# Archive | research

## 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 the the images of the first two vectors are linearly independent, and of the last three, vanish. Hence $$\dim\ker(A)=3; \dim\image(D)=2$$.

2. [30] Consider complete graph on vertices $$v_1, v_2, v_3, v_4, v_5$$. How many spanning trees (out of $$5^3=125$$) contain the edge $$(v_1,v_2)$$?

## 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 $$o(x_1,\ldots,x_n)$$) lies at the intersection of the bisectors $$H_{kl}, 1\leq k\lt l\leq n$$, hyperplanes of points equidistant from $$x_k, x_l$$.

Assume now that the points $$x_k,k=1,\ldots,n$$ are independent standard $$d$$-dimensional Brownian motions. What is the law of the centroid?

It quite easy to see that it is a continuous martingale.…

## 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 space of real functions on $latex (-\infty, infty)$ given by
$$(Af)(x)=f(x)-f(-x); (Bf)(x)=f(x)+f(-x).$$
• Find the kernel and the image of the operator of differentiation of functions on an interval.
• The same for the operator of differentiation of the functions on the unit circle.

## 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.