Archive | work

Topologically constrained models of statistical physics.

$$\def\Real{\mathbb{R}} \def\Int{\mathbb{Z}} \def\Comp{\mathbb{C}} \def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}} \def\Fun{\mathbf{Fun}} \def\e{\mathbf{e}} \def\f{\mathbf{f}} \def\bv{\mathbf{v}} \def\blob{\mathcal{B}}$$

Consider the following planar “spin model”: the state of the system is a function from $$\Int^2$$ into $$\{0,1\}$$ (on and off states). We interpret the site $$(i,j), … CDC’15 Slides are here.… Topology in Motion @ICERM Fall’16 If you are interested in Applied Topology, tend to plan far, far ahead and have some free time on your hands in Fall 2016, check this out, and let us know! math285, week of october 27 Read the textbook, chapters 9.1-9.3. Homework (due by Tuesday, 11.6): 1. Find Fourier transform of the function on \((-\pi,\pi)$$ equal to $$0$$ for $$-\pi<x<0$$ and to $$1$$ for $$0\leq x<\pi$$.
2. Find cosine expansion of $$\sin(2x)$$ on

september 23

$$\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}} \def\Fun{\mathbf{Fun}} \def\e{\mathbf{e}} \def\f{\mathbf{f}}$$

Global asymptotic stability for LTI.

Equivalent to local stability.

Hurwitz operators. Multiple eigenvalues and Jordan normal forms.

Lyapunov’s direct method

Positive definite functions.

Lyapunov functions, definition.

Lyapunov’s theorem.

Convenient Lyapunov functions: homogeneous ones.

Strict homogeneous …

dimension of the Internet

As reported in the talk on the hyperbolic geometry of maps and networks, the often claimed approximability of the Internet graph (the routing graph as seen by Border Gateway Protocol) by samples from disks of large radius in the …

september 2

$$\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}}$$

Fields, vector spaces, subspaces, linear operators, range space, null space

A field $$(F,+,·)$$ is a set with two commutative associative operations with $$0$$ and$$1$$, additive and multiplicative inverses and subject to the distributive law.

Examples (Common fields)

$$\def\Real{\mathbb{R}} \def\views{\mathbb{G}} \def\earth{\mathbb{E}} \def\hsp{\mathbb{H}}$$