The Class of k -Convex Functions
|
|
- Mervyn Bryan
- 7 years ago
- Views:
Transcription
1 The Class o k -Convex Functions Martin Kaae Jensen May 5, 0 Abstract This brie note characterizes the class o k -convex and k -concave unctions which represent a strengthening o convexity and a weakening o concavity, respectively. The note is meant primarily as a mathematical support note or Acemoglu and Jensen (0) and Jensen (0), but may be o independent interest to some readers. Keywords: k -convexity, k -concavity, HARA unctions. JEL Classiication Codes: C6, D90, E. Department o Economics, University o Birmingham. ( m.k.jensen@bham.ac.uk)
2 The class o k -Convex Functions Throughout X is assumed to be a convex subset o an ordered vector space. Deinition Let k 0. A unction : X + is said to be k -convex i: When k, the unction k [ (x)] k is convex, or equivalently, i or all α [0, ]: [α( (x)) k + ( α)( (y )) k ] k (αx + ( α)y ) () When k =, the unction log (x ) is convex (i.e. α [0, ]: is log-convex), or equivalently, i or all ( (x)) α ( (y )) α (αx + ( α)y ) () k -concavity is deined similarly by requiring the unction k [ (x)] k to be concave (or reversing the inequalities in ()-()). Notice that the let-hand sides o ()-() are weighted power means (the exponent is k and the weights are (α, α)). So k -convexity can be seen as convexity except that the usual weighted mean is allowed to be any power mean. When k = 0 we thus recoup convexity in the usual sense since the power mean with exponent is just the usual (arithmetic) mean. Notice also that by the power mean inequality, the power mean is smaller than or equal to the usual arithmetic mean whenever k 0 : α[ (x)] + ( α) (y ) [α( (x)) k + ( α)( (y )) k ] k Combining with (), we immediately see that any k -convex unction is convex (in act k -convexity implies k -convexity or all k k by the general ormulation o the power mean inequality). By the same line o reasoning, one sees that k -concavity implies k -concavity or all k k. So while k -convexity is a stronger condition than convexity, k -concavity is weaker than concavity. These statements are also true in the log-convex and log-concave cases, i.e., when k = (and here they are well known). The ollowing lemma tells us how we can conclude that a composite unction is k -convex: Lemma Consider (x ) = h(g (x),..., g n (x)) where h : n is k -convex and g i : n. Then is k -convex i or each i one o the ollowing holds: h is non-increasing in the i th argument and g i is concave. h is non-decreasing in the i th argument and g i is convex. g i is aine. We also note that k -concavity implies quasi-concavity because the power mean on the let-hand side o () converges to min{ (x ), (y )} as k (thus () with the inequality reversed precisely becomes the condition or quasi-concavity in the limit k ).
3 Proo. The statement o the lemma is well-known or convex unctions ( is convex when h is convex and the lemma s conditions on g = (g,..., g n ) hold). Use this on the composition o h(y,...) = k [h(y,..., y n )] k (which is convex) and g to conclude that h g is convex, i.e., = h g is k - convex. For a given k, the set o k -convex unctions is a very well behaved class: It orms a closed and convex cone in the topological vector space o real-valued unctions with the pointwise topology ((i)-(ii)) o the ollowing lemma). And k -convexity is preserved by integration ((iii) o the lemma). All o this is o course well known or 0- and -convex unctions (i.e., convex and log-convex unctions). Lemma (i) I α,β 0, and and g are k -convex, the weighted sum α + β g is k -convex. (ii) The pointwise limit o a sequence o k -convex unctions is k -convex. (iii) k -convexity is preserved by integration: I (x, z ) is measurable in z and k -convex in x or almost every z Z then (x, z )µ(d z ) Z is k -convex. Proo. (i) We take α = β = (the case where these weights are non-trivial adds no extra diiculties to the proo below). Let F = k k and G = k g k and assume that these are convex. We wish to show that, k ( + g ) k = [(( k )F ) k + (( k )G ) k ] k, k is then a convex unction. But this ollows directly rom the act that H(y, y ) = k [(( k )y ) k + (( k )y ) k ] k is increasing and convex or k 0 since the above unction is the composition o H with F and G (the limit case k = where is log-convex is proved by the same argument except that one now uses that H(y, y ) = log(exp(y ) + exp(y )) is convex). (ii) Let (x) = lim n n (x) where each n is k -convex. Since the pointwise limit o a sequence o convex unctions is convex, k [ (x)] k = lim n k [ n(x)] k is convex. (iii) Follows rom (i)-(ii) by a standard argument. Since is not k -concave i is k -convex (in act k -concavity cannot even be deined or ), we cannot in general subtract a k -concave unction rom a k -convex unction and expect that anything nice comes out o it (in this respect k -convex/k -concave unctions behave more like quasiconvex/quasi-concave unctions than they behave like convex/concave unctions). Using that the sum o k -convex unctions is k -convex one can however prove the ollowing subtraction result. Lemma 3 Fix k 0, and let (x) = v (x) u (x) where v : X + is k -convex, and u : X + is k -concave. Then {x X : (x) r } is convex or all r 0, i.e., is quasi-convex when restricted to the domain {x X : (x) 0}. I instead v is k -concave and u is k -convex, then is quasi-concave on the restricted domain {x X : (x) 0}. That k -convexity is preserved by integration was irst proved by Carroll and Kimball (996) in the special case where is twice dierentiable and X + (Carroll and Kimball (996), Lemma ). That proo, however, uses dierentiability in an indispensable manner. The generalization presented here plays an important role or the results in Jensen (0).
4 Proo. Fix r 0 and consider the set L r = {x X : v (x ) u (x) r }. Clearly L r = {x X : [v (x) k r ] k k [u (x)] k 0} (in particular, v (x ) r is a well-deined positive k -convex unction by (i) o lemma ). But since k [v (x ) r ] k k [u (x )] k is convex, it is quasi-convex and thereore L r is a convex set. For the second claim, simply repeat the argument or (x ) = u (x) v (x) in order to conclude that {x X : u (x) v (x) r } is convex or all r 0, which is equivalent to {x X : (x) r } convex or all r 0. Since k -convexity is equivalent to convexity o k [ (x )] k, we can use the Hessian criteria or convexity o a unction to establish k -convexity/k -concavity when the unctions involved are suiciently smooth. Note that since the irst derivative o k [ (x)] k is [ (x)] k (x) (here taking X ) we must assume that (x ) > 0 or all x X in order to be able to use such dierentiability arguments. Lemma 4 Assume that : X + is twice dierentiable and that (x ) > 0 or all x X. I X, is k -convex (k -concave) i and only i: (x) (x) ( )k ( (x)) (3) When (x ) 0 or all x X, this is in turn equivalent to the condition: (x) (x ) ( (x )) ( ) k (4) I X, will be k -convex (k -concave) i and only i the ollowing matrix is positive semideinite (negative semi-deinite): x x k ( x ) x k x x k x x k ( ) The determinant o this matrix will be non-negative i, or example: (i) The Hessian matrix o has a non-negative determinant, and (ii) satisies (assuming here in addition that x, x 0): x x x [ x ( x ) + x ( ) ] Hence will be k -convex (k -concave) i (i)-(ii) hold and one o the diagonal entries o (5) are non-negative (non-positive). 3 Proo. This is just calculus and linear algebra. The determinant o the matrix in the lemma is k [ x x x x ( x ) x x ( x ) x x ] + ( ) [ x x x x ( x x ) ]. (5) The second o these terms 3 O course i one o the diagonal entries is non-negative (non-positive) both will be non-negative (non-positive) when the determinant is non-negative. The point is that we only need to veriy this or one o them when we are aced with a concrete application. 3
5 is the determinant o the Hessian o (multiplied by a positive constant). The irst term equals k ( x ) ( ) [ x x x ] when the irst derivatives are non-zero. That (i)-(ii) imply a x x ( x ) ( x ) non-negative determinant is thereore obvious. Remark It is clear that the diagonal criterion at the end o the previous lemma is a necessary condition or k -convexity or k -concavity. (i) is also a necessary condition in the k -convex case (any k -convex unction is convex), but it is not a necessary condition in the k -concave case (concavity implies k -concavity, but the converse is alse). (ii) is not a necessary condition in either case, but it provides a very convenient way to veriy k -convexity/k -concavity. 4 Remark (The HARA Class) A unction F : X, X, with F 0, is said to exhibit Hyperbolic Absolute Risk Aversion (or lie in the HARA class) i F F (F ) = k 0 (Carroll and Kimball (996)). From the previous lemma we see that this is equivalent to assuming that the derivative o F, F = is k -convex as well as k -concave. Reerences Acemoglu, D. and M. K. Jensen (0): Robust Comparative Statics in Large Dynamic Economies, Mimeo MIT. Carroll, C.D. and M.S. Kimball (996): On the Concavity o the Consumption Function, Econometrica 64, Jensen, M.K. (0): Distributional Comparative Statics, Mimeo, University o Birmingham. 4 As may be veriied, (ii) is in act a necessary condition in the special case where s Hessian matrix has a zero determinant. This is a situation one requently encounters in applications. 4
Polynomials with non-negative coefficients
Polynomials with non-negative coeicients R. W. Barnard W. Dayawansa K. Pearce D.Weinberg Department o Mathematics, Texas Tech University, Lubbock, TX 79409 1 Introduction Can a conjugate pair o zeros be
More informationStudent Outcomes. Lesson Notes. Classwork. Discussion (10 minutes)
NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 8 Student Outcomes Students know the definition of a number raised to a negative exponent. Students simplify and write equivalent expressions that contain
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 informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column
More informationThe Stekloff Problem for Rotationally Invariant Metrics on the Ball
Revista Colombiana de Matemáticas Volumen 47(2032, páginas 8-90 The Steklo Problem or Rotationally Invariant Metrics on the Ball El problema de Steklo para métricas rotacionalmente invariantes en la bola
More information3. Convex functions. basic properties and examples. operations that preserve convexity. the conjugate function. quasiconvex functions
3. Convex functions Convex Optimization Boyd & Vandenberghe basic properties and examples operations that preserve convexity the conjugate function quasiconvex functions log-concave and log-convex functions
More information1 Solving LPs: The Simplex Algorithm of George Dantzig
Solving LPs: The Simplex Algorithm of George Dantzig. Simplex Pivoting: Dictionary Format We illustrate a general solution procedure, called the simplex algorithm, by implementing it on a very simple example.
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 informationHOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!
Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following
More informationNotes on Orthogonal and Symmetric Matrices MENU, Winter 2013
Notes on Orthogonal and Symmetric Matrices MENU, Winter 201 These notes summarize the main properties and uses of orthogonal and symmetric matrices. We covered quite a bit of material regarding these topics,
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 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 informationPractical Guide to the Simplex Method of Linear Programming
Practical Guide to the Simplex Method of Linear Programming Marcel Oliver Revised: April, 0 The basic steps of the simplex algorithm Step : Write the linear programming problem in standard form Linear
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 informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
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. Finite-Dimensional
More informationFollow links for Class Use and other Permissions. For more information send email to: permissions@pupress.princeton.edu
COPYRIGHT NOTICE: Ariel Rubinstein: Lecture Notes in Microeconomic Theory is published by Princeton University Press and copyrighted, c 2006, by Princeton University Press. All rights reserved. No part
More informationSECTION 6: FIBER BUNDLES
SECTION 6: FIBER BUNDLES In this section we will introduce the interesting class o ibrations given by iber bundles. Fiber bundles lay an imortant role in many geometric contexts. For examle, the Grassmaniann
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 informationSensitivity Analysis 3.1 AN EXAMPLE FOR ANALYSIS
Sensitivity Analysis 3 We have already been introduced to sensitivity analysis in Chapter via the geometry of a simple example. We saw that the values of the decision variables and those of the slack and
More informationMATH 4330/5330, Fourier Analysis Section 11, The Discrete Fourier Transform
MATH 433/533, Fourier Analysis Section 11, The Discrete Fourier Transform Now, instead of considering functions defined on a continuous domain, like the interval [, 1) or the whole real line R, we wish
More informationSpecial Situations in the Simplex Algorithm
Special Situations in the Simplex Algorithm Degeneracy Consider the linear program: Maximize 2x 1 +x 2 Subject to: 4x 1 +3x 2 12 (1) 4x 1 +x 2 8 (2) 4x 1 +2x 2 8 (3) x 1, x 2 0. We will first apply the
More informationPower functions: f(x) = x n, n is a natural number The graphs of some power functions are given below. n- even n- odd
5.1 Polynomial Functions A polynomial unctions is a unction o the orm = a n n + a n-1 n-1 + + a 1 + a 0 Eample: = 3 3 + 5 - The domain o a polynomial unction is the set o all real numbers. The -intercepts
More informationStochastic Inventory Control
Chapter 3 Stochastic Inventory Control 1 In this chapter, we consider in much greater details certain dynamic inventory control problems of the type already encountered in section 1.3. In addition to the
More informationPYTHAGOREAN TRIPLES KEITH CONRAD
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient
More informationNOTES ON CATEGORIES AND FUNCTORS
NOTES ON CATEGORIES AND FUNCTORS These notes collect basic definitions and facts about categories and functors that have been mentioned in the Homological Algebra course. For further reading about category
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 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 informationChapter 2: Linear Equations and Inequalities Lecture notes Math 1010
Section 2.1: Linear Equations Definition of equation An equation is a statement that equates two algebraic expressions. Solving an equation involving a variable means finding all values of the variable
More informationLecture 5 Principal Minors and the Hessian
Lecture 5 Principal Minors and the Hessian Eivind Eriksen BI Norwegian School of Management Department of Economics October 01, 2010 Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and
More informationNotes on Factoring. MA 206 Kurt Bryan
The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor
More information3. Mathematical Induction
3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)
More information5.3 The Cross Product in R 3
53 The Cross Product in R 3 Definition 531 Let u = [u 1, u 2, u 3 ] and v = [v 1, v 2, v 3 ] Then the vector given by [u 2 v 3 u 3 v 2, u 3 v 1 u 1 v 3, u 1 v 2 u 2 v 1 ] is called the cross product (or
More information0 0 such that f x L whenever x a
Epsilon-Delta Definition of the Limit Few statements in elementary mathematics appear as cryptic as the one defining the limit of a function f() at the point = a, 0 0 such that f L whenever a Translation:
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 informationDefinition and Properties of the Production Function: Lecture
Definition and Properties of the Production Function: Lecture II August 25, 2011 Definition and : Lecture A Brief Brush with Duality Cobb-Douglas Cost Minimization Lagrangian for the Cobb-Douglas Solution
More informationLecture Notes on Elasticity of Substitution
Lecture Notes on Elasticity of Substitution Ted Bergstrom, UCSB Economics 210A March 3, 2011 Today s featured guest is the elasticity of substitution. Elasticity of a function of a single variable Before
More informationChapter 6. Linear Programming: The Simplex Method. Introduction to the Big M Method. Section 4 Maximization and Minimization with Problem Constraints
Chapter 6 Linear Programming: The Simplex Method Introduction to the Big M Method In this section, we will present a generalized version of the simplex method that t will solve both maximization i and
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
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 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 information8 Square matrices continued: Determinants
8 Square matrices continued: Determinants 8. Introduction Determinants give us important information about square matrices, and, as we ll soon see, are essential for the computation of eigenvalues. You
More informationProperties of Real Numbers
16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should
More information5.1 Radical Notation and Rational Exponents
Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots
More informationSUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by
SUBGROUPS OF CYCLIC GROUPS KEITH CONRAD 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by g = {g k : k Z}. If G = g, then G itself is cyclic, with g as a generator. Examples
More informationQuotient Rings and Field Extensions
Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.
More informationSome Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.
Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,
More informationFIXED INCOME ATTRIBUTION
Sotware Requirement Speciication FIXED INCOME ATTRIBUTION Authors Risto Lehtinen Version Date Comment 0.1 2007/02/20 First Drat Table o Contents 1 Introduction... 3 1.1 Purpose o Document... 3 1.2 Glossary,
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationLecture 3: Finding integer solutions to systems of linear equations
Lecture 3: Finding integer solutions to systems of linear equations Algorithmic Number Theory (Fall 2014) Rutgers University Swastik Kopparty Scribe: Abhishek Bhrushundi 1 Overview The goal of this lecture
More informationby the matrix A results in a vector which is a reflection of the given
Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that
More informationCITY AND REGIONAL PLANNING 7230. Consumer Behavior. Philip A. Viton. March 4, 2015. 1 Introduction 2
CITY AND REGIONAL PLANNING 7230 Consumer Behavior Philip A. Viton March 4, 2015 Contents 1 Introduction 2 2 Foundations 2 2.1 Consumption bundles........................ 2 2.2 Preference relations.........................
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More information1 Homework 1. [p 0 q i+j +... + p i 1 q j+1 ] + [p i q j ] + [p i+1 q j 1 +... + p i+j q 0 ]
1 Homework 1 (1) Prove the ideal (3,x) is a maximal ideal in Z[x]. SOLUTION: Suppose we expand this ideal by including another generator polynomial, P / (3, x). Write P = n + x Q with n an integer not
More information15 Limit sets. Lyapunov functions
15 Limit sets. Lyapunov functions At this point, considering the solutions to ẋ = f(x), x U R 2, (1) we were most interested in the behavior of solutions when t (sometimes, this is called asymptotic behavior
More informationGambling Systems and Multiplication-Invariant Measures
Gambling Systems and Multiplication-Invariant Measures by Jeffrey S. Rosenthal* and Peter O. Schwartz** (May 28, 997.. Introduction. This short paper describes a surprising connection between two previously
More informationGeneral Framework for an Iterative Solution of Ax b. Jacobi s Method
2.6 Iterative Solutions of Linear Systems 143 2.6 Iterative Solutions of Linear Systems Consistent linear systems in real life are solved in one of two ways: by direct calculation (using a matrix factorization,
More informationTHE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING
THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING 1. Introduction The Black-Scholes theory, which is the main subject of this course and its sequel, is based on the Efficient Market Hypothesis, that arbitrages
More informationLOGARITHMIC FUNCTIONAL AND THE WEIL RECIPROCITY LAW
85 LOGARITHMIC FUNCTIONAL AND THE WEIL RECIPROCITY LAW KHOVANSKII A. Department o Mathematics, University o Toronto, Toronto, Ontario, Canada E-mail: askold@math.toronto.edu http://www.math.toronto.edu
More informationLecture 7: Finding Lyapunov Functions 1
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 7: Finding Lyapunov Functions 1
More informationVector Math Computer Graphics Scott D. Anderson
Vector Math Computer Graphics Scott D. Anderson 1 Dot Product The notation v w means the dot product or scalar product or inner product of two vectors, v and w. In abstract mathematics, we can talk about
More information0.8 Rational Expressions and Equations
96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to
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 informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More information3 Some Integer Functions
3 Some Integer Functions A Pair of Fundamental Integer Functions The integer function that is the heart of this section is the modulo function. However, before getting to it, let us look at some very simple
More informationc 2008 Je rey A. Miron We have described the constraints that a consumer faces, i.e., discussed the budget constraint.
Lecture 2b: Utility c 2008 Je rey A. Miron Outline: 1. Introduction 2. Utility: A De nition 3. Monotonic Transformations 4. Cardinal Utility 5. Constructing a Utility Function 6. Examples of Utility Functions
More informationMulti-variable Calculus and Optimization
Multi-variable Calculus and Optimization Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Multi-variable Calculus and Optimization 1 / 51 EC2040 Topic 3 - Multi-variable Calculus
More informationFinancial Services [Applications]
Financial Services [Applications] Tomáš Sedliačik Institute o Finance University o Vienna tomas.sedliacik@univie.ac.at 1 Organization Overall there will be 14 units (12 regular units + 2 exams) Course
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 informationSolving Systems of Linear Equations Using Matrices
Solving Systems of Linear Equations Using Matrices What is a Matrix? A matrix is a compact grid or array of numbers. It can be created from a system of equations and used to solve the system of equations.
More informationThis unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.
Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More informationBindel, Spring 2012 Intro to Scientific Computing (CS 3220) Week 3: Wednesday, Feb 8
Spaces and bases Week 3: Wednesday, Feb 8 I have two favorite vector spaces 1 : R n and the space P d of polynomials of degree at most d. For R n, we have a canonical basis: R n = span{e 1, e 2,..., e
More informationSample Induction Proofs
Math 3 Worksheet: Induction Proofs III, Sample Proofs A.J. Hildebrand Sample Induction Proofs Below are model solutions to some of the practice problems on the induction worksheets. The solutions given
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 informationLINEAR INEQUALITIES. Mathematics is the art of saying many things in many different ways. MAXWELL
Chapter 6 LINEAR INEQUALITIES 6.1 Introduction Mathematics is the art of saying many things in many different ways. MAXWELL In earlier classes, we have studied equations in one variable and two variables
More informationSECOND DERIVATIVE TEST FOR CONSTRAINED EXTREMA
SECOND DERIVATIVE TEST FOR CONSTRAINED EXTREMA This handout presents the second derivative test for a local extrema of a Lagrange multiplier problem. The Section 1 presents a geometric motivation for the
More informationI. GROUPS: BASIC DEFINITIONS AND EXAMPLES
I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called
More informationSection 4.4 Inner Product Spaces
Section 4.4 Inner Product Spaces In our discussion of vector spaces the specific nature of F as a field, other than the fact that it is a field, has played virtually no role. In this section we no longer
More informationThe Determinant: a Means to Calculate Volume
The Determinant: a Means to Calculate Volume Bo Peng August 20, 2007 Abstract This paper gives a definition of the determinant and lists many of its well-known properties Volumes of parallelepipeds are
More informationMATH10040 Chapter 2: Prime and relatively prime numbers
MATH10040 Chapter 2: Prime and relatively prime numbers Recall the basic definition: 1. Prime numbers Definition 1.1. Recall that a positive integer is said to be prime if it has precisely two positive
More information11 Ideals. 11.1 Revisiting Z
11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(
More informationChoice under Uncertainty
Choice under Uncertainty Part 1: Expected Utility Function, Attitudes towards Risk, Demand for Insurance Slide 1 Choice under Uncertainty We ll analyze the underlying assumptions of expected utility theory
More informationThe Steepest Descent Algorithm for Unconstrained Optimization and a Bisection Line-search Method
The Steepest Descent Algorithm for Unconstrained Optimization and a Bisection Line-search Method Robert M. Freund February, 004 004 Massachusetts Institute of Technology. 1 1 The Algorithm The problem
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 informationLINEAR EQUATIONS IN TWO VARIABLES
66 MATHEMATICS CHAPTER 4 LINEAR EQUATIONS IN TWO VARIABLES The principal use of the Analytic Art is to bring Mathematical Problems to Equations and to exhibit those Equations in the most simple terms that
More informationCritical points of once continuously differentiable functions are important because they are the only points that can be local maxima or minima.
Lecture 0: Convexity and Optimization We say that if f is a once continuously differentiable function on an interval I, and x is a point in the interior of I that x is a critical point of f if f (x) =
More informationName. Final Exam, Economics 210A, December 2011 Here are some remarks to help you with answering the questions.
Name Final Exam, Economics 210A, December 2011 Here are some remarks to help you with answering the questions. Question 1. A firm has a production function F (x 1, x 2 ) = ( x 1 + x 2 ) 2. It is a price
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,
More informationLinear Programming Notes V Problem Transformations
Linear Programming Notes V Problem Transformations 1 Introduction Any linear programming problem can be rewritten in either of two standard forms. In the first form, the objective is to maximize, the material
More information16.3 Fredholm Operators
Lectures 16 and 17 16.3 Fredholm Operators A nice way to think about compact operators is to show that set of compact operators is the closure of the set of finite rank operator in operator norm. In this
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 informationSYSTEMS OF EQUATIONS AND MATRICES WITH THE TI-89. by Joseph Collison
SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI-89 by Joseph Collison Copyright 2000 by Joseph Collison All rights reserved Reproduction or translation of any part of this work beyond that permitted by Sections
More informationReview of Fundamental Mathematics
Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools
More informationON FIBONACCI NUMBERS WITH FEW PRIME DIVISORS
ON FIBONACCI NUMBERS WITH FEW PRIME DIVISORS YANN BUGEAUD, FLORIAN LUCA, MAURICE MIGNOTTE, SAMIR SIKSEK Abstract If n is a positive integer, write F n for the nth Fibonacci number, and ω(n) for the number
More informationCS 103X: Discrete Structures Homework Assignment 3 Solutions
CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On well-ordering and induction: (a) Prove the induction principle from the well-ordering principle. (b) Prove the well-ordering
More informationLinear Codes. Chapter 3. 3.1 Basics
Chapter 3 Linear Codes In order to define codes that we can encode and decode efficiently, we add more structure to the codespace. We shall be mainly interested in linear codes. A linear code of length
More informationUndergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics
Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights
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 informationPart 1 Expressions, Equations, and Inequalities: Simplifying and Solving
Section 7 Algebraic Manipulations and Solving Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Before launching into the mathematics, let s take a moment to talk about the words
More information