III.32 Generating Functions, III.33 Genus, III.34 Graphs, III.35 Hamiltonians, by Terence Tao, III.36 The Heat Equation, Igor Rodnianski: Pages 214 – 216.

A generating function may be created from a sequence of numbers

f(x) = a0 + a1x1 + a2x2 +  an-1xn-1 + … + an-1xn-1 + anxn + … .

For the triangulation of an orientable surface count the vertices’s, edges and faces as: V, E and F.  Let g be the genus then the Euler characteristic is: V – E + F = 2 – 2 x g.  The genus, g, corresponds to the number of “holes” in the surface.

Graphs have nodes and links.  Graph theory is used to study these graphs or networks including the world wide web Internet.

Hamiltonians are used to tie together many otherwise diverse equations of areas of physics, including: classical and quantum mechanics, nonrelativistic and relativistic physics and particle physics with statistical mechanics.  In each of these the evolution of a physical system over time, and the steady state, is largely controlled by the Hamiltonian.

The Hamiltonian, H, for the position, u, and momenta, p, of a system is incorporated into the Hamilton equations: q/t = H/p, p/t = –H/q.

The Hamiltonian in the Schrodinger wave equation is: i Ñ dy/dt= Hy.

The classical heat equation is: u(t,x)/t – Du(t,x) = 0.

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