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midterm redo

Posted on compass. Reply here if any questions arise.

(Corrected) solutions are here.…

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Math 487, Oct. 31

  • Analytic functions.
  • Cauchy-Riemann equations.
  • Harmonic functions and CR equations.
  • Integrals along a curve.
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ECE 515, Oct. 30

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

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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.
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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 \((v_1,v_2)\)?
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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.…

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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.
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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.
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ECE 515, 9.11

Lecture notes, 3.1-3.5.

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