# On the Eigenvalues of Integral Operators

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Çanaya Üniversitesi Fen-Edebiyat Faültesi, Journal of Arts and Sciences Say : 6 / Aral 006 On the Eigenvalues of Integral Operators Yüsel SOYKAN Abstract In this paper, we obtain asymptotic estimates of the eigenvalues of certain positive integral operators Key words: Positive Integral Operators, Eigenvalues, Hardy Spaces Özet Bu çal flmada baz positive integral operatörlerin özde erlerinin asimtoti yalafl mlar n elde edece iz Anahtar Kelimeler: Pozitif ntegral Operatörleri, Özde erler, Hardy Uzaylar INTROUCTION From now on, let J be a fixed closed subinterval of the real line R Suppose that is a simply-connected domain containing the real closed interval J and ϕ is any function, which maps conformally onto, where is the open unit dis of complex plane C Let us define a function K on x by K ( ) ( z) (, z) for all, z, ( ) ( z) for either of the branches of The function K is independent of the choice of mapping function ϕ, see [, p40] By restricting the function K to the square JxJ we obtain a compact symmetric operator T on L defined by Karaelmas Üniversitesi, Fen Edebiyat Faültesi, Matemati Bölümü, Zongulda 77

2 On the Eigenvalues of Integral Operators T f() s K(,) s t f() t dt ( f L ( J), s J) J This operator is always positive in the sense of operator theory (ie Tf, f 0 for all f L ( J), see [] We shall use n( K) to denote the eigenvalues of T In this wor the following theorem shall be proved in detail THEOREM If,, are three half-planes and their boundary lines are not parallel pairwise and if contains the real closed interval J, then n( K) n( K K ) K where an b n means an O( bn) and bn O( an) To prove Theorem we will show that i) n( K ) ( ( )) O n K K K ii) ( K K K ) O( ( K )) n n This is a special case of a theorem in [, Theorem ] and we give a different proof PRELIMINARIES The space H ( ) is just the set of all bounded analytic function on with the p uniform norm For p, H ( ) is the set of all functions f analytic on such that i p sup f( re ) d 0 r 0 () p The p-th root of the left hand side of () here defines a complete norm on H ( ) For more information on this spaces see [ and ] In the case of p=, H be the familiar Hardy space of all functions analytic on with square-summable Maclaurin coefficients Let be a simply connected domain in C C and let be a Riemann mapping function for, that is, a conformal map of onto An analytic function g on is said to be of class E ( ) if there exists a function f H ( ) such that 78

3 Yüsel SOYKAN ( ) ( ) ( ) gz f z z ( z ) where is a branch of the square root of We define g f Thus, by construction, E ( ) is a Hilbert space with E ( ) H ( ) g, g f, f E H where ( ) ( ) ( ) ( ) ( ) gi z fi z z, (i =, ) and the map U : H ( ) E ( ) given by U f( z) f( ( z)) ( z) ( f H ( ), z ) is an isometric bijection For more information on this spaces see [] If rectifiable Jordan curve then the same formula V f( z) f( ( z)) ( z) ( f L ( ), z ) is a defines an isometric bijection V of L ( ) onto L ( ), the L space of normalized arc length measure on where and denote the boundary of and respectively The inverse V V : L ( ) L ( ) of V is given by Vgw g w w g L w ( ) ( ( )) ) ( ( ),, ) To prove Theorem we need the following lemma This is Corollary to Lemma in [4] LEMMA Suppose that is a disc or a codisc or a half-plane and be a circular arc (or a straight line) then for every g E ( ), g (z) dz g g (z) dz E ( ) Suppose now that contains our fixed interval J By restricting to J we obtain a linear operator S : E ( ) L ( J) defined by S f ( s) f ( s) ( f E ( ), s J) Then S is compact operator and T SS is the compact, positive integral operator on J with ernel K : () s () t K (,) s t ( s) ( t) for all s,t J This is proved in [] 79

4 On the Eigenvalues of Integral Operators EFINITION Let H and H be Hilbert spaces and suppose that T is a compact, positive operator on H If S: H H is a compact operator such that T SS, then S is called a quasi square-root of T We call H the domain space of S REMARK Suppose that,, are simply-connected domains containing J and let T, T, T be continuous positive operators on a Hilbert space L ( J ) and suppose that for each i, S i is a quasi square-root of T i with domain space E ( i ) If T T( K K ) ( ) K T K i so i T f( s) ( K ( st, ) K (, ) (, )) ( ) ( ( ), ), J st K st f tdt fl J sj that T : L ( J) L ( J) is compact, positive integral operator and T has the quasi square-root S : E ( ) E ( ) E ( ) L ( J), so that S ( f f f ) S f S f S f then S f f f s S f s S f s S f s f s f s f s f L J sj ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ( ), ) LEMMA Let T, T be compact operators on a Hilbert space H and suppose that S, S are quasi square-root of T, T with domain H, H respectively i) If there exists a continuous operator V : H H such that S VS then ( Tf, f) Tf (, f) for some >0 and so n( T) O( n( T)) ( n 0) ii) If there exists continuous operators V : H H and W : H H such that S WSV, then n( T) O( n( T)) Proof See [, page 407] PROOF OF MAIN RESULT Suppose that is a simply connected and bounded domain Let be a Riemann mapping function for and suppose that is the inverse function of An analytic function f on is said to be of class H ( ) if it is bounded on PROPOSITION If H, then H ( ) E ( ) 80

5 Proof Suppose that f H ( ) For z, define Yüsel SOYKAN gz ( ) f ( z) ( z) Then f H ( ) and H It follows that ( f ) H f E ( ) Hence f PROPOSITION i) H Suppose that is a rectifiable Jordan curve, then ii) Each function f E ( ) has a non-tangential limit f L ( ) f is an isometric isomorphism and f f (z) dz E ( ) The map iii) If is a convex region, it is a Smirnov domain iv) If is a Smirnov domain, then polynomials (thus H ( ) ) are dense in E ( ) v) E ( ) coincides with the is a Smirnov domain L ( ) closure of the polinomials if and only if Proof See [, pages 44, 70 and 7] For the definition of Smirnov domain see [, page 7] LEMMA 4 If is a disc, or codisc or half-plane, then the formula Pf( z) f ( d i x z defines a continuous linear operator P : L ( ) E ( ) with P Proof See [, page 4] From now on, suppose now that,, are three half-planes, and let contains the real closed interval J (see Figure ) For =,,, let 8

6 On the Eigenvalues of Integral Operators J Figure We shall exhibit continuous operators N : E ( ) E ( ) E ( ) E ( ) and M : E ( ) E ( ) E ( ) E ( ) To define N suppose first that G={f: f is a polynomial in E ( )} Since is convex, G E ( ) If f G, then for all z, Cauchy's Integral Formula gives f(w f(w f(w f ( z) dw dw dw i wz i wz i wz For f G and, define a function f on by f(w f ( z) dw i ( z ), w z and define a function f on by f ( ), if f ( z) if ( z ) 8

7 Yüsel SOYKAN LEMMA 5 If f G and then i) f L ( ) and f E ( ) ii) The formula Vf ( f, f, f) defines a continuous linear operator V : GE ( ) E ( ) E ( ) and so that V has an extension N by continuity to E ( ) Proof i) Let be a Riemann mapping function for and suppose V and are as in Section The map P : L ( ) E ( ) given by U P f( z) f ( d i z is a continuous linear operator with P (from Lemma 4) Since f P f, it follows that f E ( ) So ( f, f, f ) E ( ) E ( ) E ( ) Since and f P f f f( ) d f E ( ) E ( ) E ( ) L ( ) Vf ( f, f, f) E ( ) E ( ) E ( ) E ( ) E ( ) E ( ) f f f f E ( ) E ( ) E ( ) E ( ) it follows that the map f ( f, f, f) is a continuous linear operator GE ( ) E ( ) E ( ) Now suppose that N is an extension by continuity to E ( ) Note that then N If we denote F H ( ) H ( ) H ( ) then F E ( ) E ( ) E ( ) LEMMA 6 The map V : F E ( ), is given by V( f, f, f)( z) f( z) f( z) f( z), (( f, f, f) F, z ), 8

8 On the Eigenvalues of Integral Operators is a continuous operator so that V has an extension M by continuity to E ( ) E ( ) E ( ) Proof If ( f, f, f) F then by Propositions and, fi H ( ) E ( ) ( i ) and V ( f, f, f ) H ( ) E ( ) So we have (,, )) E ( ) E ( ) = f f f Real f, f E ( ) E ( ) E ( ) E ( ) V f f f f f f Real f, f Real f, f E ( ) E ( ) f f E f ( f f ( ) ( ) ( ) ( ) ) E E E E ( ) f f f f E ( ) E ( ) E ( ) E ( ) ( f f f ) E ( ) E ( ) E ( ) ( ) ( ) (by Lemma ) ( ) f f f E ( ) E ( ) E ( ) E ( ) E ( ) E ( ) 9 ( f, f, f ) Hence V 9 and V is a continuous linear operator Let now M be extension by continuity to E ( ) E ( ) E ( ) Note that then M 9 PROOF OF THEOREM i) Suppose that V is as in Lemma 5 Note that here T+ S+ S+ and T SS By definition of V, we have S f SV f for every f G Thus, by continuity of V, S f SNf for every f E ( ) and so S S N So for g L ( J),, S S g g S g N S g N S g N S S g, g SS gg, 84

9 Yüsel SOYKAN That is, S S S S Hence by Lemma as required n( SS ) n( S S ) ii) Suppose that V is as in Lemma 6 By definition of V, it follows that S+ ( f, f, f) SV( f, f, f) for every ( f, f, f) H ( ) H ( ) H ( ) Thus, by continuity of V, S ( f, f, f ) S M( f, f, f ) for every ( f, f, f ) E ( ) E ( ) E ( ) + and so S+ S M So for ie, SS g L ( J), we have SS gg, M SS gg, 9 SS g, g 9 SS Consequently, from Lemma, n( SS ) 9 n( SS ) REFERENCES Little, G, Equivalences of positive integrals operators with rational ernels, Proc London Math Soc () 6 (99), uren, P L, Theory of spaces, Academic Press, New Yor, 970 Koosis, P, Introduction to spaces, Cambridge University Press, Cambridge, Second Edition, 998 Soyan Y, An Inequality of Fejer-Riesz Type, Çanaya University, Journal of Arts and Sciences, Issue: 5, May 006,

10 On the Eigenvalues of Integral Operators 86

### Amply Fws Modules (Aflk n, Sonlu Zay f Eklenmifl Modüller)

Çankaya Üniversitesi Fen-Edebiyat Fakültesi, Journal of Arts and Sciences Say : 4 / Aral k 2005 Amply Fws Modules (Aflk n, Sonlu Zay f Eklenmifl Modüller) Gökhan B LHA * Abstract In this work amply finitely

### 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

### FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied

### BANACH AND HILBERT SPACE REVIEW

BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but

### arxiv: v1 [math.oa] 17 Dec 2016

arxiv:161.05683v1 [math.oa] 17 Dec 016 COMPLETE SPECTRAL SETS AND NUMERICAL RANGE KENNETH R. DAVIDSON, VERN I. PAULSEN, AND HUGO J. WOERDEMAN Abstract. We define the complete numerical radius norm for

### Quadratic Equations in Finite Fields of Characteristic 2

Quadratic Equations in Finite Fields of Characteristic 2 Klaus Pommerening May 2000 english version February 2012 Quadratic equations over fields of characteristic 2 are solved by the well known quadratic

### 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

### Metric Spaces. Chapter 7. 7.1. Metrics

Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some

### 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

### Chapter 5: Application: Fourier Series

321 28 9 Chapter 5: Application: Fourier Series For lack of time, this chapter is only an outline of some applications of Functional Analysis and some proofs are not complete. 5.1 Definition. If f L 1

### Holomorphic motion of fiber Julia sets

ACADEMIC REPORTS Fac. Eng. Tokyo Polytech. Univ. Vol. 37 No.1 (2014) 7 Shizuo Nakane We consider Axiom A polynomial skew products on C 2 of degree d 2. The stable manifold of a hyperbolic fiber Julia set

### Applications of Fourier series

Chapter Applications of Fourier series One of the applications of Fourier series is the evaluation of certain infinite sums. For example, n= n,, are computed in Chapter (see for example, Remark.4.). n=

### Recall that two vectors in are perpendicular or orthogonal provided that their dot

Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal

### 4. 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,

### A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS. In memory of Rou-Huai Wang

A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS XAVIER CABRÉ, MANEL SANCHÓN, AND JOEL SPRUCK In memory of Rou-Huai Wang 1. Introduction In this note we consider semistable

### 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

### Course 421: Algebraic Topology Section 1: Topological Spaces

Course 421: Algebraic Topology Section 1: Topological Spaces David R. Wilkins Copyright c David R. Wilkins 1988 2008 Contents 1 Topological Spaces 1 1.1 Continuity and Topological Spaces...............

### 1 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)

### 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

### 88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a

88 CHAPTER. VECTOR FUNCTIONS.4 Curvature.4.1 Definitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small

### Math 563 Measure Theory Project 1 (Funky Functions Group) Luis Zerón, Sergey Dyachenko

Math 563 Measure Theory Project (Funky Functions Group) Luis Zerón, Sergey Dyachenko 34 Let C and C be any two Cantor sets (constructed in Exercise 3) Show that there exists a function F: [,] [,] with

### Numerical 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

### 7. Cauchy s integral theorem and Cauchy s integral formula

7. Cauchy s integral theorem and Cauchy s integral formula 7.. Independence of the path of integration Theorem 6.3. can be rewritten in the following form: Theorem 7. : Let D be a domain in C and suppose

### MOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao yufeiz@mit.edu

Integer Polynomials June 9, 007 Yufei Zhao yufeiz@mit.edu We will use Z[x] to denote the ring of polynomials with integer coefficients. We begin by summarizing some of the common approaches used in dealing

### FUNCTIONAL ANALYSIS PIOTR HAJ LASZ

FUNCTIONAL ANALYSIS PIOTR HAJ LASZ 1. Banach and Hilbert spaces In what follows K will denote R of C. Definition. A normed space is a pair (X, ), where X is a linear space over K and : X [0, ) is a function,

### MATH : HONORS CALCULUS-3 HOMEWORK 6: SOLUTIONS

MATH 16300-33: HONORS CALCULUS-3 HOMEWORK 6: SOLUTIONS 25-1 Find the absolute value and argument(s) of each of the following. (ii) (3 + 4i) 1 (iv) 7 3 + 4i (ii) Put z = 3 + 4i. From z 1 z = 1, we have

### Reading 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

### Notes from February 11

Notes from February 11 Math 130 Course web site: www.courses.fas.harvard.edu/5811 Two lemmas Before proving the theorem which was stated at the end of class on February 8, we begin with two lemmas. The

### Compactness in metric spaces

MATHEMATICS 3103 (Functional Analysis) YEAR 2012 2013, TERM 2 HANDOUT #2: COMPACTNESS OF METRIC SPACES Compactness in metric spaces The closed intervals [a, b] of the real line, and more generally the

### Continuity of the Perron Root

Linear and Multilinear Algebra http://dx.doi.org/10.1080/03081087.2014.934233 ArXiv: 1407.7564 (http://arxiv.org/abs/1407.7564) Continuity of the Perron Root Carl D. Meyer Department of Mathematics, North

### MATH31011/MATH41011/MATH61011: FOURIER ANALYSIS AND LEBESGUE INTEGRATION. Chapter 4: Fourier Series and L 2 ([ π, π], µ) ( 1 π

MATH31011/MATH41011/MATH61011: FOURIER ANALYSIS AND LEBESGUE INTEGRATION Chapter 4: Fourier Series and L ([, π], µ) Square Integrable Functions Definition. Let f : [, π] R be measurable. We say that f

### Ri and. i=1. S i N. and. R R i

The subset R of R n is a closed rectangle if there are n non-empty closed intervals {[a 1, b 1 ], [a 2, b 2 ],..., [a n, b n ]} so that R = [a 1, b 1 ] [a 2, b 2 ] [a n, b n ]. The subset R of R n is an

### 1. Let P be the space of all polynomials (of one real variable and with real coefficients) with the norm

Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: F3B, F4Sy, NVP 005-06-15 Skrivtid: 9 14 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok

### CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE

CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verify

### Sumit Chandok and T. D. Narang INVARIANT POINTS OF BEST APPROXIMATION AND BEST SIMULTANEOUS APPROXIMATION

F A S C I C U L I M A T H E M A T I C I Nr 51 2013 Sumit Chandok and T. D. Narang INVARIANT POINTS OF BEST APPROXIMATION AND BEST SIMULTANEOUS APPROXIMATION Abstract. In this paper we generalize and extend

### Isometries of some classical function spaces among the composition operators

Contemporary Mathematics Isometries of some classical function spaces among the composition operators María J. Martín and ragan Vukotić edicated to Professor Joseph Cima on the occasion of his 7th birthday

### 6. Metric spaces. In this section we review the basic facts about metric spaces. d : X X [0, )

6. Metric spaces In this section we review the basic facts about metric spaces. Definitions. A metric on a non-empty set X is a map with the following properties: d : X X [0, ) (i) If x, y X are points

### ON DEGREE OF APPROXIMATION ON A JORDAN CURVE TO A FUNCTION ANALYTIC INTERIOR TO THE CURVE BY FUNCTIONS NOT NECESSARILY ANALYTIC INTERIOR TO THE CURVE

ON DEGREE OF APPROXIMATION ON A JORDAN CURVE TO A FUNCTION ANALYTIC INTERIOR TO THE CURVE BY FUNCTIONS NOT NECESSARILY ANALYTIC INTERIOR TO THE CURVE J. L. WALSH It is our object here to consider the subject

### Metric Spaces Joseph Muscat 2003 (Last revised May 2009)

1 Distance J Muscat 1 Metric Spaces Joseph Muscat 2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) 1 Distance A metric space can be thought of

### 5. Möbius Transformations

5. Möbius Transformations 5.1. The linear transformation and the inversion. In this section we investigate the Möbius transformation which provides very convenient methods of finding a one-to-one mapping

### Chap2: The Real Number System (See Royden pp40)

Chap2: The Real Number System (See Royden pp40) 1 Open and Closed Sets of Real Numbers The simplest sets of real numbers are the intervals. We define the open interval (a, b) to be the set (a, b) = {x

### WHICH LINEAR-FRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE?

WHICH LINEAR-FRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE? JOEL H. SHAPIRO Abstract. These notes supplement the discussion of linear fractional mappings presented in a beginning graduate course

### A NEW APPROACH TO THE CORONA THEOREM FOR DOMAINS BOUNDED BY A C 1+α CURVE

Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 40, 2015, 767 772 A NEW APPROACH TO THE CORONA THEOREM FOR DOMAINS BOUNDED BY A C 1+α CURVE José Manuel Enríquez-Salamanca and María José González

### Problem Set. Problem Set #2. Math 5322, Fall December 3, 2001 ANSWERS

Problem Set Problem Set #2 Math 5322, Fall 2001 December 3, 2001 ANSWERS i Problem 1. [Problem 18, page 32] Let A P(X) be an algebra, A σ the collection of countable unions of sets in A, and A σδ the collection

### Diploma Plus in Certificate in Advanced Engineering

Diploma Plus in Certificate in Advanced Engineering Mathematics New Syllabus from April 2011 Ngee Ann Polytechnic / School of Interdisciplinary Studies 1 I. SYNOPSIS APPENDIX A This course of advanced

### PETER G. CASAZZA AND GITTA KUTYNIOK

A GENERALIZATION OF GRAM SCHMIDT ORTHOGONALIZATION GENERATING ALL PARSEVAL FRAMES PETER G. CASAZZA AND GITTA KUTYNIOK Abstract. Given an arbitrary finite sequence of vectors in a finite dimensional Hilbert

### 6. Define log(z) so that π < I log(z) π. Discuss the identities e log(z) = z and log(e w ) = w.

hapter omplex integration. omplex number quiz. Simplify 3+4i. 2. Simplify 3+4i. 3. Find the cube roots of. 4. Here are some identities for complex conjugate. Which ones need correction? z + w = z + w,

### Solving Linear Systems, Continued and The Inverse of a Matrix

, Continued and The of a Matrix Calculus III Summer 2013, Session II Monday, July 15, 2013 Agenda 1. The rank of a matrix 2. The inverse of a square matrix Gaussian Gaussian solves a linear system by reducing

### Section 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

### The General Cauchy Theorem

Chapter 3 The General Cauchy Theorem In this chapter, we consider two basic questions. First, for a given open set Ω, we try to determine which closed paths in Ω have the property that f(z) dz = 0for every

### On an isomorphic Banach-Mazur rotation problem and maximal norms in Banach spaces

On an isomorphic Banach-Mazur rotation problem and maximal norms in Banach spaces B. Randrianantoanina (joint work with Stephen Dilworth) Department of Mathematics Miami University Ohio, USA Conference

### Sequences of Functions

Sequences of Functions Uniform convergence 9. Assume that f n f uniformly on S and that each f n is bounded on S. Prove that {f n } is uniformly bounded on S. Proof: Since f n f uniformly on S, then given

### Weak topologies. David Lecomte. May 23, 2006

Weak topologies David Lecomte May 3, 006 1 Preliminaries from general topology In this section, we are given a set X, a collection of topological spaces (Y i ) i I and a collection of maps (f i ) i I such

### Introduction to Algebraic Geometry. Bézout s Theorem and Inflection Points

Introduction to Algebraic Geometry Bézout s Theorem and Inflection Points 1. The resultant. Let K be a field. Then the polynomial ring K[x] is a unique factorisation domain (UFD). Another example of a

### 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

### Power Series. Chapter Introduction

Chapter 10 Power Series In discussing power series it is good to recall a nursery rhyme: There was a little girl Who had a little curl Right in the middle of her forehead When she was good She was very,

### Erdős 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)

### Complex Function Theory. Second Edition. Donald Sarason >AMS AMERICAN MATHEMATICAL SOCIETY

Complex Function Theory Second Edition Donald Sarason >AMS AMERICAN MATHEMATICAL SOCIETY Contents Preface to the Second Edition Preface to the First Edition ix xi Chapter I. Complex Numbers 1 1.1. Definition

### DEFINITION 5.1.1 A complex number is a matrix of the form. x y. , y x

Chapter 5 COMPLEX NUMBERS 5.1 Constructing the complex numbers One way of introducing the field C of complex numbers is via the arithmetic of matrices. DEFINITION 5.1.1 A complex number is a matrix of

### 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

### Course 221: Analysis Academic year , First Semester

Course 221: Analysis Academic year 2007-08, First Semester David R. Wilkins Copyright c David R. Wilkins 1989 2007 Contents 1 Basic Theorems of Real Analysis 1 1.1 The Least Upper Bound Principle................

### 1 The Dirichlet Problem. 2 The Poisson kernel. Math 857 Fall 2015

Math 857 Fall 2015 1 The Dirichlet Problem Before continuing to Fourier integrals, we consider first an application of Fourier series. Let Ω R 2 be open and connected (region). Recall from complex analysis

### 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

### Essential spectrum of the Cariñena operator

Theoretical Mathematics & Applications, vol.2, no.3, 2012, 39-45 ISSN: 1792-9687 (print), 1792-9709 (online) Scienpress Ltd, 2012 Essential spectrum of the Cariñena operator Y. Mensah 1 and M.N. Hounkonnou

### 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

### ON THE MULTIPLIERS OF THE INTERSECTION OF WEIGHTED FUNCTION SPACES. Serap Öztop 1. INTRODUCTION

TAIWANESE JOURNAL OF MATHEMATICS Vol. 11, No. 1, pp. 151-160, March 2007 This paper is available online at http://www.math.nthu.edu.tw/tjm/ ON THE MULTIPLIERS OF THE INTERSECTION OF WEIGHTED FUNCTION SPACES

### Math 4310 Handout - Quotient Vector Spaces

Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable

### Liouville Quantum gravity and KPZ. Scott Sheffield

Liouville Quantum gravity and KPZ Scott Sheffield Scaling limits of random planar maps Central mathematical puzzle: Show that the scaling limit of some kind of discrete quantum gravity (perhaps decorated

### 33 Cauchy Integral Formula

33 AUHY INTEGRAL FORMULA October 27, 2006 PROOF Use the theorem to write f ()d + 1 f ()d + 1 f ()d = 0 2 f ()d = f ()d f ()d. 2 1 2 This is the deformation principle; if you can continuously deform 1 to

### MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES

MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES 2016 47 4. Diophantine Equations A Diophantine Equation is simply an equation in one or more variables for which integer (or sometimes rational) solutions

### INTEGRAL OPERATORS ON THE PRODUCT OF C(K) SPACES

INTEGRAL OPERATORS ON THE PRODUCT OF C(K) SPACES FERNANDO BOMBAL AND IGNACIO VILLANUEVA Abstract. We study and characterize the integral multilinear operators on a product of C(K) spaces in terms of the

### In memory of Lars Hörmander

ON HÖRMANDER S SOLUTION OF THE -EQUATION. I HAAKAN HEDENMALM ABSTRAT. We explain how Hörmander s classical solution of the -equation in the plane with a weight which permits growth near infinity carries

### 1 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

### 1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form

1. LINEAR EQUATIONS A linear equation in n unknowns x 1, x 2,, x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b, where a 1, a 2,..., a n, b are given real numbers. For example, with x and

### arxiv:0904.4794v1 [math.ap] 30 Apr 2009

arxiv:0904.4794v [math.ap] 30 Apr 009 Reconstruction in the Calderón Problem with Partial Data Adrian Nachman and Brian Street Abstract We consider the problem of recovering the coefficient σ (x of the

### A characterization of trace zero symmetric nonnegative 5x5 matrices

A characterization of trace zero symmetric nonnegative 5x5 matrices Oren Spector June 1, 009 Abstract The problem of determining necessary and sufficient conditions for a set of real numbers to be the

### 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.,

### INVARIANT METRICS WITH NONNEGATIVE CURVATURE ON COMPACT LIE GROUPS

INVARIANT METRICS WITH NONNEGATIVE CURVATURE ON COMPACT LIE GROUPS NATHAN BROWN, RACHEL FINCK, MATTHEW SPENCER, KRISTOPHER TAPP, AND ZHONGTAO WU Abstract. We classify the left-invariant metrics with nonnegative

### HEAVISIDE CABLE, THOMSON CABLE AND THE BEST CONSTANT OF A SOBOLEV-TYPE INEQUALITY. Received June 1, 2007; revised September 30, 2007

Scientiae Mathematicae Japonicae Online, e-007, 739 755 739 HEAVISIDE CABLE, THOMSON CABLE AND THE BEST CONSTANT OF A SOBOLEV-TYPE INEQUALITY Yoshinori Kametaka, Kazuo Takemura, Hiroyuki Yamagishi, Atsushi

### Prerequsites: Math 1A-1B, 53 (lower division calculus courses)

Math 151 Prerequsites: Math 1A-1B, 53 (lower division calculus courses) Development of the rational number system. Use the number line (real line), starting with the concept of parts of a whole : fractions,

### 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

### MATH 52: MATLAB HOMEWORK 2

MATH 52: MATLAB HOMEWORK 2. omplex Numbers The prevalence of the complex numbers throughout the scientific world today belies their long and rocky history. Much like the negative numbers, complex numbers

### CONICS ON THE PROJECTIVE PLANE

CONICS ON THE PROJECTIVE PLANE CHRIS CHAN Abstract. In this paper, we discuss a special property of conics on the projective plane and answer questions in enumerative algebraic geometry such as How many

### FUNCTIONAL ANALYSIS LECTURE NOTES CHAPTER 3. BANACH SPACES

FUNCTIONAL ANALYSIS LECTURE NOTES CHAPTER 3. BANACH SPACES CHRISTOPHER HEIL 1. Elementary Properties and Examples Notation 1.1. Throughout, F will denote either the real line R or the complex plane C.

### Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.

MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called

### 2. Length and distance in hyperbolic geometry

2. Length and distance in hyperbolic geometry 2.1 The upper half-plane There are several different ways of constructing hyperbolic geometry. These different constructions are called models. In this lecture

### Acknowledgements: I would like to thank Alekos Kechris, who told me about the problem studied here.

COMPACT METRIZABLE GROUPS ARE ISOMETRY GROUPS OF COMPACT METRIC SPACES JULIEN MELLERAY Abstract. This note is devoted to proving the following result: given a compact metrizable group G, there is a compact

### VERTICES OF GIVEN DEGREE IN SERIES-PARALLEL GRAPHS

VERTICES OF GIVEN DEGREE IN SERIES-PARALLEL GRAPHS MICHAEL DRMOTA, OMER GIMENEZ, AND MARC NOY Abstract. We show that the number of vertices of a given degree k in several kinds of series-parallel labelled

### 9 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......

### 1 Quiz on Linear Equations with Answers. This is Theorem II.3.1. There are two statements of this theorem in the text.

1 Quiz on Linear Equations with Answers (a) State the defining equations for linear transformations. (i) L(u + v) = L(u) + L(v), vectors u and v. (ii) L(Av) = AL(v), vectors v and numbers A. or combine

### Mathematical Methods of Engineering Analysis

Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................

### ON TORI TRIANGULATIONS ASSOCIATED WITH TWO-DIMENSIONAL CONTINUED FRACTIONS OF CUBIC IRRATIONALITIES.

ON TORI TRIANGULATIONS ASSOCIATED WITH TWO-DIMENSIONAL CONTINUED FRACTIONS OF CUBIC IRRATIONALITIES. O. N. KARPENKOV Introduction. A series of properties for ordinary continued fractions possesses multidimensional

### Polish spaces and standard Borel spaces

APPENDIX A Polish spaces and standard Borel spaces We present here the basic theory of Polish spaces and standard Borel spaces. Standard references for this material are the books [143, 231]. A.1. Polish

### 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

### A MIXED TYPE IDENTIFICATION PROBLEM RELATED TO A PHASE-FIELD MODEL WITH MEMORY

Guidetti, D. and Lorenzi, A. Osaka J. Math. 44 (27), 579 613 A MIXED TYPE IDENTIFICATION PROBLEM RELATED TO A PHASE-FIELD MODEL WITH MEMORY DAVIDE GUIDETTI and ALFREDO LORENZI (Received January 23, 26,

### Orthogonal Diagonalization of Symmetric Matrices

MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding

### ON SUPERCYCLICITY CRITERIA. Nuha H. Hamada Business Administration College Al Ain University of Science and Technology 5-th st, Abu Dhabi, 112612, UAE

International Journal of Pure and Applied Mathematics Volume 101 No. 3 2015, 401-405 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v101i3.7

### Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.

Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.