Convergence of c k f(kx) and the Lip α class
|
|
- Robert Wilcox
- 7 years ago
- Views:
Transcription
1 Convergene of and the Lip α lass Christoph Aistleitner Abstrat By Carleson s theorem a trigonometri series k osπkx or k sin πkx is ae onvergent if < (1) Gaposhkin generalized this result to series of the form k () for funtions f satisfying f(x + 1) = f(x), 1 f(x) = 0 and belonging to the Lip α lass 0 for some α > 1/ In the ase α 1/ ondition (1) is in general no longer suffiient to guarantee the ae onvergene of () For 0 < α < 1/ Gaposhkin showed that () is ae onvergent if kk 1 α (log k) β < for some β > 1 + α (3) In this paper we show that ondition (3) an be signifiantly weakened for α [1/4, 1/) In fat, we show that in this ase the fator k 1 α (log k) β an be replaed by a fator whih is asymptotially smaller than any positive power of k 1 Introdution and statement of results In 1966 Carleson showed that trigonometri series of the form are ae onvergent if k os πkx or k sin πkx k < (4) Graz University of Tehnology, Institute of Mathematis A, Steyrergasse 30, 8010 Graz, Austria aistleitner@mathtugrazat Researh supported by the Austrian Researh Foundation (FWF), Projet S9603-N3 This paper was written while the author was a partiipant of the Oberwolfah Leibniz Fellowship Programme (OWLF) of the Mathematial Researh Institute of Oberwolfah, Germany Mathematis Subjet Classifiation: Primary 4A61, 4A0 Keywords: Almost everywhere onvergene, Lipshitz lasses 1
2 (see [7] for Carleson s paper; f also the monographs of Mozzohi [15] and Arias de Reyna [4]) Gaposhkin [11] showed that (4) also implies the ae onvergene of series of the form, (5) where, here and throughout the paper, f is a measurable funtion satisfying f(x + 1) = f(x), 1 0 f(x) dx = 0, and belonging to the Lip α lass for some α > 1/ If the ondition that f Lip α for some α > 1/ is dropped the onvergene of the sum (4) will in general no longer be suffiient to guarantee the ae onvergene of (5) One possibility to meet this fat is to onsider series of the form k f(n k x) (6) instead of (5), where (n k ) k 1 is a fast growing sequene of positive integers A typial growth ondition in this ase is Hadamard s ondition, requesting that n k+1 n k > q, k 1, for some q > 1 Under this ondition (4) is still suffiient to have ae onvergene of (6) for f belonging to the Lip α lass for any α > 0 (Ka [13]; for reent results in the field f eg Fukuyama [9], Aistleitner and Berkes [], Aistleitner [1]) An other possibility to get ae onvergene results for series of the form (5) is to impose a stronger ondition than (4) on the sequene ( k ) k 1, typially depending on the modulus of ontinuity of f For example, for f Lip 1/ Gaposhkin [10] proved the ae onvergene of (5) under the ondition k (log k)β <, β > 3 (later, Berkes and Weber [6] showed that it is suffiient to assume β > ) On the other hand, Berkes [5] showed that there exist a funtion f Lip 1/ and a sequene ( k ) k 1 satisfying k < suh that (5) is ae divergent In the ase 0 < α < 1/, Gaposhkin [10] showed that (5) is ae onvergent, provided k k1 α (log k) β < for some β > 1 + α, and Berkes proved that there exists a funtion f Lip α and a sequene ( k ) k 1 suh that (5) is ae divergent, although k (log k)γ < for all γ < 1 α
3 Reapitulating these results, we see that there is a large gap between the neessary and suffiient onditions on ( k ) k 1 to guarantee the ae onvergene of (5), where f Lip α, 0 < α 1/ In the ase α = 1/ the neessary and suffiient ondition has to be somewhere between k < and k (log k)β <, β >, and in the ase 0 < α < 1/ between and k (log k)γ <, γ [0,1 α) k k1 α (log k) β <, β > 1 + α Conerning this problem Berkes and Weber [6] wrote: It is possible that in the ase 0 < α < 1/ the ondition k (log k)γ < for a suitable γ > 0 suffies for the ae onvergene of kf(n k x), but this remains open The purpose of this paper is to give a strong improvement of Gaposhkin s result for the ase α [1/4,1/) We show that in this ase for onvergene in () the fator k 1 α in (3) an be replaed by k ε for any ε > 0, or even by ( ) log k exp (7) log log k (Remark: here, and in the sequel, we write exp(x) for e x Also, to simplify notation, we understand log x as (1, log x)) Observe that the funtion in (7) is asymptotially smaller than any positive power of k More preisely, we will prove the following theorem: Theorem 1 Let f Lip α for some α [1/4,1/) Then onverges ae provided ( ) log k k exp < log log k We note that, despite our improvement, the exat best possible ondition for ( k ) k 1 to imply ae onvergene of () remains unknown In partiular, the funtion ( ) log k exp log log k 3
4 in our theorem grows faster than (log n) β for any β R Therefore, the question whether k (log k)β < is suffiient to have the ae onvergene of () remains open (f the aforementioned remark of Berkes and Weber) Auxiliary results Lemma 1 Let f Lip α for some α [1/4,1/), and write Then for any m 1 for some onstant f a 0 + a j os πjx + b j sin πjx m+1 j=1 j= m +1 ( a j + b j) mα This is formula (33) in Zygmund [16, p 41] Lemma Let J and N be positive integers and let (n k ) 1kN be a sequenes of distint non-zero integers Then the number of solutions to j 1 n k1 = j n k with is bounded by where is a onstant 1 j 1,j J, 1 k 1,k N ( ) 5log N JN exp, log log N This is a speial ase of Harman and Dyer [8, Theorem ] (also ontained in Harmans monograph [1] as Theorem 39) We have already used this result in an earlier paper in a related ontext (f Aistleitner, Mayer and Ziegler [3]) 3 Preparations Throughout the rest of the paper we will assume that f and ( k ) k 1 are fixed, and that f Lip α for some α [1/4,1/) Wlog we will assume that f is an even funtion, ie that the Fourier series of f is a pure osine-series (the proof in the general ase is exatly the same), and that f 1 and k 1, k 1 We write f(x) a j os πjx (8) j=1 4
5 for the Fourier series of f and define ( 1 1/ f(x) = f(x) dx) Furthermore, we define f(x) = 0 a j os πjx j=1 Let K be a set of positive integers Then by the orthogonality of the trigonometri system k f(kx) k f(kx) (9) We will write for appropriate positive onstants, not always the same First we will prove the following Lemma 3 Let (n k ) k 1 be a stritly inreasing sequene of positive integers, and let M < Nbe positive integers Let Then Using this we an show K = {k : n k [M + 1,N]} and K = #K ( ) 5log K f(n k x) K exp Lemma 4 Let M < N be positive integers Let L > 1 Then ( ) 5log N log L exp M<kN 4log log N M<kN This yields k L 1 Lemma 5 Let N > N 1 be positive integers Then ( ) N 1 <MN 3log N N + exp log log N N 1 <km To dedue Theorem 1 from Lemma 5 we will finally need the following k 1/ N 1 <kn k Lemma 6 Assume that for every given ε > 0 there exists an M 0 suh that M M>M 0 ε Then is ae onvergent k=m /
6 Proof of Lemma 3: For s = 0,1,, we define r s (x) = s+1 j= s +1 a j os πjx (the a j s are defined in (8)) Then by Minkowski s inequality f(n k x) r s (n k x) s 0 (10) By Lemma 1 we have s+1 j= s +1 a j sα, (11) where is a onstant Let η [0,1] (we will hoose the exat value of η later) By Minkowski s inequality and the orthogonality of the trigonometri system r s (n k x) (1) a j os πjn k x + a j os πjn k x sα sα(1 η) <a j, sα sα(1 η) a j s <j s+1, s <j s+1 a j s+1 os πjn k x + sα sα(1 η) os πjn k x sα sα(1 η) <a j, j= s +1 s <j s+1 a j os πjn k x (13) j: sα sα(1 η) <a j + sα+sαη s+1 j= s +1 os πjn k x (14) By the orthogonality of the trigonometri system and (11) the term (13) is bounded by K sα sα(1 η) <a j, s <j s+1 a j s+1 sα+sα(1 η) K K sαη j= s +1 a j 6
7 By Lemma the term (14) is bounded by sα+sαη Therefore (1) is bounded by We hoose and see that (1) is bounded by s+1 k 1,k K j 1,j = s +1 1(j 1 n k1 = j n k ) sα+sαη s/ ( ) 5log K K exp ( ) 5log K K s(α 1/)+sαη exp K sαη + K s(α 1/)+sαη exp η = 1 1 4α, K sα+s/4 exp ( ) 5log K 1/ ( ) 5log K By (10) this implies f(n k x) ( ) 5log K K s(α 1/4) exp s 0 ( ) 5log K K exp, whih proves Lemma 3 Proof of Lemma 4 : Let M < N and L be given By (9) we have M<kN k L 1 M<kN k L 1 s: 1 s L s: 1 s L k f(kx) M<kN s s 1 k < s M<kN s 1 k < s k f(kx) f(kx) 7
8 Using Lemma 3 we get M<kN s 1 k < s f(kx) ( ) 5log K(s) K(s) exp, (s) where Trivially K(s) N M K(s) = # { M < k N : s 1 k < s} Using the Cauhy-Shwarz inequality we finally get, s ( ) 5log K(s) K(s) exp M<kN (s) s: 1 s L k L 1 ( ) 5log N log L exp s K(s) 4log log N s: 1 s L ( ) 5log N log L exp 4log log N M<kN k 1/ 1/ This proves Lemma 4 Proof of Lemma 5: Let N > N 1 be given We have N 1 MN N 1 <km N 1 MN k f(kx) + N 1 <km k N 3 N 1 MN N 1 <km k N 3 (15) The first term in (15) is bounded by N 1 MN N 1 <km k N 3 N 1 <kn k N 3 k N (16) To estimate the value of the seond term in (15) we use Lemma 4, where we hose L = N 3 8
9 Then we have N 1 <kn k N 3 ( ) 5log (log N ) 1/ N exp k 4log log N N 1 <kn 1/ (17) Imitating the proof of the Rademaher-Menshov inequality (see [14, p 13]), we an easily show that ( ) 5log N 1 MN k f(kx) (log N ) 3/ N exp 1/ k 4log log N N 1 <km N 1 <kn k N 3 ( ) 3log N exp 1/ k log log N N 1 <kn Combining this with (16) we have ( ) N 1 MN 3log N N + exp log log N N 1 <km N 1 <kn k 1/ This proves Lemma 5 Proof of Lemma 6: Assume that for every given ε > 0 there exists an M 0 suh that sup N ε (18) N>M 0 By Minkowski s inequality this implies sup Therefore and inf M N >N 1 >M 0 k=m 0 +1 N k=n 1 ε inf N sup M N >N 1 >M k=n 1 = 0 sup N >N 1 >M whih implies the ae onvergene of N k=n 1 = 0 ae, 9
10 4 Proof of Theorem 1 Assume that ( k ) k 1 satisfies ( ) log k k exp < (19) log log k As a onsequene of (19) we have for r 1 r+1 k= r +1 ( log( r ) k exp ) log log( r ) By the monotone onvergene theorem and Minkowski s inequality we have, for any m 1, sup M M> m k f(kx) = lim M sup w k= m +1 m <M m+w k f(kx) k= m +1 M lim w k f(kx) mrm+w 1 r <M r+1 k= r +1 Together with Lemma 5 this implies sup M M> m k f(kx) k= m +1 M r m r <M r+1 k f(kx) k= r +1 r m ( r ) + exp m + r m ( log( r ) ) exp log log( r ) ( 3log( r ) ) log log( r k ) r <k r+1 1/ (0) For any given ε > 0, we an hoose m so large that (0) is smaller than ε Therefore, by Lemma 6 the series is ae onvergent Referenes [1] C Aistleitner The law of the iterated logarithm for k f(n k x) In Dependene in Probability, Analysis and Number Theory A Volume in Memory of Walter Philipp, pages 1 34 Kendrik Press,
11 [] C Aistleitner and I Berkes On the entral limit theorem for f(n k x) Probab Theory Related Fields, 146(1-):67 89, 010 [3] C Aistleitner, P A Mayer, and V Ziegler Metri disrepany theory, funtions of bounded variation and gd sums Unif Distribution Theory, 5(1):95 109, 010 [4] J Arias de Reyna Pointwise onvergene of Fourier series, volume 1785 of Leture Notes in Mathematis Springer-Verlag, Berlin, 00 [5] I Berkes On the onvergene of n f(nx) and the Lip 1/ lass Trans Amer Math So, 349(10): , 1997 [6] I Berkes and M Weber On the onvergene of k f(n k x) Mem Amer Math So, 01(943):viii+7, 009 [7] L Carleson On onvergene and growth of partial sumas of Fourier series Ata Math, 116: , 1966 [8] T Dyer and G Harman Sums involving ommon divisors J London Math So (), 34(1):1 11, 1986 [9] K Fukuyama The law of the iterated logarithm for disrepanies of {θ n x} Ata Math Hungar, 118(1-): , 008 [10] V F Gapoškin On series with respet to the system {ϕ(nx)} Mat Sb (NS), 69 (111):38 353, 1966 [11] V F Gapoškin Convergene and divergene systems Mat Zametki, 4:53 60, 1968 [1] G Harman Metri number theory, volume 18 of London Mathematial Soiety Monographs New Series The Clarendon Press Oxford University Press, New York, 1998 [13] M Ka Convergene of ertain gap series Ann of Math (), 44: , 1943 [14] M Loève Probability theory II Springer-Verlag, New York, fourth edition, 1978 Graduate Texts in Mathematis, Vol 46 [15] C J Mozzohi On the pointwise onvergene of Fourier series Leture Notes in Mathematis, Vol 199 Springer-Verlag, Berlin, 1971 [16] A Zygmund Trigonometri series nd ed Vols I, II Cambridge University Press, New York,
5.2 The Master Theorem
170 CHAPTER 5. RECURSION AND RECURRENCES 5.2 The Master Theorem Master Theorem In the last setion, we saw three different kinds of behavior for reurrenes of the form at (n/2) + n These behaviors depended
More informationcos t sin t sin t cos t
Exerise 7 Suppose that t 0 0andthat os t sin t At sin t os t Compute Bt t As ds,andshowthata and B ommute 0 Exerise 8 Suppose A is the oeffiient matrix of the ompanion equation Y AY assoiated with the
More informationChapter 6 A N ovel Solution Of Linear Congruenes Proeedings NCUR IX. (1995), Vol. II, pp. 708{712 Jerey F. Gold Department of Mathematis, Department of Physis University of Utah Salt Lake City, Utah 84112
More information9 More on differentiation
Tel Aviv University, 2013 Measure and category 75 9 More on differentiation 9a Finite Taylor expansion............... 75 9b Continuous and nowhere differentiable..... 78 9c Differentiable and nowhere monotone......
More information1.3 Complex Numbers; Quadratic Equations in the Complex Number System*
04 CHAPTER Equations and Inequalities Explaining Conepts: Disussion and Writing 7. Whih of the following pairs of equations are equivalent? Explain. x 2 9; x 3 (b) x 29; x 3 () x - 2x - 22 x - 2 2 ; x
More informationn k=1 k=0 1/k! = e. Example 6.4. The series 1/k 2 converges in R. Indeed, if s n = n then k=1 1/k, then s 2n s n = 1 n + 1 +...
6 Series We call a normed space (X, ) a Banach space provided that every Cauchy sequence (x n ) in X converges. For example, R with the norm = is an example of Banach space. Now let (x n ) be a sequence
More informationUSA Mathematical Talent Search. PROBLEMS / SOLUTIONS / COMMENTS Round 3 - Year 12 - Academic Year 2000-2001
USA Mathematial Talent Searh PROBLEMS / SOLUTIONS / COMMENTS Round 3 - Year - Aademi Year 000-00 Gene A. Berg, Editor /3/. Find the smallest positive integer with the property that it has divisors ending
More informationTHE SQUARE PARTIAL SUMS OF THE FOURIER TRANSFORM OF RADIAL FUNCTIONS IN THREE DIMENSIONS
Scientiae Mathematicae Japonicae Online, Vol. 5,, 9 9 9 THE SQUARE PARTIAL SUMS OF THE FOURIER TRANSFORM OF RADIAL FUNCTIONS IN THREE DIMENSIONS CHIKAKO HARADA AND EIICHI NAKAI Received May 4, ; revised
More informationLecture 13: Martingales
Lecture 13: Martingales 1. Definition of a Martingale 1.1 Filtrations 1.2 Definition of a martingale and its basic properties 1.3 Sums of independent random variables and related models 1.4 Products of
More informationStatic Fairness Criteria in Telecommunications
Teknillinen Korkeakoulu ERIKOISTYÖ Teknillisen fysiikan koulutusohjelma 92002 Mat-208 Sovelletun matematiikan erikoistyöt Stati Fairness Criteria in Teleommuniations Vesa Timonen, e-mail: vesatimonen@hutfi
More informationUndergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics
Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights
More information10.1 The Lorentz force law
Sott Hughes 10 Marh 2005 Massahusetts Institute of Tehnology Department of Physis 8.022 Spring 2004 Leture 10: Magneti fore; Magneti fields; Ampere s law 10.1 The Lorentz fore law Until now, we have been
More informationClassical Electromagnetic Doppler Effect Redefined. Copyright 2014 Joseph A. Rybczyk
Classial Eletromagneti Doppler Effet Redefined Copyright 04 Joseph A. Rybzyk Abstrat The lassial Doppler Effet formula for eletromagneti waves is redefined to agree with the fundamental sientifi priniples
More informationCapacity at Unsignalized Two-Stage Priority Intersections
Capaity at Unsignalized Two-Stage Priority Intersetions by Werner Brilon and Ning Wu Abstrat The subjet of this paper is the apaity of minor-street traffi movements aross major divided four-lane roadways
More informationCounting Primes whose Sum of Digits is Prime
2 3 47 6 23 Journal of Integer Sequences, Vol. 5 (202), Article 2.2.2 Counting Primes whose Sum of Digits is Prime Glyn Harman Department of Mathematics Royal Holloway, University of London Egham Surrey
More informationThere are only finitely many Diophantine quintuples
There are only finitely many Diophantine quintuples Andrej Dujella Department of Mathematis, University of Zagreb, Bijenička esta 30 10000 Zagreb, Croatia E-mail: duje@math.hr Abstrat A set of m positive
More informationChannel Assignment Strategies for Cellular Phone Systems
Channel Assignment Strategies for Cellular Phone Systems Wei Liu Yiping Han Hang Yu Zhejiang University Hangzhou, P. R. China Contat: wliu5@ie.uhk.edu.hk 000 Mathematial Contest in Modeling (MCM) Meritorious
More informationNote on some explicit formulae for twin prime counting function
Notes on Number Theory and Discrete Mathematics Vol. 9, 03, No., 43 48 Note on some explicit formulae for twin prime counting function Mladen Vassilev-Missana 5 V. Hugo Str., 4 Sofia, Bulgaria e-mail:
More informationSection 4.2: The Division Algorithm and Greatest Common Divisors
Section 4.2: The Division Algorithm and Greatest Common Divisors The Division Algorithm The Division Algorithm is merely long division restated as an equation. For example, the division 29 r. 20 32 948
More informationTaylor and Maclaurin Series
Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions
More informationMATH 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
More information6. 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,
More informationx a x 2 (1 + x 2 ) n.
Limits and continuity Suppose that we have a function f : R R. Let a R. We say that f(x) tends to the limit l as x tends to a; lim f(x) = l ; x a if, given any real number ɛ > 0, there exists a real number
More informationBANACH 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
More informationSeparation Properties for Locally Convex Cones
Journal of Convex Analysis Volume 9 (2002), No. 1, 301 307 Separation Properties for Locally Convex Cones Walter Roth Department of Mathematics, Universiti Brunei Darussalam, Gadong BE1410, Brunei Darussalam
More informationα α λ α = = λ λ α ψ = = α α α λ λ ψ α = + β = > θ θ β > β β θ θ θ β θ β γ θ β = γ θ > β > γ θ β γ = θ β = θ β = θ β = β θ = β β θ = = = β β θ = + α α α α α = = λ λ λ λ λ λ λ = λ λ α α α α λ ψ + α =
More informationPowers of Two in Generalized Fibonacci Sequences
Revista Colombiana de Matemáticas Volumen 462012)1, páginas 67-79 Powers of Two in Generalized Fibonacci Sequences Potencias de dos en sucesiones generalizadas de Fibonacci Jhon J. Bravo 1,a,B, Florian
More informationCHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.
CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,
More informationON SOME ANALOGUE OF THE GENERALIZED ALLOCATION SCHEME
ON SOME ANALOGUE OF THE GENERALIZED ALLOCATION SCHEME Alexey Chuprunov Kazan State University, Russia István Fazekas University of Debrecen, Hungary 2012 Kolchin s generalized allocation scheme A law of
More informationEvery tree contains a large induced subgraph with all degrees odd
Every tree contains a large induced subgraph with all degrees odd A.J. Radcliffe Carnegie Mellon University, Pittsburgh, PA A.D. Scott Department of Pure Mathematics and Mathematical Statistics University
More informationProperties of BMO functions whose reciprocals are also BMO
Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a non-negative BMO-function w, whose reciprocal is also in BMO, belongs to p> A p,and
More informationThe Goldberg Rao Algorithm for the Maximum Flow Problem
The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }
More informationBipan Hazarika ON ACCELERATION CONVERGENCE OF MULTIPLE SEQUENCES. 1. Introduction
F A S C I C U L I M A T H E M A T I C I Nr 51 2013 Bipan Hazarika ON ACCELERATION CONVERGENCE OF MULTIPLE SEQUENCES Abstract. In this article the notion of acceleration convergence of double sequences
More informationThe Fourier Series of a Periodic Function
1 Chapter 1 he Fourier Series of a Periodic Function 1.1 Introduction Notation 1.1. We use the letter with a double meaning: a) [, 1) b) In the notations L p (), C(), C n () and C () we use the letter
More information( ) ( ) [2],[3],[4],[5],[6],[7]
( ) ( ) Lebesgue Takagi, - []., [2],[3],[4],[5],[6],[7].,. [] M. Hata and M. Yamaguti, The Takagi function and its generalization, Japan J. Appl. Math., (984), 83 99. [2] T. Sekiguchi and Y. Shiota, A
More informationMath 319 Problem Set #3 Solution 21 February 2002
Math 319 Problem Set #3 Solution 21 February 2002 1. ( 2.1, problem 15) Find integers a 1, a 2, a 3, a 4, a 5 such that every integer x satisfies at least one of the congruences x a 1 (mod 2), x a 2 (mod
More informationProgramming Basics - FORTRAN 77 http://www.physics.nau.edu/~bowman/phy520/f77tutor/tutorial_77.html
CWCS Workshop May 2005 Programming Basis - FORTRAN 77 http://www.physis.nau.edu/~bowman/phy520/f77tutor/tutorial_77.html Program Organization A FORTRAN program is just a sequene of lines of plain text.
More informationA simple criterion on degree sequences of graphs
Discrete Applied Mathematics 156 (2008) 3513 3517 Contents lists available at ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam Note A simple criterion on degree
More informationHow To Fator
CHAPTER hapter 4 > Make the Connetion 4 INTRODUCTION Developing seret odes is big business beause of the widespread use of omputers and the Internet. Corporations all over the world sell enryption systems
More informationMetric 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
More informationStationary random graphs on Z with prescribed iid degrees and finite mean connections
Stationary random graphs on Z with prescribed iid degrees and finite mean connections Maria Deijfen Johan Jonasson February 2006 Abstract Let F be a probability distribution with support on the non-negative
More informationMaximum Likelihood Estimation
Math 541: Statistical Theory II Lecturer: Songfeng Zheng Maximum Likelihood Estimation 1 Maximum Likelihood Estimation Maximum likelihood is a relatively simple method of constructing an estimator for
More informationDIOPHANTINE m-tuples FOR LINEAR POLYNOMIALS. II. EQUAL DEGREES
DIOPHANTINE m-tuples FOR LINEAR POLYNOMIALS. II. EQUAL DEGREES ANDREJ DUJELLA, CLEMENS FUCHS, AND GARY WALSH Abstract. In this paper we prove the best possible upper bounds for the number of elements in
More informationSebastián Bravo López
Transfinite Turing mahines Sebastián Bravo López 1 Introdution With the rise of omputers with high omputational power the idea of developing more powerful models of omputation has appeared. Suppose that
More information1. 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
More informationINDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS
INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem
More informationPRIME FACTORS OF CONSECUTIVE INTEGERS
PRIME FACTORS OF CONSECUTIVE INTEGERS MARK BAUER AND MICHAEL A. BENNETT Abstract. This note contains a new algorithm for computing a function f(k) introduced by Erdős to measure the minimal gap size in
More informationPolarization codes and the rate of polarization
Polarization codes and the rate of polarization Erdal Arıkan, Emre Telatar Bilkent U., EPFL Sept 10, 2008 Channel Polarization Given a binary input DMC W, i.i.d. uniformly distributed inputs (X 1,...,
More informationMathematical 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.............................
More informationSignal detection and goodness-of-fit: the Berk-Jones statistics revisited
Signal detection and goodness-of-fit: the Berk-Jones statistics revisited Jon A. Wellner (Seattle) INET Big Data Conference INET Big Data Conference, Cambridge September 29-30, 2015 Based on joint work
More informationThe Heat Equation. Lectures INF2320 p. 1/88
The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)
More informationON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP. A. K. Das and R. K. Nath
International Electronic Journal of Algebra Volume 7 (2010) 140-151 ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP A. K. Das and R. K. Nath Received: 12 October 2009; Revised: 15 December
More informationInner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More informationAsymptotics for a discrete-time risk model with Gamma-like insurance risks. Pokfulam Road, Hong Kong
Asymptotics for a discrete-time risk model with Gamma-like insurance risks Yang Yang 1,2 and Kam C. Yuen 3 1 Department of Statistics, Nanjing Audit University, Nanjing, 211815, China 2 School of Economics
More informationDSP-I DSP-I DSP-I DSP-I
DSP-I DSP-I DSP-I DSP-I Digital Signal Proessing I (8-79) Fall Semester, 005 IIR FILER DESIG EXAMPLE hese notes summarize the design proedure for IIR filters as disussed in lass on ovember. Introdution:
More informationComputer Networks Framing
Computer Networks Framing Saad Mneimneh Computer Siene Hunter College of CUNY New York Introdution Who framed Roger rabbit? A detetive, a woman, and a rabbit in a network of trouble We will skip the physial
More informationA 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
More informationHenley Business School at Univ of Reading. Pre-Experience Postgraduate Programmes Chartered Institute of Personnel and Development (CIPD)
MS in International Human Resoure Management For students entering in 2012/3 Awarding Institution: Teahing Institution: Relevant QAA subjet Benhmarking group(s): Faulty: Programme length: Date of speifiation:
More informationCONTRIBUTIONS TO ZERO SUM PROBLEMS
CONTRIBUTIONS TO ZERO SUM PROBLEMS S. D. ADHIKARI, Y. G. CHEN, J. B. FRIEDLANDER, S. V. KONYAGIN AND F. PAPPALARDI Abstract. A prototype of zero sum theorems, the well known theorem of Erdős, Ginzburg
More informationCIS570 Lecture 4 Introduction to Data-flow Analysis 3
Introdution to Data-flow Analysis Last Time Control flow analysis BT disussion Today Introdue iterative data-flow analysis Liveness analysis Introdue other useful onepts CIS570 Leture 4 Introdution to
More informationCONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12
CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.
More information1 Completeness of a Set of Eigenfunctions. Lecturer: Naoki Saito Scribe: Alexander Sheynis/Allen Xue. May 3, 2007. 1.1 The Neumann Boundary Condition
MAT 280: Laplacian Eigenfunctions: Theory, Applications, and Computations Lecture 11: Laplacian Eigenvalue Problems for General Domains III. Completeness of a Set of Eigenfunctions and the Justification
More informationPrices and Heterogeneous Search Costs
Pries and Heterogeneous Searh Costs José L. Moraga-González Zsolt Sándor Matthijs R. Wildenbeest First draft: July 2014 Revised: June 2015 Abstrat We study prie formation in a model of onsumer searh for
More informationSHAFTS: TORSION LOADING AND DEFORMATION
ECURE hird Edition SHAFS: ORSION OADING AND DEFORMAION A. J. Clark Shool of Engineering Department of Civil and Environmental Engineering 6 Chapter 3.1-3.5 by Dr. Ibrahim A. Assakkaf SPRING 2003 ENES 220
More informationChapter 1 Microeconomics of Consumer Theory
Chapter 1 Miroeonomis of Consumer Theory The two broad ategories of deision-makers in an eonomy are onsumers and firms. Eah individual in eah of these groups makes its deisions in order to ahieve some
More information) ( )( ) ( ) ( )( ) ( ) ( ) (1)
OPEN CHANNEL FLOW Open hannel flow is haraterized by a surfae in ontat with a gas phase, allowing the fluid to take on shapes and undergo behavior that is impossible in a pipe or other filled onduit. Examples
More informationVOLUME 13, ARTICLE 5, PAGES 117-142 PUBLISHED 05 OCTOBER 2005 DOI: 10.4054/DemRes.2005.13.
Demographi Researh a free, expedited, online journal of peer-reviewed researh and ommentary in the population sienes published by the Max Plank Institute for Demographi Researh Konrad-Zuse Str. 1, D-157
More informationON THE EXISTENCE AND LIMIT BEHAVIOR OF THE OPTIMAL BANDWIDTH FOR KERNEL DENSITY ESTIMATION
Statistica Sinica 17(27), 289-3 ON THE EXISTENCE AND LIMIT BEHAVIOR OF THE OPTIMAL BANDWIDTH FOR KERNEL DENSITY ESTIMATION J. E. Chacón, J. Montanero, A. G. Nogales and P. Pérez Universidad de Extremadura
More informationI. Pointwise convergence
MATH 40 - NOTES Sequences of functions Pointwise and Uniform Convergence Fall 2005 Previously, we have studied sequences of real numbers. Now we discuss the topic of sequences of real valued functions.
More information1 Sufficient statistics
1 Sufficient statistics A statistic is a function T = rx 1, X 2,, X n of the random sample X 1, X 2,, X n. Examples are X n = 1 n s 2 = = X i, 1 n 1 the sample mean X i X n 2, the sample variance T 1 =
More informationTriangle deletion. Ernie Croot. February 3, 2010
Triangle deletion Ernie Croot February 3, 2010 1 Introduction The purpose of this note is to give an intuitive outline of the triangle deletion theorem of Ruzsa and Szemerédi, which says that if G = (V,
More informationArrangements of Stars on the American Flag
Arrangements of Stars on the American Flag Dimitris Koukoulopoulos and Johann Thiel Abstract. In this article, we examine the existence of nice arrangements of stars on the American flag. We show that
More informationInequalities of Analysis. Andrejs Treibergs. Fall 2014
USAC Colloquium Inequalities of Analysis Andrejs Treibergs University of Utah Fall 2014 2. USAC Lecture: Inequalities of Analysis The URL for these Beamer Slides: Inequalities of Analysis http://www.math.utah.edu/~treiberg/inequalitiesslides.pdf
More informationLecture 24: Spinodal Decomposition: Part 3: kinetics of the
Leture 4: Spinodal Deoposition: Part 3: kinetis of the oposition flutuation Today s topis Diffusion kinetis of spinodal deoposition in ters of the onentration (oposition) flutuation as a funtion of tie:
More informationUser s Guide VISFIT: a computer tool for the measurement of intrinsic viscosities
File:UserVisfit_2.do User s Guide VISFIT: a omputer tool for the measurement of intrinsi visosities Version 2.a, September 2003 From: Multiple Linear Least-Squares Fits with a Common Interept: Determination
More informationON FIBONACCI NUMBERS WITH FEW PRIME DIVISORS
ON FIBONACCI NUMBERS WITH FEW PRIME DIVISORS YANN BUGEAUD, FLORIAN LUCA, MAURICE MIGNOTTE, SAMIR SIKSEK Abstract If n is a positive integer, write F n for the nth Fibonacci number, and ω(n) for the number
More informationAlgebraic and Transcendental Numbers
Pondicherry University July 2000 Algebraic and Transcendental Numbers Stéphane Fischler This text is meant to be an introduction to algebraic and transcendental numbers. For a detailed (though elementary)
More informationWeighting Methods in Survey Sampling
Setion on Survey Researh Methods JSM 01 Weighting Methods in Survey Sampling Chiao-hih Chang Ferry Butar Butar Abstrat It is said that a well-designed survey an best prevent nonresponse. However, no matter
More informationHenley Business School at Univ of Reading. Chartered Institute of Personnel and Development (CIPD)
MS in International Human Resoure Management (full-time) For students entering in 2015/6 Awarding Institution: Teahing Institution: Relevant QAA subjet Benhmarking group(s): Faulty: Programme length: Date
More informationErdos and the Twin Prime Conjecture: Elementary Approaches to Characterizing the Differences of Primes
Erdos and the Twin Prime Conjecture: Elementary Approaches to Characterizing the Differences of Primes Jerry Li June, 010 Contents 1 Introduction Proof of Formula 1 3.1 Proof..................................
More informationTHE NUMBER OF REPRESENTATIONS OF n OF THE FORM n = x 2 2 y, x > 0, y 0
THE NUMBER OF REPRESENTATIONS OF n OF THE FORM n = x 2 2 y, x > 0, y 0 RICHARD J. MATHAR Abstract. We count solutions to the Ramanujan-Nagell equation 2 y +n = x 2 for fixed positive n. The computational
More informationA Holistic Method for Selecting Web Services in Design of Composite Applications
A Holisti Method for Seleting Web Servies in Design of Composite Appliations Mārtiņš Bonders, Jānis Grabis Institute of Information Tehnology, Riga Tehnial University, 1 Kalu Street, Riga, LV 1658, Latvia,
More informationIn 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
More informationTHE SINE PRODUCT FORMULA AND THE GAMMA FUNCTION
THE SINE PRODUCT FORMULA AND THE GAMMA FUNCTION ERICA CHAN DECEMBER 2, 2006 Abstract. The function sin is very important in mathematics and has many applications. In addition to its series epansion, it
More information1. Let X and Y be normed spaces and let T B(X, Y ).
Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: NVP, Frist. 2005-03-14 Skrivtid: 9 11.30 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok
More informationF. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)
Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p -CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p
More informationA Three-Hybrid Treatment Method of the Compressor's Characteristic Line in Performance Prediction of Power Systems
A Three-Hybrid Treatment Method of the Compressor's Charateristi Line in Performane Predition of Power Systems A Three-Hybrid Treatment Method of the Compressor's Charateristi Line in Performane Predition
More information( 1 ) Obtain the equation of the circle passing through the points ( 5, - 8 ), ( - 2, 9 ) and ( 2, 1 ).
PROBLEMS 03 CIRCLE Page ( ) Obtain the equation of the irle passing through the points ( 5 8 ) ( 9 ) and ( ). [ Ans: x y 6x 48y 85 = 0 ] ( ) Find the equation of the irumsribed irle of the triangle formed
More informationALGORITHMIC TRADING WITH MARKOV CHAINS
June 16, 2010 ALGORITHMIC TRADING WITH MARKOV CHAINS HENRIK HULT AND JONAS KIESSLING Abstract. An order book consists of a list of all buy and sell offers, represented by price and quantity, available
More informationAsymptotics of discounted aggregate claims for renewal risk model with risky investment
Appl. Math. J. Chinese Univ. 21, 25(2: 29-216 Asymptotics of discounted aggregate claims for renewal risk model with risky investment JIANG Tao Abstract. Under the assumption that the claim size is subexponentially
More informationAdaptive Online Gradient Descent
Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650
More informationminimal polyonomial Example
Minimal Polynomials Definition Let α be an element in GF(p e ). We call the monic polynomial of smallest degree which has coefficients in GF(p) and α as a root, the minimal polyonomial of α. Example: We
More informationMATH 381 HOMEWORK 2 SOLUTIONS
MATH 38 HOMEWORK SOLUTIONS Question (p.86 #8). If g(x)[e y e y ] is harmonic, g() =,g () =, find g(x). Let f(x, y) = g(x)[e y e y ].Then Since f(x, y) is harmonic, f + f = and we require x y f x = g (x)[e
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationA Uniform Asymptotic Estimate for Discounted Aggregate Claims with Subexponential Tails
12th International Congress on Insurance: Mathematics and Economics July 16-18, 2008 A Uniform Asymptotic Estimate for Discounted Aggregate Claims with Subexponential Tails XUEMIAO HAO (Based on a joint
More information36 CHAPTER 1. LIMITS AND CONTINUITY. Figure 1.17: At which points is f not continuous?
36 CHAPTER 1. LIMITS AND CONTINUITY 1.3 Continuity Before Calculus became clearly de ned, continuity meant that one could draw the graph of a function without having to lift the pen and pencil. While this
More informationSign changes of Hecke eigenvalues of Siegel cusp forms of degree 2
Sign changes of Hecke eigenvalues of Siegel cusp forms of degree 2 Ameya Pitale, Ralf Schmidt 2 Abstract Let µ(n), n > 0, be the sequence of Hecke eigenvalues of a cuspidal Siegel eigenform F of degree
More informationIntroduction to Finite Fields (cont.)
Chapter 6 Introduction to Finite Fields (cont.) 6.1 Recall Theorem. Z m is a field m is a prime number. Theorem (Subfield Isomorphic to Z p ). Every finite field has the order of a power of a prime number
More informationInternational 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,
More informationMATE: MPLS Adaptive Traffic Engineering
MATE: MPLS Adaptive Traffi Engineering Anwar Elwalid Cheng Jin Steven Low Indra Widjaja Bell Labs EECS Dept EE Dept Fujitsu Network Communiations Luent Tehnologies Univ. of Mihigan Calteh Pearl River,
More information