ECEN 629
Shuguang Cui
Lecture 2: Convex functions
f : Rn R is convex if dom f is convex and for all x, y dom f , [0, 1]
f (x + (1 )y) f (x) + (1 )f (y)
f is concave if f is convex
x
convex
x
concave
x
neither
examples (on R)
f (x) = x2 is convex
f (x) =
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Academic Press
Encyclopedia of Physical Science and Technology
Fourier Series
James S. Walker
Department of Mathematics
University of WisconsinEau Claire
Eau Claire, WI 547024004
Phone: 7158363301
Fax: 7158362924
e-mail: [email protected]
1
Encyclopedia o
Math 554 - Fall 08
Lecture Note Set # 1 Defn. From the introductory lectures, an ordered set is a set S with a relation < which satises two properties: 1. (Trichotomy property) for any two elements a, b S, exactly one of the following hold a < b, a = b, o
Maths220 Supremum and inmum
Supremum and inmum
So we are now moving on towards properties of the real numbers, limits etc.
Unfortunately, while I do like the text it is decient in 1 topic namely one fundamental
property of the reals that we should really
The Kuratowski Closure-Complement Theorem
By Greg Strabel
The Kuratowski Closure-Complement Theorem, a result of basic point-set
topology, was first posed and proven by the Polish mathematician Kazimierz Kuratowski
in 1922. Since then, Kuratowskis Theorem
Chapter 8
Eulers Gamma function
The Gamma function plays an important role in the functional equation for (s)
that we will derive in the next chapter. In the present chapter we have collected
some properties of the Gamma function.
For t R>0 , z C, dene tz
Universal quadratic forms and the 290-Theorem
Manjul Bhargava and Jonathan Hanke
1
Introduction
In 1993, Conway formulated a remarkable conjecture regarding universal quadratic forms,
i.e., integer-coecient, positive-denite quadratic forms representing al
The Laplacian in Terms of Polar Coordinates
2
2
2
0.1. Linear dierential operators. The Laplacian = x2 + y2 + z2 is an example of a
linear dierential operator, which operates on any suciently smooth function u(x, y, z) placed
to the right of it. It is of
Chapter 1
Basic (Elementary) Inequalities
and Their Application
There are many trivial facts which are the basis for proving inequalities. Some of
them are as follows:
1.
2.
3.
4.
5.
If x y and y z then x z, for any x, y, z R.
If x y and a b then x + a y
Triple Integrals for Volumes of Some Classic Shapes
In the following pages, I give some worked out examples where triple integrals are used to nd some
classic shapes volumes (boxes, cylinders, spheres and cones) For all of these shapes, triple integrals a
1
CHAPTER 3
PLANE AND SPHERICAL TRIGONOMETRY
3.1 Introduction
It is assumed in this chapter that readers are familiar with the usual elementary formulas
encountered in introductory trigonometry. We start the chapter with a brief review of the solution
of
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Suprema/Inma Review Sheet
This is a review of some basic facts about suprema and inma. You may use
these facts in your homework assignments without proving them.
Denition 1. Consider a set A R.
(1) A number u is said to be an upper bound for A if u a for
z
Spherical Coordinates
^
r
Transforms
! r
The forward and reverse coordinate transformations are
r=
x2 + y2 + z 2
y = r sin! sin"
z = r cos !
& = arctan ( y, x )
^
!
r
x = r sin ! cos"
! = arctan " x 2 + y 2 , z$
#
%
^
"
y
x
"
where we formally take adva
sidca
2003/9
page 1
10
Chapter 1. Introduction to the Central Concepts
1.2
Useful Properties of Convex Functions
We have already mentioned that convex functions are tractable in optimization (or
minimization) problems and this is mainly because of the fol
12
Lipschitz Continuity
Calculus required continuity, and continuity was supposed to require
the innitely little, but nobody could discover what the innitely
little might be. (Russell)
12.1 Introduction
When we graph a function f (x) of a rational variabl
Math 396. Interior, closure, and boundary
We wish to develop some basic geometric concepts in metric spaces which make precise certain
intuitive ideas centered on the themes of interior and boundary of a subset of a metric space.
One warning must be given
1. ORDERED SETS
Denitions. (Partially and totally ordered sets.)
(1) The ordered pair (X, ) is called a partially ordered set
if X is a set and is a partial order relation in X.
(2) The ordered pair (X, ) is called a totally ordered set (or linear ordered
GMAT CRITICAL REASONING SAMPLE QUESTIONS
Instruction: This file contains 205 sample questions on GMAT Critical Reasoning and
explanations for 25 of them. For answers with complete explanations to other 180
questions, please order the Complete GMAT Prep Co
Questions and Answers About the New Question Types Included
in the GRE General Test Beginning in November 2007
Why did you add new question types to the test?
The addition of new question types is part of the first phase of improvements that the GRE
Progr