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Fact of the day: Brownian centroids

\(\def\Real{\mathbb{R}}\def\Z{\mathcal{Z}}\def\sg{\mathfrak{S}}\def\B{\mathbf{B}}\def\xx{\mathbf{x}}\def\ex{\mathbb{E}}
\)

Consider \(n\) points in Euclidean space, \(\xx=\{x_1,\ldots, x_n\}, x_k\in \Real^d, n\leq d+1\).

Generically, there is a unique sphere in the affine space spanned by those points, containing all of them. This centroid (which we will denote as \(o(x_1,\ldots,x_n)\)) …

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ECE 515, 9.18

Covering course notes, 3.6-8 and 2-8-10.

  • Peano-Baker series for fundamental matrices, – properties, convergence, role of commutativity;
  • Solutions for forced systems for LTV systems;
  • General classification of spaces of solutions of LTI;

Quadratic and Hermitian forms, norms, self-adjoint operators and …

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ECE 515, Homework 1, solutions.

\(\def\Real{\mathbb{R}}\def\ee{\mathbf{e}}\def\ff{\mathbf{f}}\newcommand\ket[1]{\vert{#1}\rangle}\newcommand\bra[1]{\langle{#1}\vert}\newcommand\bk[2]{\langle #1\vert#2\rangle}\def\lin{\mathtt{Lin}} \)

  1. (30) Consider the dynamical system
    $$
    \dot{x}=x-(x+y)^2/4;\\
    \dot{y}=y-(x+y)^2/4.
    $$
    1. Find equilibrium points.
      Setting right hand sides to zero, we obtain that \(x=y\), and, consequently, \(x=x^2\). Hence the equilibria are \((0,0)\) and \((1,1)\).
    2. Linearize the system near the
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Math 487, 9.17

Underlying text: DiAngelo’s book, chapter 1.

  • Fields, – definition, examples: \(\mathbb{Q, R, C, Z}_p\).
  • General linear spaces (matrices, functions with values in a field).
  • Linear operators.

Exercises:

  • Find the image of the linear operators \(A,B:U\to U\) on \(U\), the
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Math 487, 9.14

Material from chapter 9 of Stanley’s book.

  • Graphs and their Laplacians
  • Spanning trees
  • Matrix-tree theorem
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ECE 515, 9.13

Lecture notes, 3.1-3.5.

  • Solutions of non-homogenous linear systems;
  • Exponentials of diagonal and diagonalizable operators and matrices;
  • Lagrange approximations and reduction of matrix functions to polynomials;
  • Non-autonomous systems and Picard approximations.
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Math 487, 9.12

Still on Prasolov’s notes on linear algebra, chapter 1.

  • Circulant matrices and determinants.
  • Minors and applications.
    • Definitions. Cofactors.
    • Laplace expansion.
    • Binet-Cauchy theorem.
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ECE 515, 9.11

Lecture notes, 3.1-3.5.

  • Cayley-Hamilton Theorem;
  • Matrix exponentials;
  • Solutions to linear systems differential equations.
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Math 487, 9.10

Mostly following Prasolov’s notes on linear algebra, chapter 1.

  • Vandermonde determinants.
  • Applications for Lagrange interpolation.
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Fact of the day: dice are chiral

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