# Max-Min Representation of Piecewise Linear Functions

Save this PDF as:

Size: px
Start display at page:

Download "Max-Min Representation of Piecewise Linear Functions"

## Transcription

1 Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 43 (2002), No. 1, Max-Min Representation of Piecewise Linear Functions Sergei Ovchinnikov Mathematics Department, San Francisco State University San Francisco, CA Abstract. It is shown that a piecewise linear function on a convex domain in R d can be represented as a boolean polynomial in terms of its linear components. 1. Introduction Let f be a piecewise linear function on R defined by g 1 (x) = 2 x + 1, x 1, f(x) = g 2 (x) = 5 2 x, 1 x 2, g 3 (x) = 0.5 x, x 2. The graph of this function is shown in Fig. 1. y g 2 g 1 g 3 x Fig /93 \$ 2.50 c 2002 Heldermann Verlag

2 298 S. Ovchinnikov: Max-Min Representation of Piecewise Linear Functions This function can be also represented as follows. f(x) = [g 1 (x) g 2 (x)] [g 1 (x) g 3 (x)], x R, (1) where and stand for operations Min and Max, respectively. In other words, f is represented as Max-Min boolean polynomial in variables g 1, g 2, g 3 and the polynomial is written in its disjunctive normal form. The main goal of the paper is to establish this representation for piecewise linear functions on closed convex domains in R d (Theorem 2.1). We also discuss the optimization problem for this representation. 2. Representation theorem In the paper, a closed domain in R d is the closure of an open set in R d and a linear function on R d is a function in the form h(x) = a 1 x 1 + a 2 x a d x d + b, where x = (x 1,..., x d ). We begin with the following definition. Definition 2.1. Let Γ be a closed convex domain in R d. A function f : Γ R is said to be piecewise linear if there is a finite family Q of closed domains such that Γ = Q and f is linear on every domain in Q. A linear function g on R d which coincides with f on some Q Q is said to be a component of f. Clearly, any piecewise linear function on Γ is continuous. The following theorem is the main result of the paper. Theorem 2.1. Let f be a piecewise linear function on Γ and {g 1,..., g n } be the set of its distinct components. There exists a family {S j } j J of incomparable (with respect to ) subsets of {1,..., n} such that f(x) = g i (x), j J i S j x Γ. (2) Here, and are operations of maximum and minimum, respectively. The expression on the right side in (2) is a disjunctive normal form of a Max-Min polynomial in the variables g i. We begin our proof with a simple geometric observation. Lemma 2.1. Let f be a piecewise linear function on [a, b] R and {g 1,..., g n } be the set of its components. There exists k such that g k (a) f(a) and g k (b) f(b). (3)

3 S. Ovchinnikov: Max-Min Representation of Piecewise Linear Functions 299 Proof. Let {l 1,... l m } be the set of closed line segments on R 2 constituting the graph of f. We assume that these line segments are enumerated in the direction from a to b. Let g n(i) be the component defining l i, and let m be the slope of the line segment [(a, f(a)), (b, f(b))]. If the slope of g n(1) (resp. g n(m) ) is greater than or equal to m, then g n(1) (resp. g n(m) ) satisfies conditions (3). It remains to consider the case when the slopes of g n(1) and g n(m) are smaller than m (see Fig. 2). y (a, f(a)) l 1 l p l m (b, f(b)) x Fig. 2 Clearly, there is l p with the slope greater than m that intersects the line segment [(a, f(a)), (b, f(b))]. Then g n(p) satisfies conditions (3). The statement of the next lemma follows immediately from Lemma 2.1. Lemma 2.2. Let f be a piecewise linear function on Γ and {g 1,..., g n } be the set of its components. For given points a, b Γ, there is k such that g k (a) f(a) and g k (b) f(b). (4) Proof. Consider the restriction of f to [a, b] and apply Lemma 2.1. Now we proceed with the proof of Theorem 2.1. Proof. For a given a Γ, let us define and S a = {i {1,..., n} : g i (a) f(a)} (5) F a (x) = i S a g i (x). Clearly, F a (a) = f(a). By Lemma 2.2, for any b Γ, F b (a) = g i (a) f(a). i S b Hence, f(x) = g i (x), x Γ. (6) a Γ i S a Let {S j } j J be the family of distinct minimal elements (with respect to ) in the family {S a } a Γ. Clearly, (6) implies (2).

4 300 S. Ovchinnikov: Max-Min Representation of Piecewise Linear Functions Corollary 2.1. Let Γ be a star-shape domain in R d such that its boundary Γ is a polyhedral complex. Let f be a function on Γ such that its restriction to each (d 1)-dimensional polyhedron in Γ is a linear function on it. Then f admits representation (2). Proof. Let a be a central point in Γ. For x R d, x a, let x be the unique intersection point of the ray from a through x with Γ. We define { x a f( x) for x a, x a f(x) = 0 for x = a. Clearly, f is a piecewise linear function on R d and f Γ = f. Thus f admits representation (2). Formula (2) is not very effective in the sense that the same component can appear in different monomials in (2). For instance, g 2 appears twice in (1). On the other hand, (1) can be written in the form f(x) = g 1 (x) [g 2 (x) g 3 (x)], x R, where each component appears only once. Note that this representation is not in the disjunctive normal form. By modifying the technique presented in [2], one can show that in one-dimensional case any piecewise linear function admits a boolean representation in which each component appears in the formula exactly once. An example in [2] shows that it is not true in higher dimensions. The reviewer of the paper suggested a more effective boolean representation in the disjunctive normal form than that given by (2). In what follows, we describe this construction. The proof uses only a minor modification of the techniques used in the proof of Theorem 2.1 and is omitted. Let H be the set of all hyperplanes that are nonempty solution sets of the equations in the form g i (x) = g j (x) for i < j and have nonempty intersections with the interior int(γ) of Γ. We consider H as a hyperplane arrangement [1] and denote T the family of nonempty intersections of the regions of H with int(γ). We use the same name region for elements of T. It is easy to see that the components g i are linearly ordered over any region in T and that for any P T, a, b P implies S a = S b (see (5)). Consider now the pairs (g i, g j ) for i < j that satisfy the following conditions: (i) There are adjacent regions P, Q T such that g i (x) = f(x) on P and g j (x) = f(x) on Q. (ii) f(x) = g i (x) g j (x) on P Q. Let H be the hyperplane arrangement defined by these pairs. We may assume that H is nonempty. (Otherwise, f is a concave function and we have a trivial representation.) We denote T the set of regions in Γ defined by H and define S P = {i {1,..., n} : g i (x) f(x), x P, P T }. The sets S P are incomparable and define the following representation f(x) = g i (x), x Γ. i S P P T Since S P S a for a P, the above formula, in general, is more effective than (2).

5 S. Ovchinnikov: Max-Min Representation of Piecewise Linear Functions Concluding remarks 1. The statement of Theorem 2.1 also holds for piecewise linear functions from Γ to R m. Namely, let f : Γ R m be a piecewise linear function and {g 1,..., g n } be the set of its distinct components. We denote f = (f 1,..., f m ) and g k = (g (k) 1,..., g (k) m ), for 1 k n. There exists a family {S k j } j J, 1 k n of subsets of {1,..., n} such that The converse is also true. f k (x) = j J i S k j g (k) i (x), x Γ, 1 k m. 2. The convexity of Γ is an essential assumption. Consider, for instance, the domain Γ = {(x 1, x 2 ) R 2 : x 2 x 1 } and define { x 2 for min{x 1, x 2 } 0, f(x) = 0 otherwise, where x = (x 1, x 2 ). This piecewise linear function has two components, g 1 (x) = x 2 and g 2 (x) = 0, but is not representable in the form (2). 3. Likewise, (2) is not true for piecewise polynomial functions as the following example (due to B. Sturmfels) illustrates. Let Γ = R 1. We define { 0 for x 0, f(x) = x 2 for x > It follows from Theorem 2.1 that any piecewise linear function on a closed convex domain in R d can be extended to a piecewise linear function on the entire space R d. 5. By using the techniques presented in the paper, a representation theorem can be established for smooth functions on closed domains in R d [3]. In this case, the role of components is played by tangent hyperplanes. Acknowledgments. The author thanks S. Gelfand, O. Musin, B. Sturmfels, and G. Ziegler for helpful discussions on the earlier versions of the paper. He is especially grateful to the anonymous referee for constructive suggestions. This work was partly supported by NSF grant SES to J.-Cl. Falmagne at University of California, Irvine. References [1] Björner, A.; Las Vergnas, M.; Sturmfels, B.; White, N.; Ziegler, G. M.: Oriented Matroids. Second Edition, Encyclopedia of Mathematics, Vol. 46, Cambridge University Press Zbl

6 302 S. Ovchinnikov: Max-Min Representation of Piecewise Linear Functions [2] Dobkin, D.; Guibas, L.; Hershberger, J.; Snoeyink, J.: An efficient algorithm for finding the CSG representation of a simple polygon. Algorithmica 10 (1993), Zbl [3] Ovchinnikov, S.: Boolean representation of manifolds and functions. J. Math. Analysis and Appl. (submitted), e-print available at Received October 12, 2000

### Rolle s Theorem. q( x) = 1

Lecture 1 :The Mean Value Theorem We know that constant functions have derivative zero. Is it possible for a more complicated function to have derivative zero? In this section we will answer this question

### FACTORIZATION OF TROPICAL POLYNOMIALS IN ONE AND SEVERAL VARIABLES. Nathan B. Grigg. Submitted to Brigham Young University in partial fulfillment

FACTORIZATION OF TROPICAL POLYNOMIALS IN ONE AND SEVERAL VARIABLES by Nathan B. Grigg Submitted to Brigham Young University in partial fulfillment of graduation requirements for University Honors Department

### Computer Graphics. Geometric Modeling. Page 1. Copyright Gotsman, Elber, Barequet, Karni, Sheffer Computer Science - Technion. An Example.

An Example 2 3 4 Outline Objective: Develop methods and algorithms to mathematically model shape of real world objects Categories: Wire-Frame Representation Object is represented as as a set of points

### FIXED POINT SETS OF FIBER-PRESERVING MAPS

FIXED POINT SETS OF FIBER-PRESERVING MAPS Robert F. Brown Department of Mathematics University of California Los Angeles, CA 90095 e-mail: rfb@math.ucla.edu Christina L. Soderlund Department of Mathematics

### Row Ideals and Fibers of Morphisms

Michigan Math. J. 57 (2008) Row Ideals and Fibers of Morphisms David Eisenbud & Bernd Ulrich Affectionately dedicated to Mel Hochster, who has been an inspiration to us for many years, on the occasion

### Critical 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) =

### ABSTRACT. For example, circle orders are the containment orders of circles (actually disks) in the plane (see [8,9]).

Degrees of Freedom Versus Dimension for Containment Orders Noga Alon 1 Department of Mathematics Tel Aviv University Ramat Aviv 69978, Israel Edward R. Scheinerman 2 Department of Mathematical Sciences

### 1. Introduction. PROPER HOLOMORPHIC MAPPINGS BETWEEN RIGID POLYNOMIAL DOMAINS IN C n+1

Publ. Mat. 45 (2001), 69 77 PROPER HOLOMORPHIC MAPPINGS BETWEEN RIGID POLYNOMIAL DOMAINS IN C n+1 Bernard Coupet and Nabil Ourimi Abstract We describe the branch locus of proper holomorphic mappings between

### Convex 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

### Shape 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 e-mail: buttazzo@sab.sns.it Paolo GUASONI Scuola Normale Superiore

### Ideal Class Group and Units

Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals

### MATH1231 Algebra, 2015 Chapter 7: Linear maps

MATH1231 Algebra, 2015 Chapter 7: Linear maps A/Prof. Daniel Chan School of Mathematics and Statistics University of New South Wales danielc@unsw.edu.au Daniel Chan (UNSW) MATH1231 Algebra 1 / 43 Chapter

### SPERNER S LEMMA AND BROUWER S FIXED POINT THEOREM

SPERNER S LEMMA AND BROUWER S FIXED POINT THEOREM ALEX WRIGHT 1. Intoduction A fixed point of a function f from a set X into itself is a point x 0 satisfying f(x 0 ) = x 0. Theorems which establish the

### Fundamentals of Media Theory

Fundamentals of Media Theory ergei Ovchinnikov Mathematics Department an Francisco tate University an Francisco, CA 94132 sergei@sfsu.edu Abstract Media theory is a new branch of discrete applied mathematics

### Surface 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 3-manifold can fiber over the circle. In

### Irreducibility criteria for compositions and multiplicative convolutions of polynomials with integer coefficients

DOI: 10.2478/auom-2014-0007 An. Şt. Univ. Ovidius Constanţa Vol. 221),2014, 73 84 Irreducibility criteria for compositions and multiplicative convolutions of polynomials with integer coefficients Anca

### Mathematics 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

### Scheduling a sequence of tasks with general completion costs

Scheduling a sequence of tasks with general completion costs Francis Sourd CNRS-LIP6 4, place Jussieu 75252 Paris Cedex 05, France Francis.Sourd@lip6.fr Abstract Scheduling a sequence of tasks in the acceptation

### 1 Sets and Set Notation.

LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most

### Binary Space Partitions

Title: Binary Space Partitions Name: Adrian Dumitrescu 1, Csaba D. Tóth 2,3 Affil./Addr. 1: Computer Science, Univ. of Wisconsin Milwaukee, Milwaukee, WI, USA Affil./Addr. 2: Mathematics, California State

### The Henstock-Kurzweil-Stieltjes type integral for real functions on a fractal subset of the real line

The Henstock-Kurzweil-Stieltjes 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,

### Examples of Tasks from CCSS Edition Course 3, Unit 5

Examples of Tasks from CCSS Edition Course 3, Unit 5 Getting Started The tasks below are selected with the intent of presenting key ideas and skills. Not every answer is complete, so that teachers can

### x 2 f 2 e 1 e 4 e 3 e 2 q f 4 x 4 f 3 x 3

Karl-Franzens-Universitat Graz & Technische Universitat Graz SPEZIALFORSCHUNGSBEREICH F 003 OPTIMIERUNG und KONTROLLE Projektbereich DISKRETE OPTIMIERUNG O. Aichholzer F. Aurenhammer R. Hainz New Results

### o-minimality and Uniformity in n 1 Graphs

o-minimality and Uniformity in n 1 Graphs Reid Dale July 10, 2013 Contents 1 Introduction 2 2 Languages and Structures 2 3 Definability and Tame Geometry 4 4 Applications to n 1 Graphs 6 5 Further Directions

### Mean Value Coordinates

Mean Value Coordinates Michael S. Floater Abstract: We derive a generalization of barycentric coordinates which allows a vertex in a planar triangulation to be expressed as a convex combination of its

### ON FIBER DIAMETERS OF CONTINUOUS MAPS

ON FIBER DIAMETERS OF CONTINUOUS MAPS PETER S. LANDWEBER, EMANUEL A. LAZAR, AND NEEL PATEL Abstract. We present a surprisingly short proof that for any continuous map f : R n R m, if n > m, then there

### LAKE ELSINORE UNIFIED SCHOOL DISTRICT

LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1-Semester 2 Grade Level: 10-12 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:

### INTRODUCTORY SET THEORY

M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H-1088 Budapest, Múzeum krt. 6-8. CONTENTS 1. SETS Set, equal sets, subset,

### Prime Numbers and Irreducible Polynomials

Prime Numbers and Irreducible Polynomials M. Ram Murty The similarity between prime numbers and irreducible polynomials has been a dominant theme in the development of number theory and algebraic geometry.

### 3. Linear Programming and Polyhedral Combinatorics

Massachusetts Institute of Technology Handout 6 18.433: Combinatorial Optimization February 20th, 2009 Michel X. Goemans 3. Linear Programming and Polyhedral Combinatorics Summary of what was seen in the

### The 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

### ISOMETRIES 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

### Basic 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

### Integer factorization is in P

Integer factorization is in P Yuly Shipilevsky Toronto, Ontario, Canada E-mail address: yulysh2000@yahoo.ca Abstract A polynomial-time algorithm for integer factorization, wherein integer factorization

### FUZZY CLUSTERING ANALYSIS OF DATA MINING: APPLICATION TO AN ACCIDENT MINING SYSTEM

International Journal of Innovative Computing, Information and Control ICIC International c 0 ISSN 34-48 Volume 8, Number 8, August 0 pp. 4 FUZZY CLUSTERING ANALYSIS OF DATA MINING: APPLICATION TO AN ACCIDENT

### The degree of a polynomial function is equal to the highest exponent found on the independent variables.

DETAILED SOLUTIONS AND CONCEPTS - POLYNOMIAL FUNCTIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE

### THE 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

### SOME PROPERTIES OF FIBER PRODUCT PRESERVING BUNDLE FUNCTORS

SOME PROPERTIES OF FIBER PRODUCT PRESERVING BUNDLE FUNCTORS Ivan Kolář Abstract. Let F be a fiber product preserving bundle functor on the category FM m of the proper base order r. We deduce that the r-th

### Fiber Polytopes for the Projections between Cyclic Polytopes

Europ. J. Combinatorics (2000) 21, 19 47 Article No. eujc.1999.0319 Available online at http://www.idealibrary.com on Fiber Polytopes for the Projections between Cyclic Polytopes CHRISTOS A. ATHANASIADIS,

### Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs

CSE599s: Extremal Combinatorics November 21, 2011 Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs Lecturer: Anup Rao 1 An Arithmetic Circuit Lower Bound An arithmetic circuit is just like

### SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,

### 4: SINGLE-PERIOD MARKET MODELS

4: SINGLE-PERIOD 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: Single-Period Market

### RIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES

RIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES GAUTAM BHARALI AND INDRANIL BISWAS Abstract. In the study of holomorphic maps, the term rigidity refers to certain types of results that give us very specific

### Properties 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 non-negative BMO-function w, whose reciprocal is also in BMO, belongs to p> A p,and

### Classification of Cartan matrices

Chapter 7 Classification of Cartan matrices In this chapter we describe a classification of generalised Cartan matrices This classification can be compared as the rough classification of varieties in terms

### MA651 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

### On the degrees of freedom in shrinkage estimation

On the degrees of freedom in shrinkage estimation Kengo Kato Graduate School of Economics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan kato ken@hkg.odn.ne.jp October, 2007 Abstract

### Module1. x 1000. y 800.

Module1 1 Welcome to the first module of the course. It is indeed an exciting event to share with you the subject that has lot to offer both from theoretical side and practical aspects. To begin with,

### Section 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

### SOLUTIONS. Would you like to know the solutions of some of the exercises?? Here you are

SOLUTIONS Would you like to know the solutions of some of the exercises?? Here you are Your first function : remember Introduction: Text: A function is a relationship between two sets by which we assign

### Scheduling and Location (ScheLoc): Makespan Problem with Variable Release Dates

Scheduling and Location (ScheLoc): Makespan Problem with Variable Release Dates Donatas Elvikis, Horst W. Hamacher, Marcel T. Kalsch Department of Mathematics, University of Kaiserslautern, Kaiserslautern,

### Tree-representation of set families and applications to combinatorial decompositions

Tree-representation of set families and applications to combinatorial decompositions Binh-Minh Bui-Xuan a, Michel Habib b Michaël Rao c a Department of Informatics, University of Bergen, Norway. buixuan@ii.uib.no

### Catalan Numbers. Thomas A. Dowling, Department of Mathematics, Ohio State Uni- versity.

7 Catalan Numbers Thomas A. Dowling, Department of Mathematics, Ohio State Uni- Author: versity. Prerequisites: The prerequisites for this chapter are recursive definitions, basic counting principles,

### Reference: 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

### Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year.

This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra

### CHAPTER 1 Splines and B-splines an Introduction

CHAPTER 1 Splines and B-splines an Introduction In this first chapter, we consider the following fundamental problem: Given a set of points in the plane, determine a smooth curve that approximates the

### An example of a computable

An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept

### Copyrighted Material. Chapter 1 DEGREE OF A CURVE

Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two

### On the irreducibility of certain polynomials with coefficients as products of terms in an arithmetic progression

On the irreducibility of certain polynomials with coefficients as products of terms in an arithmetic progression Carrie E. Finch and N. Saradha Abstract We prove the irreducibility of ceratin polynomials

### Discernibility Thresholds and Approximate Dependency in Analysis of Decision Tables

Discernibility Thresholds and Approximate Dependency in Analysis of Decision Tables Yu-Ru Syau¹, En-Bing Lin²*, Lixing Jia³ ¹Department of Information Management, National Formosa University, Yunlin, 63201,

### Zeros of Polynomial Functions

Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction

### The Line Connectivity Problem

Konrad-Zuse-Zentrum für Informationstechnik Berlin Takustraße 7 D-14195 Berlin-Dahlem Germany RALF BORNDÖRFER MARIKA NEUMANN MARC E. PFETSCH The Line Connectivity Problem Supported by the DFG Research

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

### Figure 2.1: Center of mass of four points.

Chapter 2 Bézier curves are named after their inventor, Dr. Pierre Bézier. Bézier was an engineer with the Renault car company and set out in the early 196 s to develop a curve formulation which would

### Optimal Trajectories for Nonholonomic Mobile Robots

Optimal Trajectories for Nonholonomic Mobile Robots P. Souères J.D. Boissonnat This is the third chapter of the book: Robot Motion Planning and Control Jean-Paul Laumond (Editor) Laboratoire d Analye et

### ON THE DEGREE OF MAXIMALITY OF DEFINITIONALLY COMPLETE LOGICS

Bulletin of the Section of Logic Volume 15/2 (1986), pp. 72 79 reedition 2005 [original edition, pp. 72 84] Ryszard Ladniak ON THE DEGREE OF MAXIMALITY OF DEFINITIONALLY COMPLETE LOGICS The following definition

### To define function and introduce operations on the set of functions. To investigate which of the field properties hold in the set of functions

Chapter 7 Functions This unit defines and investigates functions as algebraic objects. First, we define functions and discuss various means of representing them. Then we introduce operations on functions

### ORIENTATIONS. Contents

ORIENTATIONS Contents 1. Generators for H n R n, R n p 1 1. Generators for H n R n, R n p We ended last time by constructing explicit generators for H n D n, S n 1 by using an explicit n-simplex which

### ON LIMIT LAWS FOR CENTRAL ORDER STATISTICS UNDER POWER NORMALIZATION. E. I. Pancheva, A. Gacovska-Barandovska

Pliska Stud. Math. Bulgar. 22 (2015), STUDIA MATHEMATICA BULGARICA ON LIMIT LAWS FOR CENTRAL ORDER STATISTICS UNDER POWER NORMALIZATION E. I. Pancheva, A. Gacovska-Barandovska Smirnov (1949) derived four

### Section 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

### Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize

### Factoring Polynomials

Factoring Polynomials Sue Geller June 19, 2006 Factoring polynomials over the rational numbers, real numbers, and complex numbers has long been a standard topic of high school algebra. With the advent

### FIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper.

FIRST YEAR CALCULUS WWLCHENW L c WWWL W L Chen, 1982, 2008. 2006. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It It is is

### Further 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

### TOPOLOGY OF SINGULAR FIBERS OF GENERIC MAPS

TOPOLOGY OF SINGULAR FIBERS OF GENERIC MAPS OSAMU SAEKI Dedicated to Professor Yukio Matsumoto on the occasion of his 60th birthday Abstract. We classify singular fibers of C stable maps of orientable

### 1 Local Brouwer degree

1 Local Brouwer degree Let D R n be an open set and f : S R n be continuous, D S and c R n. Suppose that the set f 1 (c) D is compact. (1) Then the local Brouwer degree of f at c in the set D is defined.

### parent ROADMAP MATHEMATICS SUPPORTING YOUR CHILD IN HIGH SCHOOL

parent ROADMAP MATHEMATICS SUPPORTING YOUR CHILD IN HIGH SCHOOL HS America s schools are working to provide higher quality instruction than ever before. The way we taught students in the past simply does

### ON DEGREES IN THE HASSE DIAGRAM OF THE STRONG BRUHAT ORDER

Séminaire Lotharingien de Combinatoire 53 (2006), Article B53g ON DEGREES IN THE HASSE DIAGRAM OF THE STRONG BRUHAT ORDER RON M. ADIN AND YUVAL ROICHMAN Abstract. For a permutation π in the symmetric group

### Questions. Strategies August/September Number Theory. What is meant by a number being evenly divisible by another number?

Content Skills Essential August/September Number Theory Identify factors List multiples of whole numbers Classify prime and composite numbers Analyze the rules of divisibility What is meant by a number

### Rotation Rate of a Trajectory of an Algebraic Vector Field Around an Algebraic Curve

QUALITATIVE THEORY OF DYAMICAL SYSTEMS 2, 61 66 (2001) ARTICLE O. 11 Rotation Rate of a Trajectory of an Algebraic Vector Field Around an Algebraic Curve Alexei Grigoriev Department of Mathematics, The

### On using numerical algebraic geometry to find Lyapunov functions of polynomial dynamical systems

Dynamics at the Horsetooth Volume 2, 2010. On using numerical algebraic geometry to find Lyapunov functions of polynomial dynamical systems Eric Hanson Department of Mathematics Colorado State University

### Tenacity and rupture degree of permutation graphs of complete bipartite graphs

Tenacity and rupture degree of permutation graphs of complete bipartite graphs Fengwei Li, Qingfang Ye and Xueliang Li Department of mathematics, Shaoxing University, Shaoxing Zhejiang 312000, P.R. China

### Research Article On the Rank of Elliptic Curves in Elementary Cubic Extensions

Numbers Volume 2015, Article ID 501629, 4 pages http://dx.doi.org/10.1155/2015/501629 Research Article On the Rank of Elliptic Curves in Elementary Cubic Extensions Rintaro Kozuma College of International

### The Ideal Class Group

Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentioned

### IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction

IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible

### Course Outlines. 1. Name of the Course: Algebra I (Standard, College Prep, Honors) Course Description: ALGEBRA I STANDARD (1 Credit)

Course Outlines 1. Name of the Course: Algebra I (Standard, College Prep, Honors) Course Description: ALGEBRA I STANDARD (1 Credit) This course will cover Algebra I concepts such as algebra as a language,

### MATHEMATICAL ENGINEERING TECHNICAL REPORTS. The Independent Even Factor Problem

MATHEMATICAL ENGINEERING TECHNICAL REPORTS The Independent Even Factor Problem Satoru IWATA and Kenjiro TAKAZAWA METR 2006 24 April 2006 DEPARTMENT OF MATHEMATICAL INFORMATICS GRADUATE SCHOOL OF INFORMATION

### Sets of Fibre Homotopy Classes and Twisted Order Parameter Spaces

Communications in Mathematical Physics Manuscript-Nr. (will be inserted by hand later) Sets of Fibre Homotopy Classes and Twisted Order Parameter Spaces Stefan Bechtluft-Sachs, Marco Hien Naturwissenschaftliche

### INTERPOLATION. Interpolation is a process of finding a formula (often a polynomial) whose graph will pass through a given set of points (x, y).

INTERPOLATION Interpolation is a process of finding a formula (often a polynomial) whose graph will pass through a given set of points (x, y). As an example, consider defining and x 0 =0, x 1 = π 4, x

### Fixed 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

### MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab

MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab MATH 0110 is established to accommodate students desiring non-course based remediation in developmental mathematics. This structure will

### 1 The Concept of a Mapping

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 1 The Concept of a Mapping The concept of a mapping (aka function) is important throughout mathematics. We have been dealing

### (2) the identity map id : S 1 S 1 to the symmetry S 1 S 1 : x x (here x is considered a complex number because the circle S 1 is {x C : x = 1}),

Chapter VI Fundamental Group 29. Homotopy 29 1. Continuous Deformations of Maps 29.A. Is it possible to deform continuously: (1) the identity map id : R 2 R 2 to the constant map R 2 R 2 : x 0, (2) the

### GROUPS ACTING ON A SET

GROUPS ACTING ON A SET MATH 435 SPRING 2012 NOTES FROM FEBRUARY 27TH, 2012 1. Left group actions Definition 1.1. Suppose that G is a group and S is a set. A left (group) action of G on S is a rule for

### A Short Proof that Compact 2-Manifolds Can Be Triangulated

Inventiones math. 5, 160--162 (1968) A Short Proof that Compact 2-Manifolds Can Be Triangulated P. H. DOYLE and D. A. MORAN* (East Lansing, Michigan) The result mentioned in the title of this paper was

### Convex Rationing Solutions (Incomplete Version, Do not distribute)

Convex Rationing Solutions (Incomplete Version, Do not distribute) Ruben Juarez rubenj@hawaii.edu January 2013 Abstract This paper introduces a notion of convexity on the rationing problem and characterizes

### Polynomial Operations and Factoring

Algebra 1, Quarter 4, Unit 4.1 Polynomial Operations and Factoring Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Identify terms, coefficients, and degree of polynomials.

### RESULTANT AND DISCRIMINANT OF POLYNOMIALS

RESULTANT AND DISCRIMINANT OF POLYNOMIALS SVANTE JANSON Abstract. This is a collection of classical results about resultants and discriminants for polynomials, compiled mainly for my own use. All results