PARTITIONS WITH PARTS IN A FINITE SET

Size: px
Start display at page:

Download "PARTITIONS WITH PARTS IN A FINITE SET"

Transcription

1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volue 28, Nuber 5, Pages S ( Artcle electroncally publshe on February 7, 2000 PARTITIONS WITH PARTS IN A FINITE SET MELVYN B. NATHANSON (Councate by Dav E. Rohrlch Abstract. Let A be a nonepty fnte set of relatvely pre postve ntegers, an let p A (n enote the nuber of parttons of n wth parts n A. An eleentary arthetc arguent s use to prove the asyptotc forula n k ( p A (n a A a (k! + O n. Let A be a nonepty set of postve ntegers. A partton of a postve nteger n wth parts n A s a representaton of n as a su of not necessarly stnct eleents of A. Two parttons are consere the sae f they ffer only n the orer of ther suans. The partton functon of the set A, enote p A (n, counts the nuber of parttons of n wth parts n A. If A s a fnte set of postve ntegers wth no coon factor greater than, then every suffcently large nteger can be wrtten as a su of eleents of A (see Nathanson [3] an Han, Krfel, an Nathanson [2], an so p A (n for all n n 0. In the specal case that A s the set of the frst k ntegers, t s known that p A (n nk k!(k!. Erős an Lehner [] prove that ths asyptotc forula hols unforly for k o(n /3. If A s an arbtrary fnte set of relatvely pre postve ntegers, then n k ( p A (n a A a (k!. The usual proof of ths result (Netto [4], Pólya Szegö [5, Proble 27] s base on the partal fracton ecoposton of the generatng functon for p A (n. The purpose of ths note s to gve a sple, purely arthetc proof of (. We efne p A (0. Theore. Let A {a,..., } be a set of k relatvely pre postve ntegers, that s, gc(a (a,...,. Receve by the etors June 5, Matheatcs Subect Classfcaton. Prary P8; Seconary 05A7, B34. Key wors an phrases. Partton functons, asyptotcs of parttons, atve nuber theory. Ths work was supporte n part by grants fro the PSC CUNY Research Awar Progra an the NSA Matheatcal Scences Progra. 269 c 2000 Aercan Matheatcal Socety Lcense or copyrght restrctons ay apply to restrbuton; see

2 270 MELVYN B. NATHANSON Let p A (n enote the nuber of parttons of n nto parts belongng to A. n k p A (n a A a (k! + O ( n. Proof. Let k A. The proof s by nucton on k. Ifk,thenA {} an p A (n, snce every postve nteger has a unque partton nto a su of s. Let k 2, an assue that the theore hols for k. Let For,...,k, we set (a,...,. (,. a a. A {a,...,a k } s a set of k relatvely pre postve ntegers, that s, gc(a. Snce the nucton assupton hols for A,wehave n p A (n a (! + O ( n k 3 for all nonnegatve ntegers n. Let n (. Snce (,, there exsts a unque nteger u such that 0 u an n u (o. s a nonnegatve nteger, an n u If v s any nonnegatve nteger such that O(n. n v (o, then v u (o Lcense or copyrght restrctons ay apply to restrbuton; see

3 PARTITIONS WITH PARTS IN A FINITE SET 27 an so v u (o, that s, v u + l for soe nonnegatve nteger l. If n v n (u + l 0, then 0 l [ n u ] [ ] r. We note that r O(n. Let π be a partton of n nto parts belongng to A. If π contans exactly v parts equal to,thenn v 0ann v 0(o, snce n v s a su of eleents n {a,..., }, an each of the eleents n ths set s vsble by. Therefore, v u + l, where 0 l r. Consequently, we can ve the parttons of n wth parts n A nto r + classes, where, for each l 0,,...,r, a partton belongs to class l f t contans exactly u + l parts equal to. The nuber of parttons of n wth exactly u+l parts equal to s exactly the nuber of parttons of n (u + l nto parts belongng to the set {a,..., },or, equvalently, the nuber of parttons of n (u + l nto parts belongng to A,whchsexactly n (u + lak p A p A ( l. Therefore, p A (n r p A ( l l0 r ( ( lak + O( k 3 a (! l0 r k ( l a + O(n. (! l0 To evaluate the nner su, we note that r l l0 r+ ( + + O(r an ( k + 0 k ( ( k. Lcense or copyrght restrctons ay apply to restrbuton; see

4 272 MELVYN B. NATHANSON r l0 ( l (! (! (! (! k k k (! r l0 0 ( ( l r l0 ( r ( a k + ( + + O(r ( ( ( l ( (!( + + O( ( (!!( + + O( ( (k ( +!( +! + O( k ( k + O( (k! + 0 k (k! + O(. + a + k ( + + O( Therefore, p A (n r k ( l a + O(n (! l0 ( k k a (k! + O(n + O(n ( k (n a (k! ua k k + O(n ( k (n k + O(n a (k! n k k a (k! + O(n. Ths copletes the proof. Lcense or copyrght restrctons ay apply to restrbuton; see

5 PARTITIONS WITH PARTS IN A FINITE SET 273 References [] P. Erős an J. Lehner. The strbuton of the nuber of suans n the parttons of a postve nteger. Duke Math. J., 8: , 94. MR 3:69a [2] S. Han, C. Krfel, an M. B. Nathanson. Lnear fors n fnte sets of ntegers. Raanuan J., 2:27 28, 998. MR 99h:0 [3] M. B. Nathanson. Sus of fnte sets of ntegers. Aer. Math. Monthly, 79:00 02, 972. MR 46:3440 [4] E. Netto. Lehrbuch er Cobnatork. Teubner, Lepzg, 927. [5] G. Pólya an G. Szegö. Aufgaben un Lehrsätze aus er Analyss. Sprnger Verlag, Berln, 925. Englsh translaton: Probles an Theores n Analyss, Sprnger Verlag, New York, 972. MR 49:8782 Departent of Matheatcs, Lehan College (CUNY, Bronx, New York 0468 E-al aress: nathansn@alpha.lehan.cuny.eu Current aress: School of Matheatcs, Insttute for Avance Stuy, Prnceton, New Jersey E-al aress: nathansn@as.eu Lcense or copyrght restrctons ay apply to restrbuton; see

ON THE NUMBER OF PRIMITIVE PYTHAGOREAN QUINTUPLES

ON THE NUMBER OF PRIMITIVE PYTHAGOREAN QUINTUPLES Journal of Algebra, Nuber Theory: Advances and Applcatons Volue 3, Nuber, 05, Pages 3-8 ON THE NUMBER OF PRIMITIVE PYTHAGOREAN QUINTUPLES Feldstrasse 45 CH-8004, Zürch Swtzerland e-al: whurlann@bluewn.ch

More information

On the number of regions in an m-dimensional space cut by n hyperplanes

On the number of regions in an m-dimensional space cut by n hyperplanes 6 On the nuber of regons n an -densonal space cut by n hyperplanes Chungwu Ho and Seth Zeran Abstract In ths note we provde a unfor approach for the nuber of bounded regons cut by n hyperplanes n general

More information

= m 1. sin π( ai z ) )

= m 1. sin π( ai z ) ) EXACT COVERING SYSTEMS AND THE GAUSS-LEGENDRE MULTIPLICATION FORMULA FOR THE GAMMA FUNCTION John Beeee Unversty of Alaska Anchorage July 0 199 The Gauss-Legendre ultplcaton forula for the gaa functon s

More information

A Radon-Nikodym Theorem for Completely Positive Maps

A Radon-Nikodym Theorem for Completely Positive Maps A Radon-Nody Theore for Copletely Postve Maps V P Belavn School of Matheatcal Scences, Unversty of Nottngha, Nottngha NG7 RD E-al: vpb@aths.nott.ac.u and P Staszews Insttute of Physcs, Ncholas Coperncus

More information

Another converse of Jensen s inequality

Another converse of Jensen s inequality Another converse of Jensen s nequalty Slavko Smc Abstract. We gve the best possble global bounds for a form of dscrete Jensen s nequalty. By some examples ts frutfulness s shown. 1. Introducton Throughout

More information

Slobodan Lakić. Communicated by R. Van Keer

Slobodan Lakić. Communicated by R. Van Keer Serdca Math. J. 21 (1995), 335-344 AN ITERATIVE METHOD FOR THE MATRIX PRINCIPAL n-th ROOT Slobodan Lakć Councated by R. Van Keer In ths paper we gve an teratve ethod to copute the prncpal n-th root and

More information

Finite Fields and Their Applications

Finite Fields and Their Applications Fnte Felds and Ther Applcatons 5 009 796 807 Contents lsts avalable at ScenceDrect Fnte Felds and Ther Applcatons www.elsever.co/locate/ffa Typcal prtve polynoals over nteger resdue rngs Tan Tan a, Wen-Feng

More information

GENERALIZED ARITHMETIC PROGRESSIONS

GENERALIZED ARITHMETIC PROGRESSIONS GENERALIZED ARITHMETIC ROGRESSIONS AND k Chun-Xuan an Bejn 00854,. O. Box 94,. R. Chna Lukx@publc.bta.net.cn Abstract We defne the eneralzed arthetc proressons and k, where,, k,,. If the arthetc functon

More information

On an identity for the cycle indices of rooted tree automorphism groups

On an identity for the cycle indices of rooted tree automorphism groups On an dentty for the cycle ndces of rooted tree autoorphs groups Stephan G Wagner Insttut für Analyss und Coputatonal Nuber Theory Technsche Unverstät Graz Steyrergasse 30, 800 Graz, Austra wagner@fnanzathtugrazat

More information

A Proof of a Conjecture for the Number of Ramified Coverings of the Sphere by the Torus

A Proof of a Conjecture for the Number of Ramified Coverings of the Sphere by the Torus Journal of Cobnatoral Theory, Seres A 88, 4658 (999) Artcle ID jcta99999, avalable onlne at httpwwwdealbraryco on A Proof of a Conjecture for the Nuber of Rafed Coverngs of the Sphere by the Torus I P

More information

LECTURE 8-9: THE BAKER-CAMPBELL-HAUSDORFF FORMULA

LECTURE 8-9: THE BAKER-CAMPBELL-HAUSDORFF FORMULA LECTURE 8-9: THE BAKER-CAMPBELL-HAUSDORFF FORMULA As we have seen, 1. Taylor s expanson on Le group, Y ] a(y ). So f G s an abelan group, then c(g) : G G s the entty ap for all g G. As a consequence, a()

More information

On Pfaff s solution of the Pfaff problem

On Pfaff s solution of the Pfaff problem Zur Pfaff scen Lösung des Pfaff scen Probles Mat. Ann. 7 (880) 53-530. On Pfaff s soluton of te Pfaff proble By A. MAYER n Lepzg Translated by D. H. Delpenc Te way tat Pfaff adopted for te ntegraton of

More information

This document has been downloaded from Tampub The Institutional Repository of University of Tampere. Publisher's version

This document has been downloaded from Tampub The Institutional Repository of University of Tampere. Publisher's version Ths ocument has been ownloae from Tampub The Insttutonal Repostory of Unversty of Tampere Publsher's verson Authors: Haukkanen Pentt Name of artcle: Some characterzatons of specally multplcatve functons

More information

On a one-parameter family of Riordan arrays and the weight distribution of MDS codes

On a one-parameter family of Riordan arrays and the weight distribution of MDS codes On a one-parameter famly of Roran arrays an the weght strbuton of MDS coes Paul Barry School of Scence Waterfor Insttute of Technology Irelan pbarry@wte Patrck Ftzpatrck Department of Mathematcs Unversty

More information

Applied Mathematics Letters

Applied Mathematics Letters Appled Matheatcs Letters 2 (2) 46 5 Contents lsts avalable at ScenceDrect Appled Matheatcs Letters journal hoepage: wwwelseverco/locate/al Calculaton of coeffcents of a cardnal B-splne Gradr V Mlovanovć

More information

The Parity of the Number of Irreducible Factors for Some Pentanomials

The Parity of the Number of Irreducible Factors for Some Pentanomials The Party of the Nuber of Irreducble Factors for Soe Pentanoals Wolfra Koepf 1, Ryul K 1 Departent of Matheatcs Unversty of Kassel, Kassel, F. R. Gerany Faculty of Matheatcs and Mechancs K Il Sung Unversty,

More information

International Journal of Mathematical Archive-9(3), 2018, Available online through ISSN

International Journal of Mathematical Archive-9(3), 2018, Available online through   ISSN Internatonal Journal of Matheatcal Archve-9(3), 208, 20-24 Avalable onlne through www.ja.nfo ISSN 2229 5046 CONSTRUCTION OF BALANCED INCOMPLETE BLOCK DESIGNS T. SHEKAR GOUD, JAGAN MOHAN RAO M AND N.CH.

More information

Numbers Related to Bernoulli-Goss Numbers

Numbers Related to Bernoulli-Goss Numbers ursh Journl of Anlyss n Nuber heory, 4, Vol., No., -8 Avlble onlne t htt://ubs.sceub.co/tnt///4 Scence n Eucton Publshng OI:.69/tnt---4 Nubers Relte to Bernoull-Goss Nubers Mohe Oul ouh Benough * érteent

More information

On the Calderón-Zygmund lemma for Sobolev functions

On the Calderón-Zygmund lemma for Sobolev functions arxv:0810.5029v1 [ath.ca] 28 Oct 2008 On the Calderón-Zygund lea for Sobolev functons Pascal Auscher october 16, 2008 Abstract We correct an naccuracy n the proof of a result n [Aus1]. 2000 MSC: 42B20,

More information

On Finite Rank Perturbation of Diagonalizable Operators

On Finite Rank Perturbation of Diagonalizable Operators Functonal Analyss, Approxmaton and Computaton 6 (1) (2014), 49 53 Publshed by Faculty of Scences and Mathematcs, Unversty of Nš, Serba Avalable at: http://wwwpmfnacrs/faac On Fnte Rank Perturbaton of Dagonalzable

More information

Existence of Two Conjugate Classes of A 5 within S 6. by Use of Character Table of S 6

Existence of Two Conjugate Classes of A 5 within S 6. by Use of Character Table of S 6 Internatonal Mathematcal Forum, Vol. 8, 2013, no. 32, 1591-159 HIKARI Ltd, www.m-hkar.com http://dx.do.org/10.12988/mf.2013.3359 Exstence of Two Conjugate Classes of A 5 wthn S by Use of Character Table

More information

The Order Relation and Trace Inequalities for. Hermitian Operators

The Order Relation and Trace Inequalities for. Hermitian Operators Internatonal Mathematcal Forum, Vol 3, 08, no, 507-57 HIKARI Ltd, wwwm-hkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence

More information

Physics 3A: Linear Momentum. Physics 3A: Linear Momentum. Physics 3A: Linear Momentum. Physics 3A: Linear Momentum

Physics 3A: Linear Momentum. Physics 3A: Linear Momentum. Physics 3A: Linear Momentum. Physics 3A: Linear Momentum Recall that there was ore to oton than just spee A ore coplete escrpton of oton s the concept of lnear oentu: p v (8.) Beng a prouct of a scalar () an a vector (v), oentu s a vector: p v p y v y p z v

More information

An Optimal Bound for Sum of Square Roots of Special Type of Integers

An Optimal Bound for Sum of Square Roots of Special Type of Integers The Sxth Internatonal Syposu on Operatons Research and Its Applcatons ISORA 06 Xnang, Chna, August 8 12, 2006 Copyrght 2006 ORSC & APORC pp. 206 211 An Optal Bound for Su of Square Roots of Specal Type

More information

The internal structure of natural numbers and one method for the definition of large prime numbers

The internal structure of natural numbers and one method for the definition of large prime numbers The nternal structure of natural numbers and one method for the defnton of large prme numbers Emmanul Manousos APM Insttute for the Advancement of Physcs and Mathematcs 3 Poulou str. 53 Athens Greece Abstract

More information

Quantum Particle Motion in Physical Space

Quantum Particle Motion in Physical Space Adv. Studes Theor. Phys., Vol. 8, 014, no. 1, 7-34 HIKARI Ltd, www.-hkar.co http://dx.do.org/10.1988/astp.014.311136 Quantu Partcle Moton n Physcal Space A. Yu. Saarn Dept. of Physcs, Saara State Techncal

More information

By Samuel Schechter. (1) H = i-î -1 with 1 ^ i,j g n. [a, b,j

By Samuel Schechter. (1) H = i-î -1 with 1 ^ i,j g n. [a, b,j On tne Inverson of Certan Matrces By Samuel Schechter 1. Introducton. Let a, a2,, an ; 61, 62,, 6 be 2n dstnct, but otherwse arbtrary, complex numbers. For the matrx H, of order n, (1) H = -î -1 wth 1

More information

The Impact of the Earth s Movement through the Space on Measuring the Velocity of Light

The Impact of the Earth s Movement through the Space on Measuring the Velocity of Light Journal of Appled Matheatcs and Physcs, 6, 4, 68-78 Publshed Onlne June 6 n ScRes http://wwwscrporg/journal/jap http://dxdoorg/436/jap646 The Ipact of the Earth s Moeent through the Space on Measurng the

More information

Chapter 12 Lyes KADEM [Thermodynamics II] 2007

Chapter 12 Lyes KADEM [Thermodynamics II] 2007 Chapter 2 Lyes KDEM [Therodynacs II] 2007 Gas Mxtures In ths chapter we wll develop ethods for deternng therodynac propertes of a xture n order to apply the frst law to systes nvolvng xtures. Ths wll be

More information

Perron Vectors of an Irreducible Nonnegative Interval Matrix

Perron Vectors of an Irreducible Nonnegative Interval Matrix Perron Vectors of an Irreducble Nonnegatve Interval Matrx Jr Rohn August 4 2005 Abstract As s well known an rreducble nonnegatve matrx possesses a unquely determned Perron vector. As the man result of

More information

THE ADJACENCY-PELL-HURWITZ NUMBERS. Josh Hiller Department of Mathematics and Computer Science, Adelpi University, New York

THE ADJACENCY-PELL-HURWITZ NUMBERS. Josh Hiller Department of Mathematics and Computer Science, Adelpi University, New York #A8 INTEGERS 8 (8) THE ADJACENCY-PELL-HURWITZ NUMBERS Josh Hller Departent of Matheatcs and Coputer Scence Adelp Unversty New York johller@adelphedu Yeş Aküzü Faculty of Scence and Letters Kafkas Unversty

More information

FORMULAS FOR BINOMIAL SUMS INCLUDING POWERS OF FIBONACCI AND LUCAS NUMBERS

FORMULAS FOR BINOMIAL SUMS INCLUDING POWERS OF FIBONACCI AND LUCAS NUMBERS U.P.B. Sc. Bull., Seres A, Vol. 77, Iss. 4, 015 ISSN 13-707 FORMULAS FOR BINOMIAL SUMS INCLUDING POWERS OF FIBONACCI AND LUCAS NUMBERS Erah KILIÇ 1, Iler AKKUS, Neşe ÖMÜR, Yücel Türer ULUTAŞ3 Recently

More information

Denote the function derivatives f(x) in given points. x a b. Using relationships (1.2), polynomials (1.1) are written in the form

Denote the function derivatives f(x) in given points. x a b. Using relationships (1.2), polynomials (1.1) are written in the form SET OF METHODS FO SOUTION THE AUHY POBEM FO STIFF SYSTEMS OF ODINAY DIFFEENTIA EUATIONS AF atypov and YuV Nulchev Insttute of Theoretcal and Appled Mechancs SB AS 639 Novosbrs ussa Introducton A constructon

More information

A GENERALIZATION OF JUNG S THEOREM. M. Henk

A GENERALIZATION OF JUNG S THEOREM. M. Henk A GENERALIZATION OF JUNG S THEOREM M. Henk Abstract. The theorem of Jung establshes a relaton between crcumraus an ameter of a convex boy. The half of the ameter can be nterprete as the maxmum of crcumra

More information

Limit Cycle Bifurcations in a Class of Cubic System near a Nilpotent Center *

Limit Cycle Bifurcations in a Class of Cubic System near a Nilpotent Center * Appled Mateatcs 77-777 ttp://dxdoorg/6/a75 Publsed Onlne July (ttp://wwwscrporg/journal/a) Lt Cycle Bfurcatons n a Class of Cubc Syste near a Nlpotent Center * Jao Jang Departent of Mateatcs Sanga Marte

More information

Least Squares Fitting of Data

Least Squares Fitting of Data Least Squares Fttng of Data Davd Eberly Geoetrc Tools, LLC http://www.geoetrctools.co/ Copyrght c 1998-2014. All Rghts Reserved. Created: July 15, 1999 Last Modfed: February 9, 2008 Contents 1 Lnear Fttng

More information

Chapter One Mixture of Ideal Gases

Chapter One Mixture of Ideal Gases herodynacs II AA Chapter One Mxture of Ideal Gases. Coposton of a Gas Mxture: Mass and Mole Fractons o deterne the propertes of a xture, we need to now the coposton of the xture as well as the propertes

More information

Multiplicative Functions and Möbius Inversion Formula

Multiplicative Functions and Möbius Inversion Formula Multplcatve Functons and Möbus Inverson Forula Zvezdelna Stanova Bereley Math Crcle Drector Mlls College and UC Bereley 1. Multplcatve Functons. Overvew Defnton 1. A functon f : N C s sad to be arthetc.

More information

A new eigenvalue inclusion set for tensors with its applications

A new eigenvalue inclusion set for tensors with its applications COMPUTATIONAL SCIENCE RESEARCH ARTICLE A new egenvalue ncluson set for tensors wth ts applcatons Cal Sang 1 and Janxng Zhao 1 * Receved: 30 Deceber 2016 Accepted: 12 Aprl 2017 Frst Publshed: 20 Aprl 2017

More information

SL n (F ) Equals its Own Derived Group

SL n (F ) Equals its Own Derived Group Internatonal Journal of Algebra, Vol. 2, 2008, no. 12, 585-594 SL n (F ) Equals ts Own Derved Group Jorge Macel BMCC-The Cty Unversty of New York, CUNY 199 Chambers street, New York, NY 10007, USA macel@cms.nyu.edu

More information

General Permanence Conditions for Nonlinear Difference Equations of Higher Order

General Permanence Conditions for Nonlinear Difference Equations of Higher Order JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 213, 496 510 1997 ARTICLE NO. AY975553 General Peranence Condtons for Nonlnear Dfference Equatons of Hgher Order Hassan Sedaghat* Departent of Matheatcal

More information

FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP

FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP C O L L O Q U I U M M A T H E M A T I C U M VOL. 80 1999 NO. 1 FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP BY FLORIAN K A I N R A T H (GRAZ) Abstract. Let H be a Krull monod wth nfnte class

More information

Our focus will be on linear systems. A system is linear if it obeys the principle of superposition and homogenity, i.e.

Our focus will be on linear systems. A system is linear if it obeys the principle of superposition and homogenity, i.e. SSTEM MODELLIN In order to solve a control syste proble, the descrptons of the syste and ts coponents ust be put nto a for sutable for analyss and evaluaton. The followng ethods can be used to odel physcal

More information

A study on compactness in metric spaces and topological spaces

A study on compactness in metric spaces and topological spaces Pure and Appled Matheatcs Journal 24; 3(5): 5-2 Publshed onlne October 2, 24 (http://wwwscencepublshnggroupco/j/paj) do: 648/jpaj24353 ISSN: 2326-979 (Prnt); ISSN: 2326-982 (Onlne) A study on copactness

More information

1. Statement of the problem

1. Statement of the problem Volue 14, 010 15 ON THE ITERATIVE SOUTION OF A SYSTEM OF DISCRETE TIMOSHENKO EQUATIONS Peradze J. and Tsklaur Z. I. Javakhshvl Tbls State Uversty,, Uversty St., Tbls 0186, Georga Georgan Techcal Uversty,

More information

Preference and Demand Examples

Preference and Demand Examples Dvson of the Huantes and Socal Scences Preference and Deand Exaples KC Border October, 2002 Revsed Noveber 206 These notes show how to use the Lagrange Karush Kuhn Tucker ultpler theores to solve the proble

More information

Inner Product. Euclidean Space. Orthonormal Basis. Orthogonal

Inner Product. Euclidean Space. Orthonormal Basis. Orthogonal Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,

More information

AN ANALYSIS OF A FRACTAL KINETICS CURVE OF SAVAGEAU

AN ANALYSIS OF A FRACTAL KINETICS CURVE OF SAVAGEAU AN ANALYI OF A FRACTAL KINETIC CURE OF AAGEAU by John Maloney and Jack Hedel Departent of Matheatcs Unversty of Nebraska at Oaha Oaha, Nebraska 688 Eal addresses: aloney@unoaha.edu, jhedel@unoaha.edu Runnng

More information

General Results of Local Metric Dimensions of. Edge-Corona of Graphs

General Results of Local Metric Dimensions of. Edge-Corona of Graphs Internatonal Matheatcal Foru, Vol 11, 016, no 16, 793-799 HIKARI Ltd, www-hkarco htt://dxdoorg/101988/f0166710 General Results of Local Metrc Densons of Edge-Corona of Grahs Rnurwat Deartent of Matheatcs,

More information

What is LP? LP is an optimization technique that allocates limited resources among competing activities in the best possible manner.

What is LP? LP is an optimization technique that allocates limited resources among competing activities in the best possible manner. (C) 998 Gerald B Sheblé, all rghts reserved Lnear Prograng Introducton Contents I. What s LP? II. LP Theor III. The Splex Method IV. Refneents to the Splex Method What s LP? LP s an optzaton technque that

More information

arxiv:math.nt/ v1 16 Feb 2005

arxiv:math.nt/ v1 16 Feb 2005 A NOTE ON q-bernoulli NUMBERS AND POLYNOMIALS arv:math.nt/0502333 v1 16 Feb 2005 Taekyun Km Insttute of Scence Eucaton, Kongju Natonal Unversty, Kongju 314-701, S. Korea Abstract. By usng q-ntegraton,

More information

Appendix B. Criterion of Riemann-Stieltjes Integrability

Appendix B. Criterion of Riemann-Stieltjes Integrability Appendx B. Crteron of Remann-Steltes Integrablty Ths note s complementary to [R, Ch. 6] and [T, Sec. 3.5]. The man result of ths note s Theorem B.3, whch provdes the necessary and suffcent condtons for

More information

arxiv: v5 [math.nt] 6 Sep 2008

arxiv: v5 [math.nt] 6 Sep 2008 FUNCTIONAL EQUATION FO THETA SEIES JAE-HYUN YANG arxv:0707.376v5 [ath.nt] 6 Sep 008 Abstract. In ths short paper, we fnd the transforaton forula for the theta seres under the acton of the Jacob odular

More information

Stability analysis for class of switched nonlinear systems

Stability analysis for class of switched nonlinear systems Stablty analyss for class of swtched nonlnear systems The MIT Faculty has made ths artcle openly avalable. Please share how ths access benefts you. Your story matters. Ctaton As Publshed Publsher Shaker,

More information

1 Generating functions, continued

1 Generating functions, continued Generatng functons, contnued. Generatng functons and parttons We can make use of generatng functons to answer some questons a bt more restrctve than we ve done so far: Queston : Fnd a generatng functon

More information

04 - Treaps. Dr. Alexander Souza

04 - Treaps. Dr. Alexander Souza Algorths Theory 04 - Treaps Dr. Alexander Souza The dctonary proble Gven: Unverse (U,

More information

ON SEQUENCES OF NUMBERS IN GENERALIZED ARITHMETIC AND GEOMETRIC PROGRESSIONS

ON SEQUENCES OF NUMBERS IN GENERALIZED ARITHMETIC AND GEOMETRIC PROGRESSIONS Palestine Journal of Matheatics Vol 4) 05), 70 76 Palestine Polytechnic University-PPU 05 ON SEQUENCES OF NUMBERS IN GENERALIZED ARITHMETIC AND GEOMETRIC PROGRESSIONS Julius Fergy T Rabago Counicated by

More information

Algorithm for reduction of Element Calculus to Element Algebra

Algorithm for reduction of Element Calculus to Element Algebra Algorth for reducton of Eleent Calculus to Eleent Algebra Introducton M. Manukyan, V. Harutunyan The XML databases currently act defnte nterest aong researchers of databases for the followng reasons: 1.

More information

Maximizing the number of nonnegative subsets

Maximizing the number of nonnegative subsets Maxmzng the number of nonnegatve subsets Noga Alon Hao Huang December 1, 213 Abstract Gven a set of n real numbers, f the sum of elements of every subset of sze larger than k s negatve, what s the maxmum

More information

ϕ -MEANS OF SOME BANACH SUBSPACES ON A BANACH ALGEBRA

ϕ -MEANS OF SOME BANACH SUBSPACES ON A BANACH ALGEBRA THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Seres A, OF THE ROMANIAN ACADEMY Volue 3, Nuber 4/202, pp 302 309 ϕ -MEANS OF SOME BANACH SUBSPACES ON A BANACH ALGEBRA Al GHAFFARI, Saaneh JAVADI

More information

Christian Aebi Collège Calvin, Geneva, Switzerland

Christian Aebi Collège Calvin, Geneva, Switzerland #A7 INTEGERS 12 (2012) A PROPERTY OF TWIN PRIMES Chrstan Aeb Collège Calvn, Geneva, Swtzerland chrstan.aeb@edu.ge.ch Grant Carns Department of Mathematcs, La Trobe Unversty, Melbourne, Australa G.Carns@latrobe.edu.au

More information

Xiangwen Li. March 8th and March 13th, 2001

Xiangwen Li. March 8th and March 13th, 2001 CS49I Approxaton Algorths The Vertex-Cover Proble Lecture Notes Xangwen L March 8th and March 3th, 00 Absolute Approxaton Gven an optzaton proble P, an algorth A s an approxaton algorth for P f, for an

More information

Convexity preserving interpolation by splines of arbitrary degree

Convexity preserving interpolation by splines of arbitrary degree Computer Scence Journal of Moldova, vol.18, no.1(52), 2010 Convexty preservng nterpolaton by splnes of arbtrary degree Igor Verlan Abstract In the present paper an algorthm of C 2 nterpolaton of dscrete

More information

Discussion 11 Summary 11/20/2018

Discussion 11 Summary 11/20/2018 Dscusson 11 Summary 11/20/2018 1 Quz 8 1. Prove for any sets A, B that A = A B ff B A. Soluton: There are two drectons we need to prove: (a) A = A B B A, (b) B A A = A B. (a) Frst, we prove A = A B B A.

More information

Excess Error, Approximation Error, and Estimation Error

Excess Error, Approximation Error, and Estimation Error E0 370 Statstcal Learnng Theory Lecture 10 Sep 15, 011 Excess Error, Approxaton Error, and Estaton Error Lecturer: Shvan Agarwal Scrbe: Shvan Agarwal 1 Introducton So far, we have consdered the fnte saple

More information

Department of Economics, Niigata Sangyo University, Niigata, Japan

Department of Economics, Niigata Sangyo University, Niigata, Japan Appled Matheatcs, 0, 5, 777-78 Publshed Onlne March 0 n ScRes. http://www.scrp.org/journal/a http://d.do.org/0.6/a.0.507 On Relatons between the General Recurrence Forula o the Etenson o Murase-Newton

More information

4 = (0.02) 3 13, = 0.25 because = 25. Simi-

4 = (0.02) 3 13, = 0.25 because = 25. Simi- Theore. Let b and be integers greater than. If = (. a a 2 a i ) b,then for any t N, in base (b + t), the fraction has the digital representation = (. a a 2 a i ) b+t, where a i = a i + tk i with k i =

More information

Journal of Mathematical Analysis and Applications

Journal of Mathematical Analysis and Applications J Math Anal Appl 388 0) 78 85 Contents lsts avalable at ScVerse ScenceDrect Journal of Mathematcal Analyss an Applcatons wwwelsevercom/locate/jmaa Multvarate nequaltes of Chernoff type for classcal orthogonal

More information

THE AVERAGE NORM OF POLYNOMIALS OF FIXED HEIGHT

THE AVERAGE NORM OF POLYNOMIALS OF FIXED HEIGHT THE AVERAGE NORM OF POLYNOMIALS OF FIXED HEIGHT PETER BORWEIN AND KWOK-KWONG STEPHEN CHOI Abstract. Let n be any integer and ( n ) X F n : a i z i : a i, ± i be the set of all polynoials of height and

More information

A P PL I CA TIONS OF FRACTIONAL EXTERIOR DI F F ER EN TIAL IN THR EE- DI M ENSIONAL S PAC E Ξ

A P PL I CA TIONS OF FRACTIONAL EXTERIOR DI F F ER EN TIAL IN THR EE- DI M ENSIONAL S PAC E Ξ Appled Mathematcs and Mechancs ( Englsh Edton, Vol 24, No 3, Mar 2003) Publshed by Shangha Unversty, Shangha, Chna Artcle ID : 0253-4827 (2003) 03-0256-05 A P PL I CA TIONS OF FRACTIONAL EXTERIOR DI F

More information

Two Conjectures About Recency Rank Encoding

Two Conjectures About Recency Rank Encoding Internatonal Journal of Matheatcs and Coputer Scence, 0(205, no. 2, 75 84 M CS Two Conjectures About Recency Rank Encodng Chrs Buhse, Peter Johnson, Wlla Lnz 2, Matthew Spson 3 Departent of Matheatcs and

More information

Case Study of Markov Chains Ray-Knight Compactification

Case Study of Markov Chains Ray-Knight Compactification Internatonal Journal of Contemporary Mathematcal Scences Vol. 9, 24, no. 6, 753-76 HIKAI Ltd, www.m-har.com http://dx.do.org/.2988/cms.24.46 Case Study of Marov Chans ay-knght Compactfcaton HaXa Du and

More information

Curious Bounds for Floor Function Sums

Curious Bounds for Floor Function Sums 1 47 6 11 Journal of Integer Sequences, Vol. 1 (018), Article 18.1.8 Curious Bounds for Floor Function Sus Thotsaporn Thanatipanonda and Elaine Wong 1 Science Division Mahidol University International

More information

,..., k N. , k 2. ,..., k i. The derivative with respect to temperature T is calculated by using the chain rule: & ( (5) dj j dt = "J j. k i.

,..., k N. , k 2. ,..., k i. The derivative with respect to temperature T is calculated by using the chain rule: & ( (5) dj j dt = J j. k i. Suppleentary Materal Dervaton of Eq. 1a. Assue j s a functon of the rate constants for the N coponent reactons: j j (k 1,,..., k,..., k N ( The dervatve wth respect to teperature T s calculated by usng

More information

First day August 1, Problems and Solutions

First day August 1, Problems and Solutions FOURTH INTERNATIONAL COMPETITION FOR UNIVERSITY STUDENTS IN MATHEMATICS July 30 August 4, 997, Plovdv, BULGARIA Frst day August, 997 Problems and Solutons Problem. Let {ε n } n= be a sequence of postve

More information

XII.3 The EM (Expectation-Maximization) Algorithm

XII.3 The EM (Expectation-Maximization) Algorithm XII.3 The EM (Expectaton-Maxzaton) Algorth Toshnor Munaata 3/7/06 The EM algorth s a technque to deal wth varous types of ncoplete data or hdden varables. It can be appled to a wde range of learnng probles

More information

REFINING CBS INEQUALITY FOR DIVISIONS OF MEASURABLE SPACE

REFINING CBS INEQUALITY FOR DIVISIONS OF MEASURABLE SPACE EFINING CBS INEQUALITY FO DIVISIONS OF MEASUABLE SPACE S. S. DAGOMI ; Abstract. In ths paper we establsh a re nement some reverses for CBS nequalty for the general Lebesgue ntegral on dvsons of measurable

More information

Fermat Varieties of Hodge Witt Type

Fermat Varieties of Hodge Witt Type JOURNAL OF ALGEBRA 180, 136155 1996 ARTICLE NO. 0057 Ferat Varetes of HodgeWtt Type Kesuke Tok Koyaa 3-5-5, Nera-ku, Tokyo 176, Japan Councated by Walter Fet Receved January 31, 1995 1. INTRODUCTION Let

More information

Fuzzy Logic and Zadeh Algebra

Fuzzy Logic and Zadeh Algebra Advances n Pure Matheatcs, 2017, 7, 353-365 http://wwwscrporg/journal/ap ISSN Onlne: 2160-0384 ISSN Prnt: 2160-0368 Fuzzy Logc and Zadeh Algebra Paavo Kukkuranen School of Engneerng Scence, Lappeenranta

More information

MAT 578 Functional Analysis

MAT 578 Functional Analysis MAT 578 Functonal Analyss John Qugg Fall 2008 Locally convex spaces revsed September 6, 2008 Ths secton establshes the fundamental propertes of locally convex spaces. Acknowledgment: although I wrote these

More information

Final Exam Solutions, 1998

Final Exam Solutions, 1998 58.439 Fnal Exa Solutons, 1998 roble 1 art a: Equlbru eans that the therodynac potental of a consttuent s the sae everywhere n a syste. An exaple s the Nernst potental. If the potental across a ebrane

More information

On C 0 multi-contractions having a regular dilation

On C 0 multi-contractions having a regular dilation SUDIA MAHEMAICA 170 (3) (2005) On C 0 mult-contractons havng a regular dlaton by Dan Popovc (mşoara) Abstract. Commutng mult-contractons of class C 0 and havng a regular sometrc dlaton are studed. We prove

More information

Infinitely Many Trees Have Non-Sperner Subtree Poset

Infinitely Many Trees Have Non-Sperner Subtree Poset Order (2007 24:133 138 DOI 10.1007/s11083-007-9064-2 Infinitely Many Trees Have Non-Sperner Subtree Poset Andrew Vince Hua Wang Received: 3 April 2007 / Accepted: 25 August 2007 / Published online: 2 October

More information

System in Weibull Distribution

System in Weibull Distribution Internatonal Matheatcal Foru 4 9 no. 9 94-95 Relablty Equvalence Factors of a Seres-Parallel Syste n Webull Dstrbuton M. A. El-Dacese Matheatcs Departent Faculty of Scence Tanta Unversty Tanta Egypt eldacese@yahoo.co

More information

Fermi-Dirac statistics

Fermi-Dirac statistics UCC/Physcs/MK/EM/October 8, 205 Fer-Drac statstcs Fer-Drac dstrbuton Matter partcles that are eleentary ostly have a type of angular oentu called spn. hese partcles are known to have a agnetc oent whch

More information

A MULTIDIMENSIONAL ANALOGUE OF THE RADEMACHER-GAUSSIAN TAIL COMPARISON

A MULTIDIMENSIONAL ANALOGUE OF THE RADEMACHER-GAUSSIAN TAIL COMPARISON A MULTIDIMENSIONAL ANALOGUE OF THE RADEMACHER-GAUSSIAN TAIL COMPARISON PIOTR NAYAR AND TOMASZ TKOCZ Abstract We prove a menson-free tal comparson between the Euclean norms of sums of nepenent ranom vectors

More information

NP-Completeness : Proofs

NP-Completeness : Proofs NP-Completeness : Proofs Proof Methods A method to show a decson problem Π NP-complete s as follows. (1) Show Π NP. (2) Choose an NP-complete problem Π. (3) Show Π Π. A method to show an optmzaton problem

More information

GELFAND-TSETLIN BASIS FOR THE REPRESENTATIONS OF gl n

GELFAND-TSETLIN BASIS FOR THE REPRESENTATIONS OF gl n GELFAND-TSETLIN BASIS FOR THE REPRESENTATIONS OF gl n KANG LU FINITE DIMENSIONAL REPRESENTATIONS OF gl n Let e j,, j =,, n denote the standard bass of the general lnear Le algebra gl n over the feld of

More information

A MULTIDIMENSIONAL ANALOGUE OF THE RADEMACHER-GAUSSIAN TAIL COMPARISON

A MULTIDIMENSIONAL ANALOGUE OF THE RADEMACHER-GAUSSIAN TAIL COMPARISON A MULTIDIMENSIONAL ANALOGUE OF THE RADEMACHER-GAUSSIAN TAIL COMPARISON PIOTR NAYAR AND TOMASZ TKOCZ Abstract We prove a menson-free tal comparson between the Euclean norms of sums of nepenent ranom vectors

More information

Valuated Binary Tree: A New Approach in Study of Integers

Valuated Binary Tree: A New Approach in Study of Integers Internatonal Journal of Scentfc Innovatve Mathematcal Research (IJSIMR) Volume 4, Issue 3, March 6, PP 63-67 ISS 347-37X (Prnt) & ISS 347-34 (Onlne) wwwarcournalsorg Valuated Bnary Tree: A ew Approach

More information

e companion ONLY AVAILABLE IN ELECTRONIC FORM

e companion ONLY AVAILABLE IN ELECTRONIC FORM e copanon ONLY AVAILABLE IN ELECTRONIC FORM Electronc Copanon Decson Analyss wth Geographcally Varyng Outcoes: Preference Models Illustratve Applcatons by Jay Son, Crag W. Krkwood, L. Robn Keller, Operatons

More information

Approximate Technique for Solving Class of Fractional Variational Problems

Approximate Technique for Solving Class of Fractional Variational Problems Appled Matheatcs, 5, 6, 837-846 Publshed Onlne May 5 n ScRes. http://www.scrp.org/journal/a http://dx.do.org/.436/a.5.6578 Approxate Technque for Solvng Class of Fractonal Varatonal Probles Ead M. Soloua,,

More information

COS 511: Theoretical Machine Learning

COS 511: Theoretical Machine Learning COS 5: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture #0 Scrbe: José Sões Ferrera March 06, 203 In the last lecture the concept of Radeacher coplexty was ntroduced, wth the goal of showng that

More information

LATTICE POINT SOLUTION OF THE GENERALIZED PROBLEM OF TERQUEi. AND AN EXTENSION OF FIBONACCI NUMBERS.

LATTICE POINT SOLUTION OF THE GENERALIZED PROBLEM OF TERQUEi. AND AN EXTENSION OF FIBONACCI NUMBERS. i LATTICE POINT SOLUTION OF THE GENERALIZED PROBLEM OF TERQUEi. AND AN EXTENSION OF FIBONACCI NUMBERS. C. A. CHURCH, Jr. and H. W. GOULD, W. Virginia University, Morgantown, W. V a. In this paper we give

More information

LECTURE :FACTOR ANALYSIS

LECTURE :FACTOR ANALYSIS LCUR :FACOR ANALYSIS Rta Osadchy Based on Lecture Notes by A. Ng Motvaton Dstrbuton coes fro MoG Have suffcent aount of data: >>n denson Use M to ft Mture of Gaussans nu. of tranng ponts If

More information

Gradient Descent Learning and Backpropagation

Gradient Descent Learning and Backpropagation Artfcal Neural Networks (art 2) Chrstan Jacob Gradent Descent Learnng and Backpropagaton CSC 533 Wnter 200 Learnng by Gradent Descent Defnton of the Learnng roble Let us start wth the sple case of lnear

More information

One-Shot Quantum Information Theory I: Entropic Quantities. Nilanjana Datta University of Cambridge,U.K.

One-Shot Quantum Information Theory I: Entropic Quantities. Nilanjana Datta University of Cambridge,U.K. One-Shot Quantu Inforaton Theory I: Entropc Quanttes Nlanjana Datta Unversty of Cabrdge,U.K. In Quantu nforaton theory, ntally one evaluated: optal rates of nfo-processng tasks, e.g., data copresson, transsson

More information

2.3 Nilpotent endomorphisms

2.3 Nilpotent endomorphisms s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms

More information

Simultaneous approximation of polynomials

Simultaneous approximation of polynomials Smultaneous approxmaton of polynomals Anre Kupavsk * János Pach Abstract Let P enote the famly of all polynomals of egree at most n one varable x, wth real coeffcents. A sequence of postve numbers x 1

More information