# Archive | ece490

## week 7

$$\def\Real{\mathbb{R}} \def\Comp{\mathbb{C}} \def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}} \def\Fun{\mathbf{Fun}} \def\e{\mathbf{e}} \def\f{\mathbf{f}} \def\bv{\mathbf{v}}$$

#### Convexity

Convex sets. Convex hulls. Supporting hyperplanes.

Convex functions; properties. Log-convex functions.

#### Sample problems

1. Is the set of continuous functions on $$\Real^2$$ positive at all points with rational coordinates numbers convex?

## weeks 5 and 6

$$\def\Real{\mathbb{R}} \def\Comp{\mathbb{C}} \def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}} \def\Fun{\mathbf{Fun}} \def\e{\mathbf{e}} \def\f{\mathbf{f}} \def\bv{\mathbf{v}}$$

Positive definite quadratic forms. Signatures, and relations between the indices of inertia of the (that is the number $$n_+$$ of positive, $$n_-$$ negative and $$n_0$$ ,zero coefficients in the …

## programming exercise 1.

$$\def\Real{\mathbb{R}} \def\Comp{\mathbb{C}} \def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}} \def\Fun{\mathbf{Fun}} \def\e{\mathbf{e}} \def\f{\mathbf{f}} \def\bv{\mathbf{v}}$$

Use any language or library. The assignments are due by midnight of 3.14 (Monday).

#### 3 credits students

Consider the piecewise linear function $$f$$ on $$[-4,4]$$
with nodes at the integer …

## mock midterm 1

$$\def\Real{\mathbb{R}} \def\Comp{\mathbb{C}} \def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}} \def\Fun{\mathbf{Fun}} \def\e{\mathbf{e}} \def\f{\mathbf{f}} \def\bv{\mathbf{v}}$$

1. Consider the following subsets of $$\Real^2$$:
$A=\{(x,y):xy>0\}; B=\{(x,y):x=0\}; C=\{(x,y):y=0\}; D=\{(0,0)\}.$
Which of the following sets is open? closed?
$A; B; A\cup C; A\cup B\cup C; A\cup D$

## week 4

$$\def\Real{\mathbb{R}} \def\Comp{\mathbb{C}} \def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}} \def\Fun{\mathbf{Fun}} \def\e{\mathbf{e}} \def\f{\mathbf{f}} \def\bv{\mathbf{v}}$$

### Differentials

Differentials. Slope functions.

Inverse function and implicit function theorems.

Matrix differentials

Sample problems:

• Find the differential of
$(x,y)\mapsto (x^2-y^2,2xy)$
• The function $$z(x,y)$$ is implicitly given by
\[

## week 3

$$\def\Real{\mathbb{R}} \def\Comp{\mathbb{C}} \def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}} \def\Fun{\mathbf{Fun}} \def\e{\mathbf{e}} \def\f{\mathbf{f}} \def\bv{\mathbf{v}}$$

### Background in Linear Algebra and Analysis

Topology: closed and open sets. Convergence. Compact sets. Criterion of compactness in $$\Real^n$$.

Continuity.

Sample problems:

• Find an open and set in $$\Real$$

## week 2

$$\def\Real{\mathbb{R}} \def\Comp{\mathbb{C}} \def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}} \def\Fun{\mathbf{Fun}} \def\e{\mathbf{e}} \def\f{\mathbf{f}} \def\bv{\mathbf{v}}$$

### Background in Linear Algebra and Analysis

Vector spaces. Norms.

Matrices and norms on them. Linear operators.

Sample problems:

• Find a vector in $$\Real^2$$ such that its $$l_1$$ norm is 7,

## week 1

### Introduction

Lange, Ch. 1.

Basic notions: feasible points; criterion of optimality; maxima and minima.

Existence and nonexistence of optimal points; uniqueness and nonuniqueness.

Univariate problems. Critical points. Classical inequalities.

Multivariate problems (unconstrained).

Constrained problems; Lagrange multipliers.

Sample problems:

• On