# Archive | work

## Protected: Curvilinear Origami

There is no excerpt because this is a protected post.

## Math 487, midterm, solutions

$$\def\Real{\mathbb{R}} \def\image{\mathtt{Im}}$$

1. [10] Find the dimensions of image and kernel for the operator $$A:\Real^5\to\Real^4$$ given by the matrix
$$A=\left(\begin{array}{ccccc} 0&1&2&3&4\\ 1&2&3&4&5\\ 2&3&4&5&6\\ 3&4&5&6&7 \end{array}\right).$$

If one chooses the basis
$$e’_1=e_1, e’_2=e_2, e’_3=e_3-2e_2+e_1,e’_4=e_4-2e_3+e_2,e’_5=e_5-2e_4+e_3,$$
one sees immediately, that

## 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 $$o(x_1,\ldots,x_n)$$) …

## 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

## 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.

## ECE 515, 9.11

Lecture notes, 3.1-3.5.

• Cayley-Hamilton Theorem;
• Matrix exponentials;
• Solutions to linear systems differential equations.

## Math 487, Homework 1


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


## 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.

## 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.

## 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, …