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 and i 0 with 3=1 i = 1, it is true that 1 x + 2 y + 3 z C
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 all solutions to 4 x 1 + x2 2 x1 + x2 6 3 x2 x1 , x2 0
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. Therefore, such a
relationship can be viewed as a restricted
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 set is
often called the set of feasible alternaties or, s
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 ordered pairs. Indeed, given two sets X and Y a functi
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 real vector space V . We will state many of the denitio
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 is
determined by the interaction of maximizing househol
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 (t), t1 t t2 , satisng the
phase constraints (t, x(t)
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 of order n
about a, namely
n
Tn (x) =
k=0
f (k) (a)
(x a
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)
where A is an m n matrix with real entries, x Rn , b Rm .
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 is
rm
simply r/m and so the initial investment grows ove
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 written as column vectors so that x = x1 x2 . . . xn and we
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) A = A B ; and (c) B = A B .
Exercise 2: Show that if C
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 set is often called the set of feasible alternaties or, s
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 preferences of various agents. Thus, for example, suppo