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Explicit classes of permutation polynomials of F_3^(3m)

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Science China Mathematics 2009年第4期52卷 639-647页

作者： DING CunSheng1, XIANG Qing2, YUAN Jin3 & YUAN PingZhi4 1 Department of Computer Science and Engineering, The Hong Kong University of Science and Technology, Clearwater Bay, Kowloon, Hong Kong, China 2 Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA 3 Department of Computing, Macquarie University, NSW 2109, Australia 4 School of Mathematics, South China Normal University, Guangzhou 510631, China

Permutation polynomials have been an interesting subject of study for a long time and have applications in many areas of mathematics and engineering. However, only a small number of specific classes of permutation polynomials are known so far. In this paper, six classes of linearized permutation polynomials and six classes of nonlinearized permutation polynomials over F33m are presented. These polynomials have simple shapes, and they are related to planar functions.

关键词： permutation polynomials planar functions

The Powers Sums, Bernoulli Numbers, Bernoulli polynomials Rethinked

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Applied Mathematics 2019年第3期10卷 100-112页

作者： Do Tan Si The HoChiMinh-City Physical Association HoChiMinh-City Vietnam

Utilizing translation operators we get the powers sums on arithmetic progressions and the Bernoulli polynomials of order munder the form of differential operators acting on monomials. It follows that (d/dn-d/dz) applied on a power sum has a meaning and is exactly equal to the Bernoulli polynomial of the same order. From this new property we get the formula giving powers sums in term of sums of successive derivatives of Bernoulli polynomial multiplied withprimitives of the same order of n. Then by changing the two arguments z,n into Z=z(z-1), λ where λ designed the 1st order power sums and proving that Bernoulli polynomials of odd order vanish for arguments equal to 0, 1/2, 1, we obtain easily the Faulhaber formula for powers sums in term of polynomials in λ having coefficients depending on Z. These coefficients are found to be derivatives of odd powers sums on integers expressed in Z. By the way we obtain the link between Faulhaber formulae for powers sums on integers and on arithmetic progressions. To complete the work we propose tables for calculating in easiest manners possibly the Bernoulli numbers, the Bernoulli polynomials, the powers sums and the Faulhaber formula for powers sums.

关键词： Bernoulli Numbers Bernoulli polynomials Powers Sums Faulhaber Conjecture Shift Operator Operator Calculus

Large-Integer Multiplication Based on Homogeneous polynomials

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International Journal of Communications, Network and System Sciences 2012年第8期5卷 437-445页

作者： Boris S. Verkhovsky Computer Science Department New Jersey Institute of Technology Newark USA

Several algorithms based on homogeneous polynomials for multiplication of large integers are described in the paper. The homogeneity of polynomials provides several simplifications: reduction of system of equations and elimination of necessity to evaluate polynomials in points with larger coordinates. It is demonstrated that a two-stage implementation of the proposed and Toom-Cook algorithms asymptotically require twice as many standard multiplications than their direct implementation. A multistage implementation of these algorithms is also less efficient than their direct implementation. Although the proposed algorithms as well as the corresponding Toom-Cook algorithms require numerous algebraic additions, the Generalized Horner rule for evaluation of homogeneous polynomials, provided in the paper, decrease this number twice.

关键词： Homogeneous polynomials Toom-Cook Algorithm Multidigit Integers Multi-Stage Multiplication Generalized Horner Rule Large-Integer Multiplication

Uniqueness of Meromorphic Functions Whose Differential polynomials Share One Value

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Journal of Applied Mathematics and Physics 2018年第11期6卷 2264-2272页

作者： Jin Tao Xinli Wang College of Science University of Shanghai for Science and Technology Shanghai China

In this paper, we prove a uniqueness theorem of meromorphic functions whose some nonlinear differential shares 1 IM with powers of the meromorphic functions, where the degrees of the powers are equal to those of the nonlinear differential polynomials. This result improves the corresponding one given by Zhang and Yang, and other authors.

关键词： Differential polynomials Meromorphic Functions Uniqueness Theorems

A Block Procedure with Linear Multi-Step Methods Using Legendre polynomials for Solving ODEs

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Applied Mathematics 2015年第4期6卷 717-723页

作者： Khadijah M. Abualnaja Department of Mathematics and Statistics College of Science Taif University Taif Saudi Arabia

In this article, we derive a block procedure for some K-step linear multi-step methods (for K = 1, 2 and 3), using Legendre polynomials as the basis functions. We give discrete methods used in block and implement it for solving the non-stiff initial value problems, being the continuous interpolant derived and collocated at grid and off-grid points. Numerical examples of ordinary differential equations (ODEs) are solved using the proposed methods to show the validity and the accuracy of the introduced algorithms. A comparison with fourth-order Runge-Kutta method is given. The ob-tained numerical results reveal that the proposed method is efficient.

关键词： Collocation Methods with Legendre polynomials Initial Value Problems Perturbation Function Fourth-Order Runge-Kutta Method

Associated Hermite polynomials Related to Parabolic Cylinder Functions

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Advances in Pure Mathematics 2019年第1期9卷 15-42页

作者： Alfred Wünsche Max-Planck-Group “Nonclassical Radiation” at Institut für Physik Humboldt-Universit&#228 t Berlin Berlin Germany

In analogy to the role of Lommel polynomials ?in relation to Bessel functions Jv(z) the theory of Associated Hermite polynomials in the scaled form ?with parmeter v to Parabolic Cylinder functions Dv(z) is developed. The group-theoretical background with the 3-parameter group of motions M(2) in the plane for Bessel functions and of the Heisenberg-Weyl group W(2) for Parabolic Cylinder functions is discussed and compared with formulae, in particular, for the lowering and raising operators and the eigenvalue equations. Recurrence relations for the Associated Hermite polynomials and for their derivative and the differential equation for them are derived in detail. Explicit expressions for the Associated Hermite polynomials with involved Jacobi polynomials at argument zero are given and by means of them the Parabolic Cylinder functions are represented by two such basic functions.

关键词： Bessel Functions Lommel polynomials Parabolic Cylinder Functions Associated Hermite polynomials Jacobi polynomials Recurrence Relations Lowering and Raising Operators Heisenberg-Weyl Group Motion Group of Plane Irreducible Representations

Chebyshev polynomials with Applications to Two-Dimensional Operators

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Advances in Pure Mathematics 2019年第12期9卷 990-1033页

作者： Alfred Wünsche Humboldt-Universitä t Institut für Physik Berlin Germany

A new application of Chebyshev polynomials of second kind Un(x) to functions of two-dimensional operators is derived and discussed. It is related to the Hamilton-Cayley identity for operators or matrices which allows to reduce powers and smooth functions of them to superpositions of the first N-1 powers of the considered operator in N-dimensional case. The method leads in two-dimensional case first to the recurrence relations for Chebyshev polynomials and due to initial conditions to the application of Chebyshev polynomials of second kind Un(x). Furthermore, a new general class of Generating functions for Chebyshev polynomials of first and second kind Un(x) comprising the known Generating function as special cases is constructed by means of a derived identity for operator functions f(A) of a general two-dimensional operator A. The basic results are Formulas (9.5) and (9.6) which are then specialized for different examples of functions f(x). The generalization of the theory for three-dimensional operators is started to attack and a partial problem connected with the eigenvalue problem and the Hamilton-Cayley identity is solved in an Appendix. A physical application of Chebyshev polynomials to a problem of relativistic kinematics of a uniformly accelerated system is solved. All operator calculations are made in coordinate-invariant form.

关键词： Hypergeometric Function Jacobi polynomials Ultraspherical polynomials Chebyshev polynomials Legendre polynomials Hamilton-Cayley Identity Generating Functions Fibonacci and Lucas Numbers Special Lorentz Transformations Coordinate-Invariant Methods

Dominating Sets and Domination polynomials of Square of Paths

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Open Journal of Discrete Mathematics 2013年第1期3卷 60-69页

作者： A. Vijayan K. Lal Gipson Department of Mathematics Mar Ephraem College of Engineering and Technology Kanayakumari District India Department of Mathematics Nesamony Memorial Christian College Marthandam India

Let G = (V, E) be a simple graph. A set S í V is a dominating set of G, if every vertex in V-S is adjacent to at least one vertex in S. Let be the square of the Path and let denote the family of all dominating sets of with cardinality i. Let . In this paper, we obtain a recursive formula for . Using this recursive formula, we construct the polynomial, , which we call domination polynomial of and obtain some properties of this polynomial.

关键词： Domination Set Domination Number Domination polynomials

A Fundamental Relationship of polynomials and Its Proof

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Advances in Pure Mathematics 2018年第6期8卷 559-563页

作者： Serdar Beji Faculty of Naval Architecture and Ocean Engineering Istanbul Technical University Istanbul Turkey

A fundamental algebraic relationship for a general polynomial of degree n is given and proven by mathematical induction. The stated relationship is based on the well-known property of polynomials that the nth-differen... 详细信息

关键词： polynomials of Degree n nth-Order Finite-Differences Recurrence Relationship for polynomials