On some classes of difference equations of infinite order

Save this PDF as:

Size: px
Start display at page:

Transcription

2 Vasilyev and Vasilyev Advances in Difference Equations (2015) 2015:211 Page 2 of 9 and discrete equations. In this paper we will consider the case of a continuous variable x, and a solution on the right-hand side will be considered in the space L 2 (R) for all equations. 1.1 Difference equation of a finite order with constant coefficients This is an equation of the type n a k u(x + β k )=v(x), x R, (3) k=0 and it easily can be solved by the Fourier transform: (Fu)(ξ) ũ(ξ)= e ix ξ u(x) dx. Indeed, applying the Fourier transform to (3)we obtain ũ(ξ) n a k e iβkξ = ṽ(ξ), k=0 or renaming ũ(ξ)p n (ξ)=ṽ(ξ). The function p n (ξ) is called a symbol of a difference operator on the left-hand side (3) (cf. [6]). If p n (ξ) 0, ξ R,then(3) can easily be solved, u(x)=f ξ x 1 ( p 1 n (ξ)ṽ(ξ)). 1.2 Difference equation of infinite order with constant coefficients The same arguments are applicable for the case of an unbounded sequence {β k } +.Then the difference operator with complex coefficients D : u(x) + has the following symbol: σ (ξ)= + a k e iβ kξ. a k u(x + β k ), x R, a k C, (4) Lemma 1 The operator D is a linear bounded operator L 2 (R) L 2 (R) if {a k } + l1. Proof The proof of this assertion can be obtained immediately. If we consider the operator (4)forx Z only D : u(x d ) + a k u(x d + β k ), x d Z, (5)

3 Vasilyev and Vasilyev Advances in Difference Equations (2015) 2015:211 Page 3 of 9 then its symbol can be defined by the discrete Fourier transform [7, 8] σ d (ξ)= + a k e iβ kξ, ξ [ π, π]. 1.3 Difference and discrete equations Obviously there are some relations between difference and discrete equations. Particularly, if {β k } + = Z, then the operator (5) is a discrete convolution operator. For studying discrete operators in a half-space the authors have developed a certain analytic technique [9 11]. Below we will try to enlarge this technique for more general situations. 2 General difference equations We consider the equation (Du)(x)=v(x), x R +, (6) where R + = {x R, x >0}. For studying this equation we will use methods of the theory of multi-dimensional singular integral and pseudo differential equations [3, 6, 12] which are non-usual in the theory of difference equations. Our next goal is to study multi-dimensional difference equations, and this one-dimensional variant is a model for considering other complicated situations. This approach is based on the classical Riemann boundary value problem and the theory of one-dimensional singular integral equations [13 15]. 2.1 Background The first step is the following. We will use the theory of so-called paired equations [15]of the type (ap + + bp )U = V (7) in the space L 2 (R), where a, b are convolution operators with corresponding functions a(x), b(x), x R, P ± are projectors on the half-axis R ±.Moreprecisely, (ap + U)(x)= 0 a(x y)u(y) dy, 0 (bp U)(x)= b(x y)u(y) dy. Applying the Fourier transformto (7) we obtain[12] the following one-dimensional singular integral equation [13 15]: ã(ξ)(pũ)(ξ)+ b(ξ)(qũ)(ξ)=ṽ (ξ), (8) where P, Q are two projectors related to the Hilbert transform (Hu)(x)=v.p. 1 + u(y) πi x y dy, P = 1 (I + H), 2 Q = 1 (I H). 2

4 Vasilyev and Vasilyev Advances in Difference Equations (2015) 2015:211 Page 4 of 9 Equation (8) is closely related to the Riemann boundary value problem [13, 14] forupper and lower half-planes. We now recall the statement of the problem: finding a pair of functions ± (ξ) which admit an analytic continuation on upper (C + )andlower(c ) half-planes in the complex plane C and of which their boundary values on R satisfy the following linear relation: + (ξ)=g(ξ) (ξ)+g(ξ), (9) where G(ξ), g(ξ)aregivenfunctionson R. There is a one-to-one correspondence between the Riemann boundary value problem (9) and the singular integral equation (8), and G(ξ)=ã 1 (ξ) b(ξ), Ṽ (ξ)=ã 1 (ξ)g(ξ). 2.2 Topological barrier We suppose that the symbol G(ξ) is a continuous non-vanishing function on the compactification Ṙ (G(ξ) 0, ξ Ṙ)and Ind G 1 2π d arg G(t) = 0. (10) The last condition (10), is necessary and sufficient for the unique solvability of the problem (9) inthespacel 2 (R) [13, 14]. Moreover, the unique solution of the problem (9) can be constructed with a help of the Cauchy type integral + (t)=g + (t)p ( G 1 + (t)g(t)), (t)= G 1 (t)q( G 1 + (t)g(t)), where G ± are factors of a factorization for the G(t)(seebelow), G + (t)=exp ( P ( ln G(t) )), G (t)=exp ( Q ( ln G(t) )). 2.3 Difference equations on a half-axis Equation (6) can easily be transformedinto (7) in thefollowingway. Since theright-hand side in (6) isdefinedonr + only we will continue v(x) onthewholer so that this continuation lf L 2 (R). Further we will rename the unknown function u + (x) and define the function u (x)=(lf )(x) (Du + )(x). Thus, we have the following equation: (Du + )(x)+u (x)=(lf )(x), x R, (11) which holds for the whole space R. AftertheFouriertransformwehave σ (ξ)ũ + (ξ)+ũ (ξ)= lv(ξ), where σ (ξ) iscalledasymbol of the operator D.

5 Vasilyev and Vasilyev Advances in Difference Equations (2015) 2015:211 Page 5 of 9 To describe a solving technique for (11)we recall the following (cf. [13, 14]). Definition A factorization for an elliptic symbol is called its representation if it is in the form σ (ξ)=σ + (ξ) σ (ξ), where the factors σ +, σ admit an analytic continuation into the upper and lower complex half-planes C ±,andσ ±1 ± L (R). Example 1 Let us consider the Cauchy type integral 1 + u(t) dt (z), 2πi z t z = x ± iy. It is well known this construction plays a crucial role for a decomposition L 2 (R)ontwo orthogonal subspaces, namely L 2 (R)=A + (R) A (R), where A ± (R) consists of functions admitting an analytic continuation onto C ±. The boundary values of the integral (z) satisfy the Plemelj-Sokhotskii formulas [13, 14], and thus the projectors P and Q are corresponding projectors on the spaces of analytic functions [15]. Thesimpleexampleweneedis exp(u) =exp(pu) exp(qu). Theorem 2 Let σ (ξ) C(Ṙ), Ind σ =0.Then (6) has unique solution in the space L 2 (R + ) for arbitrary right-hand side v L 2 (R + ), and its Fourier transform is given by the formula ũ(ξ)= 1 2 σ 1 (ξ) lv(ξ)+ σ + 1(ξ) σ 1 v.p. (η) lv(η) dη. 2πi ξ η Proof We have σ + (ξ)ũ + (ξ)+σ 1 (ξ)ũ (ξ)=σ 1 (ξ) lv(ξ). Further, since σ 1(ξ) lv(ξ) L 2 (R) we decompose it into two summands σ 1 (ξ) lv(ξ)=p ( σ 1 (ξ) lv(ξ) ) + Q ( σ 1 (ξ) lv(ξ) ) and write σ + (ξ)ũ + (ξ) P ( σ 1 (ξ) lv(ξ) ) = Q ( σ 1 (ξ) lv(ξ) ) σ 1 (ξ)ũ (ξ). The left-hand side of the last quality belongs to the space A + (R), but the right-hand side belongs to A (R), consequently these are zeros. Thus, σ + (ξ)ũ + (ξ) P ( σ 1 (ξ) lv(ξ) ) =0

6 Vasilyev and Vasilyev Advances in Difference Equations (2015) 2015:211 Page 6 of 9 and ũ + (ξ)=σ 1 + (ξ)p( σ 1 (ξ) lv(ξ) ), (12) or in the complete form ũ + (ξ)= 1 2 σ 1 (ξ) lv(ξ)+ σ + 1(ξ) σ 1 v.p. (η) lv(η) dη. 2πi ξ η Remark 1 This result does not depend on the continuation lv. LetusdenotebyM ± (x) the inverse Fourier images of the functions σ ± 1 (ξ). Indeed, (12) leads to the following construction: u + (x)= = ( ) M + (x y) M (y t)(lv)(t) dt dy M + (x y) 0 ( 0 ) M (y t)v(t) dt dy. Remark 2 The condition σ (ξ) C(Ṙ) is not a strong restriction. Such symbols exist for example in the case that σ (ξ) isrepresentedbyafinitesum,andβ k Q. Thenσ (ξ) isa continuous periodic function. 3 General solution Since σ (ξ) C(Ṙ), and Ind σ is an integer, we consider the case æ Ind σ N in this section. Theorem 3 Let Ind σ N. Then a general solution of (6) in the Fourier image can be written in the form ũ + (ξ)=σ 1 + (ξ)ω æ (ξ)p ( σ 1 (ξ) lv(ξ) ) +(ξ i) æ σ 1 + (ξ)p æ 1(ξ), and it depends on æ arbitrary constants. Proof The function ω(ξ)= ξ i ξ + i has the index 1 [13 15], thus the function ω æ (ξ)σ (ξ)= ( ) ξ i æ σ (ξ) ξ + i has the index 0, and we can factorize this function ω æ (ξ)σ (ξ)=σ + (ξ)σ (ξ). Further, we write after (11) ω æ (ξ)ω æ (ξ)σ (ξ)ũ + (ξ)+ũ (ξ)= lv(ξ),

7 Vasilyev and Vasilyev Advances in Difference Equations (2015) 2015:211 Page 7 of 9 factorize ω æ (ξ)σ (ξ), and rewrite ω æ (ξ)σ + (ξ)ũ + (ξ)+σ 1 (ξ)ũ (ξ)=σ 1 (ξ) lv(ξ). Taking into account our notations we have σ + (ξ)ũ + (ξ)+ω æ (ξ)σ 1 (ξ)ũ (ξ) = ω æ (ξ)p ( σ 1 (ξ) lv(ξ) ) + ω æ (ξ)q ( σ 1 (ξ) lv(ξ) ), (13) because σ 1 (ξ) lv(ξ)=p ( σ 1 (ξ) lv(ξ) ) + Q ( σ 1 (ξ) lv(ξ) ) and σ + (ξ)ũ + (ξ) ω æ (ξ)p ( σ 1 (ξ) lv(ξ) ) = ω æ (ξ)q ( σ 1 (ξ) lv(ξ) ) ω æ (ξ)σ 1 (ξ)ũ (ξ) (14) and we conclude from the last that the left-hand side and the right-hand side also are a polynomial P æ 1 (ξ) oforderæ 1. It follows from the generalized Liouville theorem [13, 14] because the left-hand side has one pole of order æ in C in the point z = i.so,wehave ũ + (ξ)=σ 1 + (ξ)ω æ (ξ)p ( σ 1 (ξ) lv(ξ) ) +(ξ i) æ σ 1 + (ξ)p æ 1(ξ). Remark 3 This result does not depend on the choice of the continuation l. Corollary 4 Let v(x) 0, æ N. Then a general solution of the homogeneous equation (6) is given by the formula ũ + (ξ)=(ξ i) æ σ 1 + (ξ)p æ 1(ξ). 4 Solvability conditions Theorem 5 Let Ind σ N. Then (6) has a solution from L 2 (R + ) iff the following conditions hold: (ξ i) k 1 σ 1 (ξ) lv(ξ) dξ =0, k =1,2,..., æ. Proof We argue as above and use the equality (14); we write it as (ξ i) æ σ + (ξ)ũ + (ξ) (ξ + i) æ P ( σ 1 (ξ) lv(ξ) ) =(ξ + i) æ Q ( σ 1 (ξ) lv(ξ) ) (ξ + i) æ σ 1 (ξ)ũ (ξ). (15) Since we work with L 2 (R) both the left-hand side and the right-hand side are equal to zero at infinity, hence these are zeros, and ( ) ξ i æ ũ + (ξ)=σ + 1 (ξ) P ( σ 1 ξ + i (ξ) lv(ξ) ).

8 Vasilyev and Vasilyev Advances in Difference Equations (2015) 2015:211 Page 8 of 9 But there is some inaccuracy. Indeed, this solution belongs to the space A + (R), but more exactly it belongs to its subspace A k + (R). This subspace consists of functions analytic in C + with zeros of the order æ in the point z = i. To obtain a solution from L 2 (R + ) we need some corrections in the last formula. Since the operator P is related to the Cauchy type integral we will use certain decomposition formulas for this integral (see also [12 14]). Let us denote σ 1(ξ) lv(ξ) g(ξ) and consider the following integral: g(η) dη, z = ξ + iτ C. z η Using a simple formula for a kernel 1 z η = æ k=1 (η i) k 1 (z i) k + (η i) æ (z i) æ (z η) we obtain the following decomposition: g(η) dη z η = æ k=1 (z i) k So, we have a following property. If the conditions (η i) k 1 (η i) æ g(η) dη g(η) dη + (z i) æ (z η). (η i) k 1 g(η) dη =0, k =1,2,..., æ, hold, then we obtain g(η) dη z η =(z i)æ (η i) æ g(η) dη. z η Hence the boundary values on R for the left-hand side and the right-hand one are equal, and thus P ( g(ξ) ) =(ξ i) æ P ( (ξ i) æ g(ξ) ). Substituting the last formula into the solution formula we write ũ + (ξ)=σ 1 + (ξ)(ξ + i)æ P ( (ξ i) æ σ 1 (ξ) lv(ξ) ). 5 Conclusion It seems this approach to difference equations may be useful for studying the case that the variable x is a discrete one.we have some experience in the theory of discrete equations [9 11], and we hope that we can be successful in this situation also. Moreover, in our opinion the developed methods might be applicable for multi-dimensional difference equations. Competing interests The authors declare that they have no competing interests.

9 Vasilyev and Vasilyev Advances in Difference Equations (2015) 2015:211 Page 9 of 9 Authors contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Author details 1 Department of Mathematical Analysis, National Research Belgorod State University, Studencheskaya 14/1, Belgorod, , Russia. 2 Chair of Pure Mathematics, Lipetsk State Technical University, Moskovskaya 30, Lipetsk, , Russia. Acknowledgements The authors are very grateful to the anonymous referees for their valuable suggestions. This work is supported by Russian Fund of Basic Research and government of Lipetsk region of Russia, project No a. Received: 4 March 2015 Accepted: 17 June 2015 References 1. Milne-Thomson, LM: The Calculus of Finite Differences. Chelsea, New York (1981) 2. Jordan, C: Calculus of Finite Differences. Chelsea, New York (1950) 3. Vasil ev, VB: Wave Factorization of Elliptic Symbols: Theory and Applications. Introduction to the Theory of Boundary Value Problems in Non-Smooth Domains. Kluwer Academic, Dordrecht (2000) 4. Vasilyev, VB: General boundary value problems for pseudo differential equations and related difference equations. Adv. Differ. Equ. 2013, 289 (2013) 5. Vasilyev, VB: On some difference equations of first order. Tatra Mt. Math. Publ. 54, (2013) 6. Mikhlin, SG, Prößdorf, S: Singular Integral Operators. Akademie Verlag, Berlin (1986) 7. Sobolev, SL: Cubature Formulas and Modern Analysis: An Introduction. Gordon & Breach, Montreux (1992) 8. Dudgeon, DE, Mersereau, RM: Multidimensional Digital Signal Processing. Prentice Hall, Englewood Cliffs (1984) 9. Vasilyev, AV, Vasilyev, VB: Discrete singular operators and equations in a half-space. Azerb. J. Math. 3(1), (2013) 10. Vasilyev, AV, Vasilyev, VB: Discrete singular integrals in a half-space. In: Current Trends in Analysis and Its Applications, Proc. 9th Congress, Krakow, Poland, August 2013, pp Birkhäuser, Basel (2015) 11. Vasilyev, AV, Vasilyev, VB: Periodic Riemann problem and discrete convolution equations. Differ. Equ. 51(5), (2015) 12. Eskin, G: Boundary Value Problems for Elliptic Pseudodifferential Equations. Am. Math. Soc., Providence (1981) 13. Gakhov, FD: Boundary Value Problems. Dover, New York (1981) 14. Muskhelishvili, NI: Singular Integral Equations. North-Holland, Amsterdam (1976) 15. Gokhberg, I, Krupnik, N: Introduction to the Theory of One-Dimensional Singular Integral Equations. Birkhäuser, Basel (2010)

Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

MICROLOCAL ANALYSIS OF THE BOCHNER-MARTINELLI INTEGRAL

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX)0000-0 MICROLOCAL ANALYSIS OF THE BOCHNER-MARTINELLI INTEGRAL NIKOLAI TARKHANOV AND NIKOLAI VASILEVSKI

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

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

DISTRIBUTIONS AND FOURIER TRANSFORM

DISTRIBUTIONS AND FOURIER TRANSFORM MIKKO SALO Introduction. The theory of distributions, or generalized functions, provides a unified framework for performing standard calculus operations on nonsmooth

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

α = u v. In other words, Orthogonal Projection

Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v

MATH 590: Meshfree Methods

MATH 590: Meshfree Methods Chapter 7: Conditionally Positive Definite Functions Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2010 fasshauer@iit.edu MATH 590 Chapter

MATH 461: Fourier Series and Boundary Value Problems

MATH 461: Fourier Series and Boundary Value Problems Chapter III: Fourier Series Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2015 fasshauer@iit.edu MATH 461 Chapter

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

LINEAR 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,

FRACTIONAL INTEGRALS AND DERIVATIVES. Theory and Applications

FRACTIONAL INTEGRALS AND DERIVATIVES Theory and Applications Stefan G. Samko Rostov State University, Russia Anatoly A. Kilbas Belorussian State University, Minsk, Belarus Oleg I. Marichev Belorussian

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

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

Understanding Basic Calculus

Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other

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

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

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

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

NON SINGULAR MATRICES. DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that

NON SINGULAR MATRICES DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that AB = I n = BA. Any matrix B with the above property is called

ORDINARY DIFFERENTIAL EQUATIONS

ORDINARY DIFFERENTIAL EQUATIONS GABRIEL NAGY Mathematics Department, Michigan State University, East Lansing, MI, 48824. SEPTEMBER 4, 25 Summary. This is an introduction to ordinary differential equations.

The 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

Taylor Polynomials and Taylor Series Math 126

Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will

4. Complex integration: Cauchy integral theorem and Cauchy integral formulas. Definite integral of a complex-valued function of a real variable

4. Complex integration: Cauchy integral theorem and Cauchy integral formulas Definite integral of a complex-valued function of a real variable Consider a complex valued function f(t) of a real variable

a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

International Journal of Information Technology, Modeling and Computing (IJITMC) Vol.1, No.3,August 2013

FACTORING CRYPTOSYSTEM MODULI WHEN THE CO-FACTORS DIFFERENCE IS BOUNDED Omar Akchiche 1 and Omar Khadir 2 1,2 Laboratory of Mathematics, Cryptography and Mechanics, Fstm, University of Hassan II Mohammedia-Casablanca,

1 Lecture: Integration of rational functions by decomposition

Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.

5 Indefinite integral

5 Indefinite integral The most of the mathematical operations have inverse operations: the inverse operation of addition is subtraction, the inverse operation of multiplication is division, the inverse

Inner Product Spaces

Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

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................

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

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

Outline of Complex Fourier Analysis. 1 Complex Fourier Series

Outline of Complex Fourier Analysis This handout is a summary of three types of Fourier analysis that use complex numbers: Complex Fourier Series, the Discrete Fourier Transform, and the (continuous) Fourier

2 Complex Functions and the Cauchy-Riemann Equations

2 Complex Functions and the Cauchy-Riemann Equations 2.1 Complex functions In one-variable calculus, we study functions f(x) of a real variable x. Likewise, in complex analysis, we study functions f(z)

Recall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula:

Chapter 7 Div, grad, and curl 7.1 The operator and the gradient: Recall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula: ( ϕ ϕ =, ϕ,..., ϕ. (7.1 x 1

Computing divisors and common multiples of quasi-linear ordinary differential equations

Computing divisors and common multiples of quasi-linear ordinary differential equations Dima Grigoriev CNRS, Mathématiques, Université de Lille Villeneuve d Ascq, 59655, France Dmitry.Grigoryev@math.univ-lille1.fr

On the representability of the bi-uniform matroid

On the representability of the bi-uniform matroid Simeon Ball, Carles Padró, Zsuzsa Weiner and Chaoping Xing August 3, 2012 Abstract Every bi-uniform matroid is representable over all sufficiently large

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,

JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson

JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials

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

1 Orthogonal projections and the approximation

Math 1512 Fall 2010 Notes on least squares approximation Given n data points (x 1, y 1 ),..., (x n, y n ), we would like to find the line L, with an equation of the form y = mx + b, which is the best fit

Vector Spaces II: Finite Dimensional Linear Algebra 1

John Nachbar September 2, 2014 Vector Spaces II: Finite Dimensional Linear Algebra 1 1 Definitions and Basic Theorems. For basic properties and notation for R N, see the notes Vector Spaces I. Definition

CONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation

Chapter 2 CONTROLLABILITY 2 Reachable Set and Controllability Suppose we have a linear system described by the state equation ẋ Ax + Bu (2) x() x Consider the following problem For a given vector x in

Degree Hypergroupoids Associated with Hypergraphs

Filomat 8:1 (014), 119 19 DOI 10.98/FIL1401119F Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Degree Hypergroupoids Associated

ANALYTICAL MATHEMATICS FOR APPLICATIONS 2016 LECTURE NOTES Series

ANALYTICAL MATHEMATICS FOR APPLICATIONS 206 LECTURE NOTES 8 ISSUED 24 APRIL 206 A series is a formal sum. Series a + a 2 + a 3 + + + where { } is a sequence of real numbers. Here formal means that we don

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

Relatively dense sets, corrected uniformly almost periodic functions on time scales, and generalizations

Wang and Agarwal Advances in Difference Equations (2015) 2015:312 DOI 10.1186/s13662-015-0650-0 R E S E A R C H Open Access Relatively dense sets, corrected uniformly almost periodic functions on time

G(s) = Y (s)/u(s) In this representation, the output is always the Transfer function times the input. Y (s) = G(s)U(s).

Transfer Functions The transfer function of a linear system is the ratio of the Laplace Transform of the output to the Laplace Transform of the input, i.e., Y (s)/u(s). Denoting this ratio by G(s), i.e.,

A 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

Again, the limit must be the same whichever direction we approach from; but now there is an infinity of possible directions.

Chapter 4 Complex Analysis 4.1 Complex Differentiation Recall the definition of differentiation for a real function f(x): f f(x + δx) f(x) (x) = lim. δx 0 δx In this definition, it is important that the

Linear Algebra Done Wrong. Sergei Treil. Department of Mathematics, Brown University

Linear Algebra Done Wrong Sergei Treil Department of Mathematics, Brown University Copyright c Sergei Treil, 2004, 2009, 2011, 2014 Preface The title of the book sounds a bit mysterious. Why should anyone

Math 124A October 06, We then use the chain rule to compute the terms of the equation (1) in these new variables.

Math 124A October 06, 2011 Viktor Grigoryan 5 Classification of second order linear PDEs Last time we derived the wave and heat equations from physical principles. We also saw that Laplace s equation describes

An Introduction to Separation of Variables with Fourier Series Math 391w, Spring 2010 Tim McCrossen Professor Haessig

An Introduction to Separation of Variables with Fourier Series Math 391w, Spring 2010 Tim McCrossen Professor Haessig Abstract: This paper aims to give students who have not yet taken a course in partial

Distributions, the Fourier Transform and Applications. Teodor Alfson, Martin Andersson, Lars Moberg

Distributions, the Fourier Transform and Applications Teodor Alfson, Martin Andersson, Lars Moberg 1 Contents 1 Distributions 2 1.1 Basic Definitions......................... 2 1.2 Derivatives of Distributions...................

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

5 Numerical Differentiation

D. Levy 5 Numerical Differentiation 5. Basic Concepts This chapter deals with numerical approximations of derivatives. The first questions that comes up to mind is: why do we need to approximate derivatives

EXISTENCE AND NON-EXISTENCE RESULTS FOR A NONLINEAR HEAT EQUATION

Sixth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 5 (7), pp. 5 65. ISSN: 7-669. UL: http://ejde.math.txstate.edu

3. INNER PRODUCT SPACES

. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

Taylor Series and Asymptotic Expansions

Taylor Series and Asymptotic Epansions The importance of power series as a convenient representation, as an approimation tool, as a tool for solving differential equations and so on, is pretty obvious.

2 Further conformal mapping

C5.6 Applied Complex Variables Draft date: 3 February 206 2 2 Further conformal mapping 2. Introduction Here we extend the ideas on conformal mapping introduced in the previous section. The Riemann Mapping

CONSTANT-SIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE p-laplacian. Pasquale Candito and Giuseppina D Aguí

Opuscula Math. 34 no. 4 2014 683 690 http://dx.doi.org/10.7494/opmath.2014.34.4.683 Opuscula Mathematica CONSTANT-SIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE p-laplacian Pasquale

7 - Linear Transformations

7 - Linear Transformations Mathematics has as its objects of study sets with various structures. These sets include sets of numbers (such as the integers, rationals, reals, and complexes) whose structure

Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

20 Applications of Fourier transform to differential equations

20 Applications of Fourier transform to differential equations Now I did all the preparatory work to be able to apply the Fourier transform to differential equations. The key property that is at use here

Asian Option Pricing Formula for Uncertain Financial Market

Sun and Chen Journal of Uncertainty Analysis and Applications (215) 3:11 DOI 1.1186/s4467-15-35-7 RESEARCH Open Access Asian Option Pricing Formula for Uncertain Financial Market Jiajun Sun 1 and Xiaowei

Notes on Application of Finite Element Method to the Solution of the Poisson Equation

Notes on Application of Finite Element Method to the Solution of the Poisson Equation Consider the arbitrary two dimensional domain, shown in Figure 1. The Poisson equation for this domain can be written

PROPERTIES AND EXAMPLES OF Z-TRANSFORMS

PROPERTIES AD EXAMPLES OF Z-TRASFORMS I. Introduction In the last lecture we reviewed the basic properties of the z-transform and the corresponding region of convergence. In this lecture we will cover

R U S S E L L L. H E R M A N

R U S S E L L L. H E R M A N A N I N T R O D U C T I O N T O F O U R I E R A N D C O M P L E X A N A LY S I S W I T H A P P L I C AT I O N S T O T H E S P E C T R A L A N A LY S I S O F S I G N A L S R.

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

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

8.1 Examples, definitions, and basic properties

8 De Rham cohomology Last updated: May 21, 211. 8.1 Examples, definitions, and basic properties A k-form ω Ω k (M) is closed if dω =. It is exact if there is a (k 1)-form σ Ω k 1 (M) such that dσ = ω.

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

Duality of linear conic problems

Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least

( ) which must be a vector

MATH 37 Linear Transformations from Rn to Rm Dr. Neal, WKU Let T : R n R m be a function which maps vectors from R n to R m. Then T is called a linear transformation if the following two properties are

ANALYSIS AND APPLICATIONS OF LAPLACE /FOURIER TRANSFORMATIONS IN ELECTRIC CIRCUIT

www.arpapress.com/volumes/vol12issue2/ijrras_12_2_22.pdf ANALYSIS AND APPLICATIONS OF LAPLACE /FOURIER TRANSFORMATIONS IN ELECTRIC CIRCUIT M. C. Anumaka Department of Electrical Electronics Engineering,

Systems of Linear Equations

Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and

1 VECTOR SPACES AND SUBSPACES

1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such

Minimize subject to. x S R

Chapter 12 Lagrangian Relaxation This chapter is mostly inspired by Chapter 16 of [1]. In the previous chapters, we have succeeded to find efficient algorithms to solve several important problems such

Riesz-Fredhölm Theory

Riesz-Fredhölm Theory T. Muthukumar tmk@iitk.ac.in Contents 1 Introduction 1 2 Integral Operators 1 3 Compact Operators 7 4 Fredhölm Alternative 14 Appendices 18 A Ascoli-Arzelá Result 18 B Normed Spaces

Let H and J be as in the above lemma. The result of the lemma shows that the integral

Let and be as in the above lemma. The result of the lemma shows that the integral ( f(x, y)dy) dx is well defined; we denote it by f(x, y)dydx. By symmetry, also the integral ( f(x, y)dx) dy is well defined;

1. Periodic Fourier series. The Fourier expansion of a 2π-periodic function f is:

CONVERGENCE OF FOURIER SERIES 1. Periodic Fourier series. The Fourier expansion of a 2π-periodic function f is: with coefficients given by: a n = 1 π f(x) a 0 2 + a n cos(nx) + b n sin(nx), n 1 f(x) cos(nx)dx

Linear Algebra Done Wrong. Sergei Treil. Department of Mathematics, Brown University

Linear Algebra Done Wrong Sergei Treil Department of Mathematics, Brown University Copyright c Sergei Treil, 2004, 2009, 2011, 2014 Preface The title of the book sounds a bit mysterious. Why should anyone

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

1. R In this and the next section we are going to study the properties of sequences of real numbers.

+a 1. R In this and the next section we are going to study the properties of sequences of real numbers. Definition 1.1. (Sequence) A sequence is a function with domain N. Example 1.2. A sequence of real

MATHEMATICAL METHODS OF STATISTICS

MATHEMATICAL METHODS OF STATISTICS By HARALD CRAMER TROFESSOK IN THE UNIVERSITY OF STOCKHOLM Princeton PRINCETON UNIVERSITY PRESS 1946 TABLE OF CONTENTS. First Part. MATHEMATICAL INTRODUCTION. CHAPTERS

Høgskolen i Narvik Sivilingeniørutdanningen STE6237 ELEMENTMETODER. Oppgaver

Høgskolen i Narvik Sivilingeniørutdanningen STE637 ELEMENTMETODER Oppgaver Klasse: 4.ID, 4.IT Ekstern Professor: Gregory A. Chechkin e-mail: chechkin@mech.math.msu.su Narvik 6 PART I Task. Consider two-point

MATH 425, PRACTICE FINAL EXAM SOLUTIONS.

MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator

Fourier Series. Chapter Some Properties of Functions Goal Preliminary Remarks

Chapter 3 Fourier Series 3.1 Some Properties of Functions 3.1.1 Goal We review some results about functions which play an important role in the development of the theory of Fourier series. These results

EXAM. Practice Questions for Exam #2. Math 3350, Spring April 3, 2004 ANSWERS

EXAM Practice Questions for Exam #2 Math 3350, Spring 2004 April 3, 2004 ANSWERS i Problem 1. Find the general solution. A. D 3 (D 2)(D 3) 2 y = 0. The characteristic polynomial is λ 3 (λ 2)(λ 3) 2. Thus,

A COUNTEREXAMPLE FOR SUBADDITIVITY OF MULTIPLIER IDEALS ON TORIC VARIETIES

Communications in Algebra, 40: 1618 1624, 2012 Copyright Taylor & Francis Group, LLC ISSN: 0092-7872 print/1532-4125 online DOI: 10.1080/00927872.2011.552084 A COUNTEREXAMPLE FOR SUBADDITIVITY OF MULTIPLIER

A matrix over a field F is a rectangular array of elements from F. The symbol

Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted

3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes

Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general

RAJALAKSHMI ENGINEERING COLLEGE MA 2161 UNIT I - ORDINARY DIFFERENTIAL EQUATIONS PART A

RAJALAKSHMI ENGINEERING COLLEGE MA 26 UNIT I - ORDINARY DIFFERENTIAL EQUATIONS. Solve (D 2 + D 2)y = 0. 2. Solve (D 2 + 6D + 9)y = 0. PART A 3. Solve (D 4 + 4)x = 0 where D = d dt 4. Find Particular Integral:

Some remarks on Phragmén-Lindelöf theorems for weak solutions of the stationary Schrödinger operator

Wan Boundary Value Problems (2015) 2015:239 DOI 10.1186/s13661-015-0508-0 R E S E A R C H Open Access Some remarks on Phragmén-Lindelöf theorems for weak solutions of the stationary Schrödinger operator

MATH 425, HOMEWORK 7, SOLUTIONS

MATH 425, HOMEWORK 7, SOLUTIONS Each problem is worth 10 points. Exercise 1. (An alternative derivation of the mean value property in 3D) Suppose that u is a harmonic function on a domain Ω R 3 and suppose

Sets and functions. {x R : x > 0}.

Sets and functions 1 Sets The language of sets and functions pervades mathematics, and most of the important operations in mathematics turn out to be functions or to be expressible in terms of functions.

MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.

MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar

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