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ECE 490, week of Jan 21

Following Chapter 2 of Guler.

Class notes (pretty incomplete, to be used just as a study guide), here and here.

Exercises:

  • Does there exist a differentiable function on unit interval with infinitely many global maxima and and global minima?
  • Does there exist a function (on real plane) with infinitely many global maxima but no local minima?
  • If a function on real line has a local minimum at \(x\); does it imply that for some \(\epsilon>0\) the function increases on \([x,x+\epsilon)\), decreases on \((x-\epsilon,x]\)?
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ECE 490, week of Jan. 14

\(\def\Real{\mathbb{R}}\)

Course outline is here.

Taylor formula

Spaces of differentiable functions on an interval \(I=[a,b]\subset\Real\):
\[
C(I,\Real), C^1(I,\Real),\ldots, C^n(I,\Real),\ldots
\]

For a function in \(C^1(I,\Real)\),
\[
f(y)=f(x)+\int_x^yf'(s)ds.
\]
Iterating (for functions in \(C^n(I,\Real)\), we obtain
\[
f(y)=f(x)+f'(x)(y-x)+\frac{f”(x)}{2!}(y-x)^2+\ldots+\frac{f^{(n-1)}(x)}{(n-1)!}(y-x)^{n-1}+
\int_{x<s_1<s_2<\ldots<s_n<y}f^{(n)}(s_1)\frac{(y-s_1)^{n-1}}{(n-1)!}ds.
\]

Implications:

  • Mean value theorem
  • If \(f”\geq 0\) on \(I\) and \(f'(x)=0\), then \(x\) is a global minimum of \(f\).

Several variables

Domain: \(U\subset\Real^n\), an open set.

Reminder: open, closed, bounded, compact sets.

Exercise: is the set \(\{|x|\leq 1, |y| \lt 1\}\) open?…

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ECE 515, mock final

Here.…

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ECE 515, week of Dec. 3

\(\def\Real{\mathbb{R}}\)

  • Pontryagin maximum principle
    • Examples
    • Lagrangians to Hamiltonians: Fenchel-Legendre transform.

      Legendre (sometimes called Legendre-Fenchel) transform \(H\) of a convex function \(L:\Real\to\Real\)) is defined as
      \[
      H(p):=\sup_x px-L(x).
      \]
      Both functions, \(L\) and \(H\) can take value \(+\infty\).

      If \(L\) is strictly convex, the gradient mapping
      \[
      G:x\mapsto \frac{dL}{dx}
      \]
      is one-to-one, and it is clear that the argmax in the definition of \(H\) solves \(p=G(x)\). Denote the functional inverse to \(G\) as \(F: p=G(x) \Leftrightarrow x=F(p)\). In this case,
      \[
      H(p)=pF(p)-L(F(p)).

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Math 487, week of Dec 3

  • Fourier series: smoothness and decay of coefficients.
  • Fourier transform (following chapter 18 of Zorich, Mathematical Analysis, 2).
    • Motivation
    • Examples
    • Properties
    • Kotelnikov formula
  • Asymptotics of integrals (chapter 19 of ibid.)
    • Laplace method
    • Localization principle
    • Standard integrals
    • Applications: Stirling formula
    • Tauberian theorems: regularly varying functions
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Math 487, Homework 5

\(\def\Res{\mathtt{Res}}\)

  1. Find function with Laplace transform equal to \[\frac{s^3 – 1}{(1 + s)^4}.\]
  2. Find Fourier coefficients of the function \(t^3-\pi^2 t\) on \([-\pi,\pi]\).
    Use Parseval’s identity to find \[\sum_{n=1}^\infty 1/n^6.\]
  3. Find general formula for the coefficients of the generating function \(W(z)\) given by \(W=1+zW^4\).
  4. Find asymptotic frequency of the numbers \(n\) such that the first digits of \(2^n\) and \(3^n\) coincide.
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ECE 515, Homework 5

\(\def\Res{\mathtt{Res}}\def\Real{\mathbb{R}}\)

    1. Consider optimal control problem for \(x\in\Real, u\in\Real\):
      \[
      \dot{x}=u,\quad J_0^T=\int_0^T\frac{u^2}{2}dt +\cos(x(T))\to\min.
      \]
      • [20]
        Find maximal \(T\) for which the cost function (referred to as \(S\) in the course notes) at time \(0\),
        \[
        V(x, 0)=inf_{\{u(t)\}_{0\leq t\leq T}, x(0)=x}\ J_0^T
        \]
        is smooth as a function of the initial point \(x\) (i.e. has continuous derivative).

        Solution. As we discussed, the cost function for the problem above is given by
        \[
        V(x,0)=\min_X C(x,X,T), \mathrm{where} C(x,X,T)=\frac{(x-X)^2}{2T}+\cos(X)
        \]
        (here \(X=x(T)\) is the potential position of the end of the trajectory, and \((x-X)^2/2T\) is the minimal cost of reaching it from \(x\)).

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Math 487, Nov. 30

  • Inverting Fourier transform.
  • “Uncertainty principle”.
  • Introduction to duality – asymptotics vs. local behavior
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ECE 515, Nov. 29

  • Kalman identity and some corollaries
  • Introduction to the minimum principle
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Math 487, Nov. 28

  • Fourier series
  • Elements of Fourier transform
  • Parseval theorem.
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