Department of Mathematical Sciences University of Delaware Prof. T. Angell September 15, 2010
Mathematics 530
Exercise Sheet 1 Exercise 1: Show that, if C Rn is a convex set then, for any x, y, z C an
Department of Mathematical Sciences University of Delaware Prof. T. Angell November 10, 2010
Mathematics 530
Test Problems
Exercise 1: Determine the extreme points of the set S , where S is the set of
Binary Relations and Equivalence Relations
Intuitively, a binary relation R on a set A is a proposition such that, for every ordered
pair (a, b) A A, one can decide if a is related to b or not. Theref
IntroductionOptimization Problems
Optimization problems in mathematics are problems in which we wish to minimize or
maximize some real-valued function relative to some set of arguments. This latter se
Functions or Maps
We are going to dene the idea of a function (or a mapping) in terms of sets. This is
not so unusual since we often think of a function in terms of its graph which cosists
of a set of
Convex Functions
Our next topic is that of convex functions. Again, we will concentrate on the context of a
map f : Rn R although the situation can be generalized immediately by replacing Rn
with any
Remarks on Ramseys Model
Ramseys model forms the basis of a lot of modern macroeconomics. It is a model based
on utility maximization and the goal is to determine the evolution of capital stock which
THE BASIC NECESSARY CONDITIONS FOR FREE PROBLEMS
Let A Rn+1 , and B R2n+2 be closed subsets. Let be a nonempty class of piecewise
continuously dierentiable functions x(t) = (x1 (t), x2 (t), . . . , xn
Some Remarks on Taylors Theorem
Suppose that f is a real-valued function of a real variable and that it has derivatives of
all orders up to n at a point a. Then we can consider the Taylor polynomial o
The Farkas-Minkowski Theorem
The results presented below, the rst of which appeared in 1902, are concerned with the
existence of non-negative solutions of the linear system
Ax
=
b,
(1.1)
x
0,
(1.2)
wh
Discounting and Present Value
We summarize here some familiar facts. When interest in an investment of amount $A is
compounded at an annual rate r over m periods during the year, the rate per period i
Preliminary MaterialBackground
Our work will be confined almost exclusively to problems in n-dimensional Euclidean space which we will denote by Rn . Vectors in the vector space Rn will always be writ
Department of Mathematical
Sciences
University of Delaware
Prof. T. Angell
August 29, 2013
Mathematics 530
Exercise Sheet 1
Exercise 1: Show that the following statements are equivalent: (a) A B ;
(b)
IntroductionOptimization Problems
Optimization problems in mathematics are problems in which we wish to minimize or maximize some real-valued function relative to some set of arguments. This latter se
Orderings on Sets
We are going to be particularly concerned with certain binary relations which are
called orderings of which there are several types. Such relations are used in economics
to describe