A Note on the Powers of Bazilevič Functions


 Erin Carroll
 2 years ago
 Views:
Transcription
1 International Journal of Mathematical Analysis Vol. 9, 015, no. 4, HIKARI Ltd, A Note on the Powers of Bailevič Functions Marjono Faculty of Mathematics and Natural Sciences Brawijaya University Malang, Jawa Timur 65145, Indonesia D. K. Thomas Department of Mathematics Swansea University, Singleton Park Swansea, SA 8PP, UK Copyright c 015 Marjono and D. K. Thomas. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract For α 0, let B 1 (α) be the set of Bailevič functions f, analytic in D = { : < 1}, given by f() = + n= a n n, and satisfying Re 1 α f () f( α > 0. For ( f() > 0, let = 1 + a n() n. We consider the problem of determining whether a n () b n (), where b n () are the coefficients of the extreme functions ( f() of for f B 1 (α). Mathematics Subject Classification: Primary 30C45; Secondary 30C50 Keywords: Univalent functions, powers, coefficients, starlike functions, Bailevič functions, FeketeSegö 1. Introduction Let S be the class of analytic normalised univalent functions f, defined in D = { : < 1} and given by
2 06 Marjono and D. K. Thomas For > 0, suppose that ( f() f() = + = 1 + a n n. (1.1) n= a n () n, and ( k() = 1 + k n () n, where k() is the Koebe function (1 ). Hayman and Hummel [3] posed the question of whether a n () k n () for n 1. De Brange s proof of the Bieberbach conjecture [] easily extends to show that a n () k n () is true when 1, whereas Hayman and Hummel [loc.cit] showed that this is false when > 1. It is known [5] that for starlike functions, a n () k n () is true for n 1, and > 0, and false for closetoconvex functions [4]. The class B 1 (α), of Bailevič functions with logarithmic growth, defined as follows, has been extensively studied see e.g. [6,7, 8].. Definition Suppose that f is analytic in D = { : < 1} and be given by (1.1). Then for α 0, f B 1 (α), if and only if, Re f () > 0. (.1) f( α α Since f B 1 (α) is a subset of the Bailevič functions [1], B 1 (α) S. For f B 1 (α), the coefficients a n exhibit irregular behaviour, and sharp bounds are known only when n =, 3, 4 [6]. In this paper we consider the validity of a n () b n () in the cases n = 1,, 3 for functions in B 1 (α), where
3 A note on the powers of Bailevič functions 063 ( f() b n () is the relevant extreme function of. We also give a FeketeSegö theorem. We first note that taking powers in the extreme functions in [6], the relevant coefficients b 1 (), b (, 1), b (, ), b 3 (, 1) and b 3 (, ) of the extreme function ( f() for are as follows. b 1 () := if α 0, (1 + α) ( + α + ) b (, 1) := if 0 α 1, (1 + α) ( + α) b (, ) := if α 1, ( + α) b 3 (, 1) := (3 + α) + 4( + 3 α α 3 α( ) if 0 α 1, 3(1 + α) 3 ( + α) 3 b 3 (, ) := if α 1. (3 + α) From (.1) we can write f () = f( α α p(), so that p P, the set of functions with positive real part in D. Let p() = 1 + p n n. We will use the following wellknown lemma. Lemma If p P with coefficients p n as above, then p n for n 1, and p 1 p 1 p 1. Without loss in generality, we can assume that α 0 and 1, since α = 0 corresponds to the starlike functions, and when = 1 there is nothing to prove.
4 064 Marjono and D. K. Thomas 3. Theorem 1 a 1 () b 1 () if α > 0 and > 0. If 0 < α 1. a () b (, 1) if 0 < < 1, and if > 1 provided α < 1 b (, ) if > 1 provided α > 1. If α 1. a () b (, ) if > 1, and if 0 < < 1 provided α > 1 b (, 1) if 0 < < 1 provided α < 1. All these inequalities for a () are sharp, and false on all complimentary intervals. The inequalities a 3 () b 3 (, 1) and a 3 () b 3 (, ) are false for all α > 0 and > 0. Proof. Using (.1) and equating coefficients we obtain a 1 () = a () = a 3 () = p 1 (1 + α), p ( + α) + (1 α)p 1 (1 + α), p 3 (3 + α) α + + ( 3α α )p 1 p (1 + α)( + α)(3 + α) + (6 + α (1 + α) 3 ( + α) 3(1 + α) (3 + α + ) + α(5 + 3 ))p 3 1 6(1 + α) 3 ( + α)(3 + α) 3. (3.1) The first inequality in Theorem 1 is trivial, since p 1 from the lemma. From (3.1), we have
5 A note on the powers of Bailevič functions 065 p a () = ( + α) + (1 α)p 1 (1 + α) 1 (1 α)( + α) = p + p ( + α) (1 + α) 1 1 = p 1 ( + α + ) ( + α) p 1 + (1 + α) p 1 1 ( 1 ( + α) p 1 ( + α + ) ) + (1 + α) p 1 := φ( p 1, α, ), where we have used the above lemma. Thus we need to maximise φ( p 1, α, ) over [0, ] for α > 0 and > 0, ignoring the case = 1. Simple consideration of the sign of p 1 in φ( p 1, α, ) gives the following. Case (i) 0 < α 1. (a) If 0 < < 1, then the maximum is a () b (, 1) is true. ( + a + ), and so (1 + α) ( + a) ( + α + ) (b) If > 1, then the maximum is again, provided α < 1, (1 + α) ( + α) and so a () b (, 1) is true provided α < 1. If > 1 and α > 1, then the maximum is is true in this case. Case (ii) α 1. (a) If 0 < < 1, the maximum is is true if α > 1. If 0 < < 1 and α < 1, the maximum is b (, 1). (b) If > 1, the maximum is ( + α), and so a () b (, ) ( + α) if α > 1 and so a () b (, ) ( + a + ) (1 + α) ( + a) and so a () ( + α), and so a () b (, ) is true.
6 066 Marjono and D. K. Thomas We note that the inequality for a 1 () is sharp when p 1 =, and the inequalities for a () are sharp on choosing either p 1 = p =, or p 1 = 0 and p = in (3.1). Choosing p 1 = p = in the expressions for a 1 () and a () in (3.1), it is a simple exercise to check that all the inequalities are false on all complementary intervals. To see that the inequalities for a 3 () are false, we choose the following values for p 1, p and p 3 in the expression for a 3 () in (3.1), noting that since the denominators are all positive, the algebraic calculations required to establish the inequalities are not difficult. Taking p 1 =, p = and p 3 =, shows that a 3 () > b 3 (, 1) when α < 1, and choosing p 1 =, p = 0 and p 3 =, that a 3 () > b 3 (, 1) when α > 1. Similarly taking p 1 = p = p 3 =, shows that a 3 () > b 3 (, ) when α < 1 and choosing p 1 = p 3 = and p = 0, that a 3 () > b 3 (, ) when α > 1. This completes the proof of Theorem 1. Finally we give a sharp FeketeSegö result, noting that since we are considering the coefficients of, the relevant coefficients are a 1 () and a (). ( f() The simple proof is omitted. 4. Theorem For f B 1 (α) with a 1 () and a () defined as above, a () µa 1 () ( + α) ( + α) + (1 α µ) (1 + α) if µ (1 α), (1 α) if µ.
7 A note on the powers of Bailevič functions 067 References [1] I. E. Bailevič, On a case of integrability in quadratures of the LöwnerKufarev equation, Mat. Sb., 37 (1955), no. 79, (in Russian) MR 17, 356. [] L. De Branges, A proof of the Bieberbach conjecture, Acta Mathematica, 154 (1985), [3] W. K. Hayman, & J. A. Hummel, Coefficients of powers of univalent functions, Complex Variables Theory Appl., 7 (1986), [4] M. Jahangiri, On the coefficients of powers of a class of Bailevič functions, Indian J. Pure Appl. Math., 17 (1986), no. 9, [5] M. Klein, Functions starlike of order Alpha, Trans. Amer. Math. Soc., 131 (1968), [6] R. Singh, On Bailevič Functions, Proc. Amer. Math. Soc., 38 (1973), no., [7] D. K. Thomas, On a Subclass of Bailevič functions, Int. Journal. Math. & Math. Sci., 8 (1985), no. 4, [8] D. K. Thomas, Proc. of an International Conference on New Trends in Geometric Function Theory and Applications, World Scientific (1991), Received: July 4, 015; Published: August 17, 015
Observation on Sums of Powers of Integers Divisible by Four
Applied Mathematical Sciences, Vol. 8, 2014, no. 45, 22192226 HIKARI Ltd, www.mhikari.com http://dx.doi.org/10.12988/ams.2014.4140 Observation on Sums of Powers of Integers Divisible by Four Djoko Suprijanto
More informationQuasi Contraction and Fixed Points
Available online at www.ispacs.com/jnaa Volume 2012, Year 2012 Article ID jnaa00168, 6 Pages doi:10.5899/2012/jnaa00168 Research Article Quasi Contraction and Fixed Points Mehdi Roohi 1, Mohsen Alimohammady
More informationA NEW APPROACH TO THE CORONA THEOREM FOR DOMAINS BOUNDED BY A C 1+α CURVE
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 40, 2015, 767 772 A NEW APPROACH TO THE CORONA THEOREM FOR DOMAINS BOUNDED BY A C 1+α CURVE José Manuel EnríquezSalamanca and María José González
More informationOn the Algebraic Structures of Soft Sets in Logic
Applied Mathematical Sciences, Vol. 8, 2014, no. 38, 18731881 HIKARI Ltd, www.mhikari.com http://dx.doi.org/10.12988/ams.2014.43127 On the Algebraic Structures of Soft Sets in Logic Burak Kurt Department
More informationF. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)
Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p
More informationCacti with minimum, secondminimum, and thirdminimum Kirchhoff indices
MATHEMATICAL COMMUNICATIONS 47 Math. Commun., Vol. 15, No. 2, pp. 4758 (2010) Cacti with minimum, secondminimum, and thirdminimum Kirchhoff indices Hongzhuan Wang 1, Hongbo Hua 1, and Dongdong Wang
More informationSome Problems of SecondOrder Rational Difference Equations with Quadratic Terms
International Journal of Difference Equations ISSN 09736069, Volume 9, Number 1, pp. 11 21 (2014) http://campus.mst.edu/ijde Some Problems of SecondOrder Rational Difference Equations with Quadratic
More informationHedging Against Foreign Exchange Risk of PesoDollar Rates Using Futures
Applied Mathematical Sciences, Vol. 8, 2014, no. 110, 54695476 HIKARI Ltd, www.mhikari.com http://dx.doi.org/10.12988/ams.2014.47595 Hedging Against Foreign Exchange Risk of PesoDollar Rates Using Futures
More informationOscillation of Higher Order Fractional Nonlinear Difference Equations
International Journal of Difference Equations ISSN 09736069, Volume 10, Number 2, pp. 201 212 (2015) http://campus.mst.edu/ijde Oscillation of Higher Order Fractional Nonlinear Difference Equations Senem
More informationAn Integrated Production Inventory System for. Perishable Items with Fixed and Linear Backorders
Int. Journal of Math. Analysis, Vol. 8, 2014, no. 32, 15491559 HIKARI Ltd, www.mhikari.com http://dx.doi.org/10.12988/ijma.2014.46176 An Integrated Production Inventory System for Perishable Items with
More informationA Factoring and Discrete Logarithm based Cryptosystem
Int. J. Contemp. Math. Sciences, Vol. 8, 2013, no. 11, 511517 HIKARI Ltd, www.mhikari.com A Factoring and Discrete Logarithm based Cryptosystem Abdoul Aziz Ciss and Ahmed Youssef Ecole doctorale de Mathematiques
More informationPRIME FACTORS OF CONSECUTIVE INTEGERS
PRIME FACTORS OF CONSECUTIVE INTEGERS MARK BAUER AND MICHAEL A. BENNETT Abstract. This note contains a new algorithm for computing a function f(k) introduced by Erdős to measure the minimal gap size in
More informationBOUNDED, ASYMPTOTICALLY STABLE, AND L 1 SOLUTIONS OF CAPUTO FRACTIONAL DIFFERENTIAL EQUATIONS. Muhammad N. Islam
Opuscula Math. 35, no. 2 (215), 181 19 http://dx.doi.org/1.7494/opmath.215.35.2.181 Opuscula Mathematica BOUNDED, ASYMPTOTICALLY STABLE, AND L 1 SOLUTIONS OF CAPUTO FRACTIONAL DIFFERENTIAL EQUATIONS Muhammad
More informationSHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH
31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,
More informationarxiv: v1 [math.ho] 7 Apr 2016
ON EXISTENCE OF A TRIANGLE WITH PRESCRIBED BISECTOR LENGTHS S. F. OSINKIN arxiv:1604.03794v1 [math.ho] 7 Apr 2016 Abstract. We suggest a geometric visualization of the process of constructing a triangle
More informationReal Roots of Quadratic Interval Polynomials 1
Int. Journal of Math. Analysis, Vol. 1, 2007, no. 21, 10411050 Real Roots of Quadratic Interval Polynomials 1 Ibraheem Alolyan Mathematics Department College of Science, King Saud University P.O. Box:
More informationResearch Article Stability Analysis for HigherOrder Adjacent Derivative in Parametrized Vector Optimization
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 510838, 15 pages doi:10.1155/2010/510838 Research Article Stability Analysis for HigherOrder Adjacent Derivative
More informationEvery Positive Integer is the Sum of Four Squares! (and other exciting problems)
Every Positive Integer is the Sum of Four Squares! (and other exciting problems) Sophex University of Texas at Austin October 18th, 00 Matilde N. Lalín 1. Lagrange s Theorem Theorem 1 Every positive integer
More informationTRIPLE POSITIVE SOLUTIONS FOR BOUNDARY VALUE PROBLEM OF A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION. Communicated by Mohammad Asadzadeh
Bulletin of the Iranian Mathematical Society Vol. 33 No. 2 (27), pp . TRIPLE POSITIVE SOLUTIONS FOR BOUNDARY VALUE PROBLEM OF A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION R. DEHGHANI AND K. GHANBARI*
More informationFactoring Dickson polynomials over finite fields
Factoring Dickson polynomials over finite fiels Manjul Bhargava Department of Mathematics, Princeton University. Princeton NJ 08544 manjul@math.princeton.eu Michael Zieve Department of Mathematics, University
More informationSCORE SETS IN ORIENTED GRAPHS
Applicable Analysis and Discrete Mathematics, 2 (2008), 107 113. Available electronically at http://pefmath.etf.bg.ac.yu SCORE SETS IN ORIENTED GRAPHS S. Pirzada, T. A. Naikoo The score of a vertex v in
More informationON A MIXED SUMDIFFERENCE EQUATION OF VOLTERRAFREDHOLM TYPE. 1. Introduction
SARAJEVO JOURNAL OF MATHEMATICS Vol.5 (17) (2009), 55 62 ON A MIXED SUMDIFFERENCE EQUATION OF VOLTERRAFREDHOLM TYPE B.. PACHPATTE Abstract. The main objective of this paper is to study some basic properties
More informationThe Factor Theorem and a corollary of the Fundamental Theorem of Algebra
Math 421 Fall 2010 The Factor Theorem and a corollary of the Fundamental Theorem of Algebra 27 August 2010 Copyright 2006 2010 by Murray Eisenberg. All rights reserved. Prerequisites Mathematica Aside
More informationFACTORING CERTAIN INFINITE ABELIAN GROUPS BY DISTORTED CYCLIC SUBSETS
International Electronic Journal of Algebra Volume 6 (2009) 95106 FACTORING CERTAIN INFINITE ABELIAN GROUPS BY DISTORTED CYCLIC SUBSETS Sándor Szabó Received: 11 November 2008; Revised: 13 March 2009
More informationA Note on Sums of Greatest (Least) Prime Factors
It. J. Cotemp. Math. Scieces, Vol. 8, 203, o. 9, 423432 HIKARI Ltd, www.mhikari.com A Note o Sums of Greatest (Least Prime Factors Rafael Jakimczuk Divisio Matemática, Uiversidad Nacioal de Luá Bueos
More informationProperties of BMO functions whose reciprocals are also BMO
Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a nonnegative BMOfunction w, whose reciprocal is also in BMO, belongs to p> A p,and
More informationON ROUGH (m, n) BIΓHYPERIDEALS IN ΓSEMIHYPERGROUPS
U.P.B. Sci. Bull., Series A, Vol. 75, Iss. 1, 2013 ISSN 12237027 ON ROUGH m, n) BIΓHYPERIDEALS IN ΓSEMIHYPERGROUPS Naveed Yaqoob 1, Muhammad Aslam 1, Bijan Davvaz 2, Arsham Borumand Saeid 3 In this
More informationarxiv:math/0606467v2 [math.co] 5 Jul 2006
A Conjectured Combinatorial Interpretation of the Normalized Irreducible Character Values of the Symmetric Group arxiv:math/0606467v [math.co] 5 Jul 006 Richard P. Stanley Department of Mathematics, Massachusetts
More informationApplied Mathematical Sciences, Vol. 7, 2013, no. 112, 55915597 HIKARI Ltd, www.mhikari.com http://dx.doi.org/10.12988/ams.2013.
Applied Mathematical Sciences, Vol. 7, 2013, no. 112, 55915597 HIKARI Ltd, www.mhikari.com http://dx.doi.org/10.12988/ams.2013.38457 Accuracy Rate of Predictive Models in Credit Screening Anirut Suebsing
More informationEvery tree contains a large induced subgraph with all degrees odd
Every tree contains a large induced subgraph with all degrees odd A.J. Radcliffe Carnegie Mellon University, Pittsburgh, PA A.D. Scott Department of Pure Mathematics and Mathematical Statistics University
More informationOn kfree Sequences of Integers
mathematics of computation, volume 26, number 119, july, 1972 On kfree Sequences of Integers By Samuel S. Wagstaff, Jr. Abstract. Let A
More informationarxiv:1112.0829v1 [math.pr] 5 Dec 2011
How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly
More informationOn Quantum Hamming Bound
On Quantum Hamming Bound Salah A. Aly Department of Computer Science, Texas A&M University, College Station, TX 778433112, USA Email: salah@cs.tamu.edu We prove quantum Hamming bound for stabilizer codes
More informationPROPERTIES OF SOME NEW SEMINORMED SEQUENCE SPACES DEFINED BY A MODULUS FUNCTION
STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume L, Number 3, September 2005 PROPERTIES OF SOME NEW SEMINORMED SEQUENCE SPACES DEFINED BY A MODULUS FUNCTION YAVUZ ALTIN AYŞEGÜL GÖKHAN HIFSI ALTINOK Abstract.
More informationAn example of a computable
An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept
More informationSumit Chandok and T. D. Narang INVARIANT POINTS OF BEST APPROXIMATION AND BEST SIMULTANEOUS APPROXIMATION
F A S C I C U L I M A T H E M A T I C I Nr 51 2013 Sumit Chandok and T. D. Narang INVARIANT POINTS OF BEST APPROXIMATION AND BEST SIMULTANEOUS APPROXIMATION Abstract. In this paper we generalize and extend
More informationSome remarks on PhragménLindelöf theorems for weak solutions of the stationary Schrödinger operator
Wan Boundary Value Problems (2015) 2015:239 DOI 10.1186/s1366101505080 R E S E A R C H Open Access Some remarks on PhragménLindelöf theorems for weak solutions of the stationary Schrödinger operator
More informationAn inequality for the group chromatic number of a graph
An inequality for the group chromatic number of a graph HongJian Lai 1, Xiangwen Li 2 and Gexin Yu 3 1 Department of Mathematics, West Virginia University Morgantown, WV 26505 USA 2 Department of Mathematics
More informationThe sum of digits of polynomial values in arithmetic progressions
The sum of digits of polynomial values in arithmetic progressions Thomas Stoll Institut de Mathématiques de Luminy, Université de la Méditerranée, 13288 Marseille Cedex 9, France Email: stoll@iml.univmrs.fr
More informationSamuel Omoloye Ajala
49 Kragujevac J. Math. 26 (2004) 49 60. SMOOTH STRUCTURES ON π MANIFOLDS Samuel Omoloye Ajala Department of Mathematics, University of Lagos, Akoka  Yaba, Lagos NIGERIA (Received January 19, 2004) Abstract.
More informationTHE DIMENSION OF A VECTOR SPACE
THE DIMENSION OF A VECTOR SPACE KEITH CONRAD This handout is a supplementary discussion leading up to the definition of dimension and some of its basic properties. Let V be a vector space over a field
More informationA STABILITY RESULT ON MUCKENHOUPT S WEIGHTS
Publicacions Matemàtiques, Vol 42 (998), 53 63. A STABILITY RESULT ON MUCKENHOUPT S WEIGHTS Juha Kinnunen Abstract We prove that Muckenhoupt s A weights satisfy a reverse Hölder inequality with an explicit
More informationPART I. THE REAL NUMBERS
PART I. THE REAL NUMBERS This material assumes that you are already familiar with the real number system and the representation of the real numbers as points on the real line. I.1. THE NATURAL NUMBERS
More informationMATHEMATICS BONUS FILES for faculty and students http://www2.onu.edu/~mcaragiu1/bonus_files.html
MATHEMATICS BONUS FILES for faculty and students http://www2onuedu/~mcaragiu1/bonus_fileshtml RECEIVED: November 1 2007 PUBLISHED: November 7 2007 Solving integrals by differentiation with respect to a
More informationThe fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992), 33 37) Bart de Smit
The fundamental group of the Hawaiian earring is not free Bart de Smit The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992),
More informationA Hajós type result on factoring finite abelian groups by subsets II
Comment.Math.Univ.Carolin. 51,1(2010) 1 8 1 A Hajós type result on factoring finite abelian groups by subsets II Keresztély Corrádi, Sándor Szabó Abstract. It is proved that if a finite abelian group is
More informationTilings of the sphere with right triangles III: the asymptotically obtuse families
Tilings of the sphere with right triangles III: the asymptotically obtuse families Robert J. MacG. Dawson Department of Mathematics and Computing Science Saint Mary s University Halifax, Nova Scotia, Canada
More informationCONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12
CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.
More information6, avenue Le Gorgeu  C.S , Brest Cedex 3, France. 136 Xuan Thuy Street, Cau Giay, Hanoi, Vietnam
NORMAL CRITERIA FOR FAMILIES OF MEROMORPHIC FUNCTIONS Gerd Dethloff a and Tran Van Tan b, and Nuyen Van Thin c a Université de Brest, LMBA, UMR CNRS 6205, 6, avenue Le Goreu  C.S. 93837, 29238 Brest Cedex
More informationA note on companion matrices
Linear Algebra and its Applications 372 (2003) 325 33 www.elsevier.com/locate/laa A note on companion matrices Miroslav Fiedler Academy of Sciences of the Czech Republic Institute of Computer Science Pod
More informationAround Hilbert s 17th Problem
Documenta Math. 433 Around Hilbert s 17th Problem Konrad Schmüdgen 2010 Mathematics Subject Classification: 14P10 Keywords and Phrases: Positive polynomials, sums of squares The starting point of the history
More informationOn the numbertheoretic functions ν(n) and Ω(n)
ACTA ARITHMETICA LXXVIII.1 (1996) On the numbertheoretic functions ν(n) and Ω(n) by Jiahai Kan (Nanjing) 1. Introduction. Let d(n) denote the divisor function, ν(n) the number of distinct prime factors,
More informationA simple criterion on degree sequences of graphs
Discrete Applied Mathematics 156 (2008) 3513 3517 Contents lists available at ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam Note A simple criterion on degree
More informationSOME RESULTS ON HYPERCYCLICITY OF TUPLE OF OPERATORS. Abdelaziz Tajmouati Mohammed El berrag. 1. Introduction
italian journal of pure and applied mathematics n. 35 2015 (487 492) 487 SOME RESULTS ON HYPERCYCLICITY OF TUPLE OF OPERATORS Abdelaziz Tajmouati Mohammed El berrag Sidi Mohamed Ben Abdellah University
More informationOn the representability of the biuniform matroid
On the representability of the biuniform matroid Simeon Ball, Carles Padró, Zsuzsa Weiner and Chaoping Xing August 3, 2012 Abstract Every biuniform matroid is representable over all sufficiently large
More informationDegree Hypergroupoids Associated with Hypergraphs
Filomat 8:1 (014), 119 19 DOI 10.98/FIL1401119F Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Degree Hypergroupoids Associated
More informationOn the largest prime factor of x 2 1
On the largest prime factor of x 2 1 Florian Luca and Filip Najman Abstract In this paper, we find all integers x such that x 2 1 has only prime factors smaller than 100. This gives some interesting numerical
More informationTHE SQUARE PARTIAL SUMS OF THE FOURIER TRANSFORM OF RADIAL FUNCTIONS IN THREE DIMENSIONS
Scientiae Mathematicae Japonicae Online, Vol. 5,, 9 9 9 THE SQUARE PARTIAL SUMS OF THE FOURIER TRANSFORM OF RADIAL FUNCTIONS IN THREE DIMENSIONS CHIKAKO HARADA AND EIICHI NAKAI Received May 4, ; revised
More informationCommentationes Mathematicae Universitatis Carolinae
Commentationes Mathematicae Universitatis Carolinae Alessandro Fedeli On the cardinality of Hausdorff spaces Commentationes Mathematicae Universitatis Carolinae, Vol. 39 (1998), No. 3, 581585 Persistent
More informationOn certain functional equation in semiprime rings and standard operator algebras
CUBO A Mathematical Journal Vol.16, N ō 01, (73 80. March 2014 On certain functional equation in semiprime rings and standard operator algebras Nejc Širovnik 1 Department of Mathematics and Computer Science,
More informationTHE ROBUSTNESS AGAINST DEPENDENCE OF NONPARAMETRIC TESTS FOR THE TWOSAMPLE LOCATION PROBLEM
APPLICATIOES MATHEMATICAE,4 (1995), pp. 469 476 P. GRZEGORZEWSKI (Warszawa) THE ROBUSTESS AGAIST DEPEDECE OF OPARAMETRIC TESTS FOR THE TWOSAMPLE LOCATIO PROBLEM Abstract. onparametric tests for the twosample
More informationCONSTANTSIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE plaplacian. Pasquale Candito and Giuseppina D Aguí
Opuscula Math. 34 no. 4 2014 683 690 http://dx.doi.org/10.7494/opmath.2014.34.4.683 Opuscula Mathematica CONSTANTSIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE plaplacian Pasquale
More informationON SUPERCYCLICITY CRITERIA. Nuha H. Hamada Business Administration College Al Ain University of Science and Technology 5th st, Abu Dhabi, 112612, UAE
International Journal of Pure and Applied Mathematics Volume 101 No. 3 2015, 401405 ISSN: 13118080 (printed version); ISSN: 13143395 (online version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v101i3.7
More informationSmooth functions statistics
Smooth functions statistics V. I. rnold To describe the topological structure of a real smooth function one associates to it the graph, formed by the topological variety, whose points are the connected
More informationMathematical Induction
Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,
More informationON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP. A. K. Das and R. K. Nath
International Electronic Journal of Algebra Volume 7 (2010) 140151 ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP A. K. Das and R. K. Nath Received: 12 October 2009; Revised: 15 December
More informationOn the greatest and least prime factors of n! + 1, II
Publ. Math. Debrecen Manuscript May 7, 004 On the greatest and least prime factors of n! + 1, II By C.L. Stewart In memory of Béla Brindza Abstract. Let ε be a positive real number. We prove that 145 1
More informationAll trees contain a large induced subgraph having all degrees 1 (mod k)
All trees contain a large induced subgraph having all degrees 1 (mod k) David M. Berman, A.J. Radcliffe, A.D. Scott, Hong Wang, and Larry Wargo *Department of Mathematics University of New Orleans New
More informationPooja Sharma and R. S. Chandel FIXED POINT THEOREMS FOR WEAKLY COMPATIBLE MAPPINGS IN FUZZY METRIC SPACES. 1. Introduction
F A S C I C U L I M A T H E M A T I C I Nr 51 2013 Pooja Sharma and R. S. Chandel FIXED POINT THEOREMS FOR WEAKLY COMPATIBLE MAPPINGS IN FUZZY METRIC SPACES Abstract. This paper presents some fixed point
More informationIf n is odd, then 3n + 7 is even.
Proof: Proof: We suppose... that 3n + 7 is even. that 3n + 7 is even. Since n is odd, there exists an integer k so that n = 2k + 1. that 3n + 7 is even. Since n is odd, there exists an integer k so that
More information(x + a) n = x n + a Z n [x]. Proof. If n is prime then the map
22. A quick primality test Prime numbers are one of the most basic objects in mathematics and one of the most basic questions is to decide which numbers are prime (a clearly related problem is to find
More informationCounting Primes whose Sum of Digits is Prime
2 3 47 6 23 Journal of Integer Sequences, Vol. 5 (202), Article 2.2.2 Counting Primes whose Sum of Digits is Prime Glyn Harman Department of Mathematics Royal Holloway, University of London Egham Surrey
More information1. Introduction. PROPER HOLOMORPHIC MAPPINGS BETWEEN RIGID POLYNOMIAL DOMAINS IN C n+1
Publ. Mat. 45 (2001), 69 77 PROPER HOLOMORPHIC MAPPINGS BETWEEN RIGID POLYNOMIAL DOMAINS IN C n+1 Bernard Coupet and Nabil Ourimi Abstract We describe the branch locus of proper holomorphic mappings between
More informationSign changes of Hecke eigenvalues of Siegel cusp forms of degree 2
Sign changes of Hecke eigenvalues of Siegel cusp forms of degree 2 Ameya Pitale, Ralf Schmidt 2 Abstract Let µ(n), n > 0, be the sequence of Hecke eigenvalues of a cuspidal Siegel eigenform F of degree
More informationBipan Hazarika ON ACCELERATION CONVERGENCE OF MULTIPLE SEQUENCES. 1. Introduction
F A S C I C U L I M A T H E M A T I C I Nr 51 2013 Bipan Hazarika ON ACCELERATION CONVERGENCE OF MULTIPLE SEQUENCES Abstract. In this article the notion of acceleration convergence of double sequences
More informationActuarial Present Values of Annuities under Stochastic Interest Rate
Int. Journal of Math. Analysis, Vol. 7, 03, no. 59, 9399 HIKARI Ltd, www.mhikari.com http://d.doi.org/0.988/ijma.03.3033 Actuarial Present Values of Annuities under Stochastic Interest Rate Zhao Xia
More informationThe positive minimum degree game on sparse graphs
The positive minimum degree game on sparse graphs József Balogh Department of Mathematical Sciences University of Illinois, USA jobal@math.uiuc.edu András Pluhár Department of Computer Science University
More informationThe Open University s repository of research publications and other research outputs
Open Research Online The Open University s repository of research publications and other research outputs The degreediameter problem for circulant graphs of degree 8 and 9 Journal Article How to cite:
More informationSOME USES OF SET THEORY IN ALGEBRA. Stanford Logic Seminar February 10, 2009
SOME USES OF SET THEORY IN ALGEBRA Stanford Logic Seminar February 10, 2009 Plan I. The Whitehead Problem early history II. Compactness and Incompactness III. Deconstruction P. Eklof and A. Mekler, Almost
More informationContinuity of the Perron Root
Linear and Multilinear Algebra http://dx.doi.org/10.1080/03081087.2014.934233 ArXiv: 1407.7564 (http://arxiv.org/abs/1407.7564) Continuity of the Perron Root Carl D. Meyer Department of Mathematics, North
More informationRECURSIVE ENUMERATION OF PYTHAGOREAN TRIPLES
RECURSIVE ENUMERATION OF PYTHAGOREAN TRIPLES DARRYL MCCULLOUGH AND ELIZABETH WADE In [9], P. W. Wade and W. R. Wade (no relation to the second author gave a recursion formula that produces Pythagorean
More informationCS 598CSC: Combinatorial Optimization Lecture date: 2/4/2010
CS 598CSC: Combinatorial Optimization Lecture date: /4/010 Instructor: Chandra Chekuri Scribe: David Morrison GomoryHu Trees (The work in this section closely follows [3]) Let G = (V, E) be an undirected
More informationON NONNEGATIVE SOLUTIONS OF NONLINEAR TWOPOINT BOUNDARY VALUE PROBLEMS FOR TWODIMENSIONAL DIFFERENTIAL SYSTEMS WITH ADVANCED ARGUMENTS
ON NONNEGATIVE SOLUTIONS OF NONLINEAR TWOPOINT BOUNDARY VALUE PROBLEMS FOR TWODIMENSIONAL DIFFERENTIAL SYSTEMS WITH ADVANCED ARGUMENTS I. KIGURADZE AND N. PARTSVANIA A. Razmadze Mathematical Institute
More informationOn the largest prime factor of the Mersenne numbers
On the largest prime factor of the Mersenne numbers Kevin Ford Department of Mathematics The University of Illinois at UrbanaChampaign Urbana Champaign, IL 61801, USA ford@math.uiuc.edu Florian Luca Instituto
More informationBOUNDS AND COMPUTATIONAL RESULTS FOR EXPONENTIAL SUMS RELATED TO CUSP FORMS
BOUNDS AND COMPUTATIONAL RESULTS FOR EXPONENTIAL SUMS RELATED TO CUSP FORMS ANNEMARIA ERNVALLHYTÖNEN AND ARTO LEPISTÖ Abstract. The aim of this paper is to present some computer data suggesting the correct
More informationRINGS WITH A POLYNOMIAL IDENTITY
RINGS WITH A POLYNOMIAL IDENTITY IRVING KAPLANSKY 1. Introduction. In connection with his investigation of projective planes, M. Hall [2, Theorem 6.2]* proved the following theorem: a division ring D in
More informationCurriculum Map by Block Geometry Mapping for Math Block Testing 20072008. August 20 to August 24 Review concepts from previous grades.
Curriculum Map by Geometry Mapping for Math Testing 20072008 Pre s 1 August 20 to August 24 Review concepts from previous grades. August 27 to September 28 (Assessment to be completed by September 28)
More informationThe Correlation and Linear Regression Analysis between Annual GDP Growth Rate and Money Laundering in Albania during the Period *
American Journal of Computational Mathematics, 3, 3, 36336 Published Online December 3 (http://www.scirp.org/journal/ajcm) http://dx.doi.org/.436/ajcm.3.344 The Correlation and Linear Regression Analysis
More informationBARROW S INEQUALITY AND SIGNED ANGLE BISECTORS. 1. Introduction
Journal of Mathematical Inequalities Volume 8, Number 3 (2014), 537 544 doi:10.7153/jmi0840 BARROW S INEQUALITY AND SIGNED ANGLE BISECTORS BNKO MALEŠEVIĆ AND MAJA PETROVIĆ (Communicated by J. Pečarić)
More informationON THE NUMBER OF REAL HYPERSURFACES HYPERTANGENT TO A GIVEN REAL SPACE CURVE
Illinois Journal of Mathematics Volume 46, Number 1, Spring 2002, Pages 145 153 S 00192082 ON THE NUMBER OF REAL HYPERSURFACES HYPERTANGENT TO A GIVEN REAL SPACE CURVE J. HUISMAN Abstract. Let C be a
More informationCONVERGENCE OF SEQUENCES OF ITERATES OF RANDOMVALUED VECTOR FUNCTIONS
C O L L O Q U I U M M A T H E M A T I C U M VOL. 97 2003 NO. 1 CONVERGENCE OF SEQUENCES OF ITERATES OF RANDOMVALUED VECTOR FUNCTIONS BY RAFAŁ KAPICA (Katowice) Abstract. Given a probability space (Ω,
More informationSOLUTIONS TO ASSIGNMENT 1 MATH 576
SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts
More informationSECRET sharing schemes were introduced by Blakley [5]
206 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 52, NO. 1, JANUARY 2006 Secret Sharing Schemes From Three Classes of Linear Codes Jin Yuan Cunsheng Ding, Senior Member, IEEE Abstract Secret sharing has
More informationInternational Journal of Information Technology, Modeling and Computing (IJITMC) Vol.1, No.3,August 2013
FACTORING CRYPTOSYSTEM MODULI WHEN THE COFACTORS DIFFERENCE IS BOUNDED Omar Akchiche 1 and Omar Khadir 2 1,2 Laboratory of Mathematics, Cryptography and Mechanics, Fstm, University of Hassan II MohammediaCasablanca,
More informationSection 3 Sequences and Limits, Continued.
Section 3 Sequences and Limits, Continued. Lemma 3.6 Let {a n } n N be a convergent sequence for which a n 0 for all n N and it α 0. Then there exists N N such that for all n N. α a n 3 α In particular
More informationSome strong sufficient conditions for cyclic homogeneous polynomial inequalities of degree four in nonnegative variables
Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 6 (013), 74 85 Research Article Some strong sufficient conditions for cyclic homogeneous polynomial inequalities of degree four in nonnegative
More informationA REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries
Acta Math. Univ. Comenianae Vol. LXVI, 2(1997), pp. 285 291 285 A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE E. T. BASKORO, M. MILLER and J. ŠIRÁŇ Abstract. It is well known that Moore digraphs do
More informationNotes from February 11
Notes from February 11 Math 130 Course web site: www.courses.fas.harvard.edu/5811 Two lemmas Before proving the theorem which was stated at the end of class on February 8, we begin with two lemmas. The
More informationBounds on the spectral radius of a Hadamard product of nonnegative or positive semidefinite matrices
Electronic Journal of Linear Algebra Volume 20 Volume 20 (2010) Article 6 2010 Bounds on the spectral radius of a Hadamard product of nonnegative or positive semidefinite matrices Roger A. Horn rhorn@math.utah.edu
More informationMean RamseyTurán numbers
Mean RamseyTurán numbers Raphael Yuster Department of Mathematics University of Haifa at Oranim Tivon 36006, Israel Abstract A ρmean coloring of a graph is a coloring of the edges such that the average
More information