In Di Nasso (2015) and Luperi Baglini (2012) it has been introduced a technique, based on nonstandard analysis, to study some problems in combinatorial numbertheory. In this paper we review such a technique and we pr...
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In Di Nasso (2015) and Luperi Baglini (2012) it has been introduced a technique, based on nonstandard analysis, to study some problems in combinatorial numbertheory. In this paper we review such a technique and we present three of its applications: the first one is a new proof of a known result regarding the algebra of beta N, namely that the center of the semigroup (beta N, circle plus) is N;the second one is a generalization of a theorem of Bergelson and Hindman on arithmetic progressions of length three;the third one regards the study of which polynomials in several variables with integers coefficients have a monochromatic solution for every finite coloring of N. We will study this last application in more detail: we will prove some algebraical properties of the set P of such polynomials and we will present a few examples of nonlinear polynomials in P. In the first part of the paper we will recall the main results of the nonstandard technique that we want to use, which is based on a characterization of ultrafilters by means of nonstandard analysis. (C) 2015 Elsevier Ltd. All rights reserved.
GeoGebra is open source mathematics education software being used in thousands of schools worldwide. It already supports equation system solving, locus equation computation and automatic geometry theorem proving by us...
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ISBN:
(纸本)9783319150819;9783319150802
GeoGebra is open source mathematics education software being used in thousands of schools worldwide. It already supports equation system solving, locus equation computation and automatic geometry theorem proving by using an embedded or outsourced CAS. GeoGebra recently changed its embedded CAS from Reduce to Giac because it fits better into the educational use. Also careful benchmarking of open source Grobner basis implementations showed that Giac is fast in algebraic computations, too, therefore it allows heavy Grobner basis calculations even in a web browser via Javascript. Grobner basis on Q for revlex ordering implementation in Giac is a modular algorithm ( E. Arnold). Each Z/pZ computation is done via the Buchberger algorithm using F4 linear algebra technics and "remake" speedups, they might be run in parallel for large examples. The output can be probabilistic or certified ( which is much slower). Experimentation shows that the probabilistic version is faster than other open-source implementations, and about 3 times slower than the Magma implementation on one processor, it also requires less memory for big examples like Cyclic9.
Many advanced models in physics use a simpler system as the foundation upon which problemspecific perturbation terms are added. There are many mathematical methods in perturbation theory which attempt to solve or at l...
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Many advanced models in physics use a simpler system as the foundation upon which problemspecific perturbation terms are added. There are many mathematical methods in perturbation theory which attempt to solve or at least approximate the solution for the advanced model based on the solution of the unperturbed system. The analytical approaches have the advantage that their approximation is an algebraic expression relating all involved quantities in the calculated solution up to a certain order. However, the complexity of the calculation often increases drastically with the number of iterations, variables, and parameters considered. On the other hand, the computer-based numerical approaches are fast once implemented, but their results are only numerical approximations without a symbolic form. A numerical integrator, for example, takes the initial values and integrates the ordinary differential equation up to the requested final state and yields the result as specific numbers. Therefore, no algebraic expression, much less a parameter dependence within the solution is given. The method presented in this work is based on the differential algebra (DA) framework, which was first developed to its current extent by Martin Berz et. al [3, 4, 5]. The used DA Normal Form Algorithm is an advancement by Martin Berz from the first arbitrary order algorithm by Forest, Berz, and Irwin [13], which was based on an DA-Lie approach. Both structures are already implemented in COSY INFINITY [18] documented in [7, 16, 17]. The result of the presented method is a numerically calculated algebraic expression of the solution up to an arbitrary truncation order. This method combines the effectiveness and automatic calculation of a computer-based numerical approximation and the algebraic relation between the involved quantities.
The class algebra and the double coset algebra are two classical commutative subalgebras of the group algebra of the symmetric group. The connexion coefficients of these two algebraic structures are important numbers ...
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The class algebra and the double coset algebra are two classical commutative subalgebras of the group algebra of the symmetric group. The connexion coefficients of these two algebraic structures are important numbers with significant applications. From a combinatorial point of view, they give the number of factorizations of a given permutation into the ordered product of permutations with specific cyclic properties and count in some cases the number of hypermaps and constellations on (locally) orientable surfaces. They are also of notable interest in the study of Schur and zonal polynomials as well as in the theory of the irreducible characters of the symmetric group and the zonal spherical functions. Furthermore as shown by Hanlon, Stanley, Stembridge (1992), the respective generating series of these coefficients in the basis of power sum symmetric functions are equal to the mathematical expectation of the trace of (XUYU*)(n) where X and Y are given symmetric (respectively hermitian) matrices, U is a random real (respectively complex) valued square matrix of standard normal distribution and n a non negative integer. This paper is devoted to the explicit evaluation of these series in terms of monomial symmetric functions. Morales and Vassilieva (2009, 2011) and Vassilieva (2013) found explicit formulas for these generating series in terms of monomial symmetric functions by introducing a bijection between partitioned hypermaps and some decorated forests and trees. Thanks to purely algebraic means, we recover the formula for the class algebra and provide a new simpler formula for the double coset algebra. As a salient ingredient, we compute an explicit formulation for zonal polynomials indexed by partitions of type [a, b, 1(n-a-b)]. (C) 2015 Elsevier B.V. All rights reserved.
Ahlswede, Khachatrian, Mauduit and Sarkozy [1] introduced the f-complexity measure ("f" for family) in order to study pseudorandom properties of large families of binary sequences. So far several families ha...
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Ahlswede, Khachatrian, Mauduit and Sarkozy [1] introduced the f-complexity measure ("f" for family) in order to study pseudorandom properties of large families of binary sequences. So far several families have been studied by this measure. In the present paper I considerably improve on my earlier result in [8], where the f-complexity measure of a family based on the Legendre symbol and polynomials over F-p is studied. This paper also extends the earlier results to a family restricted on irreducible polynomials. (C) 2014 Elsevier Inc. All rights reserved.
Nowadays fast computing and lightweight cryptography play a crucial role in the field of cryptography. Whenever we concern about the cryptography, the aspect of discrete mathematics can't be omitted. Security in c...
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ISBN:
(纸本)9781467394178
Nowadays fast computing and lightweight cryptography play a crucial role in the field of cryptography. Whenever we concern about the cryptography, the aspect of discrete mathematics can't be omitted. Security in cryptography is completely depends upon the key and the computations. Generation of key is nothing but the implementation of graph theory, discrete logarithms, linear & abstract algebra etc. These branches of modern mathematics has been playing a great role for the implementation of algorithms such as elliptic curve cryptography, stream cipher, block cipher, wireless sensor network etc. This paper is written to literally represent the key concept of such branches of modern mathematics and their implementations in the field of cryptography. This paper mainly deals the applications of Galois Field, primitive polynomials, primitive polynomials over Galois Field, number theoretic functions, Congruence Calculus or modular arithmetic's, Residue Class Rings and Prime Fields. The paper focus on these key topics to develop a mathematical tool, that are needed for the design and security analysis of a cryptosystems.
The wide number of languages and programming paradigms, as well as the heterogeneity of ‘programs’ and ‘executions’ require new generalisations of propositional dynamic logic. The dynamisation method, introduced i...
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Text. Recently, R. Dere and Y. Simsek have studied applications of umbral algebra to generating functions for the Hermite type Genocchi polynomials and numbers [6]. In this paper, we investigate some interesting prope...
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Text. Recently, R. Dere and Y. Simsek have studied applications of umbral algebra to generating functions for the Hermite type Genocchi polynomials and numbers [6]. In this paper, we investigate some interesting properties arising from umbral calculus. These properties are useful in deriving some identities of Bernoulli polynomials. (C) 2013 The Authors. Published by Elsevier Inc.
Availability of computeralgebra systems (CAS) lead to the resurrection of the resultant method for eliminating one or more variables from the polynomials system. The resultant matrix method has advantages over the Gr...
Availability of computeralgebra systems (CAS) lead to the resurrection of the resultant method for eliminating one or more variables from the polynomials system. The resultant matrix method has advantages over the Groebner basis and Ritt-Wu method due to their high complexity and storage requirement. This paper focuses on the current resultant matrix formulations and investigates their ability or otherwise towards producing optimal resultant matrices. A determinantal formula that gives exact resultant or a formulation that can minimize the presence of extraneous factors in the resultant formulation is often sought for when certain conditions that it exists can be determined. We present some applications of elimination theory via resultant formulations and examples are given to explain each of the presented settings.
Given a square matrix A with entries in a commutative ring S, the ideal of S[X] consisting of polynomials f with f (A) = 0 is called the null ideal of A. Very little is known about null ideals of matrices over general...
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Given a square matrix A with entries in a commutative ring S, the ideal of S[X] consisting of polynomials f with f (A) = 0 is called the null ideal of A. Very little is known about null ideals of matrices over general commutative rings. First, we determine a certain generating set of the null ideal of a matrix in case S = D/dD is the residue class ring of a principal ideal domain D modulo d is an element of D. After that we discuss two applications. We compute a decomposition of the S-module S[A] into cyclic S-modules and explain the strong relationship between this decomposition and the determined generating set of the null ideal of A. And finally, we give a rather explicit description of the ring Int(A, M-n(D)) of all integer-valued polynomials on A. (C) 2016 The Author. Published by Elsevier Inc.
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