# Archive | work

## midterm redo

Posted on compass. Reply here if any questions arise.

(Corrected) solutions are here.…

## Math 487, Oct. 31

• Analytic functions.
• Cauchy-Riemann equations.
• Harmonic functions and CR equations.
• Integrals along a curve.

## ECE 515, Oct. 30

Disturbance decoupling (rejection). Based on Trentelman et al, Control theory for linear systems, ch. 4.…

## Math 487, Oct. 29

• Potential forms. Closed forms that are not potential. Invariance of the integrals of closed forms over loops under the deformation of the loops.
• Example:
$$\omega=\frac{(x-a) dy-(y-b) dx}{(x-a)^2+(y-b)^2}:$$
Winding numbers around $$(a,b)$$.

Complex analysis:

• Complex analytic functions.

## Protected: Curvilinear Origami

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## 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.