(Corrected) solutions are here.…

]]>(Corrected) solutions are here.

]]>$$

\omega=\frac{(x-a) dy-(y-b) dx}{(x-a)^2+(y-b)^2}:

$$

Winding numbers around \((a,b)\).

Complex analysis:

- Complex analytic functions.

$$

\omega=\frac{(x-a) dy-(y-b) dx}{(x-a)^2+(y-b)^2}:

$$

Winding numbers around \((a,b)\).

Complex analysis:

- Complex analytic functions.

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- [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 \((v_1,v_2)\)?

- [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 \((v_1,v_2)\)?
*Spanning tree that contains an edge is the same as a spanning tree in the graph with that edge contracted to one vertex \(v_*\). In this new graph (on vertices \(v_*, v_2, v_3, v_4\)), the numbers of edges between \(v_*\) and \(v_k, k\geq 2\) is \(2\), and any pair of other vertex are connected by a single edge. Using the Cayley formula for the minor of the Laplacian obtained by excluding \(v_*\) we get*

$$

N=\det\left(

\begin{array}{ccc}

4&-1&-1\\

-1&4&-1\\

-1&-1&4

\end{array}

\right)=50.

$$ - [20] Diagonalize (if possible) the following matrix, – that is, find the decomposition \(A=MDM^{-1}, D – \mathit{diagonal}\), where

$$

A=\left(\begin{array}{cc}2&1\\1&1\end{array}\right).

$$*We find the characteristic polynomial \(pA=\lambda^2-3\lambda+1\) and the roots \(\lambda_{1,2}=(3\pm\sqrt{5})/2\). The roots are different, matrix is diagonalizable, and the eigenvectors are*

$$

v_1\in\image(A_{\lambda_2})=\left(

\begin{array}{c}

\frac{1+\sqrt{5}}{2}\\

1

\end{array}

\right);

v_2\in\image(A_{\lambda_1}); v_2=\left(

\begin{array}{c}

-1\\

\frac{1+\sqrt{5}}{2}

\end{array}

\right)

$$

Hence,

$$

A=\left(\begin{array}{cc}

\phi&-1\\

1&\phi

\end{array}

\right)

\left(

\begin{array}{cc}

\frac{3+\sqrt{5}}{2}&0\\

0&\frac{3-\sqrt{5}}{2}

\end{array}

\right)

\left(

\begin{array}{cc}

\frac{\phi}{\phi^2+1}&\frac{1}{\phi^2+1}\\

\frac{-1}{\phi^2+1}&\frac{\phi}{\phi^2+1}

\end{array}

\right),

$$

where \(\phi=(1+\sqrt{5})/2\) is the golden ratio. - [30] Find \(\exp(3B)\) for

$$

B=\left(\begin{array}{cc}2&3\\-1&2\end{array}\right).

$$*Representing \(B=2E+C\), where*

$$

C=\left(

\begin{array}{cc}

0&3\\

-1&0

\end{array}

\right)

$$

we see that \(\exp(3B)=e^6\exp(3C)\).

To find \(\exp(3C)\) we look for a linear function \(z\mapsto b_0+b_1z\) matching \(\exp(3z)\) on the spectrum of \(C\), that is on \(\pm\sqrt{3}i\).*One easily finds \(b_0=\cos(3\sqrt{3}); b_1=\sin(3\sqrt{3})/\sqrt{3}\), so that*

$$

e^6\exp(3C)=e^6(b_0E+b_1C)=

e^6\left(

\begin{array}{cc}

\cos(3\sqrt{3})&\sqrt{3}\sin(3\sqrt{3})\\

-\sin(3\sqrt{3})/\sqrt{3}&\cos(3\sqrt{3})

\end{array}

\right).

$$ - [10] Find eigenbasis for the operator \(Cp=x\frac{dp}{dx}-2p\) acting on the space of the polynomials of degree \(\leq 4\).
*In the standard basis \(e_1=1,\ldots, e_5=x^4\), the operator is diagonal with eigenvalues \(\lambda_k=(k-3)\).*

\)

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.…

]]>\)

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. What is the quadratic variation of \(o\)?

If the directions orthogonal to the affine span \(L(\xx)\) of the tuple, the answer is immediate: it is proportional to

$$

\sum_{k=1}^n p^2_k,

$$

where \(p_k\) are the barycentric coordinates of \(o\) with respect to \(x_1,\ldots,x_n\). Indeed, it is immediate that if \(dx_k^\perp\) are the components of the Brownian increments orthogonal to \(L(\xx)\), then

$$

do^\perp(\xx)=\sum_k p_k dx_k^\perp.

$$

It follows -as \(dx_k^\perp\) are independent and uncorrelated) that the component of \(o\) orthogonal to \(L(\xx)\) is a martingale with variation equal to (Euclidean norm on \(L(\xx)^\perp\) times) \(\sum_k p_k^2\).

Let’s turn to the motion of \(o(\xx)\) inside \(L(\xx)\). Let \(S(\xx)\subset L(\xx)\) be the sphere centered at the centroid and containing all the points of the sample. Let \(dx_k^\parallel=\xi_k+\eta_k\) be the decomposition of the \(L(\xx)\) component of \(dx_k\) into the vector in the tangent space \(\xi_k\in T_{x_k}S(\xx)\), and the vector orthogonal to it. If we denote by \(n_k=(x_k-o(\xx))/|x_k-o(\xx)|\) the unit norm vector from the centroid to \(k\)-th point of the sample, then \(\eta_k=s_k\cdot n_k\), where \(s_k\) is the increment of a Brownian motion independent of all all other components (as follows from the orthogonalities hardwired in the construction).

Now, the condition that as the points of the sample move, the distances to the centroid remain constant is equivalent to

$$

\langle \eta_k-\nu,n_k\rangle=\langle \eta_l-\nu,n_l\rangle,

$$

for all \(1\leq l,k\leq n\). (Here we denote by

$$

\nu:=do(\xx)^\parallel

$$

the \(L(\xx)\)-component of \(do(\xx)\).)

Equivalently, this means that \(\langle \nu, n_k-n_l\rangle=s_k-s_l\).

As \(s_k, s_l\) are independent, with unit quadratic variation, we conclude that

$$

\langle (n_k-n_l),C (n_k-n_l)\rangle=2

$$

for all \(k,l\), – here

$$

C:=\ex \nu\otimes\nu’

$$

is the quadratic variation of the centroid in the affine plane spanned by the sample.

Geometrically, an equivalent description is that \(C=AA’\), where \(A\) is the linear part of the affine transformation taking the points of the sample to the regular \(n\)-simplex with sides \(\sqrt{2}\).

]]>- 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.

- 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.

- 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.

- 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.

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

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