This is a complement to my previous article "Advanced Determinant Calculus" [C. Krattenthaler, Advanced determinant calculus, Seminaire Lotharingien Combin. 42 (1999) ("The Andrews Festschrift"), A...
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This is a complement to my previous article "Advanced Determinant Calculus" [C. Krattenthaler, Advanced determinant calculus, Seminaire Lotharingien Combin. 42 (1999) ("The Andrews Festschrift"), Article B42q, 67 pp.]. In the present article, I share with the reader my experience of applying the methods described in the previous article in order to solve a particular problem from numbertheory [G. Almkvist, C. Krattenthaler, J. Petersson, Some new formulas for pi, Experiment. Math. 12 (2003) 441-456]. Moreover, I add a list of determinant evaluations which I consider as interesting, which have been found since the appearance of the previous article, or which I failed to mention there, including several conjectures and open problems. (c) 2005 Elsevier Inc. All rights reserved.
We investigate the properties of certain operators that were used in previous papers to create geometric and algebraic intersection number functions. Using these operators we show how to construct a general intersecti...
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We investigate the properties of certain operators that were used in previous papers to create geometric and algebraic intersection number functions. Using these operators we show how to construct a general intersection theory and then we investigate properties of certain polynomials that arise naturally in this context. We indicate connections to twin primes.
There is a well-established instability index theory for linear and quadratic matrix polynomials for which the coefficient matrices are Hermitian and skew-Hermitian. This theory relates the number of negative directio...
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There is a well-established instability index theory for linear and quadratic matrix polynomials for which the coefficient matrices are Hermitian and skew-Hermitian. This theory relates the number of negative directions for the matrix coefficients which are Hermitian to the total number of unstable eigenvalues for the polynomial. Herein we extend the theory to star-even matrix polynomials of any finite degree. In particular, unlike previously known cases we show that the instability index depends upon the size of the matrices when the degree of the polynomial is greater than two. We also consider Hermitian matrix polynomials, and derive an index which counts the number of eigenvalues with nonpositive imaginary part. The results are refined if we consider the Hermitian matrix polynomial to be a perturbation of a star-even polynomials;however, this refinement requires additional assumptions on the matrix coefficients. (C) 2013 Elsevier Inc. All rights reserved.
A real algebraic integer alpha > 1 is called a Salem number if all its remaining conjugates have modulus at most I with at least one having modulus exactly 1. It is known [J.A. de la Pena, Coxeter transformations a...
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A real algebraic integer alpha > 1 is called a Salem number if all its remaining conjugates have modulus at most I with at least one having modulus exactly 1. It is known [J.A. de la Pena, Coxeter transformations and the representation theory of algebras, in: V. Dlab et al. (Eds.), Finite Dimensional algebras and Related Topics, Proceedings of the NATO Advanced Research Workshop on Representations of algebras and Related Topics, Ottawa, Canada, Kluwer, August 1018, 1992, pp. 223-253;J.F. McKee, P. Rowlinson, C.J. Smyth, Salem numbers and Pisot numbers from stars, numbertheory in progress. in: K. Gyory et al. (Eds.), Proc. Int. Conf. Banach Int. Math. Center, Diophantine problems and polynomials, vol. 1, de Gruyter, Berlin, 1999, pp. 309-319;P. Lakatos, On Coxeter polynomials of wild stars, Linear algebra Appl. 293 (1999) 159-170] that the spectral radii of Coxeter transformation defined by stars, which are neither of Dynkin nor of extended Dynkin type, are Salem numbers. We prove that the spectral radii of the Coxeter transformation of generalized stars are also Salem numbers. A generalized star is a connected graph without multiple edges and loops that has exactly one vertex of degree at least 3. (C) 2009 Elsevier Inc. All rights reserved.
The problem of computing the greatest common divisor（GCD） of multivariate polynomials, as one of the most important tasks of computeralgebra and symbolic computation in more general scope, has been studied extensiv...
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The problem of computing the greatest common divisor（GCD） of multivariate polynomials, as one of the most important tasks of computeralgebra and symbolic computation in more general scope, has been studied extensively since the beginning of the interdisciplinary of mathematics with computer science. For many real applications such as digital image restoration and enhancement,robust control theory of nonlinear systems, L1-norm convex optimization in compressed sensing techniques, as well as algebraic decoding of Reed-Solomon and BCH codes, the concept of sparse GCD plays a core role where only the greatest common divisors with much fewer terms than the original polynomials are of interest due to the nature of problems or data structures. This paper presents two methods via multivariate polynomial interpolation which are based on the variation of Zippel＇s method and Ben-Or/Tiwari algorithm, respectively. To reduce computational complexity, probabilistic techniques and randomization are employed to deal with univariate GCD computation and univariate polynomial interpolation. The authors demonstrate the practical performance of our algorithms on a significant body of examples. The implemented experiment illustrates that our algorithms are efficient for a quite wide range of input.
Using mathematical tools from numbertheory and finite fields, Applied algebra: Codes, Ciphers, and Discrete Algorithms, Second Edition presents practical methods for solving problems in data security and data integri...
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ISBN:
(数字)9781439894699
ISBN:
(纸本)9781420071429
Using mathematical tools from numbertheory and finite fields, Applied algebra: Codes, Ciphers, and Discrete Algorithms, Second Edition presents practical methods for solving problems in data security and data integrity. It is designed for an applied algebra course for students who have had prior classes in abstract or linear algebra. While the content has been reworked and improved, this edition continues to cover many algorithms that arise in cryptography and error-control codes. New to the Second EditionA CD-ROM containing an interactive version of the book that is powered by Scientific Notebook®, a mathematical word processor and easy-to-use computeralgebra systemNew appendix that reviews prerequisite topics in algebra and numbertheoryDouble the number of exercisesInstead of a general study on finite groups, the book considers finite groups of permutations and develops just enough of the theory of finite fields to facilitate construction of the fields used for error-control codes and the Advanced Encryption Standard. It also deals with integers and polynomials. Explaining the mathematics as needed, this text thoroughly explores how mathematical techniques can be used to solve practical problems. About the AuthorsDarel W. Hardy is Professor Emeritus in the Department of Mathematics at Colorado State University. His research interests include applied algebra and *** Richman is a professor in the Department of Mathematical Sciences at Florida Atlantic University. His research interests include Abelian group theory and constructive *** L. Walker is Associate Dean Emeritus in the Department of Mathematical Sciences at New Mexico State University. Her research interests include Abelian group theory, appl
This note provides a short, self-contained treatment, using linear algebra and matrix theory, for establishing maximal periods, underlying structure, and choice of starting values for shift-register and lagged-Fibonac...
This note provides a short, self-contained treatment, using linear algebra and matrix theory, for establishing maximal periods, underlying structure, and choice of starting values for shift-register and lagged-Fibonacci random number generators.
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.
On any set X may be defined the free algebra R (respectively, free commutative algebra R[X]) with coefficients in a ring R. It may also be equivalently described as the algebra of the free monoid X* (respectively, fr...
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ISBN:
(纸本)9783642544798;9783642544781
On any set X may be defined the free algebra R < X > (respectively, free commutative algebra R[X]) with coefficients in a ring R. It may also be equivalently described as the algebra of the free monoid X* (respectively, free commutative monoid M(X)). Furthermore, the algebra of differential polynomials R{X} with variables in X may be constructed. The main objective of this contribution is to provide a functorial description of this kind of objects with their relations ( including abelianization and unitarization) in the category of differential algebras, and also to introduce new structures such as the differential algebra of a semigroup, of a monoid, or the universal differential envelope of an algebra.
In this paper we address some algorithmic problems related to computations in finite-dimensional associative algebras over finite fields. Our starting point is the structure theory of finite-dimensional associative al...
In this paper we address some algorithmic problems related to computations in finite-dimensional associative algebras over finite fields. Our starting point is the structure theory of finite-dimensional associative algebras. This theory determines, mostly in a nonconstructive way, the building blocks of these algebras. Our aim is to give polynomial time algorithms to find these building blocks, the radical and the simple direct summands of the radical-free part. The radical algorithm is based on a new, tractable characterisation of the radical. The algorithm for decomposition of semisimple algebras into simple ideals involves (and generalises) factoring polynomials over the ground field. Next, we study the problem of finding zero divisors in finite algebras. We show that thisproblem is in the same complexity class as the problem of factoring polynomials over finte fields. applications include a polynomial time Las Vegas method to find a common invariant subspace of a set of linear transformations as well as an explicit isomorphism between a given finite simple algebra and a full matrix algebra over a finite field.
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