On a Polygon Equality Problem
|
|
- Rose Nicholson
- 5 years ago
- Views:
Transcription
1 JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 223, ARTICLE NO. AY O a Polygo Equality Proble L. Elser* Fakultat fur Matheatik, Uiersitat Bielefeld, Postfach 003, 3350 Bielefeld, Geray L. Ha, I. Koltracht, M. Neua, ad M. Zippi Departet of Matheatics, Uiersity of Coecticut, Storrs, Coecticut Subitted by Joh Horath Received October 4, 997 Berius ad Blachard of Bielefeld Uiversity i Geray have cojectured the followig polygo iequality: for ay two sets of vectors x,..., x ad y,..., y i, x x y y x y ij ij i, j i the 2-or ad that, oreover, equality holds i Ž. if ad oly if there exists a perutatio o, 2,..., 4 such that y x, i,...,. That Ž. i Ži. is valid is a cosequece of a iequality that holds i certai Baach spaces ad which was recetly proved by Leard, Togue, ad Westo. We therefore characterize here the case of equality i Ž., actually for vectors i the space X L Ž,., ad subsequetly use this characterizatio to coplete the proof of the BeriusBlachard cojecture cocerig the equality case i a Hilbert space. 998 Acadeic Press *Research supported i part by Soderforschugsbereich 343 Diskrete Strukture i der Matheatik, Uiversitat Bielefeld. E-ail: eua@ath.uco.edu. Peraet address: Istitute of Matheatics, The Hebrew Uiversity, Jerusale 9904, Israel. Participat i Workshop i Liear Aalysis ad Probability, Texas A & M Uiversity, 997. Partially supported by the Ladau Ceter for Research i Matheatical Aalysis, sposored by the Mierva Foudatio Ž Geray X98 $25.00 Copyright 998 by Acadeic Press All rights of reproductio i ay for reserved.
2 68 ELSNER ET AL.. INTRODUCTION I the study of learig i artificial eural etworks the followig easure of welless of represetatio has arise; see Berius : Gie a set of 2 poits i, 4, diide it ito two parts of equal sizes M,..., 4 ad M,..., 4 2 such that M represets M2 as well as possible. As a Ž also coputatioally coeiet. easure of the degree of represetatio, Berius itroduced the fuctio H,,2 i, j ij ij where is the Euclidea or i the real -diesioal space. Berius ad Blachard cojectured that H, 2 0 ad they have also raised the questio, ore iportat to the, of whe H, 2 0. Actually, it has bee kow, by a result of Leard, Togue, ad Westo 3 via a equivalece with a result of Bretagolle, Castelle, ad Krivie 2, that the iequality H, 2 0 holds i a Baach space X for every 2 ad every two sets M ad M2 of vectors if ad oly if X is isoetric to a subspace of L Ž,. for soe easure space Ž,..Itis well kow that the space L 0, p with p 2 is isoetric to a subspace of L 0, Žsee Lidestrauss ad Tzafriri 5 ad the refereces cited therei. ad hece H, 2 0 holds for ay 2 ad every two sets M ad M2 of vectors i the or of L p, p2. Berius ad Blachard forulated the followig: CONJECTURE Ž Polygo Equality Cojecture.. Let x,..., x ad y,..., y be ectors i. The x x y y x y Ž.. i j 2 i j 2 i j 2 ij ij i, j if ad oly if there exists a perutatio of, 2,..., 4 such that yi x Ži., i,...,. Two special cases of the cojecture were proved i : ad ad ad 2. As usual, for ay vector x Ž. T ad for ay positive uber r, we shall let r r ž i / i x r
3 ON A POLYGON EQUALITY PROBLEM 69 ad we shall let l r deote the space of all -vectors x T of real ubers with the or x r. The ai purpose of this paper is to prove the polygo equality cojecture. However, it turs out that the atural settig for studyig the cojecture is the space L Ž,. istead of a Hilbert space. Therefore, before provig the cojecture i Sectio 3 we will cosider, i Sectio 2, the questio of whe equality holds i Ž. i the L -or. 2. THE CASE OF EQUALITY IN L, Let us start with a siple exaple. 2 EXAMPLE 2.. Let X l 4 ad let xi i 2 deote the uit basis of X. Put y Ž,. ad y Ž 0, 0.. The x x 2 y y ad x y for all i, j 2. Hece i j 2 i j 2 2 i, j x y x x y y, but the sets x 4 2 ad y 4 2 are ot idetical. i i i i Hece the BeriusBlachard cojecture is false i L Ž,. ad we would like to deterie exactly whe for the sets M x,..., x 4 ad M y,..., y 4 2 i L equality holds. Our ai tool i studyig the equality case is the followig: LEMMA 2.2. Let s s ad t t be ay two collectios of real ubers. The t s t s s s t t. Ž 2.. i i i i, j ij Proof. We start our iductive arguet with the case 2. We ust show that t s t s t s t s t s t s s s t t 2 2, which is the sae as showig that the iequality s s t t t s t s
4 70 ELSNER ET AL. holds. But this follows fro s s2 ad t t2 by checkig. Let us pass to the iductio step. Suppose that Ž 2.. holds for ad let s s s ad t t t be give. Assue without loss of geerality that t s. The i i i i t s t s s s. It ow follows fro the triagle iequality that i i i i ts ssi sti i i i i i i t s s t s s t t. Ž 2.2. But the 2.2 ad the iductio hypothesis give that ad the proof is doe. s t s s t t i, j ij t s s t i i t s s t i i i j i, j i i i s s t t Ž s s t t. i j i j ij t s t s t s i i i i i i We are ow ready to characterize the sets of vectors x 4 4 i iad y for which equality holds i Ž. i the L -or. i i
5 ON A POLYGON EQUALITY PROBLEM 7 THEOREM 2.3. Let x 4 4 i iad yi i be two collectios of fuctios i L Ž,.. The equality holds i Ž. if ad oly if for alost eery, the uerical sets x Ž.4 ad y Ž.4 are idetical. Proof. i i i i The if part is iediate because i j i, j x Ž. y Ž. i j i j ij x Ž. x Ž. y Ž. y Ž. 0 for alost every Ž with respect to. ad by itegratig over with respect to. Let us the cosider the oly if part. Suppose that equality holds i Ž. ad, for each, let Ž,i. ad Ž,i. be perutatios Ž, 2,...,. for which ad x Ž. x Ž. Ž,. Ž,. y Ž. y Ž.. Ž,. Ž,. We will eed the icreasig order of these ubers i order to effectively use Lea 2.2. We ust prove that for every i, xž, i. Ž. yž, i. Ž. a.e. Ž 2.3. If Ž 2.3. is false, the there exist 0, 0, ad a subset with Ž. such that Ž, i. Ž, i. i for every, x Ž. y Ž.. Ž 2.4. Note that, for every, the expressios x i Ž. y j Ž. ad i, j i j i j ij Ž x Ž. x Ž. y Ž. y Ž.. are ivariat uder perutatios of x 4 ad y 4. Therefore Ž 2.4. i i i i, Lea 2.2, ad the equality x y x x y y 0 i, j ij
6 72 ELSNER ET AL. yield the iequality 0 x y x x y y i, j ij ½ H i j i, j x Ž. y Ž. ½ Ž i j i j. 5 x Ž. x Ž. y Ž. y Ž. d ij H Ž, i. Ž, j. i, j x Ž. y Ž. Ž Ž, i. Ž, j. ij x Ž. x Ž. H Ž, i. Ž, i. i y Ž. y Ž.. d Ž, i. Ž, j. 5 x Ž. y Ž. dž., a cotradictio, ad our proof is doe. Reark 2.4. Suppose that the easure space Ž,. is a copact etric space ad is a Borel easure. Assue also that x 4 i i ad y4 i i are cotiuous fuctios. The the proof of Theore 2.3 shows that the equality x y x x y y 0 i, j ij iplies that x 4 4 i i ad yi i are idetical for every. Lea 2.2 ow yields the followig result for ootoically icreasig sequeces of fuctios: COROLLARY 2.5. Let x 4 y 4 L Ž,. i i i i ad assue that x Ž. x Ž. ad y Ž. y Ž. a.e. The i i i j i j i j i i, j ij x y x y x x y y.
7 ON A POLYGON EQUALITY PROBLEM PROOF OF THE POLYGON EQUALITY CONJECTURE Let us ow cosider the polygo equality cojecture. Suppose that Xl ad x 4 y 4 X are such that Ž.. holds. Let 2 i i i i 4 S ux u. Furtherore, let deote the oralized rotatio-ivariat easure o Ži.e., Ž... The the itegral H ², y: dž y. depeds oly o ad ot o itself. Put The the ap defied by H c ² u, y: dž y., u. T : l2 LŽ,., Ž Tx.Ž. c ² x, :, is a isoetric ebeddig of l ito L Ž,. Ž 2 see Lidestrauss ad Pelczyski 4, p. 32, Prop Moreover, is a copact etric space ad, for every x l2, Tx is a cotiuous fuctio. It follows by Theore 2.3 ad Reark 2.4 that, for every, the uerical sets Tx 4 i i ad Ty 4 i i are idetical for every S. Cotiuig, sice the expressio x y x x y y i, j ij does ot deped o the orderig of the x i s ad y i s we ay assue that both sequeces are arraged by decreasig order i the ors, viz., x x ad y y. Suppose ow that y x ad let y ys. As etioed earlier, by Theore 2.3 ad Reark 2.4, the sets Tx 4 4 i i ad Tyi i are idetical, ad so for soe j, we ust have that c ² x, : Ž Tx.Ž. Ž Ty.Ž. j j ² : c y y, y c y.
8 74 ELSNER ET AL. It follows that ² : ² : y x y x, y y y, y y. j j This iplies that x y. Clearly, we ca assue without loss of geerality j that j. Sice j j j j j2 j2 j2 j2 we obtai that x y y x y y x x, i j i j i j i, j2 2ij x y x x y y. Repeatig the above procedure, we get that there is a j, 2j, such that x y. Cotiuig i this aer, we arrive at the equality j 2 i j i, j x y x x y y, which iplies, by the triagle iequality, that either x y ad x y or x y ad x y. We have therefore proved the polygo equality cojecture. We coclude with the followig coet. For ay two sets of vectors 4 4 M x,..., x ad M y,..., y i l, defie the fuctio i, j ij f M, M x y x x y y. Fro the foregoig we kow that fž M, M. 0 ad that fž M, M if ad oly if there exists a perutatio o, 2,..., 4 such that yi x Ži., i,...,. Oe ight therefore be tepted to cojecture that f, defies a etric o the subsets of cardiality of l 2. That this is ot the case is show i the followig exaple i which trasitiveess fails to hold. Let ž / ž / ž / 5 ½ M 5, 5, 5, ½ž 0 / ž 0 / ž 0 / 5 M2 3, 3, 0, 0 0 0
9 ad A coputatio ow shows that ON A POLYGON EQUALITY PROBLEM 75 ž / ž / ž / 5 ½ M3 2, 2, fž M, M2. fž M, M3. fž M 3, M REFERENCES. H. C. Berius, Traiig ud Geeralisierugsverhalte Neuroaler Netzwerke, Diploarbeit, Fakultat fur Physik, Uiversitat Bielefeld, J. Bretagolle, D. Dacuha Castelle, ad J. Krivie, Foctios de type postif sur les espaces p L, C. R. Acad. Sci. Paris Ser. I. Math. 26 Ž 965., C. J. Leard, A. M. Togue, ad A. Westo, Geeralized roudedess ad egative type, Michiga J. Math. 44 Ž 997., J. Lidestrauss ad A. Pelczyski, Absolutely suig operators i Lp-spaces ad their applicatios, Studia Math. 29 Ž 968., J. Lidestrauss ad L. Tzafriri, Classical Baach Spaces II, Spriger-Verlag, New York, 979.
Lecture 10: Bounded Linear Operators and Orthogonality in Hilbert Spaces
Lecture : Bouded Liear Operators ad Orthogoality i Hilbert Spaces 34 Bouded Liear Operator Let ( X, ), ( Y, ) i i be ored liear vector spaces ad { } X Y The, T is said to be bouded if a real uber c such
More informationOn Some Properties of Tensor Product of Operators
Global Joural of Pure ad Applied Matheatics. ISSN 0973-1768 Volue 12, Nuber 6 (2016), pp. 5139-5147 Research Idia Publicatios http://www.ripublicatio.co/gjpa.ht O Soe Properties of Tesor Product of Operators
More informationAvailable online at J. Math. Comput. Sci. 4 (2014), No. 3, ISSN:
Available olie at http://scik.org J. Math. Coput. Sci. (1, No. 3, 9-5 ISSN: 197-537 ON SYMMETRICAL FUNCTIONS WITH BOUNDED BOUNDARY ROTATION FUAD. S. M. AL SARARI 1,, S. LATHA 1 Departet of Studies i Matheatics,
More informationf(1), and so, if f is continuous, f(x) = f(1)x.
2.2.35: Let f be a additive fuctio. i Clearly fx = fx ad therefore f x = fx for all Z+ ad x R. Hece, for ay, Z +, f = f, ad so, if f is cotiuous, fx = fx. ii Suppose that f is bouded o soe o-epty ope set.
More informationBinomial transform of products
Jauary 02 207 Bioial trasfor of products Khristo N Boyadzhiev Departet of Matheatics ad Statistics Ohio Norther Uiversity Ada OH 4580 USA -boyadzhiev@ouedu Abstract Give the bioial trasfors { b } ad {
More informationGeneralized Fixed Point Theorem. in Three Metric Spaces
It. Joural of Math. Aalysis, Vol. 4, 00, o. 40, 995-004 Geeralized Fixed Poit Thee i Three Metric Spaces Kristaq Kikia ad Luljeta Kikia Departet of Matheatics ad Coputer Scieces Faculty of Natural Scieces,
More informationThe Hypergeometric Coupon Collection Problem and its Dual
Joural of Idustrial ad Systes Egieerig Vol., o., pp -7 Sprig 7 The Hypergeoetric Coupo Collectio Proble ad its Dual Sheldo M. Ross Epstei Departet of Idustrial ad Systes Egieerig, Uiversity of Souther
More informationA New Type of q-szász-mirakjan Operators
Filoat 3:8 07, 567 568 https://doi.org/0.98/fil7867c Published by Faculty of Scieces ad Matheatics, Uiversity of Niš, Serbia Available at: http://www.pf.i.ac.rs/filoat A New Type of -Szász-Miraka Operators
More information42 Dependence and Bases
42 Depedece ad Bases The spa s(a) of a subset A i vector space V is a subspace of V. This spa ay be the whole vector space V (we say the A spas V). I this paragraph we study subsets A of V which spa V
More informationA new sequence convergent to Euler Mascheroni constant
You ad Che Joural of Iequalities ad Applicatios 08) 08:7 https://doi.org/0.86/s3660-08-670-6 R E S E A R C H Ope Access A ew sequece coverget to Euler Mascheroi costat Xu You * ad Di-Rog Che * Correspodece:
More informationBertrand s postulate Chapter 2
Bertrad s postulate Chapter We have see that the sequece of prie ubers, 3, 5, 7,... is ifiite. To see that the size of its gaps is ot bouded, let N := 3 5 p deote the product of all prie ubers that are
More informationDefinition 4.2. (a) A sequence {x n } in a Banach space X is a basis for X if. unique scalars a n (x) such that x = n. a n (x) x n. (4.
4. BASES I BAACH SPACES 39 4. BASES I BAACH SPACES Sice a Baach space X is a vector space, it must possess a Hamel, or vector space, basis, i.e., a subset {x γ } γ Γ whose fiite liear spa is all of X ad
More informationON MEAN ERGODIC CONVERGENCE IN THE CALKIN ALGEBRAS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX0000-0 ON MEAN ERGODIC CONVERGENCE IN THE CALKIN ALGEBRAS MARCH T. BOEDIHARDJO AND WILLIAM B. JOHNSON 2
More informationA PROBABILITY PROBLEM
A PROBABILITY PROBLEM A big superarket chai has the followig policy: For every Euros you sped per buy, you ear oe poit (suppose, e.g., that = 3; i this case, if you sped 8.45 Euros, you get two poits,
More informationAutomated Proofs for Some Stirling Number Identities
Autoated Proofs for Soe Stirlig Nuber Idetities Mauel Kauers ad Carste Scheider Research Istitute for Sybolic Coputatio Johaes Kepler Uiversity Altebergerstraße 69 A4040 Liz, Austria Subitted: Sep 1, 2007;
More informationAVERAGE MARKS SCALING
TERTIARY INSTITUTIONS SERVICE CENTRE Level 1, 100 Royal Street East Perth, Wester Australia 6004 Telephoe (08) 9318 8000 Facsiile (08) 95 7050 http://wwwtisceduau/ 1 Itroductio AVERAGE MARKS SCALING I
More informationSOME FINITE SIMPLE GROUPS OF LIE TYPE C n ( q) ARE UNIQUELY DETERMINED BY THEIR ELEMENT ORDERS AND THEIR ORDER
Joural of Algebra, Nuber Theory: Advaces ad Applicatios Volue, Nuber, 010, Pages 57-69 SOME FINITE SIMPLE GROUPS OF LIE TYPE C ( q) ARE UNIQUELY DETERMINED BY THEIR ELEMENT ORDERS AND THEIR ORDER School
More informationRefinements of Jensen s Inequality for Convex Functions on the Co-Ordinates in a Rectangle from the Plane
Filoat 30:3 (206, 803 84 DOI 0.2298/FIL603803A Published by Faculty of Scieces ad Matheatics, Uiversity of Niš, Serbia Available at: http://www.pf.i.ac.rs/filoat Refieets of Jese s Iequality for Covex
More information1.2 AXIOMATIC APPROACH TO PROBABILITY AND PROPERTIES OF PROBABILITY MEASURE 1.2 AXIOMATIC APPROACH TO PROBABILITY AND
NTEL- robability ad Distributios MODULE 1 ROBABILITY LECTURE 2 Topics 1.2 AXIOMATIC AROACH TO ROBABILITY AND ROERTIES OF ROBABILITY MEASURE 1.2.1 Iclusio-Exclusio Forula I the followig sectio we will discuss
More informationMath 4707 Spring 2018 (Darij Grinberg): midterm 2 page 1. Math 4707 Spring 2018 (Darij Grinberg): midterm 2 with solutions [preliminary version]
Math 4707 Sprig 08 Darij Griberg: idter page Math 4707 Sprig 08 Darij Griberg: idter with solutios [preliiary versio] Cotets 0.. Coutig first-eve tuples......................... 3 0.. Coutig legal paths
More informationJacobi symbols. p 1. Note: The Jacobi symbol does not necessarily distinguish between quadratic residues and nonresidues. That is, we could have ( a
Jacobi sybols efiitio Let be a odd positive iteger If 1, the Jacobi sybol : Z C is the costat fuctio 1 1 If > 1, it has a decopositio ( as ) a product of (ot ecessarily distict) pries p 1 p r The Jacobi
More informationSOME SEQUENCE SPACES DEFINED BY ORLICZ FUNCTIONS
ARCHIVU ATHEATICU BRNO Tomus 40 2004, 33 40 SOE SEQUENCE SPACES DEFINED BY ORLICZ FUNCTIONS E. SAVAŞ AND R. SAVAŞ Abstract. I this paper we itroduce a ew cocept of λ-strog covergece with respect to a Orlicz
More informationSome remarks on the paper Some elementary inequalities of G. Bennett
Soe rears o the paper Soe eleetary iequalities of G. Beett Dag Ah Tua ad Luu Quag Bay Vieta Natioal Uiversity - Haoi Uiversity of Sciece Abstract We give soe couterexaples ad soe rears of soe of the corollaries
More informationCommon Fixed Points for Multifunctions Satisfying a Polynomial Inequality
BULETINUL Uiversităţii Petrol Gaze di Ploieşti Vol LXII No /00 60-65 Seria Mateatică - Iforatică - Fizică Coo Fixed Poits for Multifuctios Satisfyig a Polyoial Iequality Alexadru Petcu Uiversitatea Petrol-Gaze
More informationDouble Derangement Permutations
Ope Joural of iscrete Matheatics, 206, 6, 99-04 Published Olie April 206 i SciRes http://wwwscirporg/joural/ojd http://dxdoiorg/04236/ojd2066200 ouble erageet Perutatios Pooya aeshad, Kayar Mirzavaziri
More informationLecture Outline. 2 Separating Hyperplanes. 3 Banach Mazur Distance An Algorithmist s Toolkit October 22, 2009
18.409 A Algorithist s Toolkit October, 009 Lecture 1 Lecturer: Joatha Keler Scribes: Alex Levi (009) 1 Outlie Today we ll go over soe of the details fro last class ad ake precise ay details that were
More informationON ABSOLUTE MATRIX SUMMABILITY FACTORS OF INFINITE SERIES. 1. Introduction
Joural of Classical Aalysis Volue 3, Nuber 2 208), 33 39 doi:0.753/jca-208-3-09 ON ABSOLUTE MATRIX SUMMABILITY FACTORS OF INFINITE SERIES AHMET KARAKAŞ Abstract. I the preset paper, a geeral theore dealig
More informationA REMARK ON A PROBLEM OF KLEE
C O L L O Q U I U M M A T H E M A T I C U M VOL. 71 1996 NO. 1 A REMARK ON A PROBLEM OF KLEE BY N. J. K A L T O N (COLUMBIA, MISSOURI) AND N. T. P E C K (URBANA, ILLINOIS) This paper treats a property
More informationMeasure and Measurable Functions
3 Measure ad Measurable Fuctios 3.1 Measure o a Arbitrary σ-algebra Recall from Chapter 2 that the set M of all Lebesgue measurable sets has the followig properties: R M, E M implies E c M, E M for N implies
More informationDISTANCE BETWEEN UNCERTAIN RANDOM VARIABLES
MATHEMATICAL MODELLING OF ENGINEERING PROBLEMS Vol, No, 4, pp5- http://doiorg/88/ep4 DISTANCE BETWEEN UNCERTAIN RANDOM VARIABLES Yogchao Hou* ad Weicai Peg Departet of Matheatical Scieces, Chaohu Uiversity,
More informationIf a subset E of R contains no open interval, is it of zero measure? For instance, is the set of irrationals in [0, 1] is of measure zero?
2 Lebesgue Measure I Chapter 1 we defied the cocept of a set of measure zero, ad we have observed that every coutable set is of measure zero. Here are some atural questios: If a subset E of R cotais a
More informationSZEGO S THEOREM STARTING FROM JENSEN S THEOREM
UPB Sci Bull, Series A, Vol 7, No 3, 8 ISSN 3-77 SZEGO S THEOREM STARTING FROM JENSEN S THEOREM Cǎli Alexe MUREŞAN Mai îtâi vo itroduce Teorea lui Jese şi uele coseciţe ale sale petru deteriarea uǎrului
More informationGENERALIZED LEGENDRE POLYNOMIALS AND RELATED SUPERCONGRUENCES
J. Nuber Theory 0, o., 9-9. GENERALIZED LEGENDRE POLYNOMIALS AND RELATED SUPERCONGRUENCES Zhi-Hog Su School of Matheatical Scieces, Huaiyi Noral Uiversity, Huaia, Jiagsu 00, PR Chia Eail: zhihogsu@yahoo.co
More informationFuzzy n-normed Space and Fuzzy n-inner Product Space
Global Joural o Pure ad Applied Matheatics. ISSN 0973-768 Volue 3, Nuber 9 (07), pp. 4795-48 Research Idia Publicatios http://www.ripublicatio.co Fuzzy -Nored Space ad Fuzzy -Ier Product Space Mashadi
More informationJournal of Mathematical Analysis and Applications 250, doi: jmaa , available online at http:
Joural of Mathematical Aalysis ad Applicatios 5, 886 doi:6jmaa766, available olie at http:wwwidealibrarycom o Fuctioal Equalities ad Some Mea Values Shoshaa Abramovich Departmet of Mathematics, Uiersity
More informationThe log-behavior of n p(n) and n p(n)/n
Ramauja J. 44 017, 81-99 The log-behavior of p ad p/ William Y.C. Che 1 ad Ke Y. Zheg 1 Ceter for Applied Mathematics Tiaji Uiversity Tiaji 0007, P. R. Chia Ceter for Combiatorics, LPMC Nakai Uivercity
More information--- L(qj)I(Pi) G(Pi)I(qj) Inm(P.Q) Gn(P)Im(Q + In(P)Lm(Q) P e F Q e F. Gk,L k. I(Piq j) WEIGHTED ADDITIVE INFORMATION MEASURES WOLFGANG SANDER
Iterat. J. Math. & Math Sci. VOL. 13 NO. 3 (1990) 417-424 417 WEIGHTED ADDITIVE INFORMATION MEASURES WOLFGANG SANDER Istitute for Aalysis Uiversity of Brauschweig Pockelsstr. 14, D 3300 Brauschweig, Geray
More information2.1 Elementary sets and (pre)content.
10 2. PEANO-JORDAN CONTENT The soewhat dissatisfactory situatio with the Riea itegral led Cator, Peao ad particularly Jorda develop a theory of (what we ow call) the Peao-Jorda cotet (a.k.a. just as Jorda
More information2 Banach spaces and Hilbert spaces
2 Baach spaces ad Hilbert spaces Tryig to do aalysis i the ratioal umbers is difficult for example cosider the set {x Q : x 2 2}. This set is o-empty ad bouded above but does ot have a least upper boud
More informationApproximation by Superpositions of a Sigmoidal Function
Zeitschrift für Aalysis ud ihre Aweduge Joural for Aalysis ad its Applicatios Volume 22 (2003, No. 2, 463 470 Approximatio by Superpositios of a Sigmoidal Fuctio G. Lewicki ad G. Mario Abstract. We geeralize
More informationCharacterizations Of (p, α)-convex Sequences
Applied Mathematics E-Notes, 172017, 77-84 c ISSN 1607-2510 Available free at mirror sites of http://www.math.thu.edu.tw/ ame/ Characterizatios Of p, α-covex Sequeces Xhevat Zahir Krasiqi Received 2 July
More informationON THE FUZZY METRIC SPACES
The Joural of Mathematics ad Computer Sciece Available olie at http://www.tjmcs.com The Joural of Mathematics ad Computer Sciece Vol.2 No.3 2) 475-482 ON THE FUZZY METRIC SPACES Received: July 2, Revised:
More informationRiesz-Fischer Sequences and Lower Frame Bounds
Zeitschrift für Aalysis ud ihre Aweduge Joural for Aalysis ad its Applicatios Volume 1 (00), No., 305 314 Riesz-Fischer Sequeces ad Lower Frame Bouds P. Casazza, O. Christese, S. Li ad A. Lider Abstract.
More informationMath 451: Euclidean and Non-Euclidean Geometry MWF 3pm, Gasson 204 Homework 3 Solutions
Math 451: Euclidea ad No-Euclidea Geometry MWF 3pm, Gasso 204 Homework 3 Solutios Exercises from 1.4 ad 1.5 of the otes: 4.3, 4.10, 4.12, 4.14, 4.15, 5.3, 5.4, 5.5 Exercise 4.3. Explai why Hp, q) = {x
More informationarxiv: v1 [math.nt] 26 Feb 2014
FROBENIUS NUMBERS OF PYTHAGOREAN TRIPLES BYUNG KEON GIL, JI-WOO HAN, TAE HYUN KIM, RYUN HAN KOO, BON WOO LEE, JAEHOON LEE, KYEONG SIK NAM, HYEON WOO PARK, AND POO-SUNG PARK arxiv:1402.6440v1 [ath.nt] 26
More informationLecture 19. sup y 1,..., yn B d n
STAT 06A: Polyomials of adom Variables Lecture date: Nov Lecture 19 Grothedieck s Iequality Scribe: Be Hough The scribes are based o a guest lecture by ya O Doell. I this lecture we prove Grothedieck s
More informationdistinct distinct n k n k n! n n k k n 1 if k n, identical identical p j (k) p 0 if k > n n (k)
THE TWELVEFOLD WAY FOLLOWING GIAN-CARLO ROTA How ay ways ca we distribute objects to recipiets? Equivaletly, we wat to euerate equivalece classes of fuctios f : X Y where X = ad Y = The fuctios are subject
More informationJORGE LUIS AROCHA AND BERNARDO LLANO. Average atchig polyoial Cosider a siple graph G =(V E): Let M E a atchig of the graph G: If M is a atchig, the a
MEAN VALUE FOR THE MATCHING AND DOMINATING POLYNOMIAL JORGE LUIS AROCHA AND BERNARDO LLANO Abstract. The ea value of the atchig polyoial is coputed i the faily of all labeled graphs with vertices. We dee
More informationAn Algorithmist s Toolkit October 20, Lecture 11
18.409 A Algorithist s Toolkit October 20, 2009 Lecture 11 Lecturer: Joatha Keler Scribe: Chaithaya Badi 1 Outlie Today we ll itroduce ad discuss Polar of a covex body. Correspodece betwee or fuctios ad
More informationA NOTE ON LEBESGUE SPACES
Volume 6, 1981 Pages 363 369 http://topology.aubur.edu/tp/ A NOTE ON LEBESGUE SPACES by Sam B. Nadler, Jr. ad Thelma West Topology Proceedigs Web: http://topology.aubur.edu/tp/ Mail: Topology Proceedigs
More informationMATH 324 Summer 2006 Elementary Number Theory Solutions to Assignment 2 Due: Thursday July 27, 2006
MATH 34 Summer 006 Elemetary Number Theory Solutios to Assigmet Due: Thursday July 7, 006 Departmet of Mathematical ad Statistical Scieces Uiversity of Alberta Questio [p 74 #6] Show that o iteger of the
More informationA Generalization of Ince s Equation
Joural of Applied Matheatics ad Physics 7-8 Published Olie Deceber i SciRes. http://www.scirp.org/oural/ap http://dx.doi.org/.36/ap..337 A Geeralizatio of Ice s Equatio Ridha Moussa Uiversity of Wiscosi
More information1. By using truth tables prove that, for all statements P and Q, the statement
Author: Satiago Salazar Problems I: Mathematical Statemets ad Proofs. By usig truth tables prove that, for all statemets P ad Q, the statemet P Q ad its cotrapositive ot Q (ot P) are equivalet. I example.2.3
More informationOn Functions -Starlike with Respect to Symmetric Conjugate Points
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 201, 2534 1996 ARTICLE NO. 0238 O Fuctios -Starlike with Respect to Symmetric Cojugate Poits Mig-Po Che Istitute of Mathematics, Academia Siica, Nakag,
More informationSeveral properties of new ellipsoids
Appl. Math. Mech. -Egl. Ed. 008 9(7):967 973 DOI 10.1007/s10483-008-0716-y c Shaghai Uiversity ad Spriger-Verlag 008 Applied Mathematics ad Mechaics (Eglish Editio) Several properties of ew ellipsoids
More informationProduct measures, Tonelli s and Fubini s theorems For use in MAT3400/4400, autumn 2014 Nadia S. Larsen. Version of 13 October 2014.
Product measures, Toelli s ad Fubii s theorems For use i MAT3400/4400, autum 2014 Nadia S. Larse Versio of 13 October 2014. 1. Costructio of the product measure The purpose of these otes is to preset the
More informationALMOST CONVERGENCE AND SOME MATRIX TRANSFORMATIONS
It. J. Cotep. Math. Sci., Vol. 1, 2006, o. 1, 39-43 ALMOST CONVERGENCE AND SOME MATRIX TRANSFORMATIONS Qaaruddi ad S. A. Mohiuddie Departet of Matheatics, Aligarh Musli Uiversity Aligarh-202002, Idia sdqaar@rediffail.co,
More informationGROUPOID CARDINALITY AND EGYPTIAN FRACTIONS
GROUPOID CARDINALITY AND EGYPTIAN FRACTIONS JULIA E. BERGNER AND CHRISTOPHER D. WALKER Abstract. I this paper we show that two very ew questios about the cardiality of groupoids reduce to very old questios
More informationIntegrals of Functions of Several Variables
Itegrals of Fuctios of Several Variables We ofte resort to itegratios i order to deterie the exact value I of soe quatity which we are uable to evaluate by perforig a fiite uber of additio or ultiplicatio
More informationSeunghee Ye Ma 8: Week 5 Oct 28
Week 5 Summary I Sectio, we go over the Mea Value Theorem ad its applicatios. I Sectio 2, we will recap what we have covered so far this term. Topics Page Mea Value Theorem. Applicatios of the Mea Value
More informationInternational Journal of Mathematical Archive-3(4), 2012, Page: Available online through ISSN
Iteratioal Joural of Mathematical Archive-3(4,, Page: 544-553 Available olie through www.ima.ifo ISSN 9 546 INEQUALITIES CONCERNING THE B-OPERATORS N. A. Rather, S. H. Ahager ad M. A. Shah* P. G. Departmet
More informationOn the transcendence of infinite sums of values of rational functions
O the trascedece of ifiite sus of values of ratioal fuctios N. Saradha ad R. Tijdea Abstract P () = We ivestigate coverget sus T = Q() ad U = P (X), Q(X) Q[X], ad Q(X) has oly siple ratioal roots. = (
More informationECE 901 Lecture 4: Estimation of Lipschitz smooth functions
ECE 9 Lecture 4: Estiatio of Lipschitz sooth fuctios R. Nowak 5/7/29 Cosider the followig settig. Let Y f (X) + W, where X is a rado variable (r.v.) o X [, ], W is a r.v. o Y R, idepedet of X ad satisfyig
More informationRational Bounds for the Logarithm Function with Applications
Ratioal Bouds for the Logarithm Fuctio with Applicatios Robert Bosch Abstract We fid ratioal bouds for the logarithm fuctio ad we show applicatios to problem-solvig. Itroductio Let a = + solvig the problem
More informationTR/46 OCTOBER THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION A. TALBOT
TR/46 OCTOBER 974 THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION by A. TALBOT .. Itroductio. A problem i approximatio theory o which I have recetly worked [] required for its solutio a proof that the
More informationIt is often useful to approximate complicated functions using simpler ones. We consider the task of approximating a function by a polynomial.
Taylor Polyomials ad Taylor Series It is ofte useful to approximate complicated fuctios usig simpler oes We cosider the task of approximatig a fuctio by a polyomial If f is at least -times differetiable
More informationOrthogonal Functions
Royal Holloway Uiversity of odo Departet of Physics Orthogoal Fuctios Motivatio Aalogy with vectors You are probably failiar with the cocept of orthogoality fro vectors; two vectors are orthogoal whe they
More informationANSWERS TO MIDTERM EXAM # 2
MATH 03, FALL 003 ANSWERS TO MIDTERM EXAM # PENN STATE UNIVERSITY Problem 1 (18 pts). State ad prove the Itermediate Value Theorem. Solutio See class otes or Theorem 5.6.1 from our textbook. Problem (18
More informationA remark on p-summing norms of operators
A remark o p-summig orms of operators Artem Zvavitch Abstract. I this paper we improve a result of W. B. Johso ad G. Schechtma by provig that the p-summig orm of ay operator with -dimesioal domai ca be
More informationAN EFFICIENT ESTIMATION METHOD FOR THE PARETO DISTRIBUTION
Joural of Statistics: Advaces i Theory ad Applicatios Volue 3, Nuber, 00, Pages 6-78 AN EFFICIENT ESTIMATION METHOD FOR THE PARETO DISTRIBUTION Departet of Matheatics Brock Uiversity St. Catharies, Otario
More informationNEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE
UPB Sci Bull, Series A, Vol 79, Iss, 207 ISSN 22-7027 NEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE Gabriel Bercu We itroduce two ew sequeces of Euler-Mascheroi type which have fast covergece
More informationMDIV. Multiple divisor functions
MDIV. Multiple divisor fuctios The fuctios τ k For k, defie τ k ( to be the umber of (ordered factorisatios of ito k factors, i other words, the umber of ordered k-tuples (j, j 2,..., j k with j j 2...
More informationOn Orlicz N-frames. 1 Introduction. Renu Chugh 1,, Shashank Goel 2
Joural of Advaced Research i Pure Mathematics Olie ISSN: 1943-2380 Vol. 3, Issue. 1, 2010, pp. 104-110 doi: 10.5373/jarpm.473.061810 O Orlicz N-frames Reu Chugh 1,, Shashak Goel 2 1 Departmet of Mathematics,
More informationITERATIONS AND FIXED POINTS FOR THE BERNSTEIN MAX-PRODUCT OPERATOR
Fixed Poit Theory, 14(2013), No. 1, 39-52 http://www.ath.ubbcluj.ro/ odeacj/sfptcj.htl ITERATIONS AND FIXED POINTS FOR THE BERNSTEIN MAX-PRODUCT OPERATOR MIRCEA BALAJ, LUCIAN COROIANU, SORIN G. GAL AND
More informationA NOTE ON INVARIANT SETS OF ITERATED FUNCTION SYSTEMS
Acta Math. Hugar., 2007 DOI: 10.1007/s10474-007-7013-6 A NOTE ON INVARIANT SETS OF ITERATED FUNCTION SYSTEMS L. L. STACHÓ ad L. I. SZABÓ Bolyai Istitute, Uiversity of Szeged, Aradi vértaúk tere 1, H-6720
More informationInt. Journal of Math. Analysis, Vol. 6, 2012, no. 31, S. Panayappan
It Joural of Math Aalysis, Vol 6, 0, o 3, 53 58 O Power Class ( Operators S Paayappa Departet of Matheatics Goveret Arts College, Coibatore 6408 ailadu, Idia paayappa@gailco N Sivaai Departet of Matheatics
More informationREGULARIZATION OF CERTAIN DIVERGENT SERIES OF POLYNOMIALS
REGULARIZATION OF CERTAIN DIVERGENT SERIES OF POLYNOMIALS LIVIU I. NICOLAESCU ABSTRACT. We ivestigate the geeralized covergece ad sums of series of the form P at P (x, where P R[x], a R,, ad T : R[x] R[x]
More informationCSE 1400 Applied Discrete Mathematics Number Theory and Proofs
CSE 1400 Applied Discrete Mathematics Number Theory ad Proofs Departmet of Computer Scieces College of Egieerig Florida Tech Sprig 01 Problems for Number Theory Backgroud Number theory is the brach of
More informationAN EXTENSION OF SIMONS INEQUALITY AND APPLICATIONS. Robert DEVILLE and Catherine FINET
2001 vol. XIV, um. 1, 95-104 ISSN 1139-1138 AN EXTENSION OF SIMONS INEQUALITY AND APPLICATIONS Robert DEVILLE ad Catherie FINET Abstract This article is devoted to a extesio of Simos iequality. As a cosequece,
More informationCOMP 2804 Solutions Assignment 1
COMP 2804 Solutios Assiget 1 Questio 1: O the first page of your assiget, write your ae ad studet uber Solutio: Nae: Jaes Bod Studet uber: 007 Questio 2: I Tic-Tac-Toe, we are give a 3 3 grid, cosistig
More informationStream Ciphers (contd.) Debdeep Mukhopadhyay
Strea Ciphers (cotd.) Debdeep Mukhopadhyay Assistat Professor Departet of Coputer Sciece ad Egieerig Idia Istitute of Techology Kharagpur IDIA -7232 Objectives iear Coplexity Berlekap Massey Algorith ow
More information6.4 Binomial Coefficients
64 Bioial Coefficiets Pascal s Forula Pascal s forula, aed after the seveteeth-cetury Frech atheaticia ad philosopher Blaise Pascal, is oe of the ost faous ad useful i cobiatorics (which is the foral ter
More informationMAT1026 Calculus II Basic Convergence Tests for Series
MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real
More informationSequences and Series of Functions
Chapter 6 Sequeces ad Series of Fuctios 6.1. Covergece of a Sequece of Fuctios Poitwise Covergece. Defiitio 6.1. Let, for each N, fuctio f : A R be defied. If, for each x A, the sequece (f (x)) coverges
More informationOscillation and Property B for Third Order Difference Equations with Advanced Arguments
Iter atioal Joural of Pure ad Applied Mathematics Volume 3 No. 0 207, 352 360 ISSN: 3-8080 (prited versio); ISSN: 34-3395 (o-lie versio) url: http://www.ijpam.eu ijpam.eu Oscillatio ad Property B for Third
More informationFACTORS OF SUMS OF POWERS OF BINOMIAL COEFFICIENTS. Dedicated to the memory of Paul Erdős. 1. Introduction. n k. f n,a =
FACTORS OF SUMS OF POWERS OF BINOMIAL COEFFICIENTS NEIL J. CALKIN Abstract. We prove divisibility properties for sus of powers of bioial coefficiets ad of -bioial coefficiets. Dedicated to the eory of
More informationMetric Dimension of Some Graphs under Join Operation
Global Joural of Pure ad Applied Matheatics ISSN 0973-768 Volue 3, Nuber 7 (07), pp 333-3348 Research Idia Publicatios http://wwwripublicatioco Metric Diesio of Soe Graphs uder Joi Operatio B S Rawat ad
More informationOn the Fibonacci-like Sequences of Higher Order
Article Iteratioal Joural of oder atheatical Scieces, 05, 3(): 5-59 Iteratioal Joural of oder atheatical Scieces Joural hoepage: wwwoderscietificpressco/jourals/ijsaspx O the Fiboacci-like Sequeces of
More informationSolutions to Tutorial 3 (Week 4)
The Uiversity of Sydey School of Mathematics ad Statistics Solutios to Tutorial Week 4 MATH2962: Real ad Complex Aalysis Advaced Semester 1, 2017 Web Page: http://www.maths.usyd.edu.au/u/ug/im/math2962/
More informationA GENERALIZED BERNSTEIN APPROXIMATION THEOREM
Ø Ñ Å Ø Ñ Ø Ð ÈÙ Ð Ø ÓÒ DOI: 10.2478/v10127-011-0029-x Tatra Mt. Math. Publ. 49 2011, 99 109 A GENERALIZED BERNSTEIN APPROXIMATION THEOREM Miloslav Duchoň ABSTRACT. The preset paper is cocered with soe
More informationIntegrable Functions. { f n } is called a determining sequence for f. If f is integrable with respect to, then f d does exist as a finite real number
MATH 532 Itegrable Fuctios Dr. Neal, WKU We ow shall defie what it meas for a measurable fuctio to be itegrable, show that all itegral properties of simple fuctios still hold, ad the give some coditios
More informationAn Extremal Property of the Regular Simplex
Covex Geometric Aalysis MSRI Publicatios Volume 34, 1998 A Extremal Property of the Regular Simplex MICHAEL SCHMUCKENSCHLÄGER Abstract. If C is a covex body i R such that the ellipsoid of miimal volume
More informationInternational Journal of Mathematical Archive-4(9), 2013, 1-5 Available online through ISSN
Iteratioal Joural o Matheatical Archive-4(9), 03, -5 Available olie through www.ija.io ISSN 9 5046 THE CUBIC RATE OF CONVERGENCE OF GENERALIZED EXTRAPOLATED NEWTON RAPHSON METHOD FOR SOLVING NONLINEAR
More informationDomination Number of Square of Cartesian Products of Cycles
Ope Joural of Discrete Matheatics, 01,, 88-94 Published Olie October 01 i SciRes http://wwwscirporg/joural/ojd http://dxdoiorg/10436/ojd014008 Doiatio Nuber of Square of artesia Products of ycles Morteza
More informationMath 220A Fall 2007 Homework #2. Will Garner A
Math 0A Fall 007 Homewor # Will Garer Pg 3 #: Show that {cis : a o-egative iteger} is dese i T = {z œ : z = }. For which values of q is {cis(q): a o-egative iteger} dese i T? To show that {cis : a o-egative
More informationGeneralization of Contraction Principle on G-Metric Spaces
Global Joural of Pure ad Applied Mathematics. ISSN 0973-1768 Volume 14, Number 9 2018), pp. 1159-1165 Research Idia Publicatios http://www.ripublicatio.com Geeralizatio of Cotractio Priciple o G-Metric
More informationarxiv: v1 [math.st] 12 Dec 2018
DIVERGENCE MEASURES ESTIMATION AND ITS ASYMPTOTIC NORMALITY THEORY : DISCRETE CASE arxiv:181.04795v1 [ath.st] 1 Dec 018 Abstract. 1) BA AMADOU DIADIÉ AND 1,,4) LO GANE SAMB 1. Itroductio 1.1. Motivatios.
More informationSupplementary Material
Suppleetary Material Wezhuo Ya a0096049@us.edu.s Departet of Mechaical Eieeri, Natioal Uiversity of Siapore, Siapore 117576 Hua Xu pexuh@us.edu.s Departet of Mechaical Eieeri, Natioal Uiversity of Siapore,
More informationX. Perturbation Theory
X. Perturbatio Theory I perturbatio theory, oe deals with a ailtoia that is coposed Ĥ that is typically exactly solvable of two pieces: a referece part ad a perturbatio ( Ĥ ) that is assued to be sall.
More informationOn Order of a Function of Several Complex Variables Analytic in the Unit Polydisc
ISSN 746-7659, Eglad, UK Joural of Iforatio ad Coutig Sciece Vol 6, No 3, 0, 95-06 O Order of a Fuctio of Several Colex Variables Aalytic i the Uit Polydisc Rata Kuar Dutta + Deartet of Matheatics, Siliguri
More information