Enumerating permutation polynomials

Size: px
Start display at page:

Download "Enumerating permutation polynomials"

Transcription

1 Enumeating pemutation polynomials Theodoulos Gaefalakis a,1, Giogos Kapetanakis a,, a Depatment of Mathematics and Applied Mathematics, Univesity of Cete, Heaklion, Geece Abstact We conside thoblem of enumeating polynomials ove F q, that have cetain coefficients pescibed to given values and pemute cetain substuctues of F q. In paticula, we ae inteested in the goup of -th oots of unity and in the submodules of F q. We employ the techniques of Konyagin and Pappaladi to obtain esults that ae simila to thei esults in [Finite Fields and thei Applications, 11):6 37, 006]. As a consequence, wove conditions that ensue the existence of low-degeemutation polynomials of the mentioned substuctues of F q. Keywods: finite fields, pemutation polynomials 010 MSC: 11T06, 11T3 1. Intoduction Let q = p t, whe is a pime and t is a positive intege. A polynomial ove the finite field F q is called a pemutation polynomial if it induces a pemutation on F q. The study of pemutation polynomials goes back to the wok of Hemite [6], Dickson [5], and subsequently Calitz [3] and othes. Recently, inteest in pemutation polynomials has been enewed due to applications they have found in coding theoy, cyptogaphy and combinatoics. We efe to Chapte 7 of [10] fo backgound on pemutation polynomials, as well as an extensive discussion on the histoy of the subect. In a ecent wok, Coulte, Hendeson and Matthews [4] pesent a new constuction of pemutation polynomials. Thei method equies a polynomial that pemutes the goup of -th oots of unity, µ, whee q 1, and an auxiliay function T which contacts F q to µ 0} and has some additional lineaity popety. This idea was genealized by Akbay, Ghioca and Wang []. Coesponding autho addesses: tgaef@uoc.g Theodoulos Gaefalakis), gnkapet@gmail.com Giogos Kapetanakis) 1 Tel.: , Fax: Tel.: , Fax: Pepint submitted to Finite Fields and thei Applications Decembe 9, 016

2 In diffeent line of wok, Konyagin and Pappaladi [7, 8] count themutation polynomials that have given coefficients equal to zeo. Given a pemutation σ SF q ), thee exists a uniquolynomial in f σ F q [X] of degee at most q such that f σ c) = σc) fo all c F q. Fo any 0 < k 1 < < k d < q 1, they define q k 1,..., k d ) to be the numbe of pemutations σ such that the coesponding polynomial f σ has the coefficients of X ki, 1 i d, equal to zeo and pove the following main esult. Theoem 1.1 [8], Theoem 1). qk 1,..., k d ) q! ) q q k 1 1)q) q/. e In paticula, this implies that thee exist such pemutations, given that q!/ > 1 + e 1/ ) q q k 1 1)q) q/. Akbay, Ghioca and Wang [1] shapened this esult by enumeating pemutation polynomials of pescibed shape, that is, with a given set of non-zeo monomials. In thesent wok, we conside thoblem of enumeating polynomials ove F q, that have cetain coefficients fixed to given values, and pemute cetain substuctues of F q, namely the goup of -th oots of unity and submodules of F q and pove the following theoems. Theoem 1.. If!/ [q 1) k 1 )] / 1 + e 1/ ), then thee exists a polynomial of F q [X] of degee at most 1, that pemutes µ, the -th oots of unity, with the coefficients of X ki equal to a i F q, fo i = 1,..., d and 0 < k 1 < < k d <, whee q 1 and q is the minimum diviso of q with q 1. Theoem 1.3. Let F be a pope subfield of F q. Suppose!/ q / k 1 1) / 1 + e 1/ ), then thee exists a polynomial of F q [X] that pemutes F, an F [X]-submodule of F q, with its coefficients of X ki equal to a i F q, fo i = 1,..., d and 0 < k 1 < < k d <, whee = n = F, q = ρ and ρ is the ode and n is the degee of the Ode of F. We employ the techniques of Konyagin and Pappaladi to obtain esults that ae simila to those in [8]. In paticula, Theoems 1. and 1.3 can be viewed as the analoges of Theoem 1.1 fo oots of unity and submodules espectively, while they also imply the existence of low-degeolynomials that pemute these substuctues of F q, see Coollaies.1 and Enumeation of polynomials that pemute oots of unity Let q 1 and σ Sµ ) be a pemutation of µ. We define the polynomial f σ X) = 1 σc)g c X), 1) c µ

3 whee g c X) = 1 =0 c X, fo c µ. It is clea that g c c) = and g c x) = 0 fo all x µ \ c}, hence f σ ω) = σω), fo evey ω µ. Given d integes 0 < k 1 < < k d <, we denote q k, a) = σ Sµ ) the coefficient of X ki of f σ is a i, 1 i d }, whee k = k 1,..., k d ) and a = a 1,..., a d ) F d q. Fom 1), we see that the -th coefficient of f σ is equal to some a F q if and only if c µ c σc) = a, so we have q k, a) = σ Sµ ) } c ki σc) = a i, 1 i d. c µ Fo any S µ, define the sets } A S = f : µ S c ki fc) = a i, 1 i d, c µ } B S = f : µ S f is suective, c ki fc) = a i, 1 i d. c µ Also, define AS) = A S and BS) = B S. It is not had to see that since AM) = T M BT ), fo evey M µ, we have that Fo M = µ, the above implies BM) = 1) M T AT ). ) T M q k, a) = S µ 1) S AS). 3) Recall that q = p t. Set u) := e πiu/p and Tx) the absolute tace of x F q, i.e. Tx) := x + x p + + x pt 1 fo x F q. Futhe, let q be the smallest powe of p such that q 1, i.e. F q is the smallest subfield of F q containing µ. If 3

4 S µ, then AS) = 1 whee α 1,...,α d ) Fq d f:µ S = 1 α 1,...,α d ) Fq d f:µ S c µ = 1 α 1,...,α d ) F d q d T α i a i + ))) c ki fc) c µ fc) )) d α ic ki c µ T T )) d α ia i T t )) d α ic ki T d α ia i )) = S + R S, 4) R S qd 1 = qd 1 max α 1,...,α d ) F d q \0} max α 1,...,α d ) F\0} c µ c µ Moeove, the AM-GM inequality implies c µ e p T t )) d α i c ki 1 T T t )) d α ic ki )) d α ia i )) d e p T t α i c ki. c µ e p T t 1 T t q )) d α i c ki )) d α i c ki 1 k 1 ) Ttu)) u F q With the help of the well-known identity, see [9, Chapte 3], we eventually get that T t c µ Ttu)) u F q / = q S, 5) )) d ) / q 1) k1 ) S α i c ki, 4. /

5 which implies R S qd 1 ) / ) / q 1) k1 ) S q 1) k1 ) S <. 6) By woking similaly as in Eq. ), but by consideing the mappings µ µ, we conclude that S µ 1) S S =!, that combined with Equations 3), 4) and 6) and the fact that e / 1, since 1 + x e x fo all x, we get qk, a)! ) / q 1) < k1 ) ) / Summing up, we havoved that =0 ) / q 1) k1 ) ) e / 1 ) / = [q 1) k 1 )] / =0 =0 = [q 1) k 1 )] / 1 + e 1/ ). ) e 1/ ) q k, a) >! [q 1) k 1)] / 1 + e 1/ ), which implies the Theoem 1.. If we apply this esult in the case k b = 1, k b 1 = 1 1,..., k 1 = b and a i = 0 fo all i, then we end up with the following inteesting consequence. Coollay.1. With the same assumptions as in Theoem 1., if!/q b bq 1)1 + e 1/ ), then thee exists a polynomial of F q of degee less than b that pemutes µ. 3. Enumeation of polynomials that pemute additive submodules Thoughout this section, we see F q as a F [X]-module, whee F is a pope subfield of F q, unde the action f x = k i=0 f ix qi fo f = k i=0 f ix i F [X] and x F q. Futhemoe, it follows diectly fom the omal Basis Theoem, see [10, Theoem.35], that F q is a cyclic F [X]-module. Let F be an F [X]-submodule of F q, whee := F = n q. Since F [X] is a pincipal ideal domain and F is a F [X]-submodule of F q, which is cyclic, it 5

6 follows that F will be cyclic as well, see [11, Theoem 6.3]. As a consequence, thee exists some monic f F [X], of degee n, with f X m 1, such that F = x F q f x = 0}, which is known as the Ode of F. Also, fo evey x F we have that n f i x i 1 0, if x 0, = f 0, if x = 0, i=0 while f 0 0, since f X m 1. ow, fo σ SF) a pemutation of F, we define f σ X) = 1 n σc) f i X c) i 1 7) f 0 and it is clea that f σ ω) = σω) fo evey ω F. Given d integes 0 < k 1 < < k d < and a 1,..., a d ) F d q, we denote q k, a) = σ SF) the coefficient of X ki of f σ is a i, 1 i d }. Fom 7), we deduce that the -th coefficient of f σ is a if and only if hence i=0 q k, a) = σ SF) i=0 n i ) 1 f i c) i 1 σc) = f 0 a, n i ) 1 f i c i 1 k σc) = f 0 a, 1 d}. i=0 Fo any S F, define the sets A S = g : F S k } n F i c i 1 k gc) = f 0 a, 1 d, i=0 B S = g A S g is suective}, whee F i stands fo ) i 1 k fi. Define AS) = A S and BS) = B S. As with Eq. ), we can show that AM) = T M BT ), fo evey M F, hence q k, a) = S F 1) S AS). 8) Futhemoe, let ρ be the ode of f, i.e. ρ is minimal such that f X ρ 1 and let q := ρ. It follows that F q is the smallest subfield of F q containing F. Fo 6

7 S F, as in the case of Equation 4), we have AS) = 1 T gc) d =1 α )) n i=0 F ic i 1 k d )) α 1,...,α d ) Fq d g:f S T =1 f 0α a = 1 T t d )) n =1 i=0 α F i c i 1 k d )) T =1 f 0α a whee α 1,...,α d ) F d q = S + R S, 9) R S qd 1 = qd 1 max α 1,...,α d ) F d q \0} max α 1,...,α d ) F\0} T T T t t d )) n =1 i=0 α F i c i 1 k d )) =1 f 0α a d =1 i=0 n α F i c i 1 k. Also, the AM-GM inequality yields d n T t α F i c i 1 k =1 i=0 1 d n / T t α F i c i 1 k 1 q T t =1 i=0 d =1 i=0 1 1 k 1 ) e p Ttu)) u F q With the help of 5), we show that d n T t α F i c i 1 k that, in tun, yields R S qd 1 =1 i=0 n / α F i c i 1 k /. q 1 k ) / S ), q 1 k ) / S ) < q 1 k ) / S ). 10) 7

8 ow, as in the case of the oots of unity, it is clea that 1) S S =!, S F which combined with Equations 8), 9) and 10) and the fact that e / 1, gives qk, a)! < q/ 1 k ) / =0 q / 1 k ) / =0 ) / = q / k 1 1) / 1 + e 1/ ). To sum up, in this section woved that ) e / 1) / q k, a) >! q/ k 1 1) / 1 + e 1/ ) which implies Theoem 1.3. By applying this fo k b = 1, k b 1 = 1 1,..., k 1 = b and a i = 0 fo all i, we end up with the following. Coollay 3.1. With the same assumptions as in Theoem 1.3, if! qb 1)1 + e 1/ ), then thee exists a polynomial of F q of degee less than b that pemutes F. Aknowledgments The authos would like the eviewe fo his/he suggestions and coections. Refeences [1] A. Akbay, D. Ghioca and Q. Wang. On pemutation polynomials with pescibed shape. Finite Fields Appl., 15:195 06, 009. [] A. Akbay, D. Ghioca and Q. Wang. On constucting pemutations of finite fields. Finite Fields Appl., 17:51 67, 011. [3] L. Calitz. Pemutations in a finite field. Poc. Ame. Math. Soc., 4:538, [4] R. Coulte, M. Hendeson and R. Matthews. A note on constucting pemutation polynomials. Finite Fields Appl., 15: ,

9 [5] L. Dickson. The analytic epesentation of substitutions on a powe of a pime numbe of lettes with a discussion of the linea goup. Ann. of Math., 11:65 10, , [6] C. Hemite. Su les fonctions de sept lettes. C.R. Acad. Sci. Pais, 57: , [7] S. Konyagin and F. Pappaladi. Enumeating pemutation polynomials ove finite fields by degee. Finite Fields Appl., 84): , 00. [8] S. Konyagin and F. Pappaladi. Enumeating pemutation polynomials ove finite fields by degee II. Finite Fields Appl., 11):6 37, 006. [9] S. Konyagin and I. Shpalinski. Chaacte Sums with Exponential Functions and thei Applications. Cambidge Univesity Pess, Cambidge, 004. [10] R. Lidl and H. iedeeite. Finite Fields. Addison-Wesley, Reading, Mass., [11] S. Roman. Advanced Linea Algeba. Spinge-Velag, ew Yok,

SOME GENERAL NUMERICAL RADIUS INEQUALITIES FOR THE OFF-DIAGONAL PARTS OF 2 2 OPERATOR MATRICES

SOME GENERAL NUMERICAL RADIUS INEQUALITIES FOR THE OFF-DIAGONAL PARTS OF 2 2 OPERATOR MATRICES italian jounal of pue and applied mathematics n. 35 015 (433 44) 433 SOME GENERAL NUMERICAL RADIUS INEQUALITIES FOR THE OFF-DIAGONAL PARTS OF OPERATOR MATRICES Watheq Bani-Domi Depatment of Mathematics

More information

arxiv: v1 [math.co] 4 May 2017

arxiv: v1 [math.co] 4 May 2017 On The Numbe Of Unlabeled Bipatite Gaphs Abdullah Atmaca and A Yavuz Ouç axiv:7050800v [mathco] 4 May 207 Abstact This pape solves a poblem that was stated by M A Haison in 973 [] This poblem, that has

More information

A Short Combinatorial Proof of Derangement Identity arxiv: v1 [math.co] 13 Nov Introduction

A Short Combinatorial Proof of Derangement Identity arxiv: v1 [math.co] 13 Nov Introduction A Shot Combinatoial Poof of Deangement Identity axiv:1711.04537v1 [math.co] 13 Nov 2017 Ivica Matinjak Faculty of Science, Univesity of Zageb Bijenička cesta 32, HR-10000 Zageb, Coatia and Dajana Stanić

More information

New problems in universal algebraic geometry illustrated by boolean equations

New problems in universal algebraic geometry illustrated by boolean equations New poblems in univesal algebaic geomety illustated by boolean equations axiv:1611.00152v2 [math.ra] 25 Nov 2016 Atem N. Shevlyakov Novembe 28, 2016 Abstact We discuss new poblems in univesal algebaic

More information

Bounds for Codimensions of Fitting Ideals

Bounds for Codimensions of Fitting Ideals Ž. JOUNAL OF ALGEBA 194, 378 382 1997 ATICLE NO. JA966999 Bounds fo Coensions of Fitting Ideals Michał Kwiecinski* Uniwesytet Jagiellonski, Instytut Matematyki, ul. eymonta 4, 30-059, Kakow, Poland Communicated

More information

ONE-POINT CODES USING PLACES OF HIGHER DEGREE

ONE-POINT CODES USING PLACES OF HIGHER DEGREE ONE-POINT CODES USING PLACES OF HIGHER DEGREE GRETCHEN L. MATTHEWS AND TODD W. MICHEL DEPARTMENT OF MATHEMATICAL SCIENCES CLEMSON UNIVERSITY CLEMSON, SC 29634-0975 U.S.A. E-MAIL: GMATTHE@CLEMSON.EDU, TMICHEL@CLEMSON.EDU

More information

arxiv: v1 [math.co] 1 Apr 2011

arxiv: v1 [math.co] 1 Apr 2011 Weight enumeation of codes fom finite spaces Relinde Juius Octobe 23, 2018 axiv:1104.0172v1 [math.co] 1 Ap 2011 Abstact We study the genealized and extended weight enumeato of the - ay Simplex code and

More information

arxiv: v2 [math.ag] 4 Jul 2012

arxiv: v2 [math.ag] 4 Jul 2012 SOME EXAMPLES OF VECTOR BUNDLES IN THE BASE LOCUS OF THE GENERALIZED THETA DIVISOR axiv:0707.2326v2 [math.ag] 4 Jul 2012 SEBASTIAN CASALAINA-MARTIN, TAWANDA GWENA, AND MONTSERRAT TEIXIDOR I BIGAS Abstact.

More information

ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION. 1. Introduction. 1 r r. r k for every set E A, E \ {0},

ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION. 1. Introduction. 1 r r. r k for every set E A, E \ {0}, ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION E. J. IONASCU and A. A. STANCU Abstact. We ae inteested in constucting concete independent events in puely atomic pobability

More information

An Estimate of Incomplete Mixed Character Sums 1 2. Mei-Chu Chang 3. Dedicated to Endre Szemerédi for his 70th birthday.

An Estimate of Incomplete Mixed Character Sums 1 2. Mei-Chu Chang 3. Dedicated to Endre Szemerédi for his 70th birthday. An Estimate of Incomlete Mixed Chaacte Sums 2 Mei-Chu Chang 3 Dedicated to Ende Szemeédi fo his 70th bithday. 4 In this note we conside incomlete mixed chaacte sums ove a finite field F n of the fom x

More information

On decompositions of complete multipartite graphs into the union of two even cycles

On decompositions of complete multipartite graphs into the union of two even cycles On decompositions of complete multipatite gaphs into the union of two even cycles A. Su, J. Buchanan, R. C. Bunge, S. I. El-Zanati, E. Pelttai, G. Rasmuson, E. Spaks, S. Tagais Depatment of Mathematics

More information

Chapter 3: Theory of Modular Arithmetic 38

Chapter 3: Theory of Modular Arithmetic 38 Chapte 3: Theoy of Modula Aithmetic 38 Section D Chinese Remainde Theoem By the end of this section you will be able to pove the Chinese Remainde Theoem apply this theoem to solve simultaneous linea conguences

More information

Miskolc Mathematical Notes HU e-issn Tribonacci numbers with indices in arithmetic progression and their sums. Nurettin Irmak and Murat Alp

Miskolc Mathematical Notes HU e-issn Tribonacci numbers with indices in arithmetic progression and their sums. Nurettin Irmak and Murat Alp Miskolc Mathematical Notes HU e-issn 8- Vol. (0), No, pp. 5- DOI 0.85/MMN.0.5 Tibonacci numbes with indices in aithmetic pogession and thei sums Nuettin Imak and Muat Alp Miskolc Mathematical Notes HU

More information

9.1 The multiplicative group of a finite field. Theorem 9.1. The multiplicative group F of a finite field is cyclic.

9.1 The multiplicative group of a finite field. Theorem 9.1. The multiplicative group F of a finite field is cyclic. Chapte 9 Pimitive Roots 9.1 The multiplicative goup of a finite fld Theoem 9.1. The multiplicative goup F of a finite fld is cyclic. Remak: In paticula, if p is a pime then (Z/p) is cyclic. In fact, this

More information

Connectedness of Ordered Rings of Fractions of C(X) with the m-topology

Connectedness of Ordered Rings of Fractions of C(X) with the m-topology Filomat 31:18 (2017), 5685 5693 https://doi.og/10.2298/fil1718685s Published by Faculty o Sciences and Mathematics, Univesity o Niš, Sebia Available at: http://www.pm.ni.ac.s/ilomat Connectedness o Odeed

More information

A generalization of the Bernstein polynomials

A generalization of the Bernstein polynomials A genealization of the Benstein polynomials Halil Ouç and Geoge M Phillips Mathematical Institute, Univesity of St Andews, Noth Haugh, St Andews, Fife KY16 9SS, Scotland Dedicated to Philip J Davis This

More information

Introduction Common Divisors. Discrete Mathematics Andrei Bulatov

Introduction Common Divisors. Discrete Mathematics Andrei Bulatov Intoduction Common Divisos Discete Mathematics Andei Bulatov Discete Mathematics Common Divisos 3- Pevious Lectue Integes Division, popeties of divisibility The division algoithm Repesentation of numbes

More information

On the Quasi-inverse of a Non-square Matrix: An Infinite Solution

On the Quasi-inverse of a Non-square Matrix: An Infinite Solution Applied Mathematical Sciences, Vol 11, 2017, no 27, 1337-1351 HIKARI Ltd, wwwm-hikaicom https://doiog/1012988/ams20177273 On the Quasi-invese of a Non-squae Matix: An Infinite Solution Ruben D Codeo J

More information

Journal of Inequalities in Pure and Applied Mathematics

Journal of Inequalities in Pure and Applied Mathematics Jounal of Inequalities in Pue and Applied Mathematics COEFFICIENT INEQUALITY FOR A FUNCTION WHOSE DERIVATIVE HAS A POSITIVE REAL PART S. ABRAMOVICH, M. KLARIČIĆ BAKULA AND S. BANIĆ Depatment of Mathematics

More information

Asymptotically Lacunary Statistical Equivalent Sequence Spaces Defined by Ideal Convergence and an Orlicz Function

Asymptotically Lacunary Statistical Equivalent Sequence Spaces Defined by Ideal Convergence and an Orlicz Function "Science Stays Tue Hee" Jounal of Mathematics and Statistical Science, 335-35 Science Signpost Publishing Asymptotically Lacunay Statistical Equivalent Sequence Spaces Defined by Ideal Convegence and an

More information

SOME SOLVABILITY THEOREMS FOR NONLINEAR EQUATIONS

SOME SOLVABILITY THEOREMS FOR NONLINEAR EQUATIONS Fixed Point Theoy, Volume 5, No. 1, 2004, 71-80 http://www.math.ubbcluj.o/ nodeacj/sfptcj.htm SOME SOLVABILITY THEOREMS FOR NONLINEAR EQUATIONS G. ISAC 1 AND C. AVRAMESCU 2 1 Depatment of Mathematics Royal

More information

QUANTUM ALGORITHMS IN ALGEBRAIC NUMBER THEORY

QUANTUM ALGORITHMS IN ALGEBRAIC NUMBER THEORY QUANTU ALGORITHS IN ALGEBRAIC NUBER THEORY SION RUBINSTEIN-SALZEDO Abstact. In this aticle, we discuss some quantum algoithms fo detemining the goup of units and the ideal class goup of a numbe field.

More information

KOEBE DOMAINS FOR THE CLASSES OF FUNCTIONS WITH RANGES INCLUDED IN GIVEN SETS

KOEBE DOMAINS FOR THE CLASSES OF FUNCTIONS WITH RANGES INCLUDED IN GIVEN SETS Jounal of Applied Analysis Vol. 14, No. 1 2008), pp. 43 52 KOEBE DOMAINS FOR THE CLASSES OF FUNCTIONS WITH RANGES INCLUDED IN GIVEN SETS L. KOCZAN and P. ZAPRAWA Received Mach 12, 2007 and, in evised fom,

More information

arxiv: v1 [math.co] 6 Mar 2008

arxiv: v1 [math.co] 6 Mar 2008 An uppe bound fo the numbe of pefect matchings in gaphs Shmuel Fiedland axiv:0803.0864v [math.co] 6 Ma 2008 Depatment of Mathematics, Statistics, and Compute Science, Univesity of Illinois at Chicago Chicago,

More information

Measure Estimates of Nodal Sets of Polyharmonic Functions

Measure Estimates of Nodal Sets of Polyharmonic Functions Chin. Ann. Math. Se. B 39(5), 08, 97 93 DOI: 0.007/s40-08-004-6 Chinese Annals of Mathematics, Seies B c The Editoial Office of CAM and Spinge-Velag Belin Heidelbeg 08 Measue Estimates of Nodal Sets of

More information

On the ratio of maximum and minimum degree in maximal intersecting families

On the ratio of maximum and minimum degree in maximal intersecting families On the atio of maximum and minimum degee in maximal intesecting families Zoltán Lóánt Nagy Lale Özkahya Balázs Patkós Máté Vize Septembe 5, 011 Abstact To study how balanced o unbalanced a maximal intesecting

More information

ON THE INVERSE SIGNED TOTAL DOMINATION NUMBER IN GRAPHS. D.A. Mojdeh and B. Samadi

ON THE INVERSE SIGNED TOTAL DOMINATION NUMBER IN GRAPHS. D.A. Mojdeh and B. Samadi Opuscula Math. 37, no. 3 (017), 447 456 http://dx.doi.og/10.7494/opmath.017.37.3.447 Opuscula Mathematica ON THE INVERSE SIGNED TOTAL DOMINATION NUMBER IN GRAPHS D.A. Mojdeh and B. Samadi Communicated

More information

Available online through ISSN

Available online through  ISSN Intenational eseach Jounal of Pue Algeba -() 01 98-0 Available online though wwwjpainfo ISSN 8 907 SOE ESULTS ON THE GOUP INVESE OF BLOCK ATIX OVE IGHT OE DOAINS Hanyu Zhang* Goup of athematical Jidong

More information

Functions Defined on Fuzzy Real Numbers According to Zadeh s Extension

Functions Defined on Fuzzy Real Numbers According to Zadeh s Extension Intenational Mathematical Foum, 3, 2008, no. 16, 763-776 Functions Defined on Fuzzy Real Numbes Accoding to Zadeh s Extension Oma A. AbuAaqob, Nabil T. Shawagfeh and Oma A. AbuGhneim 1 Mathematics Depatment,

More information

On a generalization of Eulerian numbers

On a generalization of Eulerian numbers Notes on Numbe Theoy and Discete Mathematics Pint ISSN 1310 513, Online ISSN 367 875 Vol, 018, No 1, 16 DOI: 10756/nntdm018116- On a genealization of Euleian numbes Claudio Pita-Ruiz Facultad de Ingenieía,

More information

On Continued Fraction of Order Twelve

On Continued Fraction of Order Twelve Pue Mathematical Sciences, Vol. 1, 2012, no. 4, 197-205 On Continued Faction of Ode Twelve B. N. Dhamenda*, M. R. Rajesh Kanna* and R. Jagadeesh** *Post Gaduate Depatment of Mathematics Mahaani s Science

More information

Results on the Commutative Neutrix Convolution Product Involving the Logarithmic Integral li(

Results on the Commutative Neutrix Convolution Product Involving the Logarithmic Integral li( Intenational Jounal of Scientific and Innovative Mathematical Reseach (IJSIMR) Volume 2, Issue 8, August 2014, PP 736-741 ISSN 2347-307X (Pint) & ISSN 2347-3142 (Online) www.acjounals.og Results on the

More information

CONSTRUCTION OF EQUIENERGETIC GRAPHS

CONSTRUCTION OF EQUIENERGETIC GRAPHS MATCH Communications in Mathematical and in Compute Chemisty MATCH Commun. Math. Comput. Chem. 57 (007) 03-10 ISSN 0340-653 CONSTRUCTION OF EQUIENERGETIC GRAPHS H. S. Ramane 1, H. B. Walika * 1 Depatment

More information

On the Number of Rim Hook Tableaux. Sergey Fomin* and. Nathan Lulov. Department of Mathematics. Harvard University

On the Number of Rim Hook Tableaux. Sergey Fomin* and. Nathan Lulov. Department of Mathematics. Harvard University Zapiski Nauchn. Seminaov POMI, to appea On the Numbe of Rim Hook Tableaux Segey Fomin* Depatment of Mathematics, Massachusetts Institute of Technology Cambidge, MA 0239 Theoy of Algoithms Laboatoy SPIIRAN,

More information

Vanishing lines in generalized Adams spectral sequences are generic

Vanishing lines in generalized Adams spectral sequences are generic ISSN 364-0380 (on line) 465-3060 (pinted) 55 Geomety & Topology Volume 3 (999) 55 65 Published: 2 July 999 G G G G T T T G T T T G T G T GG TT G G G G GG T T T TT Vanishing lines in genealized Adams spectal

More information

CENTRAL INDEX BASED SOME COMPARATIVE GROWTH ANALYSIS OF COMPOSITE ENTIRE FUNCTIONS FROM THE VIEW POINT OF L -ORDER. Tanmay Biswas

CENTRAL INDEX BASED SOME COMPARATIVE GROWTH ANALYSIS OF COMPOSITE ENTIRE FUNCTIONS FROM THE VIEW POINT OF L -ORDER. Tanmay Biswas J Koean Soc Math Educ Se B: Pue Appl Math ISSNPint 16-0657 https://doiog/107468/jksmeb01853193 ISSNOnline 87-6081 Volume 5, Numbe 3 August 018, Pages 193 01 CENTRAL INDEX BASED SOME COMPARATIVE GROWTH

More information

Secret Exponent Attacks on RSA-type Schemes with Moduli N = p r q

Secret Exponent Attacks on RSA-type Schemes with Moduli N = p r q Secet Exponent Attacks on RSA-type Schemes with Moduli N = p q Alexande May Faculty of Compute Science, Electical Engineeing and Mathematics Univesity of Padebon 33102 Padebon, Gemany alexx@uni-padebon.de

More information

EM Boundary Value Problems

EM Boundary Value Problems EM Bounday Value Poblems 10/ 9 11/ By Ilekta chistidi & Lee, Seung-Hyun A. Geneal Desciption : Maxwell Equations & Loentz Foce We want to find the equations of motion of chaged paticles. The way to do

More information

On the ratio of maximum and minimum degree in maximal intersecting families

On the ratio of maximum and minimum degree in maximal intersecting families On the atio of maximum and minimum degee in maximal intesecting families Zoltán Lóánt Nagy Lale Özkahya Balázs Patkós Máté Vize Mach 6, 013 Abstact To study how balanced o unbalanced a maximal intesecting

More information

ON SPARSELY SCHEMMEL TOTIENT NUMBERS. Colin Defant 1 Department of Mathematics, University of Florida, Gainesville, Florida

ON SPARSELY SCHEMMEL TOTIENT NUMBERS. Colin Defant 1 Department of Mathematics, University of Florida, Gainesville, Florida #A8 INTEGERS 5 (205) ON SPARSEL SCHEMMEL TOTIENT NUMBERS Colin Defant Depatment of Mathematics, Univesity of Floida, Gainesville, Floida cdefant@ufl.edu Received: 7/30/4, Revised: 2/23/4, Accepted: 4/26/5,

More information

Semicanonical basis generators of the cluster algebra of type A (1)

Semicanonical basis generators of the cluster algebra of type A (1) Semicanonical basis geneatos of the cluste algeba of type A (1 1 Andei Zelevinsky Depatment of Mathematics Notheasten Univesity, Boston, USA andei@neu.edu Submitted: Jul 7, 006; Accepted: Dec 3, 006; Published:

More information

The Congestion of n-cube Layout on a Rectangular Grid S.L. Bezrukov J.D. Chavez y L.H. Harper z M. Rottger U.-P. Schroeder Abstract We consider the pr

The Congestion of n-cube Layout on a Rectangular Grid S.L. Bezrukov J.D. Chavez y L.H. Harper z M. Rottger U.-P. Schroeder Abstract We consider the pr The Congestion of n-cube Layout on a Rectangula Gid S.L. Bezukov J.D. Chavez y L.H. Hape z M. Rottge U.-P. Schoede Abstact We conside the poblem of embedding the n-dimensional cube into a ectangula gid

More information

A STUDY OF HAMMING CODES AS ERROR CORRECTING CODES

A STUDY OF HAMMING CODES AS ERROR CORRECTING CODES AGU Intenational Jounal of Science and Technology A STUDY OF HAMMING CODES AS ERROR CORRECTING CODES Ritu Ahuja Depatment of Mathematics Khalsa College fo Women, Civil Lines, Ludhiana-141001, Punjab, (India)

More information

Construction and Analysis of Boolean Functions of 2t + 1 Variables with Maximum Algebraic Immunity

Construction and Analysis of Boolean Functions of 2t + 1 Variables with Maximum Algebraic Immunity Constuction and Analysis of Boolean Functions of 2t + 1 Vaiables with Maximum Algebaic Immunity Na Li and Wen-Feng Qi Depatment of Applied Mathematics, Zhengzhou Infomation Engineeing Univesity, Zhengzhou,

More information

Complements of plane curves with logarithmic Kodaira dimension zero

Complements of plane curves with logarithmic Kodaira dimension zero J. Math. Soc. Japan Vol. 52, No. 4, 2000 Complements of plane cuves with logaithmic Kodaia dimension zeo By Hideo Kojima (Received Ap. 14, 1999) Abstact. We pove that logaithmic geometic genus of a complement

More information

ON LACUNARY INVARIANT SEQUENCE SPACES DEFINED BY A SEQUENCE OF MODULUS FUNCTIONS

ON LACUNARY INVARIANT SEQUENCE SPACES DEFINED BY A SEQUENCE OF MODULUS FUNCTIONS STUDIA UNIV BABEŞ BOLYAI, MATHEMATICA, Volume XLVIII, Numbe 4, Decembe 2003 ON LACUNARY INVARIANT SEQUENCE SPACES DEFINED BY A SEQUENCE OF MODULUS FUNCTIONS VATAN KARAKAYA AND NECIP SIMSEK Abstact The

More information

I. CONSTRUCTION OF THE GREEN S FUNCTION

I. CONSTRUCTION OF THE GREEN S FUNCTION I. CONSTRUCTION OF THE GREEN S FUNCTION The Helmohltz equation in 4 dimensions is 4 + k G 4 x, x = δ 4 x x. In this equation, G is the Geen s function and 4 efes to the dimensionality. In the vey end,

More information

STUDY OF SOLUTIONS OF LOGARITHMIC ORDER TO HIGHER ORDER LINEAR DIFFERENTIAL-DIFFERENCE EQUATIONS WITH COEFFICIENTS HAVING THE SAME LOGARITHMIC ORDER

STUDY OF SOLUTIONS OF LOGARITHMIC ORDER TO HIGHER ORDER LINEAR DIFFERENTIAL-DIFFERENCE EQUATIONS WITH COEFFICIENTS HAVING THE SAME LOGARITHMIC ORDER UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA doi: 104467/20843828AM170027078 542017, 15 32 STUDY OF SOLUTIONS OF LOGARITHMIC ORDER TO HIGHER ORDER LINEAR DIFFERENTIAL-DIFFERENCE EQUATIONS WITH COEFFICIENTS

More information

A Bijective Approach to the Permutational Power of a Priority Queue

A Bijective Approach to the Permutational Power of a Priority Queue A Bijective Appoach to the Pemutational Powe of a Pioity Queue Ia M. Gessel Kuang-Yeh Wang Depatment of Mathematics Bandeis Univesity Waltham, MA 02254-9110 Abstact A pioity queue tansfoms an input pemutation

More information

arxiv: v1 [math.nt] 28 Oct 2017

arxiv: v1 [math.nt] 28 Oct 2017 ON th COEFFICIENT OF DIVISORS OF x n axiv:70049v [mathnt] 28 Oct 207 SAI TEJA SOMU Abstact Let,n be two natual numbes and let H(,n denote the maximal absolute value of th coefficient of divisos of x n

More information

Chaos and bifurcation of discontinuous dynamical systems with piecewise constant arguments

Chaos and bifurcation of discontinuous dynamical systems with piecewise constant arguments Malaya Jounal of Matematik ()(22) 4 8 Chaos and bifucation of discontinuous dynamical systems with piecewise constant aguments A.M.A. El-Sayed, a, and S. M. Salman b a Faculty of Science, Aleandia Univesity,

More information

Compactly Supported Radial Basis Functions

Compactly Supported Radial Basis Functions Chapte 4 Compactly Suppoted Radial Basis Functions As we saw ealie, compactly suppoted functions Φ that ae tuly stictly conditionally positive definite of ode m > do not exist The compact suppot automatically

More information

arxiv: v1 [math.nt] 12 May 2017

arxiv: v1 [math.nt] 12 May 2017 SEQUENCES OF CONSECUTIVE HAPPY NUMBERS IN NEGATIVE BASES HELEN G. GRUNDMAN AND PAMELA E. HARRIS axiv:1705.04648v1 [math.nt] 12 May 2017 ABSTRACT. Fo b 2 and e 2, let S e,b : Z Z 0 be the function taking

More information

On the Poisson Approximation to the Negative Hypergeometric Distribution

On the Poisson Approximation to the Negative Hypergeometric Distribution BULLETIN of the Malaysian Mathematical Sciences Society http://mathusmmy/bulletin Bull Malays Math Sci Soc (2) 34(2) (2011), 331 336 On the Poisson Appoximation to the Negative Hypegeometic Distibution

More information

Journal of Number Theory

Journal of Number Theory Jounal of umbe Theoy 3 2 2259 227 Contents lists available at ScienceDiect Jounal of umbe Theoy www.elsevie.com/locate/jnt Sums of poducts of hypegeometic Benoulli numbes Ken Kamano Depatment of Geneal

More information

Lacunary I-Convergent Sequences

Lacunary I-Convergent Sequences KYUNGPOOK Math. J. 52(2012), 473-482 http://dx.doi.og/10.5666/kmj.2012.52.4.473 Lacunay I-Convegent Sequences Binod Chanda Tipathy Mathematical Sciences Division, Institute of Advanced Study in Science

More information

A NOTE ON ROTATIONS AND INTERVAL EXCHANGE TRANSFORMATIONS ON 3-INTERVALS KARMA DAJANI

A NOTE ON ROTATIONS AND INTERVAL EXCHANGE TRANSFORMATIONS ON 3-INTERVALS KARMA DAJANI A NOTE ON ROTATIONS AND INTERVAL EXCHANGE TRANSFORMATIONS ON 3-INTERVALS KARMA DAJANI Abstact. Wepove the conjectue that an inteval exchange tansfomation on 3-intevals with coesponding pemutation (1; 2;

More information

Fixed Argument Pairing Inversion on Elliptic Curves

Fixed Argument Pairing Inversion on Elliptic Curves Fixed Agument Paiing Invesion on Elliptic Cuves Sungwook Kim and Jung Hee Cheon ISaC & Dept. of Mathematical Sciences Seoul National Univesity Seoul, Koea {avell7,jhcheon}@snu.ac.k Abstact. Let E be an

More information

Research Article On Alzer and Qiu s Conjecture for Complete Elliptic Integral and Inverse Hyperbolic Tangent Function

Research Article On Alzer and Qiu s Conjecture for Complete Elliptic Integral and Inverse Hyperbolic Tangent Function Abstact and Applied Analysis Volume 011, Aticle ID 697547, 7 pages doi:10.1155/011/697547 Reseach Aticle On Alze and Qiu s Conjectue fo Complete Elliptic Integal and Invese Hypebolic Tangent Function Yu-Ming

More information

THE JEU DE TAQUIN ON THE SHIFTED RIM HOOK TABLEAUX. Jaejin Lee

THE JEU DE TAQUIN ON THE SHIFTED RIM HOOK TABLEAUX. Jaejin Lee Koean J. Math. 23 (2015), No. 3, pp. 427 438 http://dx.doi.og/10.11568/kjm.2015.23.3.427 THE JEU DE TAQUIN ON THE SHIFTED RIM HOOK TABLEAUX Jaejin Lee Abstact. The Schensted algoithm fist descibed by Robinson

More information

A THREE CRITICAL POINTS THEOREM AND ITS APPLICATIONS TO THE ORDINARY DIRICHLET PROBLEM

A THREE CRITICAL POINTS THEOREM AND ITS APPLICATIONS TO THE ORDINARY DIRICHLET PROBLEM A THREE CRITICAL POINTS THEOREM AND ITS APPLICATIONS TO THE ORDINARY DIRICHLET PROBLEM DIEGO AVERNA AND GABRIELE BONANNO Abstact. The aim of this pape is twofold. On one hand we establish a thee citical

More information

This aticle was oiginally published in a jounal published by Elsevie, the attached copy is povided by Elsevie fo the autho s benefit fo the benefit of the autho s institution, fo non-commecial eseach educational

More information

COLLAPSING WALLS THEOREM

COLLAPSING WALLS THEOREM COLLAPSING WALLS THEOREM IGOR PAK AND ROM PINCHASI Abstact. Let P R 3 be a pyamid with the base a convex polygon Q. We show that when othe faces ae collapsed (otated aound the edges onto the plane spanned

More information

Quasi-Randomness and the Distribution of Copies of a Fixed Graph

Quasi-Randomness and the Distribution of Copies of a Fixed Graph Quasi-Randomness and the Distibution of Copies of a Fixed Gaph Asaf Shapia Abstact We show that if a gaph G has the popety that all subsets of vetices of size n/4 contain the coect numbe of tiangles one

More information

Asymmetric Quantum Codes on Toric Surfaces

Asymmetric Quantum Codes on Toric Surfaces Institute of Mathematics Aahus Univesity Octobe 1, 2017 Abstact Asymmetic quantum eo-coecting codes ae quantum codes defined ove biased quantum channels: qubit-flip and phase-shift eos may have equal o

More information

Fixed Point Results for Multivalued Maps

Fixed Point Results for Multivalued Maps Int. J. Contemp. Math. Sciences, Vol., 007, no. 3, 119-1136 Fixed Point Results fo Multivalued Maps Abdul Latif Depatment of Mathematics King Abdulaziz Univesity P.O. Box 8003, Jeddah 1589 Saudi Aabia

More information

Chromatic number and spectral radius

Chromatic number and spectral radius Linea Algeba and its Applications 426 2007) 810 814 www.elsevie.com/locate/laa Chomatic numbe and spectal adius Vladimi Nikifoov Depatment of Mathematical Sciences, Univesity of Memphis, Memphis, TN 38152,

More information

(received April 9, 1967) Let p denote a prime number and let k P

(received April 9, 1967) Let p denote a prime number and let k P ON EXTREMAL OLYNOMIALS Kenneth S. Williams (eceived Apil 9, 1967) Let p denote a pime numbe and let k denote the finite field of p elements. Let f(x) E k [x] be of fixed degee d 2 2. We suppose that p

More information

q i i=1 p i ln p i Another measure, which proves a useful benchmark in our analysis, is the chi squared divergence of p, q, which is defined by

q i i=1 p i ln p i Another measure, which proves a useful benchmark in our analysis, is the chi squared divergence of p, q, which is defined by CSISZÁR f DIVERGENCE, OSTROWSKI S INEQUALITY AND MUTUAL INFORMATION S. S. DRAGOMIR, V. GLUŠČEVIĆ, AND C. E. M. PEARCE Abstact. The Ostowski integal inequality fo an absolutely continuous function is used

More information

THE MAXIMUM SIZE OF A PARTIAL SPREAD II: UPPER BOUNDS

THE MAXIMUM SIZE OF A PARTIAL SPREAD II: UPPER BOUNDS THE MAXIMUM SIZE OF A PARTIAL SPREAD II: UPPER BOUNDS ESMERALDA NĂSTASE MATHEMATICS DEPARTMENT XAVIER UNIVERSITY CINCINNATI, OHIO 4507, USA PAPA SISSOKHO MATHEMATICS DEPARTMENT ILLINOIS STATE UNIVERSITY

More information

H.W.GOULD West Virginia University, Morgan town, West Virginia 26506

H.W.GOULD West Virginia University, Morgan town, West Virginia 26506 A F I B O N A C C I F O R M U L A OF LUCAS A N D ITS SUBSEQUENT M A N I F E S T A T I O N S A N D R E D I S C O V E R I E S H.W.GOULD West Viginia Univesity, Mogan town, West Viginia 26506 Almost eveyone

More information

Do Managers Do Good With Other People s Money? Online Appendix

Do Managers Do Good With Other People s Money? Online Appendix Do Manages Do Good With Othe People s Money? Online Appendix Ing-Haw Cheng Haison Hong Kelly Shue Abstact This is the Online Appendix fo Cheng, Hong and Shue 2013) containing details of the model. Datmouth

More information

SPECTRAL SEQUENCES. im(er

SPECTRAL SEQUENCES. im(er SPECTRAL SEQUENCES MATTHEW GREENBERG. Intoduction Definition. Let a. An a-th stage spectal (cohomological) sequence consists of the following data: bigaded objects E = p,q Z Ep,q, a diffeentials d : E

More information

HE DI ELMONSER. 1. Introduction In 1964 H. Mink and L. Sathre [15] proved the following inequality. n, n N. ((n + 1)!) n+1

HE DI ELMONSER. 1. Introduction In 1964 H. Mink and L. Sathre [15] proved the following inequality. n, n N. ((n + 1)!) n+1 -ANALOGUE OF THE ALZER S INEQUALITY HE DI ELMONSER Abstact In this aticle, we ae inteested in giving a -analogue of the Alze s ineuality Mathematics Subject Classification (200): 26D5 Keywods: Alze s ineuality;

More information

arxiv: v1 [math.ca] 12 Mar 2015

arxiv: v1 [math.ca] 12 Mar 2015 axiv:503.0356v [math.ca] 2 Ma 205 AN APPLICATION OF FOURIER ANALYSIS TO RIEMANN SUMS TRISTRAM DE PIRO Abstact. We develop a method fo calculating Riemann sums using Fouie analysis.. Poisson Summation Fomula

More information

Analytical Solutions for Confined Aquifers with non constant Pumping using Computer Algebra

Analytical Solutions for Confined Aquifers with non constant Pumping using Computer Algebra Poceedings of the 006 IASME/SEAS Int. Conf. on ate Resouces, Hydaulics & Hydology, Chalkida, Geece, May -3, 006 (pp7-) Analytical Solutions fo Confined Aquifes with non constant Pumping using Compute Algeba

More information

The Archimedean Circles of Schoch and Woo

The Archimedean Circles of Schoch and Woo Foum Geometicoum Volume 4 (2004) 27 34. FRUM GEM ISSN 1534-1178 The Achimedean Cicles of Schoch and Woo Hioshi kumua and Masayuki Watanabe Abstact. We genealize the Achimedean cicles in an abelos (shoemake

More information

Solution to HW 3, Ma 1a Fall 2016

Solution to HW 3, Ma 1a Fall 2016 Solution to HW 3, Ma a Fall 206 Section 2. Execise 2: Let C be a subset of the eal numbes consisting of those eal numbes x having the popety that evey digit in the decimal expansion of x is, 3, 5, o 7.

More information

Application of Fractional Calculus Operators to Related Areas

Application of Fractional Calculus Operators to Related Areas Gen. Math. Notes, Vol. 7, No., Novembe 2, pp. 33-4 ISSN 229-784; Copyight ICSRS Publication, 2 www.i-css.og Available fee online at http://www.geman.in Application of Factional Calculus Opeatos to Related

More information

Numerical approximation to ζ(2n+1)

Numerical approximation to ζ(2n+1) Illinois Wesleyan Univesity Fom the SelectedWoks of Tian-Xiao He 6 Numeical appoximation to ζ(n+1) Tian-Xiao He, Illinois Wesleyan Univesity Michael J. Dancs Available at: https://woks.bepess.com/tian_xiao_he/6/

More information

Weighted least-squares estimators of parametric functions of the regression coefficients under a general linear model

Weighted least-squares estimators of parametric functions of the regression coefficients under a general linear model Ann Inst Stat Math (2010) 62:929 941 DOI 10.1007/s10463-008-0199-8 Weighted least-squaes estimatos of paametic functions of the egession coefficients unde a geneal linea model Yongge Tian Received: 9 Januay

More information

JANOWSKI STARLIKE LOG-HARMONIC UNIVALENT FUNCTIONS

JANOWSKI STARLIKE LOG-HARMONIC UNIVALENT FUNCTIONS Hacettepe Jounal of Mathematics and Statistics Volume 38 009, 45 49 JANOWSKI STARLIKE LOG-HARMONIC UNIVALENT FUNCTIONS Yaşa Polatoğlu and Ehan Deniz Received :0 :008 : Accepted 0 : :008 Abstact Let and

More information

Duality between Statical and Kinematical Engineering Systems

Duality between Statical and Kinematical Engineering Systems Pape 00, Civil-Comp Ltd., Stiling, Scotland Poceedings of the Sixth Intenational Confeence on Computational Stuctues Technology, B.H.V. Topping and Z. Bittna (Editos), Civil-Comp Pess, Stiling, Scotland.

More information

Galois points on quartic surfaces

Galois points on quartic surfaces J. Math. Soc. Japan Vol. 53, No. 3, 2001 Galois points on quatic sufaces By Hisao Yoshihaa (Received Nov. 29, 1999) (Revised Ma. 30, 2000) Abstact. Let S be a smooth hypesuface in the pojective thee space

More information

Several new identities involving Euler and Bernoulli polynomials

Several new identities involving Euler and Bernoulli polynomials Bull. Math. Soc. Sci. Math. Roumanie Tome 9107 No. 1, 016, 101 108 Seveal new identitie involving Eule and Benoulli polynomial by Wang Xiaoying and Zhang Wenpeng Abtact The main pupoe of thi pape i uing

More information

VOLUMES OF CONVEX POLYTOPES

VOLUMES OF CONVEX POLYTOPES VOLUMES OF CONVEX POLYTOPES Richad P. Stanley Depatment of Mathematics M.I.T. 2-375 Cambidge, MA 02139 stan@math.mit.edu http://www-math.mit.edu/~stan Tanspaencies available at: http://www-math.mit.edu/~stan/tans.html

More information

SUFFICIENT CONDITIONS FOR MAXIMALLY EDGE-CONNECTED AND SUPER-EDGE-CONNECTED GRAPHS DEPENDING ON THE CLIQUE NUMBER

SUFFICIENT CONDITIONS FOR MAXIMALLY EDGE-CONNECTED AND SUPER-EDGE-CONNECTED GRAPHS DEPENDING ON THE CLIQUE NUMBER Discussiones Mathematicae Gaph Theoy 39 (019) 567 573 doi:10.7151/dmgt.096 SUFFICIENT CONDITIONS FOR MAXIMALLY EDGE-CONNECTED AND SUPER-EDGE-CONNECTED GRAPHS DEPENDING ON THE CLIQUE NUMBER Lutz Volkmann

More information

On Polynomials Construction

On Polynomials Construction Intenational Jounal of Mathematical Analysis Vol., 08, no. 6, 5-57 HIKARI Ltd, www.m-hikai.com https://doi.og/0.988/ima.08.843 On Polynomials Constuction E. O. Adeyefa Depatment of Mathematics, Fedeal

More information

arxiv: v1 [math.ca] 31 Aug 2009

arxiv: v1 [math.ca] 31 Aug 2009 axiv:98.4578v [math.ca] 3 Aug 9 On L-convegence of tigonometic seies Bogdan Szal Univesity of Zielona Góa, Faculty of Mathematics, Compute Science and Econometics, 65-56 Zielona Góa, ul. Szafana 4a, Poland

More information

titrrvers:rtt t>1 NO~~H CAROLINA

titrrvers:rtt t>1 NO~~H CAROLINA titvers:tt t>1 NO~~H CAROLINA Depatment of statistics Chapel Hill, N. C. ON A BOUN.D USEFUL IN THE THEORY OF FACTORIAL DESIGNS AND ERROR CORRECTING CODES by R. C. Bose and J. N. Sivastava Apil 1963 Gant

More information

Suborbital graphs for the group Γ 2

Suborbital graphs for the group Γ 2 Hacettepe Jounal of Mathematics and Statistics Volume 44 5 2015, 1033 1044 Subobital gaphs fo the goup Γ 2 Bahadı Özgü Güle, Muat Beşenk, Yavuz Kesicioğlu, Ali Hikmet Değe Keywods: Abstact In this pape,

More information

Lecture 18: Graph Isomorphisms

Lecture 18: Graph Isomorphisms INFR11102: Computational Complexity 22/11/2018 Lectue: Heng Guo Lectue 18: Gaph Isomophisms 1 An Athu-Melin potocol fo GNI Last time we gave a simple inteactive potocol fo GNI with pivate coins. We will

More information

On absence of solutions of a semi-linear elliptic equation with biharmonic operator in the exterior of a ball

On absence of solutions of a semi-linear elliptic equation with biharmonic operator in the exterior of a ball Tansactions of NAS of Azebaijan, Issue Mathematics, 36, 63-69 016. Seies of Physical-Technical and Mathematical Sciences. On absence of solutions of a semi-linea elliptic euation with bihamonic opeato

More information

We give improved upper bounds for the number of primitive solutions of the Thue inequality

We give improved upper bounds for the number of primitive solutions of the Thue inequality NUMBER OF SOLUTIONS OF CUBIC THUE INEQUALITIES WITH POSITIVE DISCRIMINANT N SARADHA AND DIVYUM SHARMA Abstact Let F(X, Y) be an ieducible binay cubic fom with intege coefficients and positive disciminant

More information

Multiple Criteria Secretary Problem: A New Approach

Multiple Criteria Secretary Problem: A New Approach J. Stat. Appl. Po. 3, o., 9-38 (04 9 Jounal of Statistics Applications & Pobability An Intenational Jounal http://dx.doi.og/0.785/jsap/0303 Multiple Citeia Secetay Poblem: A ew Appoach Alaka Padhye, and

More information

Double sequences of interval numbers defined by Orlicz functions

Double sequences of interval numbers defined by Orlicz functions ACTA ET COENTATIONES UNIVERSITATIS TARTUENSIS DE ATHEATICA Volume 7, Numbe, June 203 Available online at www.math.ut.ee/acta/ Double sequences of inteval numbes efine by Olicz functions Ayhan Esi Abstact.

More information

Syntactical content of nite approximations of partial algebras 1 Wiktor Bartol Inst. Matematyki, Uniw. Warszawski, Warszawa (Poland)

Syntactical content of nite approximations of partial algebras 1 Wiktor Bartol Inst. Matematyki, Uniw. Warszawski, Warszawa (Poland) Syntactical content of nite appoximations of patial algebas 1 Wikto Batol Inst. Matematyki, Uniw. Waszawski, 02-097 Waszawa (Poland) batol@mimuw.edu.pl Xavie Caicedo Dep. Matematicas, Univ. de los Andes,

More information

Math 301: The Erdős-Stone-Simonovitz Theorem and Extremal Numbers for Bipartite Graphs

Math 301: The Erdős-Stone-Simonovitz Theorem and Extremal Numbers for Bipartite Graphs Math 30: The Edős-Stone-Simonovitz Theoem and Extemal Numbes fo Bipatite Gaphs May Radcliffe The Edős-Stone-Simonovitz Theoem Recall, in class we poved Tuán s Gaph Theoem, namely Theoem Tuán s Theoem Let

More information

Banach Journal of Mathematical Analysis ISSN: (electronic)

Banach Journal of Mathematical Analysis ISSN: (electronic) Banach J. Math. Anal. 4 200), no., 87 9 Banach Jounal of Mathematical Analysis ISSN: 75-8787 electonic) www.emis.de/jounals/bjma/ ON A REVERSE OF ANDO HIAI INEQUALITY YUKI SEO This pape is dedicated to

More information

Gradient-based Neural Network for Online Solution of Lyapunov Matrix Equation with Li Activation Function

Gradient-based Neural Network for Online Solution of Lyapunov Matrix Equation with Li Activation Function Intenational Confeence on Infomation echnology and Management Innovation (ICIMI 05) Gadient-based Neual Netwok fo Online Solution of Lyapunov Matix Equation with Li Activation unction Shiheng Wang, Shidong

More information