Archive | research

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)

$$(\Real,+,·)$$ is a field. Rational, Complex numbers. Matrices (when they are fields?)

Vector spaces – modules over a field. (Real definition: abelian group with an action of $$\Field$$ and several natural axioms.)

Examples

Fun($$A,\Field$$), for any $$A$$ – vector space over $$\Field$$.…

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
• $y’=\frac{2x^2-4x+3}{x-1}, y(0)=2;$
• $y’=\frac{x^2-4x+3}{x-2}, y(0)=2;$
• $y’=\frac{x-2}{x^2-4x+3}, y(0)=2;$
3. Find with precision 5%
$\int_{-\pi}^\pi\cos(x)^{100} dx.$

(hint a: $$\cos(x)=(e^{ix}+e^{-ix})/2$$; hint b: $$\cos(x)\approx 1-x^2/2$$ for small $$x$$).

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
$x\in \Real^n, u\in\Real^k, y\in\Real^m;\\ y:[0,\infty)\to\Real^m, u :[0,\infty)\to\Real^k, x :[0,\infty)\to\Real^n,$
are the (vector-valued) functions called input, state, and output of the system.

The first-order differential equations is called the state and the output equation.

When all the matrices $$A(t), B(t), C(t), D(t)$$ are constant, the system
is called a linear time-invariant (LTI) system.…

math 285, the week of August 25

Brash up your calculus. Review Chapter 1.1. Listen to the first lecture.

Homework (due by Monday, 9.1):

1. For which $$x$$ there exists a real solution to
$y^2-4xy+4=0?$
2. Simplify
$\frac{\sin(x+y)-\sin(x-y)}{\sin(y)}.$
3. Solve differential equations
$y’=\sin(2x+3)$
and
4. $y’=\frac{1}{1+t^2}$

$$\def\Real{\mathbb{R}} \def\views{\mathbb{G}} \def\earth{\mathbb{E}} \def\hsp{\mathbb{H}}$$