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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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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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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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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, Homework 1

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

  1. (15) Find all matrices
    $$
    X=\left(
    \begin{array}{cc}
    x&y\\
    z&w
    \end{array}
    \right)
    $$
    such that \(XA=BX\), where
    $$
    A=\left(
    \begin{array}{cc}
    1&2\\
    -1&0
    \end{array}
    \right), \mathrm{\ and\ }
    B=\left(
    \begin{array}{cc}
    0&1\\
    3&0
    \end{array}
    \right).
    $$
  2. (15) Same for
    $$
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ECE 515, 9.6

Following lecture notes, 2.6, 2.7:

  • Eigenvalues and eigenvectors;
  • Operators with distinct eigenvalues; diagonalization;
  • Failure to diagonalize leads to considering chains of subspaces \(V^{(k)}_\lambda:={\mathtt{Ker}} (A_\lambda)^k, A_\lambda:=A-\lambda E\), which lead to
  • Jordan normal form.
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Math 487, 9.5

Continuing to use Olver’s book, chapter 1, but also relying on Prasolov’s notes on linear algebra.

  • Determinant (as a function from \({\mathtt{Mat}}(n\times n;\mathbf{k})\): existence and properties.
  • Matrix is singular iff its determinant vanishes.
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Fact of the day: Condorset domains, tiling permutations and contractibility

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

Consider a collection of vectors \(e_1,\ldots,e_n\) in the upper half-plane, such that \(e_k=(x_k,1)\) and \(x_1>x_2\gt \ldots \gt x_n\). Minkowski sum of the segments \(s_k:=[0,e_k]\) is a zonotope \(\Z\). Rhombus in this context are the Minkowski sums \(\Z(k,l)=s_k\oplus s_l, …

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day 10

Social choice theory, reminder.

Topological version of social choice theory. Chichilnisky theory of manipulability. Arrow’s impossibility theorem as a topological result.

Topological aspects of clustering. Kleinberg’s impossibility theorem. Carlsson-Memoli’s characterization of simple link clustering.…

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day 9

\(\def\Real{\mathbb{R}}\)

Data aggregation. Averaging. Examples.

Axioms: anonymity, unanimity.

More topology: Kunneth formula.

Nonexistence of averaging on spheres.

General result (Eckmann): averagings of all orders on tame topological spaces exists only when the space is contractible.…

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