We give an explicit formula for the value of the Bernoulli polynomial B2k(t) when t is a rational number in the interval (0, 1). When t = 12, 31, 32, 41, 34, 61, 65 the value of B2k(t) is known explicitly. In 1938 Emm...
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We give an explicit formula for the value of the Bernoulli polynomial B2k(t) when t is a rational number in the interval (0, 1). When t = 12, 31, 32, 41, 34, 61, 65 the value of B2k(t) is known explicitly. In 1938 Emma Lehmer asked for the value of B2k(t) when the denominator of t is 5, 8, 10, or 12. We apply our formula to determine B2k(t) when the denominator of t is 5, 8, 10, and 12 thereby answering Lehmer's 87 year old question. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
number fields and their rings of integers, which generalize the rational numbers and the integers, are foundational objects in numbertheory. There are several computeralgebra systems and databases concerned with the...
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ISBN:
(纸本)9798400713477
number fields and their rings of integers, which generalize the rational numbers and the integers, are foundational objects in numbertheory. There are several computeralgebra systems and databases concerned with the computational aspects of these. In particular, computing the ring of integers of a given number field is one of the main tasks of computational algebraic numbertheory. In this paper, we describe a formalization in Lean 4 for certifying such computations. In order to accomplish this, we developed several data types amenable to computation. Moreover, many other underlying mathematical concepts and results had to be formalized, most of which are also of independent interest. These include resultants and discriminants, as well as methods for proving irreducibility of univariate polynomials over finite fields and over the rational numbers. To illustrate the feasibility of our strategy, we formally verified entries from the number fields section of the L-functions and modular forms database (LMFDB). These concern, for several number fields, the explicitly given integral basis of the ring of integers and the discriminant. To accomplish this, we wrote SageMath code that computes the corresponding certificates and outputs a Lean proof of the statement to be verified.
We present an alternative GPU acceleration for plane waves pseudopotentials electronic structure codes designed for systems that have small unit cells but require a large number of k points to sample the Brillouin zon...
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We present an alternative GPU acceleration for plane waves pseudopotentials electronic structure codes designed for systems that have small unit cells but require a large number of k points to sample the Brillouin zone as happens, for instance, in metals. We discuss the diagonalization of the Kohn and Sham equations and the solution of the linear system derived in density functional perturbation theory. Both problems take advantage from a rewriting of the routine that applies the Hamiltonian to the Bloch wave-functions to work simultaneously (in parallel on the GPU threads) on the wave-functions with different wave-vectors k, as many as allowed by the GPU memory. Our implementation is written in CUDA Fortran and makes extensive use of kernel routines that run on the GPU (GLOBAL routines) or can be called from inside the GPU kernel (DEVICE routines). We compare our method with the CPUs only calculation and with the approach currently implemented in Quantum ESPRESSO that uses GPU accelerated libraries for the FFT and for the linear algebra tasks such as the matrix-matrix multiplications as well as OpenACC directives for loop parallelization. We show in a realistic example that our method can give a significant improvement in the cases for which it has been designed.
This paper introduces a novel approach to understanding Galois theory, one of the foundational areas of algebra, through the lens of machine learning. By analyzing polynomial equations with machine learning techniques...
We consider the spectral properties of matrix polynomials over the max algebra. In particular, we show how the Perron-Frobenius theorem for the max algebra extends to such polynomials and illustrate the relevance of t...
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We consider the spectral properties of matrix polynomials over the max algebra. In particular, we show how the Perron-Frobenius theorem for the max algebra extends to such polynomials and illustrate the relevance of this for multistep difference equations in the max algebra. We also present a number of inequalities for the largest max eigenvalue of a matrix polynomial. (C) 2010 Elsevier Inc. All rights reserved.
Formulas which replace multiplication and division of polynomials by equivalent matrix operations are derived. The expressions not only simplify the computational process but can also be used in those fields where for...
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Formulas which replace multiplication and division of polynomials by equivalent matrix operations are derived. The expressions not only simplify the computational process but can also be used in those fields where formal manipulations with polynomials are needed. Some applications in computer arithmetic are also mentioned.
作者：
WAGNER, EGComputer Science Principles
Mathematical Sciences Department IBM Research Division T.J. Watson Research Center Yorktown Heights NY 10598 USA
This paper presents an introduction to some of the more algebraic applications of elementary category theory in computer science. Topics include: a category based look at universal algebra; the definition of polynomia...
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This paper presents an introduction to some of the more algebraic applications of elementary category theory in computer science. Topics include: a category based look at universal algebra; the definition of polynomials over arbitrary algebras and their application to the study of substitution; a development of Lawvere algebraic theories based on polynomials, and the application of such theories to algebra and to the study of iteration and recursion in programming languages.
This is a review of an object oriented computeralgebra system which is devoted to epresentation theory, invariant theory and combinatorics of the symmetric group. Moreover, it can be used for classical multivariate p...
This is a review of an object oriented computeralgebra system which is devoted to epresentation theory, invariant theory and combinatorics of the symmetric group. Moreover, it can be used for classical multivariate polynomials via the different actions of the symmetric group on the algebra of polynomials. The review contains a brief introduction to the basic methods used. Schubert polynomials are introduced, examples are given, and some applications are described. In particular, they provide a new algorithm for the evaluation of Littlewood-Richardson coefficients via symbolic computations using integer sequences instead of partitions, tableaux or lattice permutations.
We present a Sage implementation of Ore algebras. The main features for the most common instances include basic arithmetic and actions;GCRD and LCLM;D-finite closure properties;natural transformations between related ...
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ISBN:
(纸本)9783319150819;9783319150802
We present a Sage implementation of Ore algebras. The main features for the most common instances include basic arithmetic and actions;GCRD and LCLM;D-finite closure properties;natural transformations between related algebras;guessing;desingularization;solvers for polynomials, rational functions and ( generalized) power series. This paper is a tutorial on how to use the package.
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