About discrete calculus

• Three tweets (the first example appeared in the Mock exam on Wednesday, the second was mentioned on Friday, the third is new).
• Watching the Schroedinger equation: ih u' = K u, where K is the Hamiltonian: the solution U(t)=exp(-K I t), which is a complex matrix. U(t) ψ gives you the wave at time t in the Schroedinger picture. U(t) A U(t)^* for a matrix A gives you the observable A(t) in the Heisenberg picture. There is really not much more to quantum mechanics. Except that people get crazy about Schroedinger cats. In reality, quantum mechanics can be summed up in one line of code:

  K=KirchhoffMatrix[GridGraph[{9,9}]];Animate[MatrixPlot[Abs[MatrixExp[-K*I*t]]],{t,0,9}]
  

• Final Practice exams and a review sheet come soon. Midterm exams are posted on the exams page.
• Since the Friday lecture was recorded without sound (I forgot to turn on the mic), here are some slides from the past
• The easter bunny appears in this video
• Here is a document from 2012 which covers three important theorems in the discrete. An in
• this document from 2014 (talk of 2013), all single variable calculus is on a single page and all multivariable calculus on one page. it is mathematics which the Greeks could have done, if they had the concept of functions and the concept of a graph with nodes and connections.

Office hours about proofs are all in SC 530.

• Monday 2-3 PM Rosie (starting January 31th)
• Wednesday 3-4 PM Kai (starting February 2nd)
• Friday 2-3 PM Charles (starting February 4th)

About the first proof unit: Unit 3

In unit 3, you saw some pitfalls of making definitions not clear. One of the themes which appeared was the zeta function. This is the function
&zeta(s) = 1^-s + 2^-s + 3 ^{-s + ....
You might recall from calculus that this series only converges for s larger than 1. For s=1, it is the harmonic series which diverges. For s=0 it is the series
&zeta(0) = 1+1+1+ ...
Rosie has mentioned
&zeta(-1) = 1+ 2 + 3 + 4 + ...
which after some manipulation became -1/12. Actually, this is the correct value of the zeta function. In order to make sense of this, the function ζ(s) has in the complex plane analytically continued. You can enter Zeta[0] into Mathematica and get -1/2 or you can enter Zeta[-1] and get the value -1/12. A popular exposition about this is here:}