Theory of Generalized k-difference Operator and Its Application in Number Theory
|
|
- Bertram Hensley
- 5 years ago
- Views:
Transcription
1 Internationa Journa of Mathematica Anaysis Vo. 9, 2015, no. 19, HIKARI Ltd, Theory of Generaized -Difference Operator and Its Appication in Number Theory V. Chandrasear Department of Mathematics, S.K.P. Engineering Coege Tiruvannamaai , Tami Nadu, S. India G. Britto Antony Xavier Department of Mathematics, Sacred Heart Coege Tirupattur , Tami Nadu, S. India R. Vijayaraj Department of Mathematics, Arunai Engineering Coege Tiruvannamaai , Tami Nadu, S. India Copyright c 2014 V. Chandrasear, G. Britto Antony Xavier and R.Vijiyaraj. This is an open access artice distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the origina wor is propery cited. Abstract In this paper, we define the generaized -difference operator and present the discrete version of Leibnitz theorem according to the generaized -difference operator. Aso, by defining its inverse, we obtain the sum of product of arithmetic and geometric progressions in the fied of Numerica Anaysis. Mathematics Subject Cassification: 39A70, 47B39, 97N40 Keywords: Factoria -Difference Operator, Generaized Factoria, Poynomia
2 956 V. Chandrasear, G. Britto Antony Xavier and R. Vijayaraj 1 Introduction The theory of Difference equations is based on the operator defined as u) = u + 1) u), N = {0, 1, 2,...}. 1) Even though many authors 1, 7 9 have suggested the definition of the difference operator as u) = u + ) u), 0, ), 2) no significant progress too pace on this ine. But, in , by taing the definition of as given in 2), the theory of difference equations was deveoped in a different direction and many interesting resuts were obtained in Number Theory 4. Jerzy Popenda 2, whie discussing the behavior of soutions of particuar type of difference equation, defined α as α u) = u + 1) αu). 3) This definition of α is being ignored for a ong time. But, recenty, M.M.S.Manue, V.Chandrasear and G.Britto Antony Xavier 5, 6, have generaized the definition of α by α) defined as α) u) = u + ) αu) 4) for the rea vaued function u) and 0, ) and aso obtained the soutions of certain type of generaized α-difference equations, in particuar generaized cairaut s α-difference equation, generaized Euer α-difference equation and formua for the sum of severa types of arithmetic-geometric progressions in Number Theory using the generaized α-bernoui poynomia, which is a soution of the generaized α-difference equation as u + ) αu) = n n 1, for n N1). With this bacground, in this paper, we define the generaized -difference operator and obtain some resuts, reations and theorems. Aso, we derive the formua for sum of product of factorias and arithmetic progressions in the fied of Numerica Anaysis using the inverse of generaized -difference operator. Throughout this paper, the foowing assumptions are used: i) N = ){0, 1, 2, 3,...}, ii) N j) = {j, j +, j + 2,...}, n iii) r v) n! = r!n r)!, iv) 1 is the integer part of and ) u) = ) u) ) ) ) uj), j is the upper integer part of.
3 -Difference operator and its appication in number theory Generaized -Difference Operator and its Reation with and E In this section, we present some basic definitions and preiminary resuts which wi be usefu for further subsequent discussions. Definition 2.1 Let u) be a rea vaued function defined on 0, ). Then the generaized -difference operator ) is defined as ) u) = u + ) u), 0, ), 0, ). 5) Remar 2.2 i) If α is a constant, then the generaized α-difference operator α) assumes the reation α) u) = u + ) αu). ii) Repacing by ± n in 5), we obtain ) u ± n) = u ± n + ) u ± n), for appropriate choosing n N. The foowing emma is the reation between ), and E. Lemma 2.3 i) If E is the usua shift operator, then E u) = ) + )u) = 1 + ) u) = 1 + )u). 6) ii) If m is positive integer, then + m) )u) = m i=0 ) m i u). 7) i iii) If c 1, c 2 are any two non-zero scaars and u), v) are any two rea vaued functions defined on 0, ), then ) c 1 u) + c 2 v) = c 1 ) u) + c 2 ) v). iv) If 0, ) and n is a positive integer, then n )u) = n v) If is, i = 1, 2,...n are positive reas, then ) n ) r u + n r)). 8) r n))u) = n + i ))u). 9) i=1 vi) If n is a positive integer, then n) u) = 1 + ) n )u). 10)
4 958 V. Chandrasear, G. Britto Antony Xavier and R. Vijayaraj The foowing is the discrete version of Leibnitz theorem according to ). Theorem 2.4 If u) and v), 0, ), are any two rea vaued functions, then n )u)v) = n Proof. Define the shift operator E 1 and E 2 as ) n n r r r )u) ) n r v + r) ). E 1u)v) = u + )v) and E 2u)v) = u)v + ). 11) Hence, we obtain Let us denote E = E 1E 2. 12) ) ) 1 = E 1 and ) ) 2 = E 2 13) for the functions u) and v), respectivey. This impies ) = E 1E 2. From 6), we obtain ) = ) ) 1 E 2 ) 2, where E 2 1 = ) 2 2, Theorem 2.5). Hence, n )u)v) = ) 2 + ) ) 1 E 2 n u)v). 14) The proof foows by Binomia Theorem, 11), 13) and 14). Lemma 2.5 If 0, ) and n is a positive integer, then E n u) = n ) n n r r r )u). 15) Proof. The proof foows from 6) and Binomia Theorem. The foowing is discrete version of generaized Binomia theorem invoving ). Theorem 2.6 If m and n are any two positive integres, then + n) m = n r ) ) n r 1) n r n+t r + n t) m. 16) r t t=0 Proof. The proof foows by taing u) = m in 15) and using 8).
5 -Difference operator and its appication in number theory 959 Lemma 2.7 Let u), 0, ), be a rea vaued function and 0, ). Then x r ur) = e x x ) r! r e u0). 17) Proof. From the shift operator, E r u0) = ur) and binomia theorem, we find e x E x r ur) u0) =. 18) r! r The proof foows from 6) and 18). The foowing theorem is the generaized version of Montmorte s theorem with reference to ). Theorem 2.8 If the series u)x, expressed as =0 u)x = =0 =0 Proof. From the shift operator E, we have N converges, then it can be x ) u0). 19) 1 x ) +1 u)x = =0 x E u0) = 1 x E ) 1 u0). =0 19) foows from 6) and the above reation. 3 The Generaized Factoria In this section, we define the generaized factoria and obtain the reation between the generaized -difference operator and generaized factoria. Definition 3.1 If 0, ) and, ), then the Generaized Factoria denoted by )! is defined as where j =. )! = ) 2) j, 20) Theorem 3.2 Let 0, ) and, ). Then Proof. The proof foows from 5) and 20). ) )! = )! 21)
6 960 V. Chandrasear, G. Britto Antony Xavier and R. Vijayaraj Lemma 3.3 Let n) = ) 2)... n 1)) be the generaized poynomia factoria. Then ) n) Proof. The proof foows from 5) and n 1) = n 1) 1 ) n 1)). 22) = ) 2) n). Theorem 3.4 If m and n are any two positive integer, then m ) n+m i) m i r t m i t=1 m ) ) n = n + m i) r t t=1 r m+1 i i=1 r m+1 i =1 r m+1 i n+m 1) m r p p=1 Proof. The proof foows by induction on m. ) +1 Coroary 3.5 If n is any positive integer, then ) ) ) ) n n +1 =. 23) t=1 n t ) n t t. 24) Proof. The proof foows by substituting m = 1 in 23). Note that when is mutipe of, ) ) = 0 and +1) = 0, hence bothsides of 24) are zero. 4 Inverse of the Operator ) In this section, we define the inverse of the operator ) and obtaining some theorems invoving the inverse of generaized -difference operator. Definition 4.1 The inverse of the generaized -difference operator is denoted by ) is defined as foows. If )v) = u), then ) u) = v) ) ) vj), 25) where vj) is constant for a N j). Theorem 4.2 If 0, ) and, ), then ) u) j = ) r 1) u r). 26)
7 -Difference operator and its appication in number theory 961 Proof. The proof foows from Definition 4.1, and the reation ) ) r 1) u r) = u). Theorem 4.3 Let λ 1, 2 and p) be a function in. Then ) r 1) λ r P r) { } λ = λ 1) λ+ ) 2 λ 1) + λ+2 2 ) 2 3 λ 1) + λ+3 3 ) 3 4 λ 1) + P ). 4 j Proof. For the function F ), we find hence, we obtain 27) ) λ F ) = λ λ E )F ) = λ P ), 28) λ E ) 1 P ) = F ). 29) From 28) and Definition 4.1, we find ) λ P )) = λ ) F ) ) λ j F j), and hence by 29), we obtain ) λ P )) = λ λ E ) 1 ) P ) ) λ j λ E j) 1 P j). 30) The proof now foows from 26), 30) and Binomia theorem. 5 Appications In section, we estabish the formua for the sum of product of factorias and arithmetic progression and arithmetic geometric progression in number theory as an appication of ). Lemma 5.1 Let 0, ) and, ). Then Proof. The proof foows from 21) and Definition 4.1. ) )! = )!. 31)
8 962 V. Chandrasear, G. Britto Antony Xavier and R. Vijayaraj Theorem 5.2 Let 0, ) and, ). Then ) r 1) r) )! = 1 ) )! ) j )!. 32) Proof. The proof foows by taing u) = )! in 26) and 31). Exampe 5.3 Substituting = 70, = 3 and j = 1 in 32), we get 67 3 )! + 67) 1) )! + 67) 2) )! + 67) 22) )! = ) 3)! 70 3) 3 )!1) 3 )! = X Theorem 5.4 If n is any positive integer, then +1) n nc t n t t ) t=1 Proof. The proof foows by 24) and Definition 4.1. Theorem 5.5 If n is any positive integer, then n r) r+1) ) r 1) nc t n t t t=1 Proof. The proof foows from 33) and Definition 4.1. Exampe 5.6 In 34), by taing n = 2, we find that 2 r) r+1) ) r 1) nc t n t t t=1 = ) ) n. 33) = ) ) = ) ) When, = 61, = 3 and j = 1, we get 58) 20) 3 58) 0) 3 357) + 55) 19) 3 58) 1) 3 339) + + 1) 1) 3 58) 19) 3 15) = 61 3) 20) ) 20) = X Theorem 5.7 Let J = ) r 1) r) r). Then, for a N J), = ) ) ) n. 34) j 2. J j ). 35) j
9 -Difference operator and its appication in number theory 963 Proof. From the equation 5) and Definition 4.1, we have ) ) The proof foows from 26) and 36). = ). 36) Exampe 5.8 In 35), substituting = 62, = 3, j = 2 and J = 1, we get 59) 0) 3 59) 20) ) 1) 3 56) 19) ) 2) 3 53) 18) ) 20) 3 1) 0) 3 6 Concusion = )21) 3 59) 20) 3 1) 1) 3 = X By seecting arge vaue for and sma vaue for > 0, one can find the sum of severa series easiy using the Theorem mentioned above. References 1 R. P. Agarwa, Difference Equations and Inequaities, Marce Deer, New Yor, Jerzy Popenda and Bazij Smanda, On the Osciation of Soutions of Certain Difference Equations, Demonstratio Mathematica, XVII1), 1984), M. Maria Susai Manue, G. Britto Antony Xavier and E.Thandapani, Theory of Generaized Difference Operator and Its Appications, Far East Journa of Mathematica Sciences, 202) 2006), M. Maria Susai Manue, G. Britto Antony Xavier and E. Thandapani, Generaized Bernoui Poynomias Through Weighted Pochhammer Symbos, Far East Journa of Appied Mathematics, 263) 2007), M. Maria Susai Manue, V. Chandrasear and G. Britto Antony Xavier, Soutions and Appications of Certain Cass of Difference Equations, Internationa Journa of Appied Mathematics, 246) 2011), M. Maria Susai Manue, V. Chandrasear and G. Britto Antony Xavier, Theory Of Generaized α-difference Operator and its Appication in Number Theory, Advances in Differentia Equations and Contro Processes, 92) 2012),
10 964 V. Chandrasear, G. Britto Antony Xavier and R. Vijayaraj 7 Ronad E. Micens, Difference Equations, Van Nostrand Reinhod Company, New Yor, Saber N. Eaydi, An Introduction to Difference Equations, Third edition, Springer, USA, Water G. Keey, Aan C. Peterson, Difference Equations, An Introduction with Appications, Academic Press Inc., Received: December 15, 2014; Pubished: March 23, 2015
Discrete Bernoulli s Formula and its Applications Arising from Generalized Difference Operator
Int. Journa of Math. Anaysis, Vo. 7, 2013, no. 5, 229-240 Discrete Bernoui s Formua and its Appications Arising from Generaized Difference Operator G. Britto Antony Xavier 1 Department of Mathematics,
More informationLaplace - Fibonacci transform by the solution of second order generalized difference equation
Nonauton. Dyn. Syst. 017; 4: 30 Research Artice Open Access Sandra Pineas*, G.B.A Xavier, S.U. Vasantha Kumar, and M. Meganathan Lapace - Fibonacci transform by the soution of second order generaized difference
More informationOn the New q-extension of Frobenius-Euler Numbers and Polynomials Arising from Umbral Calculus
Adv. Studies Theor. Phys., Vo. 7, 203, no. 20, 977-99 HIKARI Ltd, www.m-hiari.com http://dx.doi.org/0.2988/astp.203.390 On the New -Extension of Frobenius-Euer Numbers and Poynomias Arising from Umbra
More informationGaussian Curvature in a p-orbital, Hydrogen-like Atoms
Advanced Studies in Theoretica Physics Vo. 9, 015, no. 6, 81-85 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/astp.015.5115 Gaussian Curvature in a p-orbita, Hydrogen-ike Atoms Sandro-Jose Berrio-Guzman
More informationResearch Article On the Lower Bound for the Number of Real Roots of a Random Algebraic Equation
Appied Mathematics and Stochastic Anaysis Voume 007, Artice ID 74191, 8 pages doi:10.1155/007/74191 Research Artice On the Lower Bound for the Number of Rea Roots of a Random Agebraic Equation Takashi
More informationarxiv: v1 [math.nt] 12 Feb 2019
Degenerate centra factoria numbers of the second ind Taeyun Kim, Dae San Kim arxiv:90.04360v [math.nt] Feb 09 In this paper, we introduce the degenerate centra factoria poynomias and numbers of the second
More informationResearch Article Building Infinitely Many Solutions for Some Model of Sublinear Multipoint Boundary Value Problems
Abstract and Appied Anaysis Voume 2015, Artice ID 732761, 4 pages http://dx.doi.org/10.1155/2015/732761 Research Artice Buiding Infinitey Many Soutions for Some Mode of Subinear Mutipoint Boundary Vaue
More informationImproving the Reliability of a Series-Parallel System Using Modified Weibull Distribution
Internationa Mathematica Forum, Vo. 12, 217, no. 6, 257-269 HIKARI Ltd, www.m-hikari.com https://doi.org/1.12988/imf.217.611155 Improving the Reiabiity of a Series-Parae System Using Modified Weibu Distribution
More informationAnother Class of Admissible Perturbations of Special Expressions
Int. Journa of Math. Anaysis, Vo. 8, 014, no. 1, 1-8 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/ijma.014.31187 Another Cass of Admissibe Perturbations of Specia Expressions Jerico B. Bacani
More informationSome Applications on Generalized Hypergeometric and Confluent Hypergeometric Functions
Internationa Journa of Mathematica Anaysis and Appications 0; 5(): 4-34 http://www.aascit.org/journa/ijmaa ISSN: 375-397 Some Appications on Generaized Hypergeometric and Confuent Hypergeometric Functions
More informationExtended central factorial polynomials of the second kind
Kim et a. Advances in Difference Equations 09 09:4 https://doi.org/0.86/s366-09-963- R E S E A R C H Open Access Extended centra factoria poynomias of the second ind Taeyun Kim,DaeSanKim,Gwan-WooJang and
More informationMathematical Scheme Comparing of. the Three-Level Economical Systems
Appied Mathematica Sciences, Vo. 11, 2017, no. 15, 703-709 IKAI td, www.m-hikari.com https://doi.org/10.12988/ams.2017.7252 Mathematica Scheme Comparing of the Three-eve Economica Systems S.M. Brykaov
More informationSummation of p-adic Functional Series in Integer Points
Fiomat 31:5 (2017), 1339 1347 DOI 10.2298/FIL1705339D Pubished by Facuty of Sciences and Mathematics, University of Niš, Serbia Avaiabe at: http://www.pmf.ni.ac.rs/fiomat Summation of p-adic Functiona
More informationResearch Article Numerical Range of Two Operators in Semi-Inner Product Spaces
Abstract and Appied Anaysis Voume 01, Artice ID 846396, 13 pages doi:10.1155/01/846396 Research Artice Numerica Range of Two Operators in Semi-Inner Product Spaces N. K. Sahu, 1 C. Nahak, 1 and S. Nanda
More informationQUADRATIC FORMS AND FOUR PARTITION FUNCTIONS MODULO 3
QUADRATIC FORMS AND FOUR PARTITION FUNCTIONS MODULO 3 JEREMY LOVEJOY AND ROBERT OSBURN Abstract. Recenty, Andrews, Hirschhorn Seers have proven congruences moduo 3 for four types of partitions using eementary
More informationMaejo International Journal of Science and Technology
Fu Paper Maejo Internationa Journa of Science and Technoogy ISSN 1905-7873 Avaiabe onine at www.mijst.mju.ac.th A study on Lucas difference sequence spaces (, ) (, ) and Murat Karakas * and Ayse Metin
More informationOn Some Basic Properties of Geometric Real Sequences
On Some Basic Properties of eometric Rea Sequences Khirod Boruah Research Schoar, Department of Mathematics, Rajiv andhi University Rono His, Doimukh-791112, Arunacha Pradesh, India Abstract Objective
More informationGeneralized Bell polynomials and the combinatorics of Poisson central moments
Generaized Be poynomias and the combinatorics of Poisson centra moments Nicoas Privaut Division of Mathematica Sciences Schoo of Physica and Mathematica Sciences Nanyang Technoogica University SPMS-MAS-05-43,
More informationORTHOGONAL MULTI-WAVELETS FROM MATRIX FACTORIZATION
J. Korean Math. Soc. 46 2009, No. 2, pp. 281 294 ORHOGONAL MLI-WAVELES FROM MARIX FACORIZAION Hongying Xiao Abstract. Accuracy of the scaing function is very crucia in waveet theory, or correspondingy,
More informationTHE REACHABILITY CONES OF ESSENTIALLY NONNEGATIVE MATRICES
THE REACHABILITY CONES OF ESSENTIALLY NONNEGATIVE MATRICES by Michae Neumann Department of Mathematics, University of Connecticut, Storrs, CT 06269 3009 and Ronad J. Stern Department of Mathematics, Concordia
More informationDifferential equations associated with higher-order Bernoulli numbers of the second kind
Goba Journa of Pure and Appied Mathematics. ISS 0973-768 Voume 2, umber 3 (206), pp. 2503 25 Research India Pubications http://www.ripubication.com/gjpam.htm Differentia equations associated with higher-order
More informationSome Measures for Asymmetry of Distributions
Some Measures for Asymmetry of Distributions Georgi N. Boshnakov First version: 31 January 2006 Research Report No. 5, 2006, Probabiity and Statistics Group Schoo of Mathematics, The University of Manchester
More informationWeek 6 Lectures, Math 6451, Tanveer
Fourier Series Week 6 Lectures, Math 645, Tanveer In the context of separation of variabe to find soutions of PDEs, we encountered or and in other cases f(x = f(x = a 0 + f(x = a 0 + b n sin nπx { a n
More informationTikhonov Regularization for Nonlinear Complementarity Problems with Approximative Data
Internationa Mathematica Forum, 5, 2010, no. 56, 2787-2794 Tihonov Reguarization for Noninear Compementarity Probems with Approximative Data Nguyen Buong Vietnamese Academy of Science and Technoogy Institute
More informationConvergence Property of the Iri-Imai Algorithm for Some Smooth Convex Programming Problems
Convergence Property of the Iri-Imai Agorithm for Some Smooth Convex Programming Probems S. Zhang Communicated by Z.Q. Luo Assistant Professor, Department of Econometrics, University of Groningen, Groningen,
More information(f) is called a nearly holomorphic modular form of weight k + 2r as in [5].
PRODUCTS OF NEARLY HOLOMORPHIC EIGENFORMS JEFFREY BEYERL, KEVIN JAMES, CATHERINE TRENTACOSTE, AND HUI XUE Abstract. We prove that the product of two neary hoomorphic Hece eigenforms is again a Hece eigenform
More informationarxiv: v4 [math.nt] 20 Jan 2015
arxiv:3.448v4 [math.nt] 20 Jan 205 NEW IDENTITIES ON THE APOSTOL-BERNOULLI POLYNOMIALS OF HIGHER ORDER DERIVED FROM BERNOULLI BASIS ARMEN BAGDASARYAN, SERKAN ARACI, MEHMET ACIKGOZ, AND YUAN HE Abstract.
More informationCourse 2BA1, Section 11: Periodic Functions and Fourier Series
Course BA, 8 9 Section : Periodic Functions and Fourier Series David R. Wikins Copyright c David R. Wikins 9 Contents Periodic Functions and Fourier Series 74. Fourier Series of Even and Odd Functions...........
More informationHomogeneity properties of subadditive functions
Annaes Mathematicae et Informaticae 32 2005 pp. 89 20. Homogeneity properties of subadditive functions Pá Burai and Árpád Száz Institute of Mathematics, University of Debrecen e-mai: buraip@math.kte.hu
More informationSome identities of Laguerre polynomials arising from differential equations
Kim et a. Advances in Difference Equations 2016) 2016:159 DOI 10.1186/s13662-016-0896-1 R E S E A R C H Open Access Some identities of Laguerre poynomias arising from differentia equations Taekyun Kim
More informationResearch Article The Diagonally Dominant Degree and Disc Separation for the Schur Complement of Ostrowski Matrix
Hindawi Pubishing Corporation Journa of Appied Mathematics Voume 2013, Artice ID 973152, 10 pages http://dxdoiorg/101155/2013/973152 Research Artice The Diagonay Dominant Degree and Disc Separation for
More informationGeneral Decay of Solutions in a Viscoelastic Equation with Nonlinear Localized Damping
Journa of Mathematica Research with Appications Jan.,, Vo. 3, No., pp. 53 6 DOI:.377/j.issn:95-65...7 Http://jmre.dut.edu.cn Genera Decay of Soutions in a Viscoeastic Equation with Noninear Locaized Damping
More informationPartial permutation decoding for MacDonald codes
Partia permutation decoding for MacDonad codes J.D. Key Department of Mathematics and Appied Mathematics University of the Western Cape 7535 Bevie, South Africa P. Seneviratne Department of Mathematics
More informationMARKOV CHAINS AND MARKOV DECISION THEORY. Contents
MARKOV CHAINS AND MARKOV DECISION THEORY ARINDRIMA DATTA Abstract. In this paper, we begin with a forma introduction to probabiity and expain the concept of random variabes and stochastic processes. After
More informationEstablishment of Weak Conditions for Darboux- Goursat-Beudon Theorem
Georgia Southern University Digita Commons@Georgia Southern Mathematica Sciences Facuty Pubications Department of Mathematica Sciences 2009 Estabishment of Weak Conditions for Darboux- Goursat-Beudon Theorem
More informationPAijpam.eu SOME RESULTS ON PRIME NUMBERS B. Martin Cerna Maguiña 1, Héctor F. Cerna Maguiña 2 and Harold Blas 3
Internationa Journa of Pure and Appied Mathematics Voume 118 No. 3 018, 845-851 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-ine version) ur: http://www.ijpam.eu doi:.173/ijpam.v118i3.9 PAijpam.eu
More informationOn the Number of Limit Cycles for Discontinuous Generalized Liénard Polynomial Differential Systems
Internationa Journa of Bifurcation and Chaos Vo. 25 No. 10 2015 1550131 10 pages c Word Scientific Pubishing Company DOI: 10.112/S02181271550131X On the Number of Limit Cyces for Discontinuous Generaized
More informationON THE POSITIVITY OF SOLUTIONS OF SYSTEMS OF STOCHASTIC PDES
ON THE POSITIVITY OF SOLUTIONS OF SYSTEMS OF STOCHASTIC PDES JACKY CRESSON 1,2, MESSOUD EFENDIEV 3, AND STEFANIE SONNER 3,4 On the occasion of the 75 th birthday of Prof. Dr. Dr.h.c. Wofgang L. Wendand
More informationGlobal Optimality Principles for Polynomial Optimization Problems over Box or Bivalent Constraints by Separable Polynomial Approximations
Goba Optimaity Principes for Poynomia Optimization Probems over Box or Bivaent Constraints by Separabe Poynomia Approximations V. Jeyakumar, G. Li and S. Srisatkunarajah Revised Version II: December 23,
More informationarxiv: v1 [math.ca] 6 Mar 2017
Indefinite Integras of Spherica Besse Functions MIT-CTP/487 arxiv:703.0648v [math.ca] 6 Mar 07 Joyon K. Boomfied,, Stephen H. P. Face,, and Zander Moss, Center for Theoretica Physics, Laboratory for Nucear
More informationSome Properties Related to the Generalized q-genocchi Numbers and Polynomials with Weak Weight α
Appied Mathematica Sciences, Vo. 6, 2012, no. 118, 5851-5859 Some Properties Reated to the Generaized q-genocchi Numbers and Poynomias with Weak Weight α J. Y. Kang Department of Mathematics Hannam University,
More informationIntegrating Factor Methods as Exponential Integrators
Integrating Factor Methods as Exponentia Integrators Borisav V. Minchev Department of Mathematica Science, NTNU, 7491 Trondheim, Norway Borko.Minchev@ii.uib.no Abstract. Recenty a ot of effort has been
More informationSolution of Wave Equation by the Method of Separation of Variables Using the Foss Tools Maxima
Internationa Journa of Pure and Appied Mathematics Voume 117 No. 14 2017, 167-174 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-ine version) ur: http://www.ijpam.eu Specia Issue ijpam.eu Soution
More informationNOISE-INDUCED STABILIZATION OF STOCHASTIC DIFFERENTIAL EQUATIONS
NOISE-INDUCED STABILIZATION OF STOCHASTIC DIFFERENTIAL EQUATIONS TONY ALLEN, EMILY GEBHARDT, AND ADAM KLUBALL 3 ADVISOR: DR. TIFFANY KOLBA 4 Abstract. The phenomenon of noise-induced stabiization occurs
More informationResearch Article Some Applications of Second-Order Differential Subordination on a Class of Analytic Functions Defined by Komatu Integral Operator
ISRN Mathematia Anaysis, Artie ID 66235, 5 pages http://dx.doi.org/1.1155/214/66235 Researh Artie Some Appiations of Seond-Order Differentia Subordination on a Cass of Anayti Funtions Defined by Komatu
More informationAn Extension of Almost Sure Central Limit Theorem for Order Statistics
An Extension of Amost Sure Centra Limit Theorem for Order Statistics T. Bin, P. Zuoxiang & S. Nadarajah First version: 6 December 2007 Research Report No. 9, 2007, Probabiity Statistics Group Schoo of
More informationExplicit overall risk minimization transductive bound
1 Expicit overa risk minimization transductive bound Sergio Decherchi, Paoo Gastado, Sandro Ridea, Rodofo Zunino Dept. of Biophysica and Eectronic Engineering (DIBE), Genoa University Via Opera Pia 11a,
More informationANALOG OF HEAT EQUATION FOR GAUSSIAN MEASURE OF A BALL IN HILBERT SPACE GYULA PAP
ANALOG OF HEAT EQUATION FOR GAUSSIAN MEASURE OF A BALL IN HILBERT SPACE GYULA PAP ABSTRACT. If µ is a Gaussian measure on a Hibert space with mean a and covariance operator T, and r is a} fixed positive
More informationSecure Information Flow Based on Data Flow Analysis
SSN 746-7659, Engand, UK Journa of nformation and Computing Science Vo., No. 4, 007, pp. 5-60 Secure nformation Fow Based on Data Fow Anaysis Jianbo Yao Center of nformation and computer, Zunyi Norma Coege,
More informationA Two-Parameter Trigonometric Series
A Two-Parameter Trigonometric Series Xiang-Qian Chang Xiang.Chang@bos.mcphs.edu Massachusetts Coege of Pharmacy and Heath Sciences Boston MA 25 Let us consider the foowing dua questions. A. Givenafunction
More informationC. Fourier Sine Series Overview
12 PHILIP D. LOEWEN C. Fourier Sine Series Overview Let some constant > be given. The symboic form of the FSS Eigenvaue probem combines an ordinary differentia equation (ODE) on the interva (, ) with a
More informationJENSEN S OPERATOR INEQUALITY FOR FUNCTIONS OF SEVERAL VARIABLES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Voume 128, Number 7, Pages 2075 2084 S 0002-99390005371-5 Artice eectronicay pubished on February 16, 2000 JENSEN S OPERATOR INEQUALITY FOR FUNCTIONS OF
More informationRestricted weak type on maximal linear and multilinear integral maps.
Restricted weak type on maxima inear and mutiinear integra maps. Oscar Basco Abstract It is shown that mutiinear operators of the form T (f 1,..., f k )(x) = R K(x, y n 1,..., y k )f 1 (y 1 )...f k (y
More informationAn explicit Jordan Decomposition of Companion matrices
An expicit Jordan Decomposition of Companion matrices Fermín S V Bazán Departamento de Matemática CFM UFSC 88040-900 Forianópois SC E-mai: fermin@mtmufscbr S Gratton CERFACS 42 Av Gaspard Coriois 31057
More informationDistributed average consensus: Beyond the realm of linearity
Distributed average consensus: Beyond the ream of inearity Usman A. Khan, Soummya Kar, and José M. F. Moura Department of Eectrica and Computer Engineering Carnegie Meon University 5 Forbes Ave, Pittsburgh,
More informationMath-Net.Ru All Russian mathematical portal
Math-Net.Ru A Russian mathematica porta D. Zaora, On properties of root eements in the probem on sma motions of viscous reaxing fuid, Zh. Mat. Fiz. Ana. Geom., 217, Voume 13, Number 4, 42 413 DOI: https://doi.org/1.1547/mag13.4.42
More information4 Separation of Variables
4 Separation of Variabes In this chapter we describe a cassica technique for constructing forma soutions to inear boundary vaue probems. The soution of three cassica (paraboic, hyperboic and eiptic) PDE
More informationKing Fahd University of Petroleum & Minerals
King Fahd University of Petroeum & Mineras DEPARTMENT OF MATHEMATICAL SCIENCES Technica Report Series TR 369 December 6 Genera decay of soutions of a viscoeastic equation Saim A. Messaoudi DHAHRAN 3161
More informationarxiv: v1 [math.ap] 8 Nov 2014
arxiv:1411.116v1 [math.ap] 8 Nov 014 ALL INVARIANT REGIONS AND GLOBAL SOLUTIONS FOR m-component REACTION-DIFFUSION SYSTEMS WITH A TRIDIAGONAL SYMMETRIC TOEPLITZ MATRIX OF DIFFUSION COEFFICIENTS SALEM ABDELMALEK
More informationBinomial Transform and Dold Sequences
1 2 3 47 6 23 11 Journa of Integer Sequences, Vo. 18 (2015), Artice 15.1.1 Binomia Transform and Dod Sequences Kaudiusz Wójcik Department of Mathematics and Computer Science Jagieonian University Lojasiewicza
More information2M2. Fourier Series Prof Bill Lionheart
M. Fourier Series Prof Bi Lionheart 1. The Fourier series of the periodic function f(x) with period has the form f(x) = a 0 + ( a n cos πnx + b n sin πnx ). Here the rea numbers a n, b n are caed the Fourier
More informationInvestigation on spectrum of the adjacency matrix and Laplacian matrix of graph G l
Investigation on spectrum of the adjacency matrix and Lapacian matrix of graph G SHUHUA YIN Computer Science and Information Technoogy Coege Zhejiang Wani University Ningbo 3500 PEOPLE S REPUBLIC OF CHINA
More informationReconstructions that Combine Cell Average Interpolation with Least Squares Fitting
App Math Inf Sci, No, 7-84 6) 7 Appied Mathematics & Information Sciences An Internationa Journa http://dxdoiorg/8576/amis/6 Reconstructions that Combine Ce Average Interpoation with Least Squares Fitting
More informationBP neural network-based sports performance prediction model applied research
Avaiabe onine www.jocpr.com Journa of Chemica and Pharmaceutica Research, 204, 6(7:93-936 Research Artice ISSN : 0975-7384 CODEN(USA : JCPRC5 BP neura networ-based sports performance prediction mode appied
More informationTracking Control of Multiple Mobile Robots
Proceedings of the 2001 IEEE Internationa Conference on Robotics & Automation Seou, Korea May 21-26, 2001 Tracking Contro of Mutipe Mobie Robots A Case Study of Inter-Robot Coision-Free Probem Jurachart
More informationHomotopy Perturbation Method for Solving Partial Differential Equations of Fractional Order
Int Journa of Math Anaysis, Vo 6, 2012, no 49, 2431-2448 Homotopy Perturbation Method for Soving Partia Differentia Equations of Fractiona Order A A Hemeda Department of Mathematics, Facuty of Science
More informationA Brief Introduction to Markov Chains and Hidden Markov Models
A Brief Introduction to Markov Chains and Hidden Markov Modes Aen B MacKenzie Notes for December 1, 3, &8, 2015 Discrete-Time Markov Chains You may reca that when we first introduced random processes,
More informationThe graded generalized Fibonacci sequence and Binet formula
The graded generaized Fibonacci sequence and Binet formua Won Sang Chung,, Minji Han and Jae Yoon Kim Department of Physics and Research Institute of Natura Science, Coege of Natura Science, Gyeongsang
More informationThe numerical radius of a weighted shift operator
Eectronic Journa of Linear Agebra Voume 30 Voume 30 2015 Artice 60 2015 The numerica radius of a weighted shift operator Batzorig Undrah Nationa University of Mongoia, batzorig_u@yahoo.com Hiroshi Naazato
More informationSmall generators of function fields
Journa de Théorie des Nombres de Bordeaux 00 (XXXX), 000 000 Sma generators of function fieds par Martin Widmer Résumé. Soit K/k une extension finie d un corps goba, donc K contient un éément primitif
More informationInput-to-state stability for a class of Lurie systems
Automatica 38 (2002) 945 949 www.esevier.com/ocate/automatica Brief Paper Input-to-state stabiity for a cass of Lurie systems Murat Arcak a;, Andrew Tee b a Department of Eectrica, Computer and Systems
More information6 Wave Equation on an Interval: Separation of Variables
6 Wave Equation on an Interva: Separation of Variabes 6.1 Dirichet Boundary Conditions Ref: Strauss, Chapter 4 We now use the separation of variabes technique to study the wave equation on a finite interva.
More informationFirst-Order Corrections to Gutzwiller s Trace Formula for Systems with Discrete Symmetries
c 26 Noninear Phenomena in Compex Systems First-Order Corrections to Gutzwier s Trace Formua for Systems with Discrete Symmetries Hoger Cartarius, Jörg Main, and Günter Wunner Institut für Theoretische
More informationTHINKING IN PYRAMIDS
ECS 178 Course Notes THINKING IN PYRAMIDS Kenneth I. Joy Institute for Data Anaysis and Visuaization Department of Computer Science University of Caifornia, Davis Overview It is frequenty usefu to think
More informationThe Partition Function and Ramanujan Congruences
The Partition Function and Ramanujan Congruences Eric Bucher Apri 7, 010 Chapter 1 Introduction The partition function, p(n), for a positive integer n is the number of non-increasing sequences of positive
More informationRELATIONSHIP BETWEEN QUATERNION LINEAR CANONICAL AND QUATERNION FOURIER TRANSFORMS
Proceedings of the 04 Internationa Conference on Waveet Anaysis and Pattern ecognition, Lanzhou, 3-6 Juy, 04 ELATIONSHIP BETWEEN QUATENION LINEA CANONICAL AND QUATENION FOUIE TANSFOMS MAWADI BAHI, YUICHI
More informationResearch Article Solution of Point Reactor Neutron Kinetics Equations with Temperature Feedback by Singularly Perturbed Method
Science and Technoogy of Nucear Instaations Voume 213, Artice ID 261327, 6 pages http://dx.doi.org/1.1155/213/261327 Research Artice Soution of Point Reactor Neutron Kinetics Equations with Temperature
More informationNIKOS FRANTZIKINAKIS. N n N where (Φ N) N N is any Følner sequence
SOME OPE PROBLEMS O MULTIPLE ERGODIC AVERAGES IKOS FRATZIKIAKIS. Probems reated to poynomia sequences In this section we give a ist of probems reated to the study of mutipe ergodic averages invoving iterates
More informationSmoothness equivalence properties of univariate subdivision schemes and their projection analogues
Numerische Mathematik manuscript No. (wi be inserted by the editor) Smoothness equivaence properties of univariate subdivision schemes and their projection anaogues Phiipp Grohs TU Graz Institute of Geometry
More informationALGORITHMIC SUMMATION OF RECIPROCALS OF PRODUCTS OF FIBONACCI NUMBERS. F. = I j. ^ = 1 ^ -, and K w = ^. 0) n=l r n «=1 -*/!
ALGORITHMIC SUMMATIO OF RECIPROCALS OF PRODUCTS OF FIBOACCI UMBERS Staney Rabinowitz MathPro Press, 2 Vine Brook Road, Westford, MA 0886 staney@tiac.net (Submitted May 997). ITRODUCTIO There is no known
More information4 1-D Boundary Value Problems Heat Equation
4 -D Boundary Vaue Probems Heat Equation The main purpose of this chapter is to study boundary vaue probems for the heat equation on a finite rod a x b. u t (x, t = ku xx (x, t, a < x < b, t > u(x, = ϕ(x
More informationA NOTE ON INFINITE DIVISIBILITY OF ZETA DISTRIBUTIONS
A NOTE ON INFINITE DIVISIBILITY OF ZETA DISTRIBUTIONS SHINGO SAITO AND TATSUSHI TANAKA Abstract. The Riemann zeta istribution, efine as the one whose characteristic function is the normaise Riemann zeta
More informationA nodal collocation approximation for the multidimensional P L equations. 3D applications.
XXI Congreso de Ecuaciones Diferenciaes y Apicaciones XI Congreso de Matemática Apicada Ciudad Rea, 1-5 septiembre 9 (pp. 1 8) A noda coocation approximation for the mutidimensiona P L equations. 3D appications.
More informationCS229 Lecture notes. Andrew Ng
CS229 Lecture notes Andrew Ng Part IX The EM agorithm In the previous set of notes, we taked about the EM agorithm as appied to fitting a mixture of Gaussians. In this set of notes, we give a broader view
More informationProcess Capability Proposal. with Polynomial Profile
Contemporary Engineering Sciences, Vo. 11, 2018, no. 85, 4227-4236 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ces.2018.88467 Process Capabiity Proposa with Poynomia Profie Roberto José Herrera
More information(This is a sample cover image for this issue. The actual cover is not yet available at this time.)
(This is a sampe cover image for this issue The actua cover is not yet avaiabe at this time) This artice appeared in a journa pubished by Esevier The attached copy is furnished to the author for interna
More informationUniprocessor Feasibility of Sporadic Tasks with Constrained Deadlines is Strongly conp-complete
Uniprocessor Feasibiity of Sporadic Tasks with Constrained Deadines is Strongy conp-compete Pontus Ekberg and Wang Yi Uppsaa University, Sweden Emai: {pontus.ekberg yi}@it.uu.se Abstract Deciding the feasibiity
More informationLecture Note 3: Stationary Iterative Methods
MATH 5330: Computationa Methods of Linear Agebra Lecture Note 3: Stationary Iterative Methods Xianyi Zeng Department of Mathematica Sciences, UTEP Stationary Iterative Methods The Gaussian eimination (or
More informationResearch Article Dynamic Behaviors of a Discrete Periodic Predator-Prey-Mutualist System
Discrete Dynamics in Nature and Society Voume 015, Artice ID 4769, 11 pages http://dx.doi.org/10.1155/015/4769 Research Artice Dynamic Behaviors of a Discrete Periodic Predator-Prey-Mutuaist System Liya
More information#A48 INTEGERS 12 (2012) ON A COMBINATORIAL CONJECTURE OF TU AND DENG
#A48 INTEGERS 12 (2012) ON A COMBINATORIAL CONJECTURE OF TU AND DENG Guixin Deng Schoo of Mathematica Sciences, Guangxi Teachers Education University, Nanning, P.R.China dengguixin@ive.com Pingzhi Yuan
More informationIntroduction to Simulation - Lecture 13. Convergence of Multistep Methods. Jacob White. Thanks to Deepak Ramaswamy, Michal Rewienski, and Karen Veroy
Introduction to Simuation - Lecture 13 Convergence of Mutistep Methods Jacob White Thans to Deepa Ramaswamy, Micha Rewiensi, and Karen Veroy Outine Sma Timestep issues for Mutistep Methods Loca truncation
More informationProblem set 6 The Perron Frobenius theorem.
Probem set 6 The Perron Frobenius theorem. Math 22a4 Oct 2 204, Due Oct.28 In a future probem set I want to discuss some criteria which aow us to concude that that the ground state of a sef-adjoint operator
More informationResearch Article New Iterative Method: An Application for Solving Fractional Physical Differential Equations
Abstract and Appied Anaysis Voume 203, Artice ID 6700, 9 pages http://d.doi.org/0.55/203/6700 Research Artice New Iterative Method: An Appication for Soving Fractiona Physica Differentia Equations A. A.
More informationA Solution to the 4-bit Parity Problem with a Single Quaternary Neuron
Neura Information Processing - Letters and Reviews Vo. 5, No. 2, November 2004 LETTER A Soution to the 4-bit Parity Probem with a Singe Quaternary Neuron Tohru Nitta Nationa Institute of Advanced Industria
More informationAkaike Information Criterion for ANOVA Model with a Simple Order Restriction
Akaike Information Criterion for ANOVA Mode with a Simpe Order Restriction Yu Inatsu * Department of Mathematics, Graduate Schoo of Science, Hiroshima University ABSTRACT In this paper, we consider Akaike
More informationIntroduction to Simulation - Lecture 14. Multistep Methods II. Jacob White. Thanks to Deepak Ramaswamy, Michal Rewienski, and Karen Veroy
Introduction to Simuation - Lecture 14 Mutistep Methods II Jacob White Thans to Deepa Ramaswamy, Micha Rewiensi, and Karen Veroy Outine Sma Timestep issues for Mutistep Methods Reminder about LTE minimization
More informationA UNIVERSAL METRIC FOR THE CANONICAL BUNDLE OF A HOLOMORPHIC FAMILY OF PROJECTIVE ALGEBRAIC MANIFOLDS
A UNIERSAL METRIC FOR THE CANONICAL BUNDLE OF A HOLOMORPHIC FAMILY OF PROJECTIE ALGEBRAIC MANIFOLDS DROR AROLIN Dedicated to M Saah Baouendi on the occasion of his 60th birthday 1 Introduction In his ceebrated
More informationRapid and Stable Determination of Rotation Matrices between Spherical Harmonics by Direct Recursion
Chemistry Pubications Chemistry 11-1999 Rapid and Stabe Determination of Rotation Matrices between Spherica Harmonics by Direct Recursion Cheo Ho Choi Iowa State University Joseph Ivanic Iowa State University
More informationStochastic Automata Networks (SAN) - Modelling. and Evaluation. Paulo Fernandes 1. Brigitte Plateau 2. May 29, 1997
Stochastic utomata etworks (S) - Modeing and Evauation Pauo Fernandes rigitte Pateau 2 May 29, 997 Institut ationa Poytechnique de Grenobe { IPG Ecoe ationae Superieure d'informatique et de Mathematiques
More informationPath planning with PH G2 splines in R2
Path panning with PH G2 spines in R2 Laurent Gajny, Richard Béarée, Eric Nyiri, Oivier Gibaru To cite this version: Laurent Gajny, Richard Béarée, Eric Nyiri, Oivier Gibaru. Path panning with PH G2 spines
More information