Posted on compass. Reply here if any questions arise.

(Corrected) solutions are here.…

Posted on compass. Reply here if any questions arise.

(Corrected) solutions are here.…

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

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

- 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.

There is no excerpt because this is a protected post.

\(\def\Real{\mathbb{R}} \def\image{\mathtt{Im}}\)

- [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 \(\dim\ker(A)=3; \dim\image(D)=2\). - [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 \((v_1,v_2)\)?

\(\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.…

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.

- 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.

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.

Lecture notes, 3.1-3.5.

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

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