{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,...


 Kelly Douglas
 1 years ago
 Views:
Transcription
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 is recovered if the range space is a normed vector space or the real numbers. The text has some theorems pointing out that the limit of sum is the sum of the limits, and that when they make sense, the limit of a product is the product of the limits and the limit of a quotient is the quotient of the limits. Compactness and connectedness are preserved by continuous functions. Theorem If M 1 is connected and f : M 1 M 2 is continuous, then f(m 1 ) is connected. Proof. Suppose not. Let f(m 1 ) = V 1 V 2. Then M 1 = f 1 (V 1 ) f 1 (V 2 ) is a separation of M 1, which doesn t exist. Theorem Let A be a subset of a metric space (M 1, d 1 ), and suppose f : A (M 2, d 2 ) is a continuous function. If A is compact, then f(a) is compact. Proof. Let F be a family of open sets covering f(a), with U F. Then {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 )}, so is a cover of f(a). {U 1,..., U n } Proposition Let A be a subset of a metric space (S 1, d 1 ), and suppose f : A (S 2, d 2 ) is a continuous function. If A is compact, and f is onetoone, then f 1 is continuous. Proof. Since f : A f(a) is onetoone we have a well defined function g = f 1 : f(a) A, and g 1 = (f 1 ) 1 = f. Suppose K is closed in A. Since A is compact, so is K. Then g 1 (K) = f(k) is compact in f(a). But this means g 1 (K) is closed, and so g 1 (K) is closed for every closed set K A.
2 4.1. MAPPINGS FROM ONE METRIC SPACE TO ANOTHER 45 Here are some theorems generalizing results used frequently in Calculus. Theorem If K M is compact and f : K R is continuous, then f has a maximum and a minimum on K. Proof. We know that f(k) R is compact, so bounded. Moreover the set f(k) is closed, so it contains sup x K f(x) and inf x K f(x), which is the claim. In the introductory remarks about connectednes we noted that a version of the Intermediate Value Theorem would hold for continuous functions f : M R if (M, d) was a path connected metric space. A somewhat more general result is true. Theorem Suppose (M, d) is a metric space, f : A M R is continuous, and A is connected. If x, y A, then for every real number c between f(x) and f(y) there is some z A with f(z) = c. Proof. The argument is by contradiction. If not, then the range of f : A R is separated by the open sets U = (, c) and V = (c, ). The open sets f 1 (U), f 1 (V ) provide a separation of A, which can t exist. An other concept that we encountered in our earlier analysis course is the notion of uniform continuity. In terms of ǫ δ language the idea is that in some situations we can pick δ independent of x. We say f : M 1 M 2 is uniformly continuous if for every ǫ > 0 there is a δ > 0, independent of x M 1, such that d 2 (f(x), f(y)) < ǫ if d 1 (x, y) < δ. In Calculus this property is commonly seen when we have a function with a bounded continuous derivative, since f(y) f(x) = y It s useful to know that compactness is enough. x f (t) dt y x sup f (t). t Theorem Suppose (M 1, d 1 ) and (M 2, d 2 ) are metric spaces, and f : M 1 M 2 is continuous. If M 1 is compact, then f is uniformly continuous.
3 46 CHAPTER 4. CONTINUOUS FUNCTIONS Proof. Given ǫ > 0 and x M 1, find disks D(x, δ x ) such that d 1 (x, y) < δ x implies d 2 (f(x), f(y)) < ǫ/2. Shrink the radii, looking at sets D(x, δ x /2). This open cover of M 1 has a finite subcover by sets D(x n, δ n /2). Take δ = min n δ n /2. Now suppose d 1 (x, y) < δ. Since every point is in a set of the cover, x is within δ < δ n /2 of one of the previously selected points x n. Then d 1 (x n, y) d 1 (x n, x) + d 1 (x, y) < δ n, so d 2 (f(x), f(y)) d 2 (f(x), f(x n )) + d 2 (f(y), f(x n )) < ǫ. Here are a few words about treating subsets of metric spaces as independent metric spaces. Suppose (M, d) is a metric space and A M. Then the metric subspace generated by A is the metric space (A, d A ), where d A is the restriction of d to A A. Define a set U A to be open in A if U A is open as a subset of the metric space (A, d A ). Define a set F A to be closed in A if F A is closed as a subset of the metric space (A, d A ). Theorem a) U A is open in A if and only if there is an open set U M such that U A = U A. b) F A is closed in A if and only if there is an closed set F M such that F A = F A. Proof. a) First observe that for x A, D A (x, ǫ) = D M (x, ǫ) A. Suppose U A is open in A, and let x U A. Then there is some ǫ > 0 such that D A (x, ǫ) U A. We have U A = x UA D A (x, ǫ x ) = A [ x UA D M (x, ǫ x )] = A U. Suppose U A = A U, where U is open in M. For every x U A there is an ǫ > 0 such that D M (x, ǫ x ) U. But then we have D A (x, ǫ x ) = A D M (x, ǫ x ) A U = U A. b) Suppose F A is closed in A. Then A \ F A is open in A, so by part (a) A \ F A = A U
4 4.1. MAPPINGS FROM ONE METRIC SPACE TO ANOTHER 47 for some open U in M. Taking complements in A gives F A = A \ (A U) = A [M \ U], which shows that F A = A F for a closed set F M. Suppose that F A = A F for a closed set F M. Then and F A is closed in A. A \ F A = A [M \ F] = A U,
5 48 CHAPTER 4. CONTINUOUS FUNCTIONS
6 Chapter 5 Uniform Convergence 5.1 Uniform Convergence Power series provide one example of the following situation. We represent a function f(x) by a sequence or series of elementary functions. The elementary functions have good properties, like ease of integration or differentiation, and we want these operations to extend to the limit function. Usually this is handled with the concept of uniform convergence. We distinguish two definitions. Suppose A is a set and (M 1, d) is a metric space. Assume f n : A M is a sequence of functions. The sequence {f n } converges pointwise to f : A M if for every ǫ > 0 and every x A there is an N(x, ǫ) such that d(f(x), f n (x)) < ǫ, n N. The sequence {f n } converges uniformly to f : A M if for every ǫ > 0 and every x S there is an N(ǫ) such that d(f(x), f n (x)) < ǫ, n N, x S. Draw an epsilon tube about f to illustrate the definitions. Very little can be said about the pointwise limits of functions. For instance we can find sliding tents f n : [0, 1] R such that f n 0 pointwise, but 1 f n = 1, 0 49
7 50 CHAPTER 5. UNIFORM CONVERGENCE or 1 f n. 0 Similarly, we can construct a sequence of infinitely differentiable functions g n (x) = tan 1 (nx), which converges pointwise to the function g = π/2, x > 0, π/2, x < 0, 0, x = 0.
Metric 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 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 information9 More on differentiation
Tel Aviv University, 2013 Measure and category 75 9 More on differentiation 9a Finite Taylor expansion............... 75 9b Continuous and nowhere differentiable..... 78 9c Differentiable and nowhere monotone......
More informationLectures 56: Taylor Series
Math 1d Instructor: Padraic Bartlett Lectures 5: Taylor Series Weeks 5 Caltech 213 1 Taylor Polynomials and Series As we saw in week 4, power series are remarkably nice objects to work with. In particular,
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 information4. Expanding dynamical systems
4.1. Metric definition. 4. Expanding dynamical systems Definition 4.1. Let X be a compact metric space. A map f : X X is said to be expanding if there exist ɛ > 0 and L > 1 such that d(f(x), f(y)) Ld(x,
More informationTHE CENTRAL LIMIT THEOREM TORONTO
THE CENTRAL LIMIT THEOREM DANIEL RÜDT UNIVERSITY OF TORONTO MARCH, 2010 Contents 1 Introduction 1 2 Mathematical Background 3 3 The Central Limit Theorem 4 4 Examples 4 4.1 Roulette......................................
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 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 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 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 informationChapter 5. Banach Spaces
9 Chapter 5 Banach Spaces Many linear equations may be formulated in terms of a suitable linear operator acting on a Banach space. In this chapter, we study Banach spaces and linear operators acting on
More informationSection 3.7. Rolle s Theorem and the Mean Value Theorem. Difference Equations to Differential Equations
Difference Equations to Differential Equations Section.7 Rolle s Theorem and the Mean Value Theorem The two theorems which are at the heart of this section draw connections between the instantaneous rate
More informationMathematics for Econometrics, Fourth Edition
Mathematics for Econometrics, Fourth Edition Phoebus J. Dhrymes 1 July 2012 1 c Phoebus J. Dhrymes, 2012. Preliminary material; not to be cited or disseminated without the author s permission. 2 Contents
More informationA ProblemText in Advanced Calculus. John M. Erdman Portland State University. Version July 1, 2014
A ProblemText in Advanced Calculus John M. Erdman Portland State University Version July 1, 2014 c 2005 John M. Erdman Email address: erdman@pdx.edu. ii To Argentina Contents PREFACE FOR STUDENTS: HOW
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 informationISU Department of Mathematics. Graduate Examination Policies and Procedures
ISU Department of Mathematics Graduate Examination Policies and Procedures There are four primary criteria to be used in evaluating competence on written or oral exams. 1. Knowledge Has the student demonstrated
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 informationTaylor and Maclaurin Series
Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions
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 information1 = (a 0 + b 0 α) 2 + + (a m 1 + b m 1 α) 2. for certain elements a 0,..., a m 1, b 0,..., b m 1 of F. Multiplying out, we obtain
Notes on realclosed fields These notes develop the algebraic background needed to understand the model theory of realclosed fields. To understand these notes, a standard graduate course in algebra is
More informationSome stability results of parameter identification in a jump diffusion model
Some stability results of parameter identification in a jump diffusion model D. Düvelmeyer Technische Universität Chemnitz, Fakultät für Mathematik, 09107 Chemnitz, Germany Abstract In this paper we discuss
More information1 The Brownian bridge construction
The Brownian bridge construction The Brownian bridge construction is a way to build a Brownian motion path by successively adding finer scale detail. This construction leads to a relatively easy proof
More informationFurther Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1
Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1 J. Zhang Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing
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 informationUNIFORMLY DISCONTINUOUS GROUPS OF ISOMETRIES OF THE PLANE
UNIFORMLY DISCONTINUOUS GROUPS OF ISOMETRIES OF THE PLANE NINA LEUNG Abstract. This paper discusses 2dimensional locally Euclidean geometries and how these geometries can describe musical chords. Contents
More informationRandom graphs with a given degree sequence
Sourav Chatterjee (NYU) Persi Diaconis (Stanford) Allan Sly (Microsoft) Let G be an undirected simple graph on n vertices. Let d 1,..., d n be the degrees of the vertices of G arranged in descending order.
More informationThe Heat Equation. Lectures INF2320 p. 1/88
The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)
More informationChapter 20. Vector Spaces and Bases
Chapter 20. Vector Spaces and Bases In this course, we have proceeded stepbystep through lowdimensional Linear Algebra. We have looked at lines, planes, hyperplanes, and have seen that there is no limit
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 informationF. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)
Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p
More informationTiers, Preference Similarity, and the Limits on Stable Partners
Tiers, Preference Similarity, and the Limits on Stable Partners KANDORI, Michihiro, KOJIMA, Fuhito, and YASUDA, Yosuke February 7, 2010 Preliminary and incomplete. Do not circulate. Abstract We consider
More informationChapter 7. Continuity
Chapter 7 Continuity There are many processes and eects that depends on certain set of variables in such a way that a small change in these variables acts as small change in the process. Changes of this
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 informationThe Steepest Descent Algorithm for Unconstrained Optimization and a Bisection Linesearch Method
The Steepest Descent Algorithm for Unconstrained Optimization and a Bisection Linesearch Method Robert M. Freund February, 004 004 Massachusetts Institute of Technology. 1 1 The Algorithm The problem
More informationPoint Set Topology. A. Topological Spaces and Continuous Maps
Point Set Topology A. Topological Spaces and Continuous Maps Definition 1.1 A topology on a set X is a collection T of subsets of X satisfying the following axioms: T 1.,X T. T2. {O α α I} T = α IO α T.
More informationNotes from Week 1: Algorithms for sequential prediction
CS 683 Learning, Games, and Electronic Markets Spring 2007 Notes from Week 1: Algorithms for sequential prediction Instructor: Robert Kleinberg 2226 Jan 2007 1 Introduction In this course we will be looking
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 informationDISINTEGRATION OF MEASURES
DISINTEGRTION OF MESURES BEN HES Definition 1. Let (, M, λ), (, N, µ) be sigmafinite measure spaces and let T : be a measurable map. (T, µ)disintegration is a collection {λ y } y of measures on M such
More informationOptimal Onlinelist Batch Scheduling
Optimal Onlinelist Batch Scheduling Jacob Jan Paulus a,, Deshi Ye b, Guochuan Zhang b a University of Twente, P.O. box 217, 7500AE Enschede, The Netherlands b Zhejiang University, Hangzhou 310027, China
More informationRANDOM INTERVAL HOMEOMORPHISMS. MICHA L MISIUREWICZ Indiana University Purdue University Indianapolis
RANDOM INTERVAL HOMEOMORPHISMS MICHA L MISIUREWICZ Indiana University Purdue University Indianapolis This is a joint work with Lluís Alsedà Motivation: A talk by Yulij Ilyashenko. Two interval maps, applied
More informationGeometrical Characterization of RNoperators between Locally Convex Vector Spaces
Geometrical Characterization of RNoperators between Locally Convex Vector Spaces OLEG REINOV St. Petersburg State University Dept. of Mathematics and Mechanics Universitetskii pr. 28, 198504 St, Petersburg
More informationRate of convergence towards Hartree dynamics
Rate of convergence towards Hartree dynamics Benjamin Schlein, LMU München and University of Cambridge Universitá di Milano Bicocca, October 22, 2007 Joint work with I. Rodnianski 1. Introduction boson
More informationLecture 13 Linear quadratic Lyapunov theory
EE363 Winter 289 Lecture 13 Linear quadratic Lyapunov theory the Lyapunov equation Lyapunov stability conditions the Lyapunov operator and integral evaluating quadratic integrals analysis of ARE discretetime
More informationA new continuous dependence result for impulsive retarded functional differential equations
CADERNOS DE MATEMÁTICA 11, 37 47 May (2010) ARTIGO NÚMERO SMA#324 A new continuous dependence result for impulsive retarded functional differential equations M. Federson * Instituto de Ciências Matemáticas
More informationON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE. 1. Introduction
ON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE J.M. CALABUIG, J. RODRÍGUEZ, AND E.A. SÁNCHEZPÉREZ Abstract. Let m be a vector measure taking values in a Banach space X. We prove that
More informationReal Number Computability and Domain Theory
Real Number Computability and Domain Theory Pietro Di Gianantonio dipartimento di Matematica e Informatica, Università di Udine via delle Scienze 206 I33100 Udine Italy Email: pietro@dimi.uniud.it Abstract
More informationINTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 24
INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 24 RAVI VAKIL Contents 1. Degree of a line bundle / invertible sheaf 1 1.1. Last time 1 1.2. New material 2 2. The sheaf of differentials of a nonsingular curve
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 information1 Sufficient statistics
1 Sufficient statistics A statistic is a function T = rx 1, X 2,, X n of the random sample X 1, X 2,, X n. Examples are X n = 1 n s 2 = = X i, 1 n 1 the sample mean X i X n 2, the sample variance T 1 =
More informationFixed Point Theorems in Topology and Geometry
Fixed Point Theorems in Topology and Geometry A Senior Thesis Submitted to the Department of Mathematics In Partial Fulfillment of the Requirements for the Departmental Honors Baccalaureate By Morgan Schreffler
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 informationMATH 132: CALCULUS II SYLLABUS
MATH 32: CALCULUS II SYLLABUS Prerequisites: Successful completion of Math 3 (or its equivalent elsewhere). Math 27 is normally not a sufficient prerequisite for Math 32. Required Text: Calculus: Early
More informationNOV  30211/II. 1. Let f(z) = sin z, z C. Then f(z) : 3. Let the sequence {a n } be given. (A) is bounded in the complex plane
Mathematical Sciences Paper II Time Allowed : 75 Minutes] [Maximum Marks : 100 Note : This Paper contains Fifty (50) multiple choice questions. Each question carries Two () marks. Attempt All questions.
More informationPersuasion by Cheap Talk  Online Appendix
Persuasion by Cheap Talk  Online Appendix By ARCHISHMAN CHAKRABORTY AND RICK HARBAUGH Online appendix to Persuasion by Cheap Talk, American Economic Review Our results in the main text concern the case
More informationOn a conjecture by Palis
1179 2000 103108 103 On a conjecture by Palis (Shuhei Hayashi) Introduction Let $M$ be a smooth compact manifold without boundary and let Diff1 $(M)$ be the set of $C^{1}$ diffeomorphisms with the $C^{1}$
More informationVerifying Numerical Convergence Rates
1 Order of accuracy Verifying Numerical Convergence Rates We consider a numerical approximation of an exact value u. Te approximation depends on a small parameter, suc as te grid size or time step, and
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 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 information1. (First passage/hitting times/gambler s ruin problem:) Suppose that X has a discrete state space and let i be a fixed state. Let
Copyright c 2009 by Karl Sigman 1 Stopping Times 1.1 Stopping Times: Definition Given a stochastic process X = {X n : n 0}, a random time τ is a discrete random variable on the same probability space as
More information1 Formulating The Low Degree Testing Problem
6.895 PCP and Hardness of Approximation MIT, Fall 2010 Lecture 5: Linearity Testing Lecturer: Dana Moshkovitz Scribe: Gregory Minton and Dana Moshkovitz In the last lecture, we proved a weak PCP Theorem,
More informationLogic, Algebra and Truth Degrees 2008. Siena. A characterization of rst order rational Pavelka's logic
Logic, Algebra and Truth Degrees 2008 September 811, 2008 Siena A characterization of rst order rational Pavelka's logic Xavier Caicedo Universidad de los Andes, Bogota Under appropriate formulations,
More informationSequences and Series
Sequences and Series Consider the following sum: 2 + 4 + 8 + 6 + + 2 i + The dots at the end indicate that the sum goes on forever. Does this make sense? Can we assign a numerical value to an infinite
More informationMathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson
Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement
More informationHedging bounded claims with bounded outcomes
Hedging bounded claims with bounded outcomes Freddy Delbaen ETH Zürich, Department of Mathematics, CH892 Zurich, Switzerland Abstract. We consider a financial market with two or more separate components
More informationEstimating the Degree of Activity of jumps in High Frequency Financial Data. joint with Yacine AïtSahalia
Estimating the Degree of Activity of jumps in High Frequency Financial Data joint with Yacine AïtSahalia Aim and setting An underlying process X = (X t ) t 0, observed at equally spaced discrete times
More informationInterpersonal Comparisons of Utility: An Algebraic Characterization of Projective Preorders and Some Welfare Consequences
DISCUSSION PAPER SERIES IZA DP No. 2594 Interpersonal Comparisons of Utility: An Algebraic Characterization of Projective Preorders and Some Welfare Consequences Juan Carlos Candeal Esteban Induráin José
More informationTOPOLOGY: THE JOURNEY INTO SEPARATION AXIOMS
TOPOLOGY: THE JOURNEY INTO SEPARATION AXIOMS VIPUL NAIK Abstract. In this journey, we are going to explore the so called separation axioms in greater detail. We shall try to understand how these axioms
More informationTuring Degrees and Definability of the Jump. Theodore A. Slaman. University of California, Berkeley. CJuly, 2005
Turing Degrees and Definability of the Jump Theodore A. Slaman University of California, Berkeley CJuly, 2005 Outline Lecture 1 Forcing in arithmetic Coding and decoding theorems Automorphisms of countable
More informationMath/Stats 425 Introduction to Probability. 1. Uncertainty and the axioms of probability
Math/Stats 425 Introduction to Probability 1. Uncertainty and the axioms of probability Processes in the real world are random if outcomes cannot be predicted with certainty. Example: coin tossing, stock
More informationDIRICHLET S PROBLEM WITH ENTIRE DATA POSED ON AN ELLIPSOIDAL CYLINDER. 1. Introduction
DIRICHLET S PROBLEM WITH ENTIRE DATA POSED ON AN ELLIPSOIDAL CYLINDER DMITRY KHAVINSON, ERIK LUNDBERG, HERMANN RENDER. Introduction A function u is said to be harmonic if u := n j= 2 u = 0. Given a domain
More informationMINIMUM VERTEX DEGREE CONDITIONS FOR LOOSE HAMILTON CYCLES IN 3UNIFORM HYPERGRAPHS., where
MINIMUM VERTEX DEGREE CONDITIONS FOR LOOSE HAMILTON CYCLES IN UNIFORM HYPERGRAPHS ENNO BUSS, HIỆP HÀN, AND MATHIAS SCHACHT Abstract. We investigate minimum vertex degree conditions for uniform hypergraphs
More informationarxiv:0805.1169v3 [math.oc] 12 Oct 2008
GEOMETRIC APPROACH TO PONTRYAGIN S MAXIMUM PRINCIPLE arxiv:0805.1169v3 [math.oc] 12 Oct 2008 M. BarberoLiñán, M. C. MuñozLecanda Departamento de Matemática Aplicada IV Edificio C 3, Campus Norte UPC.
More informationLebesgue Measure on R n
8 CHAPTER 2 Lebesgue Measure on R n Our goal is to construct a notion of the volume, or Lebesgue measure, of rather general subsets of R n that reduces to the usual volume of elementary geometrical sets
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 informationA simple criterion on degree sequences of graphs
Discrete Applied Mathematics 156 (2008) 3513 3517 Contents lists available at ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam Note A simple criterion on degree
More informationINTRODUCTORY SET THEORY
M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H1088 Budapest, Múzeum krt. 68. CONTENTS 1. SETS Set, equal sets, subset,
More informationA power series about x = a is the series of the form
POWER SERIES AND THE USES OF POWER SERIES Elizabeth Wood Now we are finally going to start working with a topic that uses all of the information from the previous topics. The topic that we are going to
More informationErdős on polynomials
Erdős on polynomials Vilmos Totik University of Szeged and University of South Florida totik@mail.usf.edu Vilmos Totik (SZTE and USF) Polynomials 1 / * Erdős on polynomials Vilmos Totik (SZTE and USF)
More informationExtension of measure
1 Extension of measure Sayan Mukherjee Dynkin s π λ theorem We will soon need to define probability measures on infinite and possible uncountable sets, like the power set of the naturals. This is hard.
More informationReference: Introduction to Partial Differential Equations by G. Folland, 1995, Chap. 3.
5 Potential Theory Reference: Introduction to Partial Differential Equations by G. Folland, 995, Chap. 3. 5. Problems of Interest. In what follows, we consider Ω an open, bounded subset of R n with C 2
More informationReading material on the limit set of a Fuchsian group
Reading material on the limit set of a Fuchsian group Recommended texts Many books on hyperbolic geometry and Kleinian and Fuchsian groups contain material about limit sets. The presentation given here
More informationLINEAR PROGRAMMING WITH ONLINE LEARNING
LINEAR PROGRAMMING WITH ONLINE LEARNING TATSIANA LEVINA, YURI LEVIN, JEFF MCGILL, AND MIKHAIL NEDIAK SCHOOL OF BUSINESS, QUEEN S UNIVERSITY, 143 UNION ST., KINGSTON, ON, K7L 3N6, CANADA EMAIL:{TLEVIN,YLEVIN,JMCGILL,MNEDIAK}@BUSINESS.QUEENSU.CA
More information17.3.1 Follow the Perturbed Leader
CS787: Advanced Algorithms Topic: Online Learning Presenters: David He, Chris Hopman 17.3.1 Follow the Perturbed Leader 17.3.1.1 Prediction Problem Recall the prediction problem that we discussed in class.
More informationProperties of BMO functions whose reciprocals are also BMO
Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a nonnegative BMOfunction w, whose reciprocal is also in BMO, belongs to p> A p,and
More informationAnalytic cohomology groups in top degrees of Zariski open sets in P n
Analytic cohomology groups in top degrees of Zariski open sets in P n Gabriel Chiriacescu, Mihnea Colţoiu, Cezar Joiţa Dedicated to Professor Cabiria Andreian Cazacu on her 80 th birthday 1 Introduction
More informationCHAPTER 5. Product Measures
54 CHAPTER 5 Product Measures Given two measure spaces, we may construct a natural measure on their Cartesian product; the prototype is the construction of Lebesgue measure on R 2 as the product of Lebesgue
More informationGambling Systems and MultiplicationInvariant Measures
Gambling Systems and MultiplicationInvariant Measures by Jeffrey S. Rosenthal* and Peter O. Schwartz** (May 28, 997.. Introduction. This short paper describes a surprising connection between two previously
More informationCHAPTER 1 BASIC TOPOLOGY
CHAPTER 1 BASIC TOPOLOGY Topology, sometimes referred to as the mathematics of continuity, or rubber sheet geometry, or the theory of abstract topological spaces, is all of these, but, above all, it is
More informationThe HenstockKurzweilStieltjes type integral for real functions on a fractal subset of the real line
The HenstockKurzweilStieltjes type integral for real functions on a fractal subset of the real line D. Bongiorno, G. Corrao Dipartimento di Ingegneria lettrica, lettronica e delle Telecomunicazioni,
More informationThe positive minimum degree game on sparse graphs
The positive minimum degree game on sparse graphs József Balogh Department of Mathematical Sciences University of Illinois, USA jobal@math.uiuc.edu András Pluhár Department of Computer Science University
More informationE3: PROBABILITY AND STATISTICS lecture notes
E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................
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 informationLECTURE 15: AMERICAN OPTIONS
LECTURE 15: AMERICAN OPTIONS 1. Introduction All of the options that we have considered thus far have been of the European variety: exercise is permitted only at the termination of the contract. These
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 informationCycles and cliqueminors in expanders
Cycles and cliqueminors in expanders Benny Sudakov UCLA and Princeton University Expanders Definition: The vertex boundary of a subset X of a graph G: X = { all vertices in G\X with at least one neighbor
More informationUniversal Algorithm for Trading in Stock Market Based on the Method of Calibration
Universal Algorithm for Trading in Stock Market Based on the Method of Calibration Vladimir V yugin Institute for Information Transmission Problems, Russian Academy of Sciences, Bol shoi Karetnyi per.
More informationThe Convolution Operation
The Convolution Operation Convolution is a very natural mathematical operation which occurs in both discrete and continuous modes of various kinds. We often encounter it in the course of doing other operations
More informationALGEBRAIC APPROACH TO COMPOSITE INTEGER FACTORIZATION
ALGEBRAIC APPROACH TO COMPOSITE INTEGER FACTORIZATION Aldrin W. Wanambisi 1* School of Pure and Applied Science, Mount Kenya University, P.O box 55350100, Kakamega, Kenya. Shem Aywa 2 Department of Mathematics,
More informationA Uniform Asymptotic Estimate for Discounted Aggregate Claims with Subexponential Tails
12th International Congress on Insurance: Mathematics and Economics July 1618, 2008 A Uniform Asymptotic Estimate for Discounted Aggregate Claims with Subexponential Tails XUEMIAO HAO (Based on a joint
More information