ON THE LEHMER CONSTANT OF FINITE CYCLIC GROUPS
|
|
- Helena McGee
- 6 years ago
- Views:
Transcription
1 ON THE LEHMER CONSTANT OF FINITE CYCLIC GROUPS NORBERT KAIBLINGER Abstract. Results of Lid o Lehmer s problem iclude the value of the Lehmer costat of the fiite cyclic group Z/Z, for 5 ad all odd. By complemetary observatios we determie the Lehmer costat of Z/Z, for all except for multiples of Itroductio Let be a positive iteger. Give a polyomial with iteger coefficiets, f Z[x], deote by m (f) its logarithmic Mahler measure over Z/Z, defied by m (f) = 1 1 log f(e 2πik/ ). k=0 By λ > 0 we deote the Lehmer costat of Z/Z, λ = mi m (f), f Z[x], m (f)>0 see [11]. We otice later that the miimum is ideed attaied, ad that it is the same if deg f 1 is assumed. Lid [11] has give a upper boud for λ, see below, ad he obtaied the values λ 1 = log 2, λ 2 = 1 2 log 3, λ 4 = 1 4 log 3, ad λ = 1 log 2 for all odd. We sharpe his result, complemet it by a lower boud, ad obtai the value of λ for all except for multiples of 420. The mai result is formulated i Sectio 2 ad it is proved i Sectio Mai result { } { } ρ() prime umber For a positive iteger, let ρ deote the smallest () positive iteger that does ot divide. We write p k whe p k is a pricipal divisor of, 2010 Mathematics Subject Classificatio. Primary 11R09; Secodary 11B83, 11C08, 11T22. Key words ad phrases. Lid s Lehmer costats, logarithmic Mahler measure, fiite cyclic group, cyclotomic polyomial. Supported by the Austria Sciece Fud FWF grat P
2 2 NORBERT KAIBLINGER that is, if p is a prime ad k is a positive iteger such that p k ad p k+1. Let ( ρ () = mi mi p, mi p pk) ( = mi ρ(), mi p pk). p p k p k Lid proved that λ 1 log ρ(), for all. Extedig his result we obtai the followig theorem, our mai result. Theorem 1. The Lehmer costat of Z/Z is of the form λ = 1 log Λ, with a iteger Λ 2 ot dividig ad i the rage ρ () Λ ρ (). For all = 1,..., 419 (mod 420), we have Λ = ρ () = ρ (). Example 1. Example ew values are λ 6 = 1 log 4, λ 6 8 = 1 log 3, or more 8 geerally, λ = 1 log 3 if, ad oly if, = 2k with 3 k, λ = 1 log 4 if, ad oly if, = 6k with odd k. Remark 1. (i) Theorem 1 yields the exact value of λ whe ρ () = ρ() or more geerally, whe ρ () = ρ (). Thus it also icludes certai multiples of 420. For example, let = 6 k 420 with 11 k. The ρ () = ρ() = 11 ad thus λ = 1 log 11. (ii) By Theorem 1 the kow upper boud λ 1 log ρ() is sharpeed strictly for all = 6 (mod 12), where it yields the exact value for λ, ad also for certai multiples of 420. For example, let = The the theorem implies λ = 1 log Λ with Λ {8, 9, 16}, while ρ() = 17. Ope questio: Determie λ = 1 log Λ for = 420. By Theorem 1 we have Λ 420 {8, 9, 11}. We have, for f Z[x], 3. Proof of Theorem 1 (1) m (f) = 1 1 log (f) with (f) = f(e 2πik/ ). The umber (f) is always a iteger, ad there is a elemetary way to see that. To this ed we recall the determiatal relatio of [13], readily exteded here to f of arbitrary degree. If deg f 1, write f(x) = a 0 + a 1 x+ +a 1 x 1, with zero coefficiets where ecessary. If a polyomial of higher degree is give, with coefficiets a 0, a 1,..., replace it first with f as above by defiig a k = l=k (mod ) a l. Let C a deote the k=0
3 ON THE LEHMER CONSTANT OF FINITE CYCLIC GROUPS 3 iteger circulat matrix with first row a = (a 0,..., a 1 ). The det C a = 1 k=0 f(e2πik/ ) ad it implies (2) (f) = det C a. Hece, (f) is ideed a iteger. Observe that expressig m (f) i terms of the iteger (f) justifies the defiitio of the Lehmer costat λ as a miimum, ot just a ifimum. We will also use the expressio of (f) as a resultat, for example see [2, 5, 11]. Ideed sice Res ( x 1, f(x) ) = 1 k=0 f(e2πik/ ), we have (3) (f) = Res ( x 1, f(x) ). The more commoly used expressio Res ( f(x), x 1 ), with iterchaged argumets, works as well, as log as oly absolute values are cosidered. Ideed the sig of the determiat i (2) or of the resultat i (3) is irrelevat for m (f). We remark that the opposite sig is obtaied for the polyomial f (x) = x 1 f(1/x), with coefficiet sequece ( a 1, a 2,..., a 0 ), the egative of the usual reciprocal polyomial. Remark 2. (i) Lehmer ad Pierce [10, 13] ivestigated the sequeces { 1 (f), 2 (f),... }, for f Z[x]. For example, f(x) = 2 x yields (f) = 2 1, the Mersee umbers; we refer to [6, 7, 8, 9]. For Lehmer s problem, formulated i [10], we refer to [3, 14] ad the spectacular solutio for odd coefficiets i [2]. Lid s Lehmer costats λ relate to the family { (f): f Z[x] }, for fixed. (ii) Our approach highlights ad makes use of the fact that fidig possible (or miimal) values of the logarithmic Mahler measure over Z/Z is equivalet to fidig possible (or miimal) values of iteger circular determiats, a ope problem attributed to Taussky-Todd [12]. Call f Z[x] cyclotomic if all its zeros lie o the complex uit circle. As a prelimiary observatio we determie, for all, the exact value of a cyclotomic variat of Lid s Lehmer costats. Lemma 1. For cyclotomic polyomials f Z[x], the miimal possible value of m (f) > 0 is determied by (4) mi m (f) = 1 f Z[x] cyclotomic log ρ (). m (f)>0 Proof. First, Kroecker s theorem implies that ay cyclotomic polyomial f Z[x] is the product of some of Φ 1, Φ 2,... ad a costat, if ecessary; here Φ m Z[x] deotes the m-th cyclotomic polyomial, i.e., the moic polyomial whose zeros are the primitive m-th roots of uity. Sice always (5) (f 1 f 2 ) = (f 1 ) (f 2 )
4 4 NORBERT KAIBLINGER ad cosequetly, m (f 1 f 2 ) m (f 1 ) + m (f 2 ), we thus obtai (6) mi m (f) = f Z[x] cyclotomic m (f)>0 mi m (Φ m ). m=1,2,... m (Φ m)>0 Let ϕ() deote Euler s totiet of. We poit out that (7) (Φ m ) = Res ( x 1, Φ m (x) ) 0 if m, 1 if at least two distict primes divide m/ gcd(m, ), p ϕ(q) if m/ gcd(m, ) is the power of a prime p = here we write gcd(m, ) = q, p ϕ(q)pk if m/ gcd(m, ) is the power of a prime p here we factorize gcd(m, ) = p k q with p k. We remark that by our approach o egative sig is eeded here, for ay m,. This formula is obtaied from [1, proof of Theorem 2], where it is used for a short proof of the formula for Res ( Φ m1 (x), Φ m2 (x) ) ; secodly, sice (8) Res ( x 1, Φ m (x) ) = Res ( Φ 1 (x ), Φ m (x) ), the formula (7) also follows from applyig [4, Propositio 14]; a third, coveiet ad direct source is [5, Theorem 3]. Notice that (7) implies for ay, m, that particularly ( (9) (Φ m ) = 0, 1, or (Φ m ) mi Sice (7) also yields (10) mi p (Φ p ) = p for p, ad (Φ p k+1) = p pk for p k, we coclude that the iequality i (9) is sharp, that is, (11) mi (Φ m ) = ρ (). m=1,2,... (Φ m) 2 p, mi p pk) = ρ (). p k Fially, sice m (Φ m ) = 1 log (Φ m ), the statemet of the lemma follows by combiig (6) ad (11). Lemma 2. Let satisfy 6 (mod 12) ad 0 (mod 420). The ρ() = ρ (), that is, the least o-divisor of is a prime (ad ot a prime power). Remark 3. The example give i Remark 1(i) shows that the implicatio of Lemma 2 caot be reversed.
5 ON THE LEHMER CONSTANT OF FINITE CYCLIC GROUPS 5 Proof of Lemma 2. Suppose that 6 (mod 12) ad ρ () < ρ(); we verify that it implies 420. First, if 6, the either ρ () = ρ() = 2 or ρ () = ρ() = 3. This cotradicts the assumptio ρ () < ρ(). Hece, we have = 6k, for some k. The case k odd is excluded by the assumptio 6 (mod 12), so we obtai k eve. I other words, = 12k, for some k. If 5 k, the we have ρ () = ρ() = 5, i cotradictio to the assumptio ρ () < ρ(). Therefore, we have = 60k, for some k. Fially, if 7 k, the ρ () = ρ() = 7, agai i cotradictio to ρ () < ρ(). Thus we coclude that = 420k, for some k. Proof of Theorem 1. Step I: First otice that ideed λ = 1 log Λ for a iteger Λ 2; i fact, (12) Λ = mi (f). f Z[x] (f) 2 Therefore, Λ = (f 0 ), for some f 0 Z[x] with deg f 0 = 1. Upo replacig f 0 with f 0 defied above, if ecessary, we ca assume that Λ = (f 0 ). Step II: We show that Λ. Suppose that Λ divides. The there exists a prime p dividig both Λ ad. Let p m Λ ad p k. Sice Λ we otice that m k. O the other had, let C a be the iteger circulat matrix whose first row cosists of the coefficiets of f 0, so that (13) Λ = (f 0 ) = det C a. The we have p k ad p m det C a, ad a result by Newma [12, Theorem 2] thus implies that m k + 1, so we obtai a cotradictio. Step III: The previous step yields that the positive iteger Λ does ot divide. By defiitio, ρ () is the smallest umber with this property. We thus obtai the lower boud ρ () Λ. Step IV: The upper boud Λ ρ () is a cosequece of Lemma 1. Step V: Suppose that = 6 (mod 12). The 2 ad 3, while 4. Hece, ρ () = 4. O the other had, (14) mi p pk = 2 21 = 4, p k ad thus ρ () = 4; otice that ρ() 5. Therefore i Theorem 1 the lower ad upper boud coicide, ad we obtai Λ = ρ () = ρ () = 4. Step VI: Suppose that 6 (mod 12) ad 0 (mod 420). By Lemma 2 these coditios o imply that ρ() = ρ (). Sice always ρ () ρ () ρ(), we coclude that ρ () = ρ (), Thus the lower ad upper boud i Theorem 1 coicide ad we obtai Λ = ρ () = ρ ().
6 6 NORBERT KAIBLINGER Refereces [1] T. M. Apostol, Resultats of cyclotomic polyomials, Proc. Amer. Math. Soc. 24 (1970), [2] P. Borwei, E. Dobrowolski, ad M. J. Mossighoff, Lehmer s problem for polyomials with odd coefficiets, A. of Math. 166 (2007), [3] D. W. Boyd, Mahler s measure ad special values of L-fuctios, Experimet. Math. 7 (1998), [4] C. C. Cheg, J. H. McKay, ad S. S.-S. Wag, Resultats of cyclotomic polyomials, Proc. Amer. Math. Soc. 123 (1995), [5] J. E. Cremoa, Uimodular iteger circulats, Math. Comp. 77 (2008), [6] M. Eisiedler, G. Everest, ad T. Ward, Primes i sequeces associated to polyomials (after Lehmer), LMS J. Comput. Math. 3 (2000), [7] G. Everest, P. Rogers, ad T. Ward, A higher-rak Mersee problem, i: Algorithmic Number Theory, W. L. Mag, C. Fieker, ad D. R. Kohel (eds.), Spriger, Berli, 2002, [8] A. Flatters, Primitive divisors of some Lehmer-Pierce sequeces, J. Number Theory 129 (2009), [9] C. J. Hillar ad L. Levie, Polyomial recurreces ad cyclic resultats, Proc. Amer. Math. Soc. 135 (2007), [10] D. H. Lehmer, Factorizatio of certai cyclotomic fuctios, A. of Math. 34 (1933), [11] D. Lid, Lehmer s problem for compact abelia groups, Proc. Amer. Math. Soc. 133 (2005), [12] M. Newma, O a problem suggested by Olga Taussky-Todd, Illiois J. Math. 24 (1980), [13] T. A. Pierce, The umerical factors of the arithmetic forms i=1 (1 ± αm i ), A. of Math. 18 (1916), [14] C. Smyth, The Mahler measure of algebraic umbers: a survey, i: Number Theory ad Polyomials, J. McKee ad C. Smyth (eds.), Cambridge Uiv. Press, 2008, Faculty of Mathematics Uiversity of Viea Nordbergstraße Viea, Austria orbert.kaibliger@uivie.ac.at
The 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 informationMath 61CM - Solutions to homework 3
Math 6CM - Solutios to homework 3 Cédric De Groote October 2 th, 208 Problem : Let F be a field, m 0 a fixed oegative iteger ad let V = {a 0 + a x + + a m x m a 0,, a m F} be the vector space cosistig
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationDirichlet s Theorem on Arithmetic Progressions
Dirichlet s Theorem o Arithmetic Progressios Athoy Várilly Harvard Uiversity, Cambridge, MA 0238 Itroductio Dirichlet s theorem o arithmetic progressios is a gem of umber theory. A great part of its beauty
More information(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3
MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special
More informationk-generalized FIBONACCI NUMBERS CLOSE TO THE FORM 2 a + 3 b + 5 c 1. Introduction
Acta Math. Uiv. Comeiaae Vol. LXXXVI, 2 (2017), pp. 279 286 279 k-generalized FIBONACCI NUMBERS CLOSE TO THE FORM 2 a + 3 b + 5 c N. IRMAK ad M. ALP Abstract. The k-geeralized Fiboacci sequece { F (k)
More informationThe 4-Nicol Numbers Having Five Different Prime Divisors
1 2 3 47 6 23 11 Joural of Iteger Sequeces, Vol. 14 (2011), Article 11.7.2 The 4-Nicol Numbers Havig Five Differet Prime Divisors Qiao-Xiao Ji ad Mi Tag 1 Departmet of Mathematics Ahui Normal Uiversity
More informationThe value of Banach limits on a certain sequence of all rational numbers in the interval (0,1) Bao Qi Feng
The value of Baach limits o a certai sequece of all ratioal umbers i the iterval 0, Bao Qi Feg Departmet of Mathematical Scieces, Ket State Uiversity, Tuscarawas, 330 Uiversity Dr. NE, New Philadelphia,
More informationand each factor on the right is clearly greater than 1. which is a contradiction, so n must be prime.
MATH 324 Summer 200 Elemetary Number Theory Solutios to Assigmet 2 Due: Wedesday July 2, 200 Questio [p 74 #6] Show that o iteger of the form 3 + is a prime, other tha 2 = 3 + Solutio: If 3 + is a prime,
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 informationA Note on the Symmetric Powers of the Standard Representation of S n
A Note o the Symmetric Powers of the Stadard Represetatio of S David Savitt 1 Departmet of Mathematics, Harvard Uiversity Cambridge, MA 0138, USA dsavitt@mathharvardedu Richard P Staley Departmet of Mathematics,
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 informationBertrand s Postulate. Theorem (Bertrand s Postulate): For every positive integer n, there is a prime p satisfying n < p 2n.
Bertrad s Postulate Our goal is to prove the followig Theorem Bertrad s Postulate: For every positive iteger, there is a prime p satisfyig < p We remark that Bertrad s Postulate is true by ispectio for,,
More information1+x 1 + α+x. x = 2(α x2 ) 1+x
Math 2030 Homework 6 Solutios # [Problem 5] For coveiece we let α lim sup a ad β lim sup b. Without loss of geerality let us assume that α β. If α the by assumptio β < so i this case α + β. By Theorem
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 informationAn analog of the arithmetic triangle obtained by replacing the products by the least common multiples
arxiv:10021383v2 [mathnt] 9 Feb 2010 A aalog of the arithmetic triagle obtaied by replacig the products by the least commo multiples Bair FARHI bairfarhi@gmailcom MSC: 11A05 Keywords: Al-Karaji s triagle;
More informationSequences and Limits
Chapter Sequeces ad Limits Let { a } be a sequece of real or complex umbers A ecessary ad sufficiet coditio for the sequece to coverge is that for ay ɛ > 0 there exists a iteger N > 0 such that a p a q
More informationFundamental Theorem of Algebra. Yvonne Lai March 2010
Fudametal Theorem of Algebra Yvoe Lai March 010 We prove the Fudametal Theorem of Algebra: Fudametal Theorem of Algebra. Let f be a o-costat polyomial with real coefficiets. The f has at least oe complex
More informationThe normal subgroup structure of ZM-groups
arxiv:1502.04776v1 [math.gr] 17 Feb 2015 The ormal subgroup structure of ZM-groups Marius Tărăuceau February 17, 2015 Abstract The mai goal of this ote is to determie ad to cout the ormal subgroups 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 informationMATH 312 Midterm I(Spring 2015)
MATH 3 Midterm I(Sprig 05) Istructor: Xiaowei Wag Feb 3rd, :30pm-3:50pm, 05 Problem (0 poits). Test for covergece:.. 3.. p, p 0. (coverges for p < ad diverges for p by ratio test.). ( coverges, sice (log
More informationA Simplified Binet Formula for k-generalized Fibonacci Numbers
A Simplified Biet Formula for k-geeralized Fiboacci Numbers Gregory P. B. Dresde Departmet of Mathematics Washigto ad Lee Uiversity Lexigto, VA 440 dresdeg@wlu.edu Zhaohui Du Shaghai, Chia zhao.hui.du@gmail.com
More informationAssignment 5: Solutions
McGill Uiversity Departmet of Mathematics ad Statistics MATH 54 Aalysis, Fall 05 Assigmet 5: Solutios. Let y be a ubouded sequece of positive umbers satisfyig y + > y for all N. Let x be aother sequece
More informationTHE ASYMPTOTIC COMPLEXITY OF MATRIX REDUCTION OVER FINITE FIELDS
THE ASYMPTOTIC COMPLEXITY OF MATRIX REDUCTION OVER FINITE FIELDS DEMETRES CHRISTOFIDES Abstract. Cosider a ivertible matrix over some field. The Gauss-Jorda elimiatio reduces this matrix to the idetity
More informationStochastic Matrices in a Finite Field
Stochastic Matrices i a Fiite Field Abstract: I this project we will explore the properties of stochastic matrices i both the real ad the fiite fields. We first explore what properties 2 2 stochastic matrices
More informationHoggatt and King [lo] defined a complete sequence of natural numbers
REPRESENTATIONS OF N AS A SUM OF DISTINCT ELEMENTS FROM SPECIAL SEQUENCES DAVID A. KLARNER, Uiversity of Alberta, Edmoto, Caada 1. INTRODUCTION Let a, I deote a sequece of atural umbers which satisfies
More informationMa 530 Introduction to Power Series
Ma 530 Itroductio to Power Series Please ote that there is material o power series at Visual Calculus. Some of this material was used as part of the presetatio of the topics that follow. What is a Power
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 informationSOLVED EXAMPLES
Prelimiaries Chapter PELIMINAIES Cocept of Divisibility: A o-zero iteger t is said to be a divisor of a iteger s if there is a iteger u such that s tu I this case we write t s (i) 6 as ca be writte as
More informationMATH 304: MIDTERM EXAM SOLUTIONS
MATH 304: MIDTERM EXAM SOLUTIONS [The problems are each worth five poits, except for problem 8, which is worth 8 poits. Thus there are 43 possible poits.] 1. Use the Euclidea algorithm to fid the greatest
More informationPolynomial Functions and Their Graphs
Polyomial Fuctios ad Their Graphs I this sectio we begi the study of fuctios defied by polyomial expressios. Polyomial ad ratioal fuctios are the most commo fuctios used to model data, ad are used extesively
More informationThe multiplicative structure of finite field and a construction of LRC
IERG6120 Codig for Distributed Storage Systems Lecture 8-06/10/2016 The multiplicative structure of fiite field ad a costructio of LRC Lecturer: Keeth Shum Scribe: Zhouyi Hu Notatios: We use the otatio
More informationMath 104: Homework 2 solutions
Math 04: Homework solutios. A (0, ): Sice this is a ope iterval, the miimum is udefied, ad sice the set is ot bouded above, the maximum is also udefied. if A 0 ad sup A. B { m + : m, N}: This set does
More informationA Hadamard-type lower bound for symmetric diagonally dominant positive matrices
A Hadamard-type lower boud for symmetric diagoally domiat positive matrices Christopher J. Hillar, Adre Wibisoo Uiversity of Califoria, Berkeley Jauary 7, 205 Abstract We prove a ew lower-boud form of
More informationMath 4400/6400 Homework #7 solutions
MATH 4400 problems. Math 4400/6400 Homewor #7 solutios 1. Let p be a prime umber. Show that the order of 1 + p modulo p 2 is exactly p. Hit: Expad (1 + p) p by the biomial theorem, ad recall from MATH
More informationThe Structure of Z p when p is Prime
LECTURE 13 The Structure of Z p whe p is Prime Theorem 131 If p > 1 is a iteger, the the followig properties are equivalet (1) p is prime (2) For ay [0] p i Z p, the equatio X = [1] p has a solutio i Z
More informationMath 140A Elementary Analysis Homework Questions 1
Math 14A Elemetary Aalysis Homewor Questios 1 1 Itroductio 1.1 The Set N of Natural Numbers 1 Prove that 1 2 2 2 2 1 ( 1(2 1 for all atural umbers. 2 Prove that 3 11 (8 5 4 2 for all N. 4 (a Guess a formula
More informationSome remarks for codes and lattices over imaginary quadratic
Some remarks for codes ad lattices over imagiary quadratic fields Toy Shaska Oaklad Uiversity, Rochester, MI, USA. Caleb Shor Wester New Eglad Uiversity, Sprigfield, MA, USA. shaska@oaklad.edu Abstract
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 informationACO Comprehensive Exam 9 October 2007 Student code A. 1. Graph Theory
1. Graph Theory Prove that there exist o simple plaar triagulatio T ad two distict adjacet vertices x, y V (T ) such that x ad y are the oly vertices of T of odd degree. Do ot use the Four-Color Theorem.
More informationConvergence of random variables. (telegram style notes) P.J.C. Spreij
Covergece of radom variables (telegram style otes).j.c. Spreij this versio: September 6, 2005 Itroductio As we kow, radom variables are by defiitio measurable fuctios o some uderlyig measurable space
More information(2) F j+2 = F J+1 + F., F ^ = 0, F Q = 1 (j = -1, 0, 1, 2, ). Editorial notes This is not our standard Fibonacci sequence.
ON THE LENGTH OF THE EUCLIDEAN ALGORITHM E. P. IV1ERKES ad DAVSD MEYERS Uiversity of Ciciati, Ciciati, Ohio Throughout this article let a ad b be itegers, a > b >. The Euclidea algorithm geerates fiite
More informationProc. Amer. Math. Soc. 139(2011), no. 5, BINOMIAL COEFFICIENTS AND THE RING OF p-adic INTEGERS
Proc. Amer. Math. Soc. 139(2011, o. 5, 1569 1577. BINOMIAL COEFFICIENTS AND THE RING OF p-adic INTEGERS Zhi-Wei Su* ad Wei Zhag Departmet of Mathematics, Naig Uiversity Naig 210093, People s Republic of
More informationLemma Let f(x) K[x] be a separable polynomial of degree n. Then the Galois group is a subgroup of S n, the permutations of the roots.
15 Cubics, Quartics ad Polygos It is iterestig to chase through the argumets of 14 ad see how this affects solvig polyomial equatios i specific examples We make a global assumptio that the characteristic
More informationOn the pure Jacobi sums
ACTA ARITHMETICA LXXV.2 (1996) O the pure Jacobi sums by Shigeki Akiyama (Niigata) Let p be a odd prime ad F q be the field of q = p 2 elemets. We cosider the Jacobi sum over F q : J(χ, ψ) = x F q χ(x)ψ(1
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 informationPUTNAM TRAINING INEQUALITIES
PUTNAM TRAINING INEQUALITIES (Last updated: December, 207) Remark This is a list of exercises o iequalities Miguel A Lerma Exercises If a, b, c > 0, prove that (a 2 b + b 2 c + c 2 a)(ab 2 + bc 2 + ca
More informationBertrand s Postulate
Bertrad s Postulate Lola Thompso Ross Program July 3, 2009 Lola Thompso (Ross Program Bertrad s Postulate July 3, 2009 1 / 33 Bertrad s Postulate I ve said it oce ad I ll say it agai: There s always a
More informationFINITE GROUPS WITH THREE RELATIVE COMMUTATIVITY DEGREES. Communicated by Ali Reza Ashrafi. 1. Introduction
Bulleti of the Iraia Mathematical Society Vol. 39 No. 2 203), pp 27-280. FINITE GROUPS WITH THREE RELATIVE COMMUTATIVITY DEGREES R. BARZGAR, A. ERFANIAN AND M. FARROKHI D. G. Commuicated by Ali Reza Ashrafi
More informationOn Some Properties of Digital Roots
Advaces i Pure Mathematics, 04, 4, 95-30 Published Olie Jue 04 i SciRes. http://www.scirp.org/joural/apm http://dx.doi.org/0.436/apm.04.46039 O Some Properties of Digital Roots Ilha M. Izmirli Departmet
More informationAxioms of Measure Theory
MATH 532 Axioms of Measure Theory Dr. Neal, WKU I. The Space Throughout the course, we shall let X deote a geeric o-empty set. I geeral, we shall ot assume that ay algebraic structure exists o X so that
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 informationPRIME RECIPROCALS AND PRIMES IN ARITHMETIC PROGRESSION
PRIME RECIPROCALS AND PRIMES IN ARITHMETIC PROGRESSION DANIEL LITT Abstract. This aer is a exository accout of some (very elemetary) argumets o sums of rime recirocals; though the statemets i Proositios
More informationBenaissa Bernoussi Université Abdelmalek Essaadi, ENSAT de Tanger, B.P. 416, Tanger, Morocco
EXTENDING THE BERNOULLI-EULER METHOD FOR FINDING ZEROS OF HOLOMORPHIC FUNCTIONS Beaissa Beroussi Uiversité Abdelmalek Essaadi, ENSAT de Tager, B.P. 416, Tager, Morocco e-mail: Beaissa@fstt.ac.ma Mustapha
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 informationZeros of Polynomials
Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree
More informationMAS111 Convergence and Continuity
MAS Covergece ad Cotiuity Key Objectives At the ed of the course, studets should kow the followig topics ad be able to apply the basic priciples ad theorems therei to solvig various problems cocerig covergece
More informationarxiv: v1 [math.co] 3 Feb 2013
Cotiued Fractios of Quadratic Numbers L ubomíra Balková Araka Hrušková arxiv:0.05v [math.co] Feb 0 February 5 0 Abstract I this paper we will first summarize kow results cocerig cotiued fractios. The we
More informationM17 MAT25-21 HOMEWORK 5 SOLUTIONS
M17 MAT5-1 HOMEWORK 5 SOLUTIONS 1. To Had I Cauchy Codesatio Test. Exercise 1: Applicatio of the Cauchy Codesatio Test Use the Cauchy Codesatio Test to prove that 1 diverges. Solutio 1. Give the series
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 informationJournal of Ramanujan Mathematical Society, Vol. 24, No. 2 (2009)
Joural of Ramaua Mathematical Society, Vol. 4, No. (009) 199-09. IWASAWA λ-invariants AND Γ-TRANSFORMS Aupam Saikia 1 ad Rupam Barma Abstract. I this paper we study a relatio betwee the λ-ivariats of a
More informationCommutativity in Permutation Groups
Commutativity i Permutatio Groups Richard Wito, PhD Abstract I the group Sym(S) of permutatios o a oempty set S, fixed poits ad trasiet poits are defied Prelimiary results o fixed ad trasiet poits are
More informationPAijpam.eu ON DERIVATION OF RATIONAL SOLUTIONS OF BABBAGE S FUNCTIONAL EQUATION
Iteratioal Joural of Pure ad Applied Mathematics Volume 94 No. 204, 9-20 ISSN: 3-8080 (prited versio); ISSN: 34-3395 (o-lie versio) url: http://www.ijpam.eu doi: http://dx.doi.org/0.2732/ijpam.v94i.2 PAijpam.eu
More informationMaximal sets of integers not containing k + 1 pairwise coprimes and having divisors from a specified set of primes
EuroComb 2005 DMTCS proc. AE, 2005, 335 340 Maximal sets of itegers ot cotaiig k + 1 pairwise coprimes ad havig divisors from a specified set of primes Vladimir Bliovsky 1 Bielefeld Uiversity, Math. Dept.,
More informationsin(n) + 2 cos(2n) n 3/2 3 sin(n) 2cos(2n) n 3/2 a n =
60. Ratio ad root tests 60.1. Absolutely coverget series. Defiitio 13. (Absolute covergece) A series a is called absolutely coverget if the series of absolute values a is coverget. The absolute covergece
More informationTWO INEQUALITIES ON THE AREAL MAHLER MEASURE
Illiois Joural of Mathematics Volume 56, Number 3, Fall 0, Pages 85 834 S 009-08 TWO INEQUALITIES ON THE AREAL MAHLER MEASURE KWOK-KWONG STEPHEN CHOI AND CHARLES L. SAMUELS Abstract. Recet wor of Pritser
More information3.2 Properties of Division 3.3 Zeros of Polynomials 3.4 Complex and Rational Zeros of Polynomials
Math 60 www.timetodare.com 3. Properties of Divisio 3.3 Zeros of Polyomials 3.4 Complex ad Ratioal Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered
More informationTRIGONOMETRIC POLYNOMIALS WITH MANY REAL ZEROS AND A LITTLEWOOD-TYPE PROBLEM. Peter Borwein and Tamás Erdélyi. 1. Introduction
TRIGONOMETRIC POLYNOMIALS WITH MANY REAL ZEROS AND A LITTLEWOOD-TYPE PROBLEM Peter Borwei ad Tamás Erdélyi Abstract. We examie the size of a real trigoometric polyomial of degree at most havig at least
More informationBeurling Integers: Part 2
Beurlig Itegers: Part 2 Isomorphisms Devi Platt July 11, 2015 1 Prime Factorizatio Sequeces I the last article we itroduced the Beurlig geeralized itegers, which ca be represeted as a sequece of real umbers
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 informationResolution Proofs of Generalized Pigeonhole Principles
Resolutio Proofs of Geeralized Pigeohole Priciples Samuel R. Buss Departmet of Mathematics Uiversity of Califoria, Berkeley Győrgy Turá Departmet of Mathematics, Statistics, ad Computer Sciece Uiversity
More informationFourier Analysis, Stein and Shakarchi Chapter 8 Dirichlet s Theorem
Fourier Aalysis, Stei ad Shakarchi Chapter 8 Dirichlet s Theorem 208.05.05 Abstract Durig the course Aalysis II i NTU 208 Sprig, this solutio file is latexed by the teachig assistat Yug-Hsiag Huag with
More informationOn values of d(n!)/m!, φ(n!)/m! and σ(n!)/m!
O values of d!/m!, φ!/m! ad σ!/m! by Daiel Baczkowski, Michael Filaseta, Floria Luca ad Ogia Trifoov 1 Itroductio I [5], the third author established that for a fixed r Q, there are oly a fiite umber of
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 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 information11. FINITE FIELDS. Example 1: The following tables define addition and multiplication for a field of order 4.
11. FINITE FIELDS 11.1. A Field With 4 Elemets Probably the oly fiite fields which you ll kow about at this stage are the fields of itegers modulo a prime p, deoted by Z p. But there are others. Now although
More information[ 11 ] z of degree 2 as both degree 2 each. The degree of a polynomial in n variables is the maximum of the degrees of its terms.
[ 11 ] 1 1.1 Polyomial Fuctios 1 Algebra Ay fuctio f ( x) ax a1x... a1x a0 is a polyomial fuctio if ai ( i 0,1,,,..., ) is a costat which belogs to the set of real umbers ad the idices,, 1,...,1 are atural
More informationDIVISIBILITY PROPERTIES OF GENERALIZED FIBONACCI POLYNOMIALS
DIVISIBILITY PROPERTIES OF GENERALIZED FIBONACCI POLYNOMIALS VERNER E. HOGGATT, JR. Sa Jose State Uiversity, Sa Jose, Califoria 95192 ad CALVIN T. LONG Washigto State Uiversity, Pullma, Washigto 99163
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationBasic Sets. Functions. MTH299 - Examples. Example 1. Let S = {1, {2, 3}, 4}. Indicate whether each statement is true or false. (a) S = 4. (e) 2 S.
Basic Sets Example 1. Let S = {1, {2, 3}, 4}. Idicate whether each statemet is true or false. (a) S = 4 (b) {1} S (c) {2, 3} S (d) {1, 4} S (e) 2 S. (f) S = {1, 4, {2, 3}} (g) S Example 2. Compute the
More informationChapter 6 Infinite Series
Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat
More informationSOME GENERALIZATIONS OF OLIVIER S THEOREM
SOME GENERALIZATIONS OF OLIVIER S THEOREM Alai Faisat, Sait-Étiee, Georges Grekos, Sait-Étiee, Ladislav Mišík Ostrava (Received Jauary 27, 2006) Abstract. Let a be a coverget series of positive real umbers.
More informationExpected Norms of Zero-One Polynomials
DRAFT: Caad. Math. Bull. July 4, 08 :5 File: borwei80 pp. Page Sheet of Caad. Math. Bull. Vol. XX (Y, ZZZZ pp. 0 0 Expected Norms of Zero-Oe Polyomials Peter Borwei, Kwok-Kwog Stephe Choi, ad Idris Mercer
More informationArkansas Tech University MATH 2924: Calculus II Dr. Marcel B. Finan
Arkasas Tech Uiversity MATH 94: Calculus II Dr Marcel B Fia 85 Power Series Let {a } =0 be a sequece of umbers The a power series about x = a is a series of the form a (x a) = a 0 + a (x a) + a (x a) +
More informationAMS Mathematics Subject Classification : 40A05, 40A99, 42A10. Key words and phrases : Harmonic series, Fourier series. 1.
J. Appl. Math. & Computig Vol. x 00y), No. z, pp. A RECURSION FOR ALERNAING HARMONIC SERIES ÁRPÁD BÉNYI Abstract. We preset a coveiet recursive formula for the sums of alteratig harmoic series of odd order.
More informationSquare-Congruence Modulo n
Square-Cogruece Modulo Abstract This paper is a ivestigatio of a equivalece relatio o the itegers that was itroduced as a exercise i our Discrete Math class. Part I - Itro Defiitio Two itegers are Square-Cogruet
More information2.1. The Algebraic and Order Properties of R Definition. A binary operation on a set F is a function B : F F! F.
CHAPTER 2 The Real Numbers 2.. The Algebraic ad Order Properties of R Defiitio. A biary operatio o a set F is a fuctio B : F F! F. For the biary operatios of + ad, we replace B(a, b) by a + b ad a b, respectively.
More informationREAL ANALYSIS II: PROBLEM SET 1 - SOLUTIONS
REAL ANALYSIS II: PROBLEM SET 1 - SOLUTIONS 18th Feb, 016 Defiitio (Lipschitz fuctio). A fuctio f : R R is said to be Lipschitz if there exists a positive real umber c such that for ay x, y i the domai
More informationTEACHER CERTIFICATION STUDY GUIDE
COMPETENCY 1. ALGEBRA SKILL 1.1 1.1a. ALGEBRAIC STRUCTURES Kow why the real ad complex umbers are each a field, ad that particular rigs are ot fields (e.g., itegers, polyomial rigs, matrix rigs) Algebra
More informationMath Solutions to homework 6
Math 175 - Solutios to homework 6 Cédric De Groote November 16, 2017 Problem 1 (8.11 i the book): Let K be a compact Hermitia operator o a Hilbert space H ad let the kerel of K be {0}. Show that there
More informationBINOMIAL PREDICTORS. + 2 j 1. Then n + 1 = The row of the binomial coefficients { ( n
BINOMIAL PREDICTORS VLADIMIR SHEVELEV arxiv:0907.3302v2 [math.nt] 22 Jul 2009 Abstract. For oegative itegers, k, cosider the set A,k = { [0, 1,..., ] : 2 k ( ). Let the biary epasio of + 1 be: + 1 = 2
More informationAppendix to Quicksort Asymptotics
Appedix to Quicksort Asymptotics James Alle Fill Departmet of Mathematical Scieces The Johs Hopkis Uiversity jimfill@jhu.edu ad http://www.mts.jhu.edu/~fill/ ad Svate Jaso Departmet of Mathematics Uppsala
More informationOn Nonsingularity of Saddle Point Matrices. with Vectors of Ones
Iteratioal Joural of Algebra, Vol. 2, 2008, o. 4, 197-204 O Nosigularity of Saddle Poit Matrices with Vectors of Oes Tadeusz Ostrowski Istitute of Maagemet The State Vocatioal Uiversity -400 Gorzów, Polad
More informationSection 11.8: Power Series
Sectio 11.8: Power Series 1. Power Series I this sectio, we cosider geeralizig the cocept of a series. Recall that a series is a ifiite sum of umbers a. We ca talk about whether or ot it coverges ad i
More informationThe Random Walk For Dummies
The Radom Walk For Dummies Richard A Mote Abstract We look at the priciples goverig the oe-dimesioal discrete radom walk First we review five basic cocepts of probability theory The we cosider the Beroulli
More informationSymmetric Division Deg Energy of a Graph
Turkish Joural of Aalysis ad Number Theory, 7, Vol, No 6, -9 Available olie at http://pubssciepubcom/tat//6/ Sciece ad Educatio Publishig DOI:69/tat--6- Symmetric Divisio Deg Eergy of a Graph K N Prakasha,
More informationSequences and Series
Sequeces ad Series Sequeces of real umbers. Real umber system We are familiar with atural umbers ad to some extet the ratioal umbers. While fidig roots of algebraic equatios we see that ratioal umbers
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 informationPRELIM PROBLEM SOLUTIONS
PRELIM PROBLEM SOLUTIONS THE GRAD STUDENTS + KEN Cotets. Complex Aalysis Practice Problems 2. 2. Real Aalysis Practice Problems 2. 4 3. Algebra Practice Problems 2. 8. Complex Aalysis Practice Problems
More information7.1 Convergence of sequences of random variables
Chapter 7 Limit Theorems Throughout this sectio we will assume a probability space (, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite
More information