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arrieregarde battles of postmodernism

The epicenter of the battle around Salaita’s offer withdrawal seems to be located at the question, whether or not political considerations are admissible when appointing a scholar. I checked what Dr. Salaita himself writes about the matter in a 2008

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september 9

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

Linear operators; normal forms

Linear operators between different spaces are easy to normalize: the rank is the only invariant; the classification is discrete.

For endomorphisms, the situation is more involved.

Diagonalization corresponds to splitting the …

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math 285: the week of september 8

Review Chapter 1.6, 2.2 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 Monday,

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ece515, homework1 (due 9.11).

\(\def\Real{\mathbb{R}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}
\)
Consider the space \(V\) of polynomials (with real coefficients) of degree \(\leq 4\). There exists a natural basis in \(V\), comprised of monomials – we will denote this basis as \(\e\)).

  1. Is
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september 5

  1. Solve separable ODEs:
    1. \[
      \sqrt{y^2+1}=xyy’;
      \]
    2. \[
      y’=\cos(y-x)
      \]
    3. \[
      y’-y=2x-3
      \]
  2. Solve
    1. \[
      (x-y)+(x+y)y’=0
      \]
    2. \[
      2x^3y’=y(2x^2-y^2)
      \]
    3. \[
      xy’-y=x\tan{y/x}.
      \]
  3. Sketch the slope fields and a few trajectories for
    1. \[
      y’=y^2-x;
      \]
    2. \[
      y’=y-x^2;
      \]
    3. \[
      y’=y^2+x^2-3.
      \]
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september 4

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

Linear operators

Their range=image, null-space=kernel (both linear subspaces).

Rank=dimension of the image (if finite-dimensional).

Important: dimension of the kernel + rank=dimension of the domain…

Example:

\(V=\Fun(S,\Field)\), where, as before \(S=\{0,1,2,3,4,5\}\subset\Comp\). Let \(Z\) is the operator …

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math 285: the week of september 1

Review Chapter 1.2-1.5. Listen to the lectures (four videos altogether).

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

Homework (due by Monday, 9.8):

  1. Find general
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september 2

\(\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\)

Fields, vector spaces, subspaces, linear operators, range space, null space

A field \((F,+,·)\) is a set with two commutative associative operations with \(0\) and\(1\), additive and multiplicative inverses and subject to the distributive law.

Examples (Common fields)
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class of 8.29

    • express \(\sin(3x)\) in term of \(\sin(x)\).
    • express \(\cos(4x)\) in terms of \(\cos(x)\).
    • simplify
      \[
      \sum_{k=-20}^{k=20} e^{k\phi}.
      \]
  1. solve
    • \[
      y’=\frac{2t}{1+t^2}, y(0)=0;
      \]
    • \[
      y’=\frac{2t}{1-t^2}, y(0)=0;
      \]
    • \[
      y’=\frac{2t}{1-t^2}, y(2)=0.
      \]
  2. solve
    • \[
      y’=\frac{2x^2-4x+3}{x-1}, y(0)=2;
      \]
    • \[
      y’=\frac{x^2-4x+3}{x-2}, y(0)=2;
      \]
    • \[
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august 28

\(\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\)

What this course deals with?

Linear control systems. A continuous-time state-space linear system is defined by the following equations:
The signals
\[
x'(t) = A(t)x(t) + B(t)u(t), y(t) = C(t)x(t) + D(t)u(t),
\]
where
\[
x\in \Real^n, …

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