------------------------------------------------------------ | Simson +------------------------------------------------------------ Simson Simson Robert (1687-1768) +------------------------------------------------------------ | Avicenna +------------------------------------------------------------ Avicenna Sina ibn +------------------------------------------------------------ | Sinan +------------------------------------------------------------ Sinan Sinan ibn Thabit (880-943) +------------------------------------------------------------ | Sintsov +------------------------------------------------------------ Sintsov Sintsov Dmitrii (1867-1946) +------------------------------------------------------------ | Sitter +------------------------------------------------------------ Sitter Sitter Willem de (1872-1934) +------------------------------------------------------------ | Skolem +------------------------------------------------------------ Skolem Skolem Thoralf (1887-1963) +------------------------------------------------------------ | Slaught +------------------------------------------------------------ Slaught Slaught Herbert (1861-1937) +------------------------------------------------------------ | Sleszynski +------------------------------------------------------------ Sleszynski Sleszynski Ivan (1854-1931) +------------------------------------------------------------ | Slutsky +------------------------------------------------------------ Slutsky Slutsky Evgeny (1880-1948) +------------------------------------------------------------ | Sluze +------------------------------------------------------------ Sluze Sluze Ren'e de (1622-1685) +------------------------------------------------------------ | Smale +------------------------------------------------------------ Smale Smale Stephen +------------------------------------------------------------ | Smirnov +------------------------------------------------------------ Smirnov Smirnov Vladimir (1887-1974) +------------------------------------------------------------ | Smith +------------------------------------------------------------ Smith Smith Henry (1826-1883) +------------------------------------------------------------ | Sneddon +------------------------------------------------------------ Sneddon Sneddon Ian (1919-2000) +------------------------------------------------------------ | Snell +------------------------------------------------------------ Snell Snell Willebrord (1580-1626) +------------------------------------------------------------ | Snyder +------------------------------------------------------------ Snyder Snyder Virgil (1869-1950) +------------------------------------------------------------ | Sobolev +------------------------------------------------------------ Sobolev Sobolev Sergei (1908-1989) +------------------------------------------------------------ | Sokhotsky +------------------------------------------------------------ Sokhotsky Sokhotsky Yulian-Karl (1842-1927) +------------------------------------------------------------ | Sokolov +------------------------------------------------------------ Sokolov Sokolov Yurii (1896-1971) +------------------------------------------------------------ | Somerville +------------------------------------------------------------ Somerville Somerville Mary (1780-1872) +------------------------------------------------------------ | Sommerfeld +------------------------------------------------------------ Sommerfeld Sommerfeld Arnold (1868-1951) +------------------------------------------------------------ | Sommerville +------------------------------------------------------------ Sommerville Sommerville Duncan (1879-1934) +------------------------------------------------------------ | Somov +------------------------------------------------------------ Somov Somov Osip (1815-1876) +------------------------------------------------------------ | Sonin +------------------------------------------------------------ Sonin Sonin Nikolay (1849-1915) +------------------------------------------------------------ | Spanier +------------------------------------------------------------ Spanier Spanier Edwin (1921-1996) +------------------------------------------------------------ | Spence +------------------------------------------------------------ Spence Spence William (1777-1815) +------------------------------------------------------------ | Sporus +------------------------------------------------------------ Sporus Sporus of Nicaea (240-300) +------------------------------------------------------------ | Spottiswoode +------------------------------------------------------------ Spottiswoode Spottiswoode William (1825-1883) +------------------------------------------------------------ | Sridhara +------------------------------------------------------------ Sridhara Sridhara Sridhara (870-930) +------------------------------------------------------------ | Sripati +------------------------------------------------------------ Sripati Sripati (1019-1066) +------------------------------------------------------------ | Stackel +------------------------------------------------------------ Stackel St"ackel Paul (1862-1919) +------------------------------------------------------------ | Stampioen +------------------------------------------------------------ Stampioen Stampioen Jan (1610-1690) +------------------------------------------------------------ | Steenrod +------------------------------------------------------------ Steenrod Steenrod Norman (1910-1971) +------------------------------------------------------------ | StefanJosef +------------------------------------------------------------ StefanJosef Stefan Josef (1835-1893) +------------------------------------------------------------ | StefanPeter +------------------------------------------------------------ StefanPeter Stefan Peter (1941-1978) +------------------------------------------------------------ | Steiner +------------------------------------------------------------ Steiner Steiner Jakob (1796-1863) +------------------------------------------------------------ | Steinhaus +------------------------------------------------------------ Steinhaus Steinhaus Hugo (1887-1972) +------------------------------------------------------------ | Steinitz +------------------------------------------------------------ Steinitz Steinitz Ernst (1871-1928) +------------------------------------------------------------ | Steklov +------------------------------------------------------------ Steklov Steklov Vladimir A (1864-1926) +------------------------------------------------------------ | Stepanov +------------------------------------------------------------ Stepanov Stepanov Vyacheslaw V (1889-1950) +------------------------------------------------------------ | Stevin +------------------------------------------------------------ Stevin Stevin Simon (1548-1620) +------------------------------------------------------------ | Stewart +------------------------------------------------------------ Stewart Stewart Matthew (1717-1785) +------------------------------------------------------------ | Stewartson +------------------------------------------------------------ Stewartson Stewartson Keith (1925-1983) +------------------------------------------------------------ | Stieltjes +------------------------------------------------------------ Stieltjes Stieltjes Thomas Jan (1856-1894) +------------------------------------------------------------ | Stifel +------------------------------------------------------------ Stifel Stifel Michael (1487-1567) +------------------------------------------------------------ | Stirling +------------------------------------------------------------ Stirling Stirling James (1692-1770) +------------------------------------------------------------ | Stokes +------------------------------------------------------------ Stokes Stokes George Gabriel (1819-1903) +------------------------------------------------------------ | Stolz +------------------------------------------------------------ Stolz Stolz Otto (1842-1905) +------------------------------------------------------------ | Stone +------------------------------------------------------------ Stone Stone Marshall (1903-1989) +------------------------------------------------------------ | Stott +------------------------------------------------------------ Stott Stott Alicia Boole (1860-1940) +------------------------------------------------------------ | Struik +------------------------------------------------------------ Struik Struik Dirk (1894-2000) +------------------------------------------------------------ | Rayleigh +------------------------------------------------------------ Rayleigh Strutt (1842-1919) +------------------------------------------------------------ | Study +------------------------------------------------------------ Study Study Eduard (1862-1930) +------------------------------------------------------------ | Sturm +------------------------------------------------------------ Sturm Sturm J Charles-Francois (1803-1855) +------------------------------------------------------------ | SturmRudolf +------------------------------------------------------------ SturmRudolf Sturm Rudolf (1841-1919) +------------------------------------------------------------ | Subbotin +------------------------------------------------------------ Subbotin Subbotin Mikhail (1893-1966) +------------------------------------------------------------ | Suetuna +------------------------------------------------------------ Suetuna Suetuna Zyoiti (1898-1970) +------------------------------------------------------------ | Suter +------------------------------------------------------------ Suter Suter Heinrich (1848-1922) +------------------------------------------------------------ | Suvorov +------------------------------------------------------------ Suvorov Suvorov Georgii (1919-1984) +------------------------------------------------------------ | Swain +------------------------------------------------------------ Swain Swain Lorna (1891-1936) +------------------------------------------------------------ | Sylow +------------------------------------------------------------ Sylow Sylow Ludwig (1832-1918) +------------------------------------------------------------ | Sylvester +------------------------------------------------------------ Sylvester Sylvester James Joseph (1814-1897) +------------------------------------------------------------ | Synge +------------------------------------------------------------ Synge Synge John (1897-1995) +------------------------------------------------------------ | Szasz +------------------------------------------------------------ Szasz Sz"asz Otto (1884-1952) +------------------------------------------------------------ | Szego +------------------------------------------------------------ Szego Szeg"o G"abor (1895-1985) +------------------------------------------------------------ | Tacquet +------------------------------------------------------------ Tacquet Tacquet Andrea (1612-1660) +------------------------------------------------------------ | Al-Baghdadi +------------------------------------------------------------ Al-Baghdadi Tahir ibn +------------------------------------------------------------ | Tait +------------------------------------------------------------ Tait Tait Peter Guthrie (1831-1901) +------------------------------------------------------------ | Takagi +------------------------------------------------------------ Takagi Takagi Teiji (1875-1960) +------------------------------------------------------------ | Seki +------------------------------------------------------------ Seki Takakazu (1642-1708) +------------------------------------------------------------ | Talbot +------------------------------------------------------------ Talbot Talbot Henry Fox (1800-1877) +------------------------------------------------------------ | Taniyama +------------------------------------------------------------ Taniyama Taniyama Yutaka (1927-1958) +------------------------------------------------------------ | TanneryPaul +------------------------------------------------------------ TanneryPaul Tannery Paul (1843-1904) +------------------------------------------------------------ | TanneryJules +------------------------------------------------------------ TanneryJules Tannery Jules (1848-1910) +------------------------------------------------------------ | Tarry +------------------------------------------------------------ Tarry Tarry Gaston (1843-1913) +------------------------------------------------------------ | Tarski +------------------------------------------------------------ Tarski Tarski Alfred (1902-1983) +------------------------------------------------------------ | Tartaglia +------------------------------------------------------------ Tartaglia Tartaglia Niccolo Fontana (1500-1557) +------------------------------------------------------------ | Tauber +------------------------------------------------------------ Tauber Tauber Alfred (1866-1942) +------------------------------------------------------------ | Taurinus +------------------------------------------------------------ Taurinus Taurinus Franz (1794-1874) +------------------------------------------------------------ | Taussky-Todd +------------------------------------------------------------ Taussky-Todd Taussky-Todd Olga +------------------------------------------------------------ | TaylorGeoffrey +------------------------------------------------------------ TaylorGeoffrey Taylor Geoffrey (1886-1975) +------------------------------------------------------------ | Taylor +------------------------------------------------------------ Taylor Taylor Brook (1685-1731) +------------------------------------------------------------ | Teichmuller +------------------------------------------------------------ Teichmuller Teichm"uller Oswald (1913-1943) +------------------------------------------------------------ | Temple +------------------------------------------------------------ Temple Temple George (1901-1992) +------------------------------------------------------------ | LeTenneur +------------------------------------------------------------ LeTenneur Tenneur Jacques (1610-1660) +------------------------------------------------------------ | Tetens +------------------------------------------------------------ Tetens Tetens Johannes (1736-1807) +------------------------------------------------------------ | Thabit +------------------------------------------------------------ Thabit Thabit ibn Qurra Abu'l (826-901) +------------------------------------------------------------ | Thales +------------------------------------------------------------ Thales Thales of Miletus (624BC-546BC) +------------------------------------------------------------ | Theaetetus +------------------------------------------------------------ Theaetetus Theaetetus of Athens (415BC-369BC) +------------------------------------------------------------ | Theodorus +------------------------------------------------------------ Theodorus Theodorus of Cyrene (465BC-398BC) +------------------------------------------------------------ | Theodosius +------------------------------------------------------------ Theodosius Theodosius of Bithynia (160BC-90BC) +------------------------------------------------------------ | TheonofSmyrna +------------------------------------------------------------ TheonofSmyrna Theon of Smyrna (70AD-135) +------------------------------------------------------------ | Theon +------------------------------------------------------------ Theon Theon of Alexandria (335-395) +------------------------------------------------------------ | Thiele +------------------------------------------------------------ Thiele Thiele Thorvald (1838-1910) +------------------------------------------------------------ | Thom +------------------------------------------------------------ Thom Thom Ren'e +------------------------------------------------------------ | Thomae +------------------------------------------------------------ Thomae Thomae Johannes (1840-1921) +------------------------------------------------------------ | Thomason +------------------------------------------------------------ Thomason Thomason Bob (1952-1995) +------------------------------------------------------------ | ThompsonJohn +------------------------------------------------------------ ThompsonJohn Thompson John +------------------------------------------------------------ | ThompsonD'Arcy +------------------------------------------------------------ ThompsonD'Arcy Thompson D'Arcy W (1860-1948) +------------------------------------------------------------ | Thomson +------------------------------------------------------------ Thomson Thomson W (1824-1907) +------------------------------------------------------------ | Thue +------------------------------------------------------------ Thue Thue Axel (1863-1922) +------------------------------------------------------------ | Thurston +------------------------------------------------------------ Thurston Thurston Bill +------------------------------------------------------------ | Thymaridas +------------------------------------------------------------ Thymaridas Thymaridas (400BC-350BC) +------------------------------------------------------------ | Tibbon +------------------------------------------------------------ Tibbon Tibbon Jacob ben (1236-1312) +------------------------------------------------------------ | Tietze +------------------------------------------------------------ Tietze Tietze Heinrich (1880-1964) +------------------------------------------------------------ | Tilly +------------------------------------------------------------ Tilly Tilly Joseph de (1837-1906) +------------------------------------------------------------ | Tinbergen +------------------------------------------------------------ Tinbergen Tinbergen Jan (1903-1994) +------------------------------------------------------------ | Tinseau +------------------------------------------------------------ Tinseau Tinseau Charles (1748-1822) +------------------------------------------------------------ | Tisserand +------------------------------------------------------------ Tisserand Tisserand F'elix (1845-1896) +------------------------------------------------------------ | Titchmarsh +------------------------------------------------------------ Titchmarsh Titchmarsh Edward (1899-1963) +------------------------------------------------------------ | Todd +------------------------------------------------------------ Todd Todd John (1908-1994) +------------------------------------------------------------ | Todhunter +------------------------------------------------------------ Todhunter Todhunter Isaac (1820-1884) +------------------------------------------------------------ | Toeplitz +------------------------------------------------------------ Toeplitz Toeplitz Otto (1881-1940) +------------------------------------------------------------ | Torricelli +------------------------------------------------------------ Torricelli Torricelli Evangelista (1608-1647) +------------------------------------------------------------ | Trail +------------------------------------------------------------ Trail Trail William (1746-1831) +------------------------------------------------------------ | Tricomi +------------------------------------------------------------ Tricomi Tricomi Francesco (1897-1978) +------------------------------------------------------------ | Troughton +------------------------------------------------------------ Troughton Troughton Edward (1753-1836) +------------------------------------------------------------ | Tsu +------------------------------------------------------------ Tsu Tsu Ch'ung Chi (430-501) +------------------------------------------------------------ | Tukey +------------------------------------------------------------ Tukey Tukey John (1915-2000) +------------------------------------------------------------ | Tunstall +------------------------------------------------------------ Tunstall Tunstall Cuthbert (1474-1559) +------------------------------------------------------------ | Turan +------------------------------------------------------------ Turan Tur"an Paul (1910-1976) +------------------------------------------------------------ | Turing +------------------------------------------------------------ Turing Turing Alan (1912-1954) +------------------------------------------------------------ | Turnbull +------------------------------------------------------------ Turnbull Turnbull Herbert (1885-1961) +------------------------------------------------------------ | Turner +------------------------------------------------------------ Turner Turner Peter (1586-1652) +------------------------------------------------------------ | Al-TusiSharaf +------------------------------------------------------------ Al-TusiSharaf Tusi Sharaf al (1135-1213) +------------------------------------------------------------ | Al-TusiNasir +------------------------------------------------------------ Al-TusiNasir Tusi Nasir al (1201-1274) +------------------------------------------------------------ | Tikhonov +------------------------------------------------------------ Tikhonov Tychonoff Andrey (1906-1993) +------------------------------------------------------------ | UhlenbeckKaren +------------------------------------------------------------ UhlenbeckKaren Uhlenbeck Karen +------------------------------------------------------------ | Uhlenbeck +------------------------------------------------------------ Uhlenbeck Uhlenbeck George (1900-1988) +------------------------------------------------------------ | Ulam +------------------------------------------------------------ Ulam Ulam Stanislaw (1909-1984) +------------------------------------------------------------ | UlughBeg +------------------------------------------------------------ UlughBeg Ulugh Beg (1393-1449) +------------------------------------------------------------ | Al-Umawi +------------------------------------------------------------ Al-Umawi Umawi Abu al (1400-1489) +------------------------------------------------------------ | Upton +------------------------------------------------------------ Upton Upton Francis (1852-1921) +------------------------------------------------------------ | Al-Uqlidisi +------------------------------------------------------------ Al-Uqlidisi Uqlidisi Abu'l al (920-980) +------------------------------------------------------------ | Urysohn +------------------------------------------------------------ Urysohn Urysohn Pavel (1898-1924) +------------------------------------------------------------ | Vacca +------------------------------------------------------------ Vacca Vacca Giovanni (1872-1953) +------------------------------------------------------------ | Vailati +------------------------------------------------------------ Vailati Vailati Giovanni (1863-1909) +------------------------------------------------------------ | DuVal +------------------------------------------------------------ DuVal Val Patrick du (1903-1987) +------------------------------------------------------------ | Valerio +------------------------------------------------------------ Valerio Valerio Luca (1552-1618) +------------------------------------------------------------ | ValleePoussin +------------------------------------------------------------ ValleePoussin Vall'ee Poussin C de (1866-1962) +------------------------------------------------------------ | Vandermonde +------------------------------------------------------------ Vandermonde Vandermonde Alexandre (1735-1796) +------------------------------------------------------------ | Vandiver +------------------------------------------------------------ Vandiver Vandiver Harry (1882-1973) +------------------------------------------------------------ | Varahamihira +------------------------------------------------------------ Varahamihira Varahamihira Varahamihira (505-587) +------------------------------------------------------------ | Varignon +------------------------------------------------------------ Varignon Varignon Pierre (1654-1722) +------------------------------------------------------------ | Veblen +------------------------------------------------------------ Veblen Veblen Oswald (1880-1960) +------------------------------------------------------------ | Saint-Venant +------------------------------------------------------------ Saint-Venant Venant Adh'emar de St- (1797-1886) +------------------------------------------------------------ | Venn +------------------------------------------------------------ Venn Venn John (1834-1923) +------------------------------------------------------------ | Verhulst +------------------------------------------------------------ Verhulst Verhulst Pierre (1804-1849) +------------------------------------------------------------ | Vernier +------------------------------------------------------------ Vernier Vernier Pierre (1584-1637) +------------------------------------------------------------ | Veronese +------------------------------------------------------------ Veronese Veronese Giuseppe (1854-1917) +------------------------------------------------------------ | LeVerrier +------------------------------------------------------------ LeVerrier Verrier Urbain Le (1811-1877) +------------------------------------------------------------ | Vessiot +------------------------------------------------------------ Vessiot Vessiot Ernest (1865-1952) +------------------------------------------------------------ | Viete +------------------------------------------------------------ Viete Vi`ete Francois (1540-1603) +------------------------------------------------------------ | Vijayanandi +------------------------------------------------------------ Vijayanandi Vijayanandi +------------------------------------------------------------ | Saint-Vincent +------------------------------------------------------------ Saint-Vincent Vincent Gregorius Saint- (1584-1667) +------------------------------------------------------------ | Leonardo +------------------------------------------------------------ Leonardo Vinci Leonardo da (1452-1519) +------------------------------------------------------------ | Vinogradov +------------------------------------------------------------ Vinogradov Vinogradov Ivan (1891-1983) +------------------------------------------------------------ | Vitali +------------------------------------------------------------ Vitali Vitali Giuseppe (1875-1932) +------------------------------------------------------------ | Viviani +------------------------------------------------------------ Viviani Viviani Vincenzo (1622-1703) +------------------------------------------------------------ | Vlacq +------------------------------------------------------------ Vlacq Vlacq Adriaan (1600-1667) +------------------------------------------------------------ | VanVleck +------------------------------------------------------------ VanVleck Vleck Edward van (1863-1943) +------------------------------------------------------------ | Volterra +------------------------------------------------------------ Volterra Volterra Vito (1860-1940) +------------------------------------------------------------ | Voronoy +------------------------------------------------------------ Voronoy Voronoy Georgy (1868-1908) +------------------------------------------------------------ | Vranceanu +------------------------------------------------------------ Vranceanu Vranceanu Gheorghe (1900-1979) +------------------------------------------------------------ | VanderWaerden +------------------------------------------------------------ VanderWaerden Waerden Bartel van der (1903-1996) +------------------------------------------------------------ | Abu'l-Wafa +------------------------------------------------------------ Abu'l-Wafa Wafa al-Buzjani Abu'l (940-998) +------------------------------------------------------------ | Wald +------------------------------------------------------------ Wald Wald Abraham (1902-1950) +------------------------------------------------------------ | WalkerJohn +------------------------------------------------------------ WalkerJohn Walker John (1825-1900) +------------------------------------------------------------ | WalkerArthur +------------------------------------------------------------ WalkerArthur Walker Geoffrey (1909-2001) +------------------------------------------------------------ | Wall +------------------------------------------------------------ Wall Wall C Terence +------------------------------------------------------------ | Wallace +------------------------------------------------------------ Wallace Wallace William (1768-1843) +------------------------------------------------------------ | Wallis +------------------------------------------------------------ Wallis Wallis John (1616-1703) +------------------------------------------------------------ | Wang +------------------------------------------------------------ Wang Wang Hsien Chung (1918-1978) +------------------------------------------------------------ | Wangerin +------------------------------------------------------------ Wangerin Wangerin Albert (1844-1933) +------------------------------------------------------------ | Wantzel +------------------------------------------------------------ Wantzel Wantzel Pierre (1814-1894) +------------------------------------------------------------ | Waring +------------------------------------------------------------ Waring Waring Edward (1734-1798) +------------------------------------------------------------ | Watson +------------------------------------------------------------ Watson Watson G N (1886-1965) +------------------------------------------------------------ | WatsonHenry +------------------------------------------------------------ WatsonHenry Watson Henry (1827-1903) +------------------------------------------------------------ | Wazewski +------------------------------------------------------------ Wazewski Wazewski Tadeusz (1896-1972) +------------------------------------------------------------ | Weatherburn +------------------------------------------------------------ Weatherburn Weatherburn Charles (1884-1974) +------------------------------------------------------------ | Weber +------------------------------------------------------------ Weber Weber Wilhelm (1804-1891) +------------------------------------------------------------ | WeberHeinrich +------------------------------------------------------------ WeberHeinrich Weber Heinrich Martin (1842-1913) +------------------------------------------------------------ | Wedderburn +------------------------------------------------------------ Wedderburn Wedderburn Joseph (1882-1948) +------------------------------------------------------------ | Weierstrass +------------------------------------------------------------ Weierstrass Weierstrass Karl (1815-1897) +------------------------------------------------------------ | Weil +------------------------------------------------------------ Weil Weil Andr'e (1906-1998) +------------------------------------------------------------ | Weingarten +------------------------------------------------------------ Weingarten Weingarten Julius (1836-1910) +------------------------------------------------------------ | Weinstein +------------------------------------------------------------ Weinstein Weinstein Alexander (1897-1979) +------------------------------------------------------------ | Weisbach +------------------------------------------------------------ Weisbach Weisbach Julius (1806-1871) +------------------------------------------------------------ | Weldon +------------------------------------------------------------ Weldon Weldon Raphael (1860-1906) +------------------------------------------------------------ | Werner +------------------------------------------------------------ Werner Werner Johann (1468-1522) +------------------------------------------------------------ | Wessel +------------------------------------------------------------ Wessel Wessel Caspar (1745-1818) +------------------------------------------------------------ | West +------------------------------------------------------------ West West John (1756-1817) +------------------------------------------------------------ | Weyl +------------------------------------------------------------ Weyl Weyl Hermann (1885-1955) +------------------------------------------------------------ | Weyr +------------------------------------------------------------ Weyr Weyr Emil (1848-1894) +------------------------------------------------------------ | Wheeler +------------------------------------------------------------ Wheeler Wheeler Anna J Pell (1883-1966) +------------------------------------------------------------ | Whiston +------------------------------------------------------------ Whiston Whiston William (1667-1752) +------------------------------------------------------------ | White +------------------------------------------------------------ White White Henry (1861-1943) +------------------------------------------------------------ | Whitehead +------------------------------------------------------------ Whitehead Whitehead Alfred N (1861-1947) +------------------------------------------------------------ | WhiteheadHenry +------------------------------------------------------------ WhiteheadHenry Whitehead J Henry C (1904-1960) +------------------------------------------------------------ | Whitney +------------------------------------------------------------ Whitney Whitney Hassler (1907-1989) +------------------------------------------------------------ | Whittaker +------------------------------------------------------------ Whittaker Whittaker Edmund (1873-1956) +------------------------------------------------------------ | WhittakerJohn +------------------------------------------------------------ WhittakerJohn Whittaker John (1905-1984) +------------------------------------------------------------ | Whyburn +------------------------------------------------------------ Whyburn Whyburn Gordon (1904-1969) +------------------------------------------------------------ | Widman +------------------------------------------------------------ Widman Widman Johannes (1462-1498) +------------------------------------------------------------ | Wielandt +------------------------------------------------------------ Wielandt Wielandt Helmut (1910-2001) +------------------------------------------------------------ | Wien +------------------------------------------------------------ Wien Wien Wilhelm (1864-1928) +------------------------------------------------------------ | WienerNorbert +------------------------------------------------------------ WienerNorbert Wiener Norbert (1894-1964) +------------------------------------------------------------ | WienerChristian +------------------------------------------------------------ WienerChristian Wiener Christian (1826-1896) +------------------------------------------------------------ | Wigner +------------------------------------------------------------ Wigner Wigner Eugene (1902-1995) +------------------------------------------------------------ | Wilczynski +------------------------------------------------------------ Wilczynski Wilczynski Ernest (1876-1932) +------------------------------------------------------------ | Wiles +------------------------------------------------------------ Wiles Wiles Andrew +------------------------------------------------------------ | Wilkins +------------------------------------------------------------ Wilkins Wilkins John (1614-1672) +------------------------------------------------------------ | Wilkinson +------------------------------------------------------------ Wilkinson Wilkinson Jim (1919-1986) +------------------------------------------------------------ | Wilks +------------------------------------------------------------ Wilks Wilks Samuel (1906-1964) +------------------------------------------------------------ | Ockham +------------------------------------------------------------ Ockham William of Ockham (1285-1349) +------------------------------------------------------------ | WilsonEdwin +------------------------------------------------------------ WilsonEdwin Wilson Edwin (1879-1964) +------------------------------------------------------------ | WilsonJohn +------------------------------------------------------------ WilsonJohn Wilson John (1741-1793) +------------------------------------------------------------ | WilsonAlexander +------------------------------------------------------------ WilsonAlexander Wilson Alexander (1714-1786) +------------------------------------------------------------ | Winkler +------------------------------------------------------------ Winkler Winkler Wilhelm (1884-1984) +------------------------------------------------------------ | Wintner +------------------------------------------------------------ Wintner Wintner Aurel (1903-1958) +------------------------------------------------------------ | Wirtinger +------------------------------------------------------------ Wirtinger Wirtinger Wilhelm (1865-1945) +------------------------------------------------------------ | Wishart +------------------------------------------------------------ Wishart Wishart John (1898-1956) +------------------------------------------------------------ | DeWitt +------------------------------------------------------------ DeWitt Witt Johan de (1625-1672) +------------------------------------------------------------ | Witt +------------------------------------------------------------ Witt Witt Ernst (1911-1991) +------------------------------------------------------------ | Witten +------------------------------------------------------------ Witten Witten Edward +------------------------------------------------------------ | Wittgenstein +------------------------------------------------------------ Wittgenstein Wittgenstein Ludwig (1889-1951) +------------------------------------------------------------ | Wolf +------------------------------------------------------------ Wolf Wolf Rudolph (1816-1893) +------------------------------------------------------------ | Wolfowitz +------------------------------------------------------------ Wolfowitz Wolfowitz Jacob (1910-1981) +------------------------------------------------------------ | Wolstenholme +------------------------------------------------------------ Wolstenholme Wolstenholme Joseph (1829-1891) +------------------------------------------------------------ | Woodhouse +------------------------------------------------------------ Woodhouse Woodhouse Robert (1773-1827) +------------------------------------------------------------ | Woodward +------------------------------------------------------------ Woodward Woodward Robert (1849-1924) +------------------------------------------------------------ | Wren +------------------------------------------------------------ Wren Wren Sir Christopher (1632-1723) +------------------------------------------------------------ | Wronski +------------------------------------------------------------ Wronski Wronski Ho"en'e (1778-1853) +------------------------------------------------------------ | Xenocrates +------------------------------------------------------------ Xenocrates Xenocrates of Chalcedon (396BC-314BC) +------------------------------------------------------------ | Yang +------------------------------------------------------------ Yang Yang Hui (1238-1298) +------------------------------------------------------------ | Yates +------------------------------------------------------------ Yates Yates Frank (1902-1994) +------------------------------------------------------------ | Yativrsabha +------------------------------------------------------------ Yativrsabha Yativrsabha (500-570) +------------------------------------------------------------ | Yau +------------------------------------------------------------ Yau Yau Shing-Tung +------------------------------------------------------------ | Yavanesvara +------------------------------------------------------------ Yavanesvara Yavanesvara (120-180) +------------------------------------------------------------ | Yoccoz +------------------------------------------------------------ Yoccoz Yoccoz Jean-Christophe +------------------------------------------------------------ | Youden +------------------------------------------------------------ Youden Youden William (1900-1971) +------------------------------------------------------------ | Young +------------------------------------------------------------ Young Young William (1863-1942) +------------------------------------------------------------ | YoungAlfred +------------------------------------------------------------ YoungAlfred Young Alfred (1873-1940) +------------------------------------------------------------ | ChisholmYoung +------------------------------------------------------------ ChisholmYoung Young Grace Chisholm (1868-1944) +------------------------------------------------------------ | Yule +------------------------------------------------------------ Yule Yule George (1871-1951) +------------------------------------------------------------ | Yunus +------------------------------------------------------------ Yunus Yunus Abu'l-Hasan ibn +------------------------------------------------------------ | Yushkevich +------------------------------------------------------------ Yushkevich Yushkevich Adolph P (1906-1993) +------------------------------------------------------------ | Ahmed +------------------------------------------------------------ Ahmed Yusuf Ahmed ibn (835-912) +------------------------------------------------------------ | QadiZada +------------------------------------------------------------ QadiZada Zada al-Rumi Qadi (1364-1436) +------------------------------------------------------------ | Vashchenko +------------------------------------------------------------ Vashchenko Zakharchenko M V- (1825-1912) +------------------------------------------------------------ | Zarankiewicz +------------------------------------------------------------ Zarankiewicz Zarankiewicz Kazimierz (1902-1959) +------------------------------------------------------------ | Zaremba +------------------------------------------------------------ Zaremba Zaremba Stanislaw (1863-1942) +------------------------------------------------------------ | Zariski +------------------------------------------------------------ Zariski Zariski Oscar (1899-1986) +------------------------------------------------------------ | Zassenhaus +------------------------------------------------------------ Zassenhaus Zassenhaus Hans (1912-1991) +------------------------------------------------------------ | Zeckendorf +------------------------------------------------------------ Zeckendorf Zeckendorf Edouard (1901-1983) +------------------------------------------------------------ | Zeeman +------------------------------------------------------------ Zeeman Zeeman Chris +------------------------------------------------------------ | Zelmanov +------------------------------------------------------------ Zelmanov Zelmanov Efim +------------------------------------------------------------ | ZenoofSidon +------------------------------------------------------------ ZenoofSidon Zeno of Sidon (150BC-70BC) +------------------------------------------------------------ | ZenoofElea +------------------------------------------------------------ ZenoofElea Zeno of Elea (490BC-430BC) +------------------------------------------------------------ | Zenodorus +------------------------------------------------------------ Zenodorus Zenodorus (200BC-140BC) +------------------------------------------------------------ | Zermelo +------------------------------------------------------------ Zermelo Zermelo Ernst (1871-1951) +------------------------------------------------------------ | Zeuthen +------------------------------------------------------------ Zeuthen Zeuthen Hieronymous (1839-1920) +------------------------------------------------------------ | Chu +------------------------------------------------------------ Chu Zhu Shie-jie (1270-1330) +------------------------------------------------------------ | Zhukovsky +------------------------------------------------------------ Zhukovsky Zhukovsky Nikolay (1847-1921) +------------------------------------------------------------ | Zolotarev +------------------------------------------------------------ Zolotarev Zolotarev Egor (1847-1878) +------------------------------------------------------------ | Zorn +------------------------------------------------------------ Zorn Zorn Max (1906-1993) +------------------------------------------------------------ | Zuse +------------------------------------------------------------ Zuse Zuse Konrad (1910-1995) +------------------------------------------------------------ | Zygmund +------------------------------------------------------------ Zygmund Zygmund Antoni (1900-1992) +------------------------------------------------------------ | D'Alembert +------------------------------------------------------------ D'Alembert d'Alembert Jean (1717-1783) +------------------------------------------------------------ | BudandeBoislaurent +------------------------------------------------------------ BudandeBoislaurent de Boislaurent Budan (1761-1840) +------------------------------------------------------------ | Coulomb +------------------------------------------------------------ Coulomb de Coulomb Charles (1736-1806) +------------------------------------------------------------ | DeBeaune +------------------------------------------------------------ DeBeaune de Beaune Florimond (1601-1652) +------------------------------------------------------------ | Carcavi +------------------------------------------------------------ Carcavi de Carcavi Pierre (1600-1684) +------------------------------------------------------------ | Broglie +------------------------------------------------------------ Broglie de Broglie Louis duc (1892-1987) +------------------------------------------------------------ | Bougainville +------------------------------------------------------------ Bougainville de Bougainville Louis (1729-1811) +------------------------------------------------------------ | Billy +------------------------------------------------------------ Billy de Billy Jacques (1602-1679) +------------------------------------------------------------ | Coriolis +------------------------------------------------------------ Coriolis de Coriolis Gustave (1792-1843) +------------------------------------------------------------ | Hunayn +------------------------------------------------------------ Hunayn ibn Ishaq Hunayn (808-873) +------------------------------------------------------------ | Lansberge +------------------------------------------------------------ Lansberge van Lansberge Philip (1561-1632) +------------------------------------------------------------ | Roomen +------------------------------------------------------------ Roomen van Roomen Adriaan (1561-1615) +------------------------------------------------------------ | Schooten +------------------------------------------------------------ Schooten van Schooten Frans (1615-1660) +------------------------------------------------------------ | Dantzig +------------------------------------------------------------ Dantzig van Dantzig David (1900-1959) +------------------------------------------------------------ | Heuraet +------------------------------------------------------------ Heuraet van Heuraet Hendrik (1633-1660) +------------------------------------------------------------ | VanCeulen +------------------------------------------------------------ VanCeulen van Ceulen Ludolph (1540-1610) +------------------------------------------------------------ | Amringe +------------------------------------------------------------ Amringe van Amringe Howard (1835-1915) +------------------------------------------------------------ | Geiringer +------------------------------------------------------------ Geiringer von Mises Hilda Geiringer (1893-1973) +------------------------------------------------------------ | Helmholtz +------------------------------------------------------------ Helmholtz von Helmholtz Hermann (1821-1894) +------------------------------------------------------------ | Lindemann +------------------------------------------------------------ Lindemann von Lindemann Carl (1852-1939) +------------------------------------------------------------ | Eotvos +------------------------------------------------------------ Eotvos von E"otv"os Roland (1848-1919) +------------------------------------------------------------ | Segner +------------------------------------------------------------ Segner von Segner Johann (1704-1777) +------------------------------------------------------------ | Seidel +------------------------------------------------------------ Seidel von Seidel Philipp (1821-1896) +------------------------------------------------------------ | VonNeumann +------------------------------------------------------------ VonNeumann von Neumann John (1903-1957) +------------------------------------------------------------ | Vega +------------------------------------------------------------ Vega von Vega Georg (1754-1802) +------------------------------------------------------------ | Mises +------------------------------------------------------------ Mises von Mises Richard (1883-1953) +------------------------------------------------------------ | VonDyck +------------------------------------------------------------ VonDyck von Dyck Walther (1856-1934) +------------------------------------------------------------ | Koch +------------------------------------------------------------ Koch von Koch Helge (1870-1924) +------------------------------------------------------------ | Tschirnhaus +------------------------------------------------------------ Tschirnhaus von Tschirnhaus E (1651-1708) +------------------------------------------------------------ | Karman +------------------------------------------------------------ Karman von K"arm"an Theodore (1881-1963) +------------------------------------------------------------ | Leibniz +------------------------------------------------------------ Leibniz von Leibniz Gottfried (1646-1719) +------------------------------------------------------------ | VonStaudt +------------------------------------------------------------ VonStaudt von Staudt Karl (1798-1867) +------------------------------------------------------------ | VonBrill +------------------------------------------------------------ VonBrill von Brill Alexander (1842-1935) This file is part of the Sofia project sponsored by the Provost's fund for teaching and learning at Harvard university. There are 1508 entries in this file. COUNT: 1508 ENTRY MATH MOVIES Author: Oliver Knill: March 2000 -March 2004 Literature: actual DVD's and corresponding movie websites +------------------------------------------------------------ | Enigma +------------------------------------------------------------ Enigma is an espionage thriller set during WW II. Most of the story is fictional. The main character Tom Jericho who serves with the British "Government Communication Headquarters" at Bletchley Park and played a significant role in breaking the German "Enigma" codes using a machine called "Collossus" to decipher the Enigma codes. The story is inspired by the lif of the mathematician Alan Turing who indeed contributed to the deciphering of Enigma during WW II. +------------------------------------------------------------ | A beautiful mind +------------------------------------------------------------ The movie A beautiful mind describes the life of the Mathematician John Nash. Nash is introduced while entering Princeton as a young graduate student. The movie shows how Nash was struggling writing his PhD with the title "Non-cooperative games", a work which later would give him the Nobel prize. Nash is described as an impossible college teacher. In a calculus class, he introduces the following problem: Find a subset X of three dimensional space which has the property that if V is the set of vector fields F on the complement of X, which satisfy curl(F)=0 and W is the set of vector fields F which are conservative F= grad(f). Then, the space V/W should be 8 dimensional. +------------------------------------------------------------ | Good will hunting +------------------------------------------------------------ The movie Good will hunting shows a math prodigy Will Hunting who grew up in a succession of orphanages in South Boston. Working as a janitor at MIT, he has taught himself mathematics. He would anonymously solve complex math problems which were left overnight on blackboards. From an AMS review: "The mathematics referred to later on ranges from basic linear algebra, through simple graph theory, to Parseval's theorem, Fourier analysis, and on to what seem to be some deeper graph theoretical results. Mathematics is referred to constantly, but in no scene is it presented coherently." +------------------------------------------------------------ | Cube +------------------------------------------------------------ Cube Six strangers wake up in a maze of cubes equiped with movie traps and have to find their way out. Each room is equipped with a triple of numbers and colored. If all numbers are simultaneously not prime, then the room is trapped and entering it would kill the person entering it. +------------------------------------------------------------ | Hypercube +------------------------------------------------------------ Hypercube In this horror movie, eight strangers wake up in a bizare cube-shaped room not knowing how they got there or how to escape. They soon learn that their "hypercube" operates in the fourth dimension and shifts into an endless maze of danger and in the end everyone dies. The movie is the sequel to the 1999 cult hit cube. Cube 2 was directed by Andrzej Sekula. +------------------------------------------------------------ | Sneakers +------------------------------------------------------------ Sneakers An espionage thriller with Robert Redford. A hunt for a futuristic device which allows to decrypt secret messages. The device was built by a "genious Mathematician" who appears in the movie giving a pompeous lecture on factorization algorithms. The movie which appeared in 1992 is not totally unrealistic from the mathematical point of view. Shortly after the movie was released, in the year 1993, mathematicians have shown that in principle, a quantum computer could break the factorization difficulty which is the fundament for many modern encryption algorithms. An other interpretation for the device would be that a new algorithm for factoring large integers would be found secretly and be hardwired into a chip. +------------------------------------------------------------ | Pi +------------------------------------------------------------ Pi In the movie Pi, the pursuit of the infinite takes on a deeper meaning. Max Cohen is a number theorist living in New York obsessed with a potentially unsolvable problem. Yet, what the story and the age-old problem uncovers is the deeper link between the mysteries of life and other topics of consciousness as seemingly disparate as the stock market, the Kaballah, technology, the DNA and the stars in the sky. "11:15 Restate my assumptions: Mathematics is the language of nature. Everything around us can be represented and understood through numbers. If you graph these numbers, patterns emerge. Therefore: There are patterns everywhere in nature." Max Cohen in Pi +------------------------------------------------------------ | Old School +------------------------------------------------------------ Old School In the college comedy "old school", three men, disenchanted with their life try to recapture their college life and wild youth by opening a frat house. In the movie, some aereal shots of Harvard appear evenso the movie seems have no scenes at all taken in Cambridge. At one point, the fraternaty members have to take a test in which they are asked about Hariotts method to solve cubics. This file is part of the Sofia project sponsored by the Provost's fund for teaching and learning at Harvard university. There are 8 entries in this file. COUNT: 8 ENTRY MEASURE THEORY Authors: Oliver Knill: 2003 Literature: measure theory +------------------------------------------------------------ | analytic set +------------------------------------------------------------ An analytic set in a complete seperable metric space is the continuous image of a Borel set. Also called A-set. Any A-set is Lebesgue measurable. Any uncountable A-set topologically contains a perfect Cantor set. Suslins criterium tells that an analytic set is a Borel set if and only if its complement is also an analytic set. +------------------------------------------------------------ | atom +------------------------------------------------------------ An atom is a measurable set Y of positive measure in a measure space such that every subset Z of Y has either zero or the same measure. Often an atom consists of only one point. More generally, an atom is minimal, non-zero element in a Boolean algebra. +------------------------------------------------------------ | atom +------------------------------------------------------------ A property which holds up to a set of measure zero is said to hold almost everywhere (= almost surely). +------------------------------------------------------------ | Banach-Tarski theorem +------------------------------------------------------------ The Banach-Tarski theorem: a ball in Euclidean space of dimension 3 can be decomposed into finitely many sets and rearranged by rigid motion to obtain two balls. The +------------------------------------------------------------ | barycentre +------------------------------------------------------------ The barycentre of a Lebesgue measurable set S in an Euclidean space is the point int_S x dx. +------------------------------------------------------------ | Boolean algebra +------------------------------------------------------------ A Boolean algebra is a set S with two binary operations + and * which are commutative monoids (S,+,0), (S,*,1) and satisfy the two distributive laws (x*(y+z)=x*y + y*z, x+(y*z) =(x+y)*(x+z) as well as the complementary laws x*x=1, y+y=0. A Boolean algebra is especially an algebra. Examples are the algebra of classes, where + is the union and * is the intersection or the algebra of propositions, for which + is and and * is or. +------------------------------------------------------------ | Boolean ring +------------------------------------------------------------ A Boolean ring is a ring in which every member is idempotent. +------------------------------------------------------------ | Borel-Cantelli lemma +------------------------------------------------------------ The Borel-Cantelli lemma: if Y_n are events in a probability space and the sum of their probabilities is finite, then the probability that infinitely many events occur is zero. If the events are independent and the sum of their probabilities is infinite, then the probability that infinitely many events occur is one. +------------------------------------------------------------ | Borel measure +------------------------------------------------------------ A Borel measure is a measure on the sigma-algebra of Borel sets. +------------------------------------------------------------ | Borel set +------------------------------------------------------------ A Borel set (=Borel measurable set) in a topological space is an element in the smallest sigma-algebra which contains all compact sets. Borel sets are also called B-sets. One can say that a B-set is a set which can be obtained of not more than a countable number of operations of union and intersection of closed open sets in a topological space. Borel sets are special cases of analytic sets. +------------------------------------------------------------ | Borel set +------------------------------------------------------------ The smallest sigma-algebra A of subsets of a topological space (X,O) containing O is called a Borel sigma-algebra. +------------------------------------------------------------ | absolutely continuous +------------------------------------------------------------ A measure mu is absolutely continuous to a measure nu if nu(Y)=0 implies mu(Y)=0. +------------------------------------------------------------ | centre of mass +------------------------------------------------------------ The centre of mass (=barycentre) of a Borel measure mu in a Euclidean space X is the point overlinex= int_X x mu(x). For example, if mu is supported on finitely many points x_i and m_i = mu(x_i) then overlinex = sum_i m_i x_i. If mu is the mass distribution of a body, then its centre of mass is called the centre of gravity. +------------------------------------------------------------ | abstract integral +------------------------------------------------------------ abstract integral. Denote by L,L^+ the set of measureable maps from a measure space (X,A,mu) to the real line (R,B), where B is the Borel sigma-algebra on R,R^+. For f in S= f=sum_i=1^n alpha_i . 1_A_i alpha_i in R , define int_X f dmu := sum_a in f(X) a . muX=a. For f in L^+ define int_X f dmu = sup_g in S int_X g dmu . For f in L finally define int f = int f^+ - int f^-, where f^+(x)=max(f(x),0) and f^-(x)=-(-f)^+(x). +------------------------------------------------------------ | abstract integral +------------------------------------------------------------ A sigma-additive function mu: A arrow 0,infinity on a measurable space (X,A) is called a measure. It is called a finite measure if mu(X)0, there is a constant mu>1 such that for all coprime a,b and c=a+b, then max(|a|,|b|,|c|) leq mu N(a,b,c)^(1+epsilon). +------------------------------------------------------------ | irreducible polynomial +------------------------------------------------------------ A root of an irreducible polynomial with integer coefficients is called an algebraic number. +------------------------------------------------------------ | amicable +------------------------------------------------------------ Two integers are called amicable if each is the sum of the distinct proper factors of the other. For example: 220 and 284 are amicable. Amicable numbers are 2-periodic orbits of the sigma function sigma(n) which is the sum of the divisors of n. +------------------------------------------------------------ | Apery's theorem +------------------------------------------------------------ Apery's theorem: the value of the zeta function at z=3 is rational. +------------------------------------------------------------ | arithmetic function +------------------------------------------------------------ An arithmetic function is a function f(n) whose domain is the set of positive integers. An important class of arithmetic functions are multiplicative functions f(n m) = f(n) f(m). An example is the M"obius function mu(n) defined by mu(1)=1, mu(n)=(-1)^r if n is the product of r distinct primes and mu(n)=0 otherwise. +------------------------------------------------------------ | Artinian conjecture +------------------------------------------------------------ Artinian conjecture: a quantitative form of the conjecture that every non-square integer is a primitive root of infinitely many primes. % Beal's conjecture If a^x + b^y = c^z, where a,b,c,x,y,z are positive integers and x,y,z>2, then a,b,c must have a common factor. It is known that for every x,y,z, there are only finitely many solutions. The Beal conjecture is a ageneralization of Fermat's last theorem. The conjecture was announced in December 1997. The prize is now 100'000 Dollars for either a proof or a counterexample. The conjecture was discovered by the Texan number theory enthusiast and banker Andrew Beal. +------------------------------------------------------------ | Bertrands postulate +------------------------------------------------------------ Bertrands postulate tells that for any integer n greater than 3, there is a prime between n and 2n-2. The postulate is a theorem, proven by Tchebychef in 1850. +------------------------------------------------------------ | Bezout's lemma +------------------------------------------------------------ Bezout's lemma tells that if f and g are polynomials over a field K and d is the greatest common divisor of f and g, then d = a f + b g, where a,b are two other polynomials. This generalizes Euclid's theorem for integers. +------------------------------------------------------------ | Brun's constant +------------------------------------------------------------ The Brun's constant is the sum of the reciprocals of all the prime twins. It is estimated to be about 1.9021605824. While one does not know, whether infinitely many prime twins exist, the sum of their reciprocals is known to be finite. This has been proven by the Norwegian Mathematician Viggo Brun (1885-1978) in 1919. +------------------------------------------------------------ | Carmichael numbers +------------------------------------------------------------ Carmichael numbers are natural numbers which are Fermat pseudoprime to any base. Named after R.D. Carmichael who discoved them in 1909. It is known that there are infinitely many Carmichael numbers. The Carmichael numbers under 100'000 are 561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341, 41041, 46657, 52633, 62745, 63973, and 75361. +------------------------------------------------------------ | Chineese reminder problem +------------------------------------------------------------ The Chineese reminder problem tells that if n_1, dots, n_k are natural numbers which are pairwise relatively prime and if a_1, dots, a_k are any integers, then there exists an integer x which solves simultaneously the congruences x == a_1 (mod n_1), dots, x == a_k (mod n_k). All solutions are congruent to a given solution modulo prod_j=1^k n_j. The theorem was established by Qin Jiushao in 1247. +------------------------------------------------------------ | coprime +------------------------------------------------------------ Two numbers a,b are called coprime if their greatest common divisor is 1. +------------------------------------------------------------ | ElGamal +------------------------------------------------------------ ElGamal The ElGamal cryptosystem is based on the difficulty to solve the discrete logarithm problem modulo a large number n=p^r, where p is a prime and and r is a positive integer: solving g^x = b mod n for x is computationally hard. Suppose you want to send a message encoded as an integer m to Alice. A large integer n and a base g are chosen and public. Alice who has a secrete integer a has published the integer c=g^k mod(n) as her public key. Everybody knows n,g,c. To send Alice a message m, we chose an integer k at random and send Alice the pair (A,B) = (g^m,k g^a m) modulo n. (We can compute g^am = (g^a)^m=c^m using the publically availble information only.) Alice can recover from this the secret message m=A^a/B. However, somebody intercepting the message is not able to recover m without knowing a. He would have to find the discrete logarithm of g^m with base g to do so but this is believed to be a computationally difficult problem. +------------------------------------------------------------ | Farey Sequence +------------------------------------------------------------ The Farey Sequence of order n is the finite sequence of rational numbers a/b, with 0 leq a leq b leq n such that a,b have no common divisor different from 1 and which are arranged in increasing order. F_1= (0/1,1/1) F_2= (0/1,1/2,1) F_3= (0/1,1/3,1/2,2/3,1/1) F_4= (0/1,1/4,1/3,1/2,2/3,3/4) F_5= (0/1,1/5,1/4,1/3,2/5,1/2,3/5,2/3,3/4/4/5) +------------------------------------------------------------ | Number +------------------------------------------------------------ Number: N: natural numbers e.g. 1,2,3,4,... Z: integers e.g. -1,0,1,2,... Q: rational numbers e.g. 5/6, 5,-8/10 R: real numbers e.g. 1,pi, e, sqrt(2), 5/4 C: complex numbers e.g. i, 2, e+i pi/2, 4pi/e +------------------------------------------------------------ | natural numbers +------------------------------------------------------------ The numbers 1,2,3, ... are called the natural numbers. +------------------------------------------------------------ | Fermats last theorem +------------------------------------------------------------ Fermats last theorem. For any ineger n bigger than 2, the equation x^n+y^n=z^n has no solutions in positive integers. This theorem was proven in 1995 by Andrew Wiles with the assistance of Richard Taylor. The theorem has a long history: in an annotation of his copy "Diophantus", Fermat wrote a note: "On the other hand, it is impossible to seperate a cube into two cubes, or a biquadrate into two biquadrates, or generally any power except a squre into two powers with the same exponent. I have discovered a truely marvelous proof of this which however the margin is not large enough to contain." +------------------------------------------------------------ | Fermat-Catalan Conjecture +------------------------------------------------------------ Fermat-Catalan Conjecture There are only finitely many triples of coprime integer powers x^q,y^q,z^r for which x^p+y^q=z^r with 1/p+1/q+1/r<1. +------------------------------------------------------------ | Fermats little theorem +------------------------------------------------------------ Fermats little theorem If p is prime and a is an integer which is not a multiple of p, then a^(p-1)=1 mod p. Example: 2^4 = 16 = 1 ( mod 5). Fermats little theorem is a consequence of the Lagrange theorem in algebra, which says that for finite groups, the order of a subgroup divides the order of the group. Fermats theorem is the special case, when the finite group is the cyclic group with p-1 elements. Fermats little theorem is somtimes also stated in the form: for every integer a and prime number p, the number a^p-a is a multiple of p. +------------------------------------------------------------ | Fermat numbers +------------------------------------------------------------ Numbers F_n = 2^(2^n)+1 are called Fermat numbers. Examples are F_0 = 3 prime F_1 = 5 prime F_2 = 17 prime F_3 = 257 prime F_4 = 65537 prime F_5 = 641 . 6700417 composite No other prime Fermat number beside the first 5 had been found so far. +------------------------------------------------------------ | fundamental theorem of arithmetic +------------------------------------------------------------ The fundamental theorem of arithmetic assures that every natural number n has a unique prime factorization. In other words, there is only one way in which one can write a number as a product of prime numbers if the order of the product does not matter. For example, 84 = 2 2 3 7 is the prime factorization of 84. +------------------------------------------------------------ | Goldbach's conjecture +------------------------------------------------------------ Goldbach's conjecture: Every even integer n greater than two is the sum of two primes. For example: 8=5+3 or 20=13+7. The conjecture has not been proven yet. +------------------------------------------------------------ | greatest common divisor +------------------------------------------------------------ The greatest common divisor of two integers n and m is the largest integer d such that d divides n and d divides m. One writes d=gcd(n,m). For example, gcd(6,9)=3. There are few recursive algorithms for gcd: one of them is the Euclidean algorithm: gcd(m,n)=k=m mod n; if (k==0) return(n); else return(gcd(n,k)) . +------------------------------------------------------------ | prime number or prime +------------------------------------------------------------ A positive integer n is called a prime number or prime, if it is divisible by 1 and n only. The first prime numbers are 2,3,5,7,11,13,17. An example of a non prime number is 12 because it is divisible by 3. There are infinitely many primes. Every natural number n can be factorized uniquely into primes: for example 42=2 3 7. +------------------------------------------------------------ | prime factorization +------------------------------------------------------------ The prime factorization of a positive integer n is a sequence of primes whose product is n. For example: 18 = 3 . 3 . 2 or 100=2 . 2 . 5 . 5 or 17 = 17. Every integer has a unique prime factorization. +------------------------------------------------------------ | Pells equation +------------------------------------------------------------ Fermat claimed first that Pells equation d y^2+1 = x^2, where d is an integer has always integer solutions x and y. The name "Pell equation" was given by Euler evenso Pell seems nothing have to do with the equation. Lagrange was the first to prove the existence of solutions. One can find solutions by performing the Continued fraction expansion of the square root of d. +------------------------------------------------------------ | Fermat number +------------------------------------------------------------ A Fermat number is an integer of the form F_k=2^(2^k)+1. The first Fermat numbers are F_0=3,F_1=5,F_2=17,F_3=257,F_4=65537. They are all primes and called Fermat primes. Fermat had claimed that all F_k are primes. Euler disproved that showing that 641 divides F_5. The Fermat numbers F_5 until F_9 are known to be not prime and also have been factored. Fermat numbers play a role in constructing regular polygons with ruler and compass. The factorization of Fermat numbers serves as a challenge to factorization algorithms. +------------------------------------------------------------ | Fermat prime +------------------------------------------------------------ A Fermat prime is a Fermat number which is prime. +------------------------------------------------------------ | Mersenne number +------------------------------------------------------------ An integer 2^n-1 is called a Mersenne number. If a Mersenne number is prime, it is called a Mersenne prime. +------------------------------------------------------------ | Mersenne number +------------------------------------------------------------ If a Mersenne number 2^n-1 is prime, it is called a Mersenne prime. In that case, n must be prime. Known examples are n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377. It is not known whether there are infinitely many Mersenne primes. +------------------------------------------------------------ | perfect number +------------------------------------------------------------ An integer n is called a perfect number if it is equal to the sum of its proper divisors. For example 6=1+2+3 or 28=1+2+4+7+14 are perfect numbers. Also, if 2^n-1 is prime then 2^n-1 (2^n-1) is perfect because 1+2+4+...+2^(n-1)=2^n-1. This was known already to Euclid. Every even perfect number is of the form p(p+1)/2, where p = 2^n-1 is a Mersenne prime. It is not known whether there is an odd perfect number, nor whether there are infinitely many Mersenne primes. +------------------------------------------------------------ | partition +------------------------------------------------------------ The partition of a number n is a decomposition of n into a sum of integers. Examples are 5=1+2+2. The number of partitions of a number n is denoted by p(n) and plays a role in the theory of representations of finite groups. For example: p(4)=5 because of the following partitions 4=3+1=2+2=1+1+2=1+1+1+1 Euler introduced the Power series f(x)= sum p(n) x^n which is (1-x)^-1 (1-x^2)^-1 ... . The algebra of formal power series leads to powerful identites like (1+x)^-1(1+x^2)^-1(1+x^3)^-1... = (1+x) (1-x) (1+x^2) (1-x^2) ... /(1-x) (1-x^2) ... = (1-x^2)(1-x^4) ... /(1-x)(1-x^2)... = (1-x)^-1(1-x^3)^-1(1-x^5)^-1 ... . The left hand side is the generating function for a(n), the number of partitions of n into distinct numbers. The right hand side is the generating function of b(n), the number of partitions of n into an odd number of summands. The algebraic identity has shown that a(n)=b(n). For example, for n=5, one has a(5)=3 decomposition 5=5=4+1=3+2 into different summands and also b(5)=3 decompositions into an odd number of summands 5=5=2+2+1=1+1+1+1+1. +------------------------------------------------------------ | prime number +------------------------------------------------------------ A prime number is an positive integer which is divisible only by 1 or itself. For example, 12 is not a prime number because it is divisible by 3 but the integer 13 is a prime number. The first prime numbers are 2,3,5,7,11,13,17,23 .... There are infinitely many prime numbers because if there were only finitely many, their product p_1 p_2 ... p_k = n has the property that n+1 is not divisible by any p_i. Therefore, n+1 would either be a new prime number of be divisible by a new prime number. This contradicts the assumption that there are only finitely many. +------------------------------------------------------------ | prime twin +------------------------------------------------------------ prime twin Two positive integers p,p+2 are called prime twins if both p and p+2 are prime numbers. For example (3,5), (11,13) and 17,19 are prime twins. It is unknown, whether there are infinitely many prime twins. One knows that sum_i 1/p_i, where (p_i,p_i+2) are prime times is finite. +------------------------------------------------------------ | Pythagorean triple +------------------------------------------------------------ Three integers x,y,z form a Pythagorean triple if x^2+y^2=z^2. An example is 3^2+4^2=5^2. Pythagorean triples define triangles with a right angle and integer side lengths x,y,z. They were known and useful already by the Babylonians and used to triangulate rectangular regions. The Pythagorean triples with even x can be parameterized with p>q and x=2 p q, y=p^2-q^2, z=p^2+q^2. Each Pythagrean triple corresponds to rational points on the unit circle: X^2+Y^2=1, where X=x/z,Y=y/z. +------------------------------------------------------------ | relatively prime +------------------------------------------------------------ Two integers n and m are relatively prime if their greatest common divisor gcd(n,m) is 1. In other words, one does not find a common factor of n and m other than 1. +------------------------------------------------------------ | Sieve of Eratosthenes +------------------------------------------------------------ The Sieve of Eratosthenes allows to construct prime numbers. By sieving away all multiplies of 2,3, ..., N and listing what is left, one obtains a list of all the prime numbers smaller than N^2. For example: multiples of 2: 4,6,8,10,12,14,16,18,20,22,24,26,... multiplis of 3: 6,9,12,15,18,21,24,... multiples of 5: 10,15,20,25,... The numbers 2,3,5,7,11,13,17,19,23 do not appear in this list and are all the prime numbers smaller or equal than 5^2. To list all the prime number up to N^2, one would have to list all the multiples of k for k leq N. +------------------------------------------------------------ | quadratic residue +------------------------------------------------------------ A square modulo m, then n is called a A quadratic residue modulo m is an integer n which is a square modulo m. That is one can find an integer x such that n = x^2 (mod m). If m is not a quadratic residue, it is called a quadratic non-residue modulo n. Examples: 2 is a quadratic residue modulo 7 because 3^2 = 2 modulo 7. If p is an odd prime, then there are (p-1)/2 quadratic residues and (p-1)/2 quadratic nonresidues modulo p. +------------------------------------------------------------ | Legendre symbol +------------------------------------------------------------ The Legendre symbol encodes, whether n is a quadratic residue modulo a prime number p or not: Legendre(n,p)=1 if n is a quadratic residue and Legendre(n,p)=-1 if n is not a quadratic residue. If p is a prime number, then Legendre(-1,p) = (-1)^((p-1)/2) and Legendre(2,p) = (-1)^((p^2-1)/8). +------------------------------------------------------------ | law of quadratic reciprocity +------------------------------------------------------------ The law of quadratic reciprocity tells that if p,q are distinct odd prime numbers, then Legendre(p,q) . Legendre(q,p) = (-1)^n, where n = (p-1) (q-1)/4. Gauss called this result the "queen of number theory". The theorem implies that if p=3 (mod 4) and q=3 (mod 4), then exactly one of the two congruences x^2=p (mod q) or x^2=q (mod p) is solvable. Otherwise, either both or none is solvable. +------------------------------------------------------------ | Jacobi symbol +------------------------------------------------------------ Jacobi symbol If m=p_1...p_k is the prime factorization of m, define Jacobi(n,m) as the product of Legendre(n,p_i), where Legendre(n,p) denotes the Legendre symbol of n and p and m=p_1 .s p_k is the prime factorization of m. +------------------------------------------------------------ | Jacobi symbol +------------------------------------------------------------ A positive integer a which generates the multiplicative group modulo a prime number p is called a primitive root of p. Examples: a=2 is a primitive root modulo p=5, because 1=a^0,2=a^1,4=a^2,3=a^3 is already the list of elements in the multiplicative group of 5. a=4 is not a primitive root modulo p-5 because 4^2=1 mod 5. +------------------------------------------------------------ | Liouville +------------------------------------------------------------ An irrational number a is called Liouville if there exists for every integer m a sequence p_n/q_n of irreducible fractions such that lim_n to infinity q_n^m |a-p_n/q_n| = 0. Liouville numbers form a class of irrational numbers which can be approximated well by rational numbers. An example of a Liouville number is 0.10100100001000000001... ,where the number of zeros between the 1's grows exponentially. +------------------------------------------------------------ | M\"obius function +------------------------------------------------------------ The M"obius function mu is an example of multiplicative arithmetic function. It is defined as mu(n)= 1 n=1 (-1)^r n=p_1 . ... p_r, p_i0 such that for all 0 1 is connected but not locally connected because small neighborhoods of the point (0,1) are not connected. +------------------------------------------------------------ | Hausdorff +------------------------------------------------------------ A topological space (X,T) is called Hausdorff if for every two points x,y in X, there are disjoint open sets U,V in T such that x in U and y in V. This is refined through seperation axioms, T0, ..., T4. Hausdorff is also called T2. Any metric space is Hausdorff: if d is the distance betwen x and y, then balls of radius d/3 around x and y seperate the points. The plane X with semimetric d(x,y) = |x_1-y_1| is not Hausdorff: the points x=(0,-1) and y=(0,1) can not be seperated by open sets. +------------------------------------------------------------ | seperation axioms +------------------------------------------------------------ seperation axioms define classes of topological spaces with decreasing seperability properties: T4 Rightarrow T3 Rightarrow T2 Rightarrow T1 Rightarrow T0. T0 space: for two different points x,y in X one of the points has an open neighborhood U not containing the other point. T1 space: for two different points x,y in X there exists an open neighborhood U of x and an open neighborhood V of y. such that x is not in V and y is not in U. T2 space: also called Hausdorff" two different points x,y can be seperated with disjoint neighborhoods U,V. T3 space: T1 and regular: any point x and any closed set F not containing x can be seperated by two disjoint neighborhood. T4 space: T1 and normal: any two disjoint sets F,G can be separated by two disjoint open sets. It is known that a T4 space with a countable basis is metrizable. +------------------------------------------------------------ | Hausdorff topology +------------------------------------------------------------ The Hausdorff topology is a metric on the set of closed bounded subsets of a complete metric space. The distance between two sets A and B is the infimum over all r for which A is contained in a r-neighborhood of B and B is contained in a r-neighborhood of A. +------------------------------------------------------------ | Lindeloef +------------------------------------------------------------ A topological space is called Lindeloef if every open cover of X contains a countable subcover. +------------------------------------------------------------ | compact +------------------------------------------------------------ A topological space is called compact if every open cover of X contains a finite subcover. Examples of results known about compactnes: Heine-Borel theorem: a closed interval in the real line is compact. If f: X to Y is continuous and onto and X is compact, then Y is compact. As a consequence, a continouous function on a compact subspace has both a maximum and a minimum. In a Hausdorff space, compact sets are closed. In a metric space, compact sets are closed and bounded. Closed subsets of compact spaces are compact. Tychonof theorem: the product of a collection of compact spaces is compact. +------------------------------------------------------------ | countably compact +------------------------------------------------------------ A topological space is called countably compact if every countable open cover of X contains a finite subcover. +------------------------------------------------------------ | locally compact +------------------------------------------------------------ A topological space is called locally compact if every point has a neighborhood, which has a compact closure. Examples. The real line is compact but not locally compact. A compact Hausdorff space is locally compact. The n-dimensional Euclidean space R^n is lcally compact but not compact. +------------------------------------------------------------ | locally compact +------------------------------------------------------------ A set U of open sets in a topological space (X,O) is called locally finite if every point x in X has a neighborhood V, such that V has a nonempty intersection with only finitely many elements in U. +------------------------------------------------------------ | paracompact +------------------------------------------------------------ A topological space (X,O) is called paracompact if every open cover has a countable, locally finite subcover. +------------------------------------------------------------ | relatively compact +------------------------------------------------------------ A subset A of a topological space (X,T) is called relatively compact if the closure of A is compact. +------------------------------------------------------------ | filter +------------------------------------------------------------ A filter on a nonempty set X is a set of subsets F satisfying X is in F, but the empty set emptyset is not in F. If A and B are in F, then their intersection is in F. If A is in F and B is a subset of A, then B is in F. Examples: Principal filter for a nonempty subset A consists of all subsets of X which contain A. Frechet filter for an infinite set consists of all subsets of X such that their complement is finite. Neighborhood filter of a point x in a topological space (X,T) is the set of open neighborhoods of x. Elementary filter for a sequence x_n in X consists of all sets A in X such that x_n is in A for large enough n. +------------------------------------------------------------ | converges +------------------------------------------------------------ A sequence x_n in a topological space converges to a point x, if for every neighborhood U of x, there exists an integer m, such that for n>m one has x_n in U. +------------------------------------------------------------ | Filter convergence +------------------------------------------------------------ Filter convergence A filter F converges to x in a topological space (X,T) if F contains the neighborhood filter G of x, that is if F contains all neighborhoods of x. For example, an elementary filter to a sequence x_n converges to a point x, if and only if x_n converges to x. +------------------------------------------------------------ | accumulation point +------------------------------------------------------------ A point y is called an accumulation point of a filter F, if there exists a filter G containing F such that G converges to x. +------------------------------------------------------------ | directed +------------------------------------------------------------ A set M is called directed if there exists a partial order (M,<) on M satisfying for every two points a,b in M there exists c, with a0, B_r(x)= y | d(x,y) X, where D is a directed set. For example: if D is the set of natural numbers, then a net is a sequence. A net defines a filter F: it is the set of all sets A such that x_t is eventually in A. A net x_t converges to a point x if and only if the associated filter converges to x. +------------------------------------------------------------ | open cover +------------------------------------------------------------ open cover A subset U of O, where (X,O) is a topological space is called an open cover of X if the union of all elements in U is X. If U and V are open covers and V subset U, then V is called a subcover of U. +------------------------------------------------------------ | product space +------------------------------------------------------------ The product space between topological spaces is defined as (X x Y, O x P), where X x Y is the set of all pairs (x,y), x in X, y in Y and O x P is the coarsest topological space which contains all products A x B, where A in O and B in P. For example, if (X,O)=(Y,P) are both the real line with the topology generated by d(x,y) = |x-y|, then the product space is homeomorphic to the plane with the metric d(x,y) = sqrt(x_1-x_2)^2+(y_1-y_2)^2. +------------------------------------------------------------ | second countable +------------------------------------------------------------ A topological space is called second countable, if it has a countable basis. Example. Every seperable metric space is second countable. Especially, every finite-dimensional Euclidean space is second countable. +------------------------------------------------------------ | metrizable +------------------------------------------------------------ A topological space is called metrizable if there exists a metric d on the set X that induces the topology of X. Any regular space with a countable basis is metrizable. +------------------------------------------------------------ | homotopic +------------------------------------------------------------ homotopic If f and g are continuous maps from the topological space X to a topological space Y, we say that f is homotopic to g if there is a continuous map F from X x I to Y, such that F(x,0) = f(x) and F(x,1) = g(x) for all x. For example, the maps f(x) = x^2 and g(x) = sin(x) on the real line are homotopic, because we can define F(x,t) = (1-t)x^2 + t sin(x). The maps f(x) = x and g(x) = sin(2 pi x) on the circle are not homotopic. While g is homotopic to the constant function h(x)=0, the map f(x) can not be deformed to a constant without breaking continuity. +------------------------------------------------------------ | induced topology +------------------------------------------------------------ The induced topology on a subset A of X, where (X,T) is a topological spoace is the the topological space (A, Y cap A _Y in T). +------------------------------------------------------------ | path homotopic +------------------------------------------------------------ path homotopic If f and g are and continuous homotopic maps from an interval to a space X, we say f and g are path homotopic if their images have the same end points. For instance, the maps f(x) = x^2 and g(x) = x^3 are path homotopic on the closed interval from 0 to 1. The maps f(x)=2 x^2 and g(x)=x^3 are homotopic on the unit interval but not path homotopic. +------------------------------------------------------------ | loop +------------------------------------------------------------ A loop is a path in a topological space that begins and ends at the same point. A loop is also called a closed curve. Loops play a role in definitions like simply connected: a topological space is simply connected if every loop is homotopic to a constant loop which is a fancy way telling that every closed path can be collapsed inside X to a point. +------------------------------------------------------------ | fundamental group +------------------------------------------------------------ The fundamental group of a topological space at a point is the set of homotopy classes of loops based at that point. +------------------------------------------------------------ | Topologist's Sine Curve +------------------------------------------------------------ The Topologist's Sine Curve is the union S of the graph of the function sin(1/x) on the positive real axes R^+ with the y-axes. It an example of a topological space which is connected but not path-connected. Proof: if S were path-connected, there would exist a path r(t)=(x(t),y(t)) connecting the two points (0,1) and (0,pi). The set t | r(t) in S is closed. Let T be the largest t in that set for which r(t) is in the y-axes. Then x(T)=0 and r(t)=(x(t),sin(1/x(t)) for t>T. Because there are times t_n >t_n-1>T, t_n to T for which y(t_n)= (-1)^n, the function r(t) can not be continuous at t=T. +------------------------------------------------------------ | Urysohn lemma +------------------------------------------------------------ The Urysohn lemma tells that if X is a normal space and A and B are disjoint closed subsets of X, then there exists a continuous map f from X to the unit interval such that f(x) = 0 for all x in A, and f(x) = 1 for all x in B. Proof: use the normality of X to construct a family U_p of open sets of X indexed by the rational numbers P in the unit interval so that for p