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math285: the week of september 22

There will be no new homework: review the material covered thus far for the upcoming midterm (on 9.29). The midterm problems will be similar to those addressed in homework and in the class.…

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dimension of the Internet

As reported in the talk on the hyperbolic geometry of maps and networks, the often claimed approximability of the Internet graph (the routing graph as seen by Border Gateway Protocol) by samples from disks of large radius in the …

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september 19

  1. Sketch the vector fields for each value of the parameter; indicate equilibria and their stability:
    • \[
      x’=3\sin(x)+x+a, a=-3,0,3;
      \]
    • \[
      x’=\tan(x)-3x+a, a=-5,0,5;
      \]
    • \[
      x’=e^x-x+a, a=-1,0,1;
      \]
  2. Find the three consecutive Picard iterations for the equation.
    Solve the differential equation;
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ece515: homework 2

\(\def\Real{\mathbb{R}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}
\)
Due by midnight of september 25.

  1. Consider the space \(V\) of polynomials (with real coefficients) of degree \(\leq 4\), and the operator \(D:V\to V\) of differentiation. Compute
    \[
    \exp(tD).
    \]
  2. Let
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september 18

\(\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}\)

Remarks
  • What operators can appear as state transition (fundamental) ones, \(\Phi(t)\)?
  • For LTVs, any operator \(\Phi\) with \(\det(\Phi)>0\) can be a fundamental solution; for LTIs the situation is far more complicated (say, the

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september 16

\(\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}\)

linear time dependent systems

Generalizing the notion of state transition matrix, we can address also the time-dependent systems
\[
\dot{x}(t)=A(t)x(t).
\]
Here, we dove the equation on \([t_1,t_2]\) with \(x(t_1)=x\); the value of solution at …

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math285: the week of september 15

Review sections 2.2, 2.4, 2.6 of the textbook (Edwards and Penney). Listen to the video lectures.

Additional reading (highly recommended!): chapter 1 of Ordinary differential equations : a practical guide by
Bernd J. Schroers (CUP, 2011).

Homework (due by …

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september 12

  1. Solve linear ODE of 1st order:
    1. \[
      y’+y\tan{x}=\sec{x};
      \]
    2. \[
      xy+e^x=xy’;
      \]
    3. \[
      (xy’-1)\ln{x}=2y.
      \]
  2. Reduce the ODE to a linear one, and solve:
    1. \[
      (x+1)(y’+y^2)=-y;
      \]
    2. \[
      xy^2y’=^2+y^3;
      \]
    3. \[
      xyy’=y^2+x.
      \]
  3. Reduce order of the equation, and solve:
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quick sort

  • Intro: sorting; bubble sort
  • Lower bound: information based
    \[
    \log{n!}\approx n\log{n}.
    \]
  • Idea: divide and conquer (Hoare).
  • Quicksort and partition.
  • Average case analysis. Address the number of comparisons
  • Connection to binary search trees.
  • Expected number of comparisons: fundamental recursion
    \[
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september 11

\(\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}\)

matrix exponential

which is instrumental in finding solutions to linear systems of differential equations.

Computing the matrix exponentials can be done by just taking power series (not advised),

or by using the Jordan normal forms…

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