Convexity in R N Main Notes 1


 Harvey Nelson
 1 years ago
 Views:
Transcription
1 John Nachbar Washington University December 16, 2016 Convexity in R N Main Notes 1 1 Introduction. These notes establish basic versions of the Supporting Hyperplane Theorem (Theorem 5) and the Separating Hyperplane Theorem (Theorem 6) for sets in R N. The supplemental notes on convexity in R N give more powerful versions, establishing conditions that are necessary as well as sufficient. All of the results here carry over immediately to general finitedimensional vector spaces; see also Remark 2. For extensions to infinitedimensional normed vector spaces, see the notes on the HahnBanach theorem. 2 Convex Sets. Definition 1. A set A R N is convex iff for any a, b A and any θ [0, 1] the point θa + (1 θ)b is also in A. Geometrically, A is convex iff A contains the line segment joining any two points in A. By an induction argument, A is convex iff for any (nonempty) finite subset {a 1,..., a m }, and weights θ i with θ i 0 and m i=1 θ i = 1, the weighted sum m i=1 θ ia i is also in A. m i=1 θ ia i is called a convex combination of a 1,..., a m. Definition 2. Let A, B R N. Let θ R. A + B = {x R N : a A, b B such that x = a + b}. θa = {x R N : a A such that x = θa}. Remark 1. A B means A + ( B) not A \ B = {x A : x / B}. For example, if A = {1} and B = {2}, then A B = { 1} but A \ B =. Theorem 1. Let A be a set of convex sets in R N, let A, B A, and let θ R. 1. A + B is convex. 2. θa is convex. 1 cbna. This work is licensed under the Creative Commons AttributionNonCommercial ShareAlike 4.0 License. 1
2 3. A A A is convex. 4. A is convex. Proof. I prove that A is convex. I leave the others as exercises. Consider any a, â A and any θ [0, 1]. Let x = θa + (1 θ)â. Take sequences (a t ) and (â t ) in A such that a t a and â t â. (If a A then one can take a t = a and similarly for â.) Let x t = θa t + (1 θ)â t. Since A is convex, x t A. By continuity, x t x. Since A is closed, x A. Theorem 2. For any x R N and any ε > 0, N ε (x) is convex. Proof. This is just a special case of a general fact mentioned in the notes on metric spaces: in any normed vector space, N ε (x) is convex. In R, it is easy to prove that A is convex iff it is an interval (possibly infinite, not necessarily containing one or both endpoints), which holds iff A is connected. In R N, one can show that convex sets are connected, but not necessarily conversely. 3 A Basic Separation Result. Definition 3. A, B R N can be strictly separated iff there exists a v R N and an r R such that v a > r > v b for every a A, b B. Note that since v a > r, v 0. Geometrically, strict separation means the following. Any N 1 dimensional plane in R N can be represented as a set {x R N : v x = r}, where v R N, v 0 and r R. The vector v gives the tilt of the plane and the number r determines the position of the plane. If r = 0, then the plane passes through the origin; in this case, I denote the plane as T v : T v = {x R N : v x = 0}. Thus, the plane T v is the set of points orthogonal to v. For r 0, the plane is a parallel copy of T v, shifted away from the origin. Strict separation means that there is a plane such that the set A lies strictly on one side of the plane and B lies strictly on the other side. Many of the results on convex sets are corollaries of the following theorem. Theorem 3 (The Basic Separation Theorem BST). Let A R N be nonempty, closed, and convex. If the origin is not in A, then A and the origin can be strictly separated. 2
3 Proof. Take v to be the element of A that has smallest norm. That is, v is a solution (in fact, the solution) to min x. x A To see that this problem has a solution, note that the norm is continuous. A solution would therefore exist if A were compact. A is not necessarily compact, since it may not be bounded, but this is not really a problem. Since A is not empty, there is some a A. Since 0 A, but a A, a 0. Take Â = A N a (0). Then Â is nonempty (it contains a) and compact (it is the intersection of a closed set and a compact set) and so the modified problem min x x Â has a solution. Call this solution v. By construction, the norm of v is smaller than the norm of any x in A \ Â, hence v solves the original minimization problem as well. Since 0 / A, but v A, v 0. Since v 0, v > 0. I claim that v a v v for any a A, which proves the result for any r (0, v v), since then v a > r > 0 = v 0. I argue by contraposition. Suppose that there is an x A such that v x < v v, hence v (x v) < 0. The directional derivative of the Euclidean norm at the point v in the direction x v is, by definition, v + θ(x v) v D x v v = lim. θ 0 θ On the other hand, D x v v = D v (x v). (See the notes on Multivariate Differentiation.) One can compute that hence, D v = 1 v v, v + θ(x v) v lim = 1 v (x v), θ 0 θ v which is strictly negative. This implies that for θ close to 0, setting x θ = v + θ(x v) = θx + (1 θ)v, x θ v < 0, θ hence x θ < v. Since v minimizes norm on A, this implies that x θ / A. Since x, v A, this shows that A is not convex. Since A is, in fact, convex, the proof follows by contraposition. Geometrically, the proof works as follows. Figure 1 provides illustration. Let v 3
4 T v v S S Separation No Separation v T v + {v} x x θ 0 T v Figure 1: Proof of BST. be the point in A that minimizes norm and let S be the sphere defined by S = {x R N : x = v }. Since S is a level set of the norm, S and since the gradient of the norm (namely v/ v ) is positively collinear with v, the set of vectors tangent to S at v is the plane, v T T x* v + {x*} T v + {v} v = {x R N : v x = v v}, See the illustration. T v + {v} splits R N into two halfspaces, one containing the origin and the other away from the origin. The proof verifies that if v x < v v, so that x lies on the halfspace containing the origin, then the line segment between x and v must cut through S, implying the existence of a point x θ = θx + (1 θ)v / A for which x θ < v. Since v minimizes norm on A, this implies that x θ / A. But if x A, this implies that A is not convex. By contraposition, v a v v for all a A: all of A must lie in the halfspace away from the origin. A B The next example shows that convexity is not necessary for BST. For simplicity, I set all of the next examples in N = 1. Similar examples can be constructed for any N. Note that when N = 1, v a = va. Example 1. Let N = 1 and let A = {1, 2}. Then A is closed and the origin is not in A but A not convex. Nevertheless, taking v = 1 A, va vv (i.e., a 1) for all a A, so the conclusion of Theorem 3 holds. E co(e) E On the other hand, some condition is needed, beyond A is closed and 0 / A, as the next example illustrates. Example 2. Let N = 1 and let A = { 1, 1}. Then A is closed and the origin is not in A but A cannot be strictly separated from the origin: if v > 0 then v( 1) < 0 while v(1) > 0; of course, v(0) = 0. The mirror image problem arises if v < 0. Similarly, it is not necessary that A be closed, but again some condition is needed, as the next example illustrates. Example 3. Let N = 1 and let A = (0, 1]. Then A is convex and the origin is not in A but A cannot be strictly separated from the origin. In particular, if v > 0 then va > v0 = 0 for every a A. But there is no r such that va > r > 0 for every a A. 4
5 The supplemental notes on convexity address these examples by providing conditions for separation that are necessary as well as sufficient. Remark 2. Although the results in these notes are stated for R N, they hold in any finitedimensional vector space V. In particular, if V is a vector subspace of R N, then note that in the proof of Theorem 3, which is the foundation of all of the other support and separation theorems in these notes, the vector v is an element of A and hence an element of the vector space V that contains A. 4 The Sum of Closed Sets. For later use, I need the following fact about the sum of closed (but not necessarily convex) sets. Theorem 4. Let A, B R N be closed. If at least one set is compact, then A + B is closed. Proof of Theorem 4. Let C = A + B and suppose that (c t ) is a sequence in C that converges to c R N. I must show that c C. Since c t C there is an a t A and a b t B such that c t = a t + b t. For concreteness, suppose that A is compact. Then (a t ) has a subsequence that converges to a point of A. Call this subsequential limit a. Along this subsequence, continuity implies that b t = c t a t converges to c a. Define b = c a. Since B is closed, b B. Therefore, c = a + b A + B = C, as was to be shown. Example 4. If A and B are closed but neither is compact, then C = A + B may not be closed. For example, let A = {x R 2 : x 1 > 0 and x 2 1/x 1 } and let B = {x R 2 : x 2 = 0}. Then A and B are closed but C = A + B is not closed. In fact, C is the half space {(x 1, x 2 ) R 2 : x 2 > 0}; every point on the x 1 axis is a limit point of C but no point on the x 1 axis is an element of C. 5 A Support Theorem. Definition 4. A R N is supported at x iff v R N, v 0, such that v a v x for all a A. That is, a set A is supported at x if the set lies to one side of (but possibly touching) a plane that passes through x. The following theorem gives a sufficient condition for a set A to be supported at x. The theorem belongs to a class of results called supporting hyperplane theorems. For us, hyperplane is just another word for plane. Theorem 5 (Supporting Hyperplane Theorem). Let A R N be nonempty, closed, and convex. If x is not interior to A, then A is supported at x. 5
6 Proof. If A = then the result holds trivially (take any v 0). Henceforth assume that A. Since x is not interior to A, for any ε > 0 there is a x ε N ε (x ) A c. This implies that there is a sequence (x t ) in A c converging to x. For each x t, consider the set C t = A {x t }. Since A is closed and convex and {x t } is (trivially) compact and convex, C t is closed (Theorem 4) and convex (Theorem 1). Moreover, since x t / A, 0 / C t. Apply Theorem 3 to get v t R N and r t > 0 such that v t c > r t > 0 for all c C t. This implies that for all a A, v (a x t ) > 0, hence v t a > v t x t. Although x t x, there is no guarantee that v t converges to anything. This is easily fixed, however. Since v t 0 (again, v t c > r > 0 implies v t 0), ˆv t = v t / v t is well defined. Since v t a > v t x t, ˆv t a > ˆv t x t. Since ˆv t belongs to a compact set, namely the unit sphere, (ˆv t ) has a convergent subsequence, converging to, say, v on the unit sphere. Since v is on the unit sphere, v 0. By continuity of inner product it follows that, for all a A, which is what I needed to show. v a v x, Remark 3. In the Supporting Hyperplane Theorem, the hard case is when x is in A but not interior to A (i.e., x is on the boundary of A). If x A then the Separating Hyperplane Theorem (Theorem 6) implies that x and A can be strictly separated. The Supporting Hyperplane Theorem can be generalized to handle A that is not closed. I address this more difficult case in the supplemental notes on convexity. The hurdle is that I need to prove that if x is not interior to A then it is not interior to A. This is true if A is convex but not in general. 6 A Separation Theorem. The following theorem gives a sufficient condition for two sets to be strictly separated. The theorem can be viewed as a generalization of BST. The theorem belongs to a class of results called separating hyperplane theorems. Again, for us, hyperplane is just another word for plane. Theorem 6 (Separating Hyperplane Theorem). Suppose that A, B R N are nonempty, closed, convex, and disjoint. If at least one is compact, then A and B can be strictly separated. 6
7 Proof. Consider C = A B. C is nonempty, convex (Theorem 1), and closed (Theorem 4). Since A B =, 0 C. Applying Theorem 3, there is a v R N and a ˆr > 0 such that v c > ˆr > 0 for all c C. This implies that v a > v b + ˆr for all a A and all b B. Define r = inf v a a A r = sup v b. b B r and r are well defined because, for example, the set of v a is bounded below by any v b and hence the inf exists. Then r r + ˆr > r. Choose any r (r, r). Then v a > r > v b for any a A, b B, as was to be shown. As established in the supplemental notes on convexity, one can relax many of the conditions on A and B and still separate them, but not necessarily strictly. Example 5. As in Example 4, suppose A = {x R 2 : x 1 > 0 and x 2 1/x 1 } and B = {x R 2 : x 2 = 0}. Then A and B are closed, convex, and disjoint but neither is compact. To separate A and B, v must be collinear with (0, 1). Without loss of generality, suppose v = (0, 1). Then v b = 0 for all b B. On the other hand, inf a A v a = 0. This implies that there cannot be strict separation. Example 6. Strict separation can also fail if one of the sets is not closed, even if that set is bounded, the other set is compact, and the sets are disjoint. For example, let A = (1, 2) and let B = [0, 1]. Clearly we cannot have strict separation. 7
Vector Spaces II: Finite Dimensional Linear Algebra 1
John Nachbar September 2, 2014 Vector Spaces II: Finite Dimensional Linear Algebra 1 1 Definitions and Basic Theorems. For basic properties and notation for R N, see the notes Vector Spaces I. Definition
More information1. Prove that the empty set is a subset of every set.
1. Prove that the empty set is a subset of every set. Basic Topology Written by MenGen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since
More informationDuality of linear conic problems
Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
More informationPOWER SETS AND RELATIONS
POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty
More information{f 1 (U), U F} is an open cover of A. Since A is compact there is a finite subcover of A, {f 1 (U 1 ),...,f 1 (U n )}, {U 1,...
44 CHAPTER 4. CONTINUOUS FUNCTIONS In Calculus we often use arithmetic operations to generate new continuous functions from old ones. In a general metric space we don t have arithmetic, but much of it
More information(Basic definitions and properties; Separation theorems; Characterizations) 1.1 Definition, examples, inner description, algebraic properties
Lecture 1 Convex Sets (Basic definitions and properties; Separation theorems; Characterizations) 1.1 Definition, examples, inner description, algebraic properties 1.1.1 A convex set In the school geometry
More informationNo: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics
No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results
More informationMath212a1010 Lebesgue measure.
Math212a1010 Lebesgue measure. October 19, 2010 Today s lecture will be devoted to Lebesgue measure, a creation of Henri Lebesgue, in his thesis, one of the most famous theses in the history of mathematics.
More informationMathematical Methods of Engineering Analysis
Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................
More informationMetric Spaces Joseph Muscat 2003 (Last revised May 2009)
1 Distance J Muscat 1 Metric Spaces Joseph Muscat 2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) 1 Distance A metric space can be thought of
More informationBANACH AND HILBERT SPACE REVIEW
BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number
More informationINCIDENCEBETWEENNESS GEOMETRY
INCIDENCEBETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full
More informationMetric Spaces. Chapter 1
Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence
More informationFUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES
FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied
More informationSOLUTIONS TO ASSIGNMENT 1 MATH 576
SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts
More informationMath 4310 Handout  Quotient Vector Spaces
Math 4310 Handout  Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable
More information6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium
6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium Asu Ozdaglar MIT February 18, 2010 1 Introduction Outline PricingCongestion Game Example Existence of a Mixed
More information1 Norms and Vector Spaces
008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)
More information2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationLectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain
Lectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n realvalued matrix A is said to be an orthogonal
More informationPART I. THE REAL NUMBERS
PART I. THE REAL NUMBERS This material assumes that you are already familiar with the real number system and the representation of the real numbers as points on the real line. I.1. THE NATURAL NUMBERS
More informationAdaptive Online Gradient Descent
Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650
More informationSOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties
SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces
More informationMatrix Representations of Linear Transformations and Changes of Coordinates
Matrix Representations of Linear Transformations and Changes of Coordinates 01 Subspaces and Bases 011 Definitions A subspace V of R n is a subset of R n that contains the zero element and is closed under
More information1 VECTOR SPACES AND SUBSPACES
1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such
More informationTHE BANACH CONTRACTION PRINCIPLE. Contents
THE BANACH CONTRACTION PRINCIPLE ALEX PONIECKI Abstract. This paper will study contractions of metric spaces. To do this, we will mainly use tools from topology. We will give some examples of contractions,
More informationRi and. i=1. S i N. and. R R i
The subset R of R n is a closed rectangle if there are n nonempty closed intervals {[a 1, b 1 ], [a 2, b 2 ],..., [a n, b n ]} so that R = [a 1, b 1 ] [a 2, b 2 ] [a n, b n ]. The subset R of R n is an
More informationIntroduction to Topology
Introduction to Topology Tomoo Matsumura November 30, 2010 Contents 1 Topological spaces 3 1.1 Basis of a Topology......................................... 3 1.2 Comparing Topologies.......................................
More informationSeries Convergence Tests Math 122 Calculus III D Joyce, Fall 2012
Some series converge, some diverge. Series Convergence Tests Math 22 Calculus III D Joyce, Fall 202 Geometric series. We ve already looked at these. We know when a geometric series converges and what it
More informationHomework Exam 1, Geometric Algorithms, 2016
Homework Exam 1, Geometric Algorithms, 2016 1. (3 points) Let P be a convex polyhedron in 3dimensional space. The boundary of P is represented as a DCEL, storing the incidence relationships between the
More informationCHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.
CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,
More informationWe call this set an ndimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P.
Volumes of parallelograms 1 Chapter 8 Volumes of parallelograms In the present short chapter we are going to discuss the elementary geometrical objects which we call parallelograms. These are going to
More informationMATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.
MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar
More informationTHE DIMENSION OF A VECTOR SPACE
THE DIMENSION OF A VECTOR SPACE KEITH CONRAD This handout is a supplementary discussion leading up to the definition of dimension and some of its basic properties. Let V be a vector space over a field
More informationMAT 1341: REVIEW II SANGHOON BAEK
MAT 1341: REVIEW II SANGHOON BAEK 1. Projections and Cross Product 1.1. Projections. Definition 1.1. Given a vector u, the rectangular (or perpendicular or orthogonal) components are two vectors u 1 and
More informationGeometric models of the card game SET
Geometric models of the card game SET Cherith Tucker April 17, 2007 Abstract The card game SET can be modeled by fourdimensional vectors over Z 3. These vectors correspond to points in the affine fourspace
More informationINDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS
INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem
More informationTHE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING
THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING 1. Introduction The BlackScholes theory, which is the main subject of this course and its sequel, is based on the Efficient Market Hypothesis, that arbitrages
More informationSec 4.1 Vector Spaces and Subspaces
Sec 4. Vector Spaces and Subspaces Motivation Let S be the set of all solutions to the differential equation y + y =. Let T be the set of all 2 3 matrices with real entries. These two sets share many common
More informationBasic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely
More informationThe Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if
More informationFoundations of Geometry 1: Points, Lines, Segments, Angles
Chapter 3 Foundations of Geometry 1: Points, Lines, Segments, Angles 3.1 An Introduction to Proof Syllogism: The abstract form is: 1. All A is B. 2. X is A 3. X is B Example: Let s think about an example.
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According
More informationx a x 2 (1 + x 2 ) n.
Limits and continuity Suppose that we have a function f : R R. Let a R. We say that f(x) tends to the limit l as x tends to a; lim f(x) = l ; x a if, given any real number ɛ > 0, there exists a real number
More informationShape Optimization Problems over Classes of Convex Domains
Shape Optimization Problems over Classes of Convex Domains Giuseppe BUTTAZZO Dipartimento di Matematica Via Buonarroti, 2 56127 PISA ITALY email: buttazzo@sab.sns.it Paolo GUASONI Scuola Normale Superiore
More informationEuclidean Geometry. We start with the idea of an axiomatic system. An axiomatic system has four parts:
Euclidean Geometry Students are often so challenged by the details of Euclidean geometry that they miss the rich structure of the subject. We give an overview of a piece of this structure below. We start
More informationx if x 0, x if x < 0.
Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the
More information4.5 Linear Dependence and Linear Independence
4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete
More informationFixed Point Theorems
Fixed Point Theorems Definition: Let X be a set and let T : X X be a function that maps X into itself. (Such a function is often called an operator, a transformation, or a transform on X, and the notation
More informationOpen and Closed Sets
Open and Closed Sets Definition: A subset S of a metric space (X, d) is open if it contains an open ball about each of its points i.e., if x S : ɛ > 0 : B(x, ɛ) S. (1) Theorem: (O1) and X are open sets.
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More information6.2 Permutations continued
6.2 Permutations continued Theorem A permutation on a finite set A is either a cycle or can be expressed as a product (composition of disjoint cycles. Proof is by (strong induction on the number, r, of
More informationChapter 1. Metric Spaces. Metric Spaces. Examples. Normed linear spaces
Chapter 1. Metric Spaces Metric Spaces MA222 David Preiss d.preiss@warwick.ac.uk Warwick University, Spring 2008/2009 Definitions. A metric on a set M is a function d : M M R such that for all x, y, z
More informationMathematical Induction
Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,
More informationSo let us begin our quest to find the holy grail of real analysis.
1 Section 5.2 The Complete Ordered Field: Purpose of Section We present an axiomatic description of the real numbers as a complete ordered field. The axioms which describe the arithmetic of the real numbers
More informationMath 5311 Gateaux differentials and Frechet derivatives
Math 5311 Gateaux differentials and Frechet derivatives Kevin Long January 26, 2009 1 Differentiation in vector spaces Thus far, we ve developed the theory of minimization without reference to derivatives.
More informationLEARNING OBJECTIVES FOR THIS CHAPTER
CHAPTER 2 American mathematician Paul Halmos (1916 2006), who in 1942 published the first modern linear algebra book. The title of Halmos s book was the same as the title of this chapter. FiniteDimensional
More informationLecture 3: Linear Programming Relaxations and Rounding
Lecture 3: Linear Programming Relaxations and Rounding 1 Approximation Algorithms and Linear Relaxations For the time being, suppose we have a minimization problem. Many times, the problem at hand can
More information1. Determine all real numbers a, b, c, d that satisfy the following system of equations.
altic Way 1999 Reykjavík, November 6, 1999 Problems 1. etermine all real numbers a, b, c, d that satisfy the following system of equations. abc + ab + bc + ca + a + b + c = 1 bcd + bc + cd + db + b + c
More informationChapter 3. Cartesian Products and Relations. 3.1 Cartesian Products
Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing
More informationWhat is Linear Programming?
Chapter 1 What is Linear Programming? An optimization problem usually has three essential ingredients: a variable vector x consisting of a set of unknowns to be determined, an objective function of x to
More informationTOPIC 4: DERIVATIVES
TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the
More informationProblem Set I: Preferences, W.A.R.P., consumer choice
Problem Set I: Preferences, W.A.R.P., consumer choice Paolo Crosetto paolo.crosetto@unimi.it Exercises solved in class on 18th January 2009 Recap:,, Definition 1. The strict preference relation is x y
More informationLinear Algebra I. Ronald van Luijk, 2012
Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.
More informationSection 1.4. Lines, Planes, and Hyperplanes. The Calculus of Functions of Several Variables
The Calculus of Functions of Several Variables Section 1.4 Lines, Planes, Hyperplanes In this section we will add to our basic geometric understing of R n by studying lines planes. If we do this carefully,
More informationIntersecting Families
Intersecting Families Extremal Combinatorics Philipp Zumstein 1 The ErdsKoRado theorem 2 Projective planes Maximal intersecting families 4 Hellytype result A familiy of sets is intersecting if any two
More informationPractice with Proofs
Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using
More informationDate: April 12, 2001. Contents
2 Lagrange Multipliers Date: April 12, 2001 Contents 2.1. Introduction to Lagrange Multipliers......... p. 2 2.2. Enhanced Fritz John Optimality Conditions...... p. 12 2.3. Informative Lagrange Multipliers...........
More informationInner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 34 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,
More informationalternate interior angles
alternate interior angles two nonadjacent angles that lie on the opposite sides of a transversal between two lines that the transversal intersects (a description of the location of the angles); alternate
More informationOn Lexicographic (Dictionary) Preference
MICROECONOMICS LECTURE SUPPLEMENTS Hajime Miyazaki File Name: lexico95.usc/lexico99.dok DEPARTMENT OF ECONOMICS OHIO STATE UNIVERSITY Fall 993/994/995 Miyazaki.@osu.edu On Lexicographic (Dictionary) Preference
More information2 Complex Functions and the CauchyRiemann Equations
2 Complex Functions and the CauchyRiemann Equations 2.1 Complex functions In onevariable calculus, we study functions f(x) of a real variable x. Likewise, in complex analysis, we study functions f(z)
More informationThe Envelope Theorem 1
John Nachbar Washington University April 2, 2015 1 Introduction. The Envelope Theorem 1 The Envelope theorem is a corollary of the KarushKuhnTucker theorem (KKT) that characterizes changes in the value
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More informationWalrasian Demand. u(x) where B(p, w) = {x R n + : p x w}.
Walrasian Demand Econ 2100 Fall 2015 Lecture 5, September 16 Outline 1 Walrasian Demand 2 Properties of Walrasian Demand 3 An Optimization Recipe 4 First and Second Order Conditions Definition Walrasian
More informationNOTES ON LINEAR TRANSFORMATIONS
NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all
More informationNotes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand
Notes V General Equilibrium: Positive Theory In this lecture we go on considering a general equilibrium model of a private ownership economy. In contrast to the Notes IV, we focus on positive issues such
More informationSolving Systems of Linear Equations
LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how
More informationConvex analysis and profit/cost/support functions
CALIFORNIA INSTITUTE OF TECHNOLOGY Division of the Humanities and Social Sciences Convex analysis and profit/cost/support functions KC Border October 2004 Revised January 2009 Let A be a subset of R m
More information4: SINGLEPERIOD MARKET MODELS
4: SINGLEPERIOD MARKET MODELS Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney Semester 2, 2015 B. Goldys and M. Rutkowski (USydney) Slides 4: SinglePeriod Market
More informationt := maxγ ν subject to ν {0,1,2,...} and f(x c +γ ν d) f(x c )+cγ ν f (x c ;d).
1. Line Search Methods Let f : R n R be given and suppose that x c is our current best estimate of a solution to P min x R nf(x). A standard method for improving the estimate x c is to choose a direction
More informationALMOST COMMON PRIORS 1. INTRODUCTION
ALMOST COMMON PRIORS ZIV HELLMAN ABSTRACT. What happens when priors are not common? We introduce a measure for how far a type space is from having a common prior, which we term prior distance. If a type
More informationSeparation Properties for Locally Convex Cones
Journal of Convex Analysis Volume 9 (2002), No. 1, 301 307 Separation Properties for Locally Convex Cones Walter Roth Department of Mathematics, Universiti Brunei Darussalam, Gadong BE1410, Brunei Darussalam
More informationSurface bundles over S 1, the Thurston norm, and the Whitehead link
Surface bundles over S 1, the Thurston norm, and the Whitehead link Michael Landry August 16, 2014 The Thurston norm is a powerful tool for studying the ways a 3manifold can fiber over the circle. In
More informationInner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More informationOnline Convex Optimization
E0 370 Statistical Learning heory Lecture 19 Oct 22, 2013 Online Convex Optimization Lecturer: Shivani Agarwal Scribe: Aadirupa 1 Introduction In this lecture we shall look at a fairly general setting
More informationISOMETRIES OF R n KEITH CONRAD
ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x
More informationArrangements And Duality
Arrangements And Duality 3.1 Introduction 3 Point configurations are tbe most basic structure we study in computational geometry. But what about configurations of more complicated shapes? For example,
More informationNotes on metric spaces
Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.
More informationMA651 Topology. Lecture 6. Separation Axioms.
MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples
More informationEXTENSIONS OF MAPS IN SPACES WITH PERIODIC HOMEOMORPHISMS
BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 78, Number 4, July 1972 EXTENSIONS OF MAPS IN SPACES WITH PERIODIC HOMEOMORPHISMS BY JAN W. JAWOROWSKI Communicated by Victor Klee, November 29, 1971
More informationA Problem With The Rational Numbers
Solvability of Equations Solvability of Equations 1. In fields, linear equations ax + b = 0 are solvable. Solvability of Equations 1. In fields, linear equations ax + b = 0 are solvable. 2. Quadratic equations
More information6.207/14.15: Networks Lecture 15: Repeated Games and Cooperation
6.207/14.15: Networks Lecture 15: Repeated Games and Cooperation Daron Acemoglu and Asu Ozdaglar MIT November 2, 2009 1 Introduction Outline The problem of cooperation Finitelyrepeated prisoner s dilemma
More informationSolutions to Homework 10
Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x
More informationMATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets.
MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets. Norm The notion of norm generalizes the notion of length of a vector in R n. Definition. Let V be a vector space. A function α
More information