Article
Mathematics, Applied
J. Alahmadi, H. Alqahtani, M. S. Pranic, L. Reichel
Summary: This paper focuses on the approximation of matrix functionals and introduces Gauss-Laurent quadrature rules for this purpose. The performance of the rules is demonstrated through computed examples.
NUMERICAL ALGORITHMS
(2021)
Article
Mathematics
Dmitry Yu Pochekutov
Summary: In this paper, branch points of complete q-diagonals of Laurent series for rational functions in several complex variables are described using the logarithmic Gauss mapping. A sufficient condition for non-algebraicity of such a diagonal is proven.
JOURNAL OF SIBERIAN FEDERAL UNIVERSITY-MATHEMATICS & PHYSICS
(2021)
Article
Mathematics, Applied
J. Alahmadi, M. Pranic, L. Reichel
Summary: This paper focuses on computing approximations of matrix functionals of the form F(A) := v(T) f (A)v, where A is a large symmetric positive definite matrix, v is a vector, and f is a Stieltjes function. The paper proposes using rational Gauss quadrature rules and develops rational Gauss-Radau and rational anti-Gauss rules. These rules can be used to determine upper and lower bounds, or approximate upper and lower bounds, for F(A). In cases where the function f has singularities close to the spectrum of A, the use of rational Gauss rules is beneficial.
NUMERISCHE MATHEMATIK
(2022)
Article
Mathematics, Applied
A. H. Bentbib, K. Jbilou, L. Reichel, M. El Ghomari
Summary: This paper describes methods for computing upper and lower bounds estimates of matrix function elements based on the extended symmetric block Lanczos process. The methods use block Gauss-Laurent and block anti-Gauss-Laurent quadrature rules to determine the desired estimates and generalize previous methods by allowing the application of extended block Krylov subspaces. Computed examples illustrate the effectiveness of the proposed methods.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Sadulla Z. Jafarov
Summary: This study examines the approximation properties of the Faber-Laurent rational series expansions in variable exponent Morrey spaces, as well as proving direct theorems of approximation theory in variable exponent Morrey-Smirnov classes defined in domains with a Dini-smooth boundary.
COMPUTATIONAL METHODS AND FUNCTION THEORY
(2022)
Article
Mathematics
William Craig, Mircea Merca
Summary: The study of Ramanujan-type congruences for functions specific to additive number theory has a long and rich history. Motivated by recent connections between divisor sums and overpartitions via congruences in arithmetic progressions, we investigate the existence and classification of Ramanujan-type congruences for functions in multiplicative number theory.
REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS
(2022)
Article
Mathematics
Alexander Dyachenko, Dmitrii Karp
Summary: In this paper, we consider the ratio of Gauss hypergeometric functions and derive a formula for ImR(x +/- i0) in terms of real hypergeometric polynomial P, beta density, and the absolute value of the Gauss hypergeometric function. We construct explicit integral representations for R when the asymptotic behavior at unity is mild and the denominator does not vanish. Several examples are provided to illustrate the results.
Article
Automation & Control Systems
Hamed Taghavian, Ross Drummond, Mikael Johansson
Summary: This paper examines the category of logarithmically completely monotonic (LCM) functions and their importance in characterizing externally positive linear systems. It proposes conditions for ensuring a rational function is LCM, which expands the space of linear continuous-time externally positive systems and allows for the development of an efficient and optimal pole-placement procedure for the monotonic tracking controller synthesis problem. These conditions are less conservative than existing approaches and computationally tractable.
Article
Mathematics, Applied
Wenjia Guo, Xiaoge Liu, Tianping Zhang
Summary: This study focuses on Dirichlet characters of rational polynomials, obtaining new identities by relying on the properties of character and Gauss sums. By introducing new ingredients, we have derived some new identities regarding the fourth power mean.
Article
Mathematics
Raul E. Curto, In Sung Hwang, Woo Young Lee
Summary: The article demonstrates that every inner divisor of the operator-valued coordinate function is a Blaschke-Potapov factor, and introduces the concept of operator-valued rational function. It further proves that delta is two-sided inner and rational if and only if it can be represented as a finite Blaschke-Potapov product.
JOURNAL OF FUNCTIONAL ANALYSIS
(2022)
Article
Mathematics, Applied
Hans-Peter Schroecker, Zbynek Sir
Summary: All rational parametric curves with prescribed polynomial tangent direction form a vector space, including the important case of rational Pythagorean hodograph curves. The vector subspaces defined by fixing the denominator polynomial are studied, and the construction of canonical bases for them is described. It is also shown that any element of the vector space can be obtained as a finite sum of curves with single roots at the denominator, analogous to the fraction decomposition of rational functions. These results provide insight into the structure of these spaces, clarify the role of polynomial and non-polynomial curves, and suggest applications to interpolation problems.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
A. A. Tuganbaev
Summary: The skew Laurent series ring A((x,φ)) is a right serial ring if and only if A is a right serial right Artinian ring.
JOURNAL OF ALGEBRA AND ITS APPLICATIONS
(2021)
Article
Mathematics, Applied
Sadulla Z. Jafarov
Summary: In this paper, we investigate the properties of approximation by partial sums of the Fourier series in a finite Jordan domain G bounded by a Dini smooth curve Gamma in the complex plane C. We prove a direct theorem for approximation by polynomials in the subspace of Morrey spaces associated with grand Lebesgue spaces. Additionally, we study the approximation properties of the Faber-Laurent rational series expansions in spaces Lp),lambda (Gamma). We also prove direct theorems of approximation theory in grand Morrey-Smirnov classes defined in domains with a Dini-smooth boundary.
Article
Computer Science, Theory & Methods
Raul Perez-Fernandez
Summary: Ordered Weighted Averaging (OWA) functions are widely used for aggregating real values in various fields. This paper presents a unified perspective by introducing different classes of multivariate OWA functions under a common framework and discussing their respective properties.
FUZZY SETS AND SYSTEMS
(2023)
Article
Mathematics
D. Y. Pochekutov, A. Senashov
Summary: This article discusses the Laurent series of a rational function in n complex variables and the n-dimensional sequence of its coefficients. The diagonal subsequence of this sequence generates the complete diagonal of the Laurent series, and a new integral representation for the complete diagonal is provided. Based on this representation, a sufficient condition for a diagonal to be algebraic is given.
SIBERIAN ELECTRONIC MATHEMATICAL REPORTS-SIBIRSKIE ELEKTRONNYE MATEMATICHESKIE IZVESTIYA
(2022)
Article
Mathematics, Applied
Armin Straub
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2019)
Article
Mathematics, Applied
Drew Lewis, Kaitlyn Perry, Armin Straub
JOURNAL OF PURE AND APPLIED ALGEBRA
(2019)
Article
Mathematics, Applied
Karl Dilcher, Armin Straub, Christophe Vignat
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2019)
Article
Mathematics, Applied
Sam Formichella, Armin Straub
ANNALS OF COMBINATORICS
(2019)
Article
Mathematics
Tewodros Amdeberhan, Victor H. Moll, Armin Straub, Christophe Vignat
Summary: The paper establishes a triple integral and an equivalent polylogarithmic double sum, discussing their connection to zeta(3). By reviewing duality and using computer algebra methods, it reveals the complex relationships between polylogarithms.
INTERNATIONAL JOURNAL OF NUMBER THEORY
(2021)
Article
Mathematics, Applied
Hannah Burson, Simone Sisneros-Thiry, Armin Straub
Summary: In this study, the authors extend the work of Nath and Sellers by considering the number of parts of partitions, d-distinct partitions, and more general (s, ms +/- r)-core partitions. As a result, they are able to determine the average and maximum number of parts in these core partitions.
ELECTRONIC JOURNAL OF COMBINATORICS
(2021)
Article
Mathematics
Marc Chamberland, Armin Straub
Summary: The article presents Apery's proof of the irrationality of zeta(3) and explores various methods for representing zeta(3) as the limit of the quotient of two rational solutions. Connections to continued fractions and the theorems of Poincare and Perron on difference equations are highlighted. An experimental mathematics approach is advocated, starting with a simple motivating example, and various open problems are discussed at the end.
AMERICAN MATHEMATICAL MONTHLY
(2021)
Article
Mathematics
Armin Straub, Wadim Zudilin
Summary: The Apery limits for the sums of powers of binomial coefficients are explicitly determined, and as an application, a weak version of Franel's conjecture on the order of recurrences for these sequences is proved.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2023)
Article
Mathematics
Armin Straub
Summary: Rowland and Zeilberger developed an algorithm to determine the modulo p(r) reductions of values of combinatorial sequences, and proposed different types of schemes. The usefulness of these schemes is demonstrated by their application in Motzkin numbers.
RESEARCH IN NUMBER THEORY
(2022)
Article
Mathematics, Applied
Joel A. Henningsen, Armin Straub
Summary: We observe that a sequence satisfies Lucas congruences modulo p if and only if its values modulo p can be described by a linear p-scheme with a single state. This simple observation suggests natural generalizations of the notion of Lucas congruences and proves explicit generalized Lucas congruences for certain integer sequences.
ADVANCES IN APPLIED MATHEMATICS
(2022)
Article
Mathematics
Wouter Castryck, Marc Houben, Frederik Vercauteren, Benjamin Wesolowski
Summary: This paper introduces how the Weil pairing can be used to evaluate the assigned characters of an imaginary quadratic order in an unknown ideal class. The method is simpler and faster when applied to ordinary elliptic curves over finite fields. The main implication is breaking the decisional Diffie-Hellman problem for practically all oriented elliptic curves.
RESEARCH IN NUMBER THEORY
(2022)
Article
Mathematics
Marc Houben, Marco Streng
Summary: The Hilbert class polynomial is related to the j-invariants of elliptic curves with a given imaginary quadratic order. It can be used to compute elliptic curves over finite fields with a specific number of points. Weber's functions are the best known alternative modular functions, reducing the size by a factor of 72. In this paper, a generalization of class polynomials is introduced, with reduction factors not limited by the Broker-Stevenhagen bound. Examples are provided that match Weber's reduction factor, and for an infinite family of discriminants, the reduction factors surpass all previously known modular functions by at least a factor of 2.
RESEARCH IN NUMBER THEORY
(2022)
Article
Mathematics
Alin Bostan, Armin Straub, Sergey Yurkevich
Summary: This paper presents the characteristics and classification of constant term sequences, studying constant term sequences in linear recurrence sequences and hypergeometric sequences.
JOURNAL OF NUMBER THEORY
(2023)
Proceedings Paper
Computer Science, Information Systems
Wouter Castryck, Thomas Decru, Marc Houben, Frederik Vercauteren
Summary: This paper addresses three main open problems related to the computation of long chains of isogenies between elliptic curves over finite fields using radical isogenies. It presents an interpolation method for finding radical isogeny formulae and pushes the range of available formulae. It derives optimized versions of these formulae to improve computation speed. It also solves the problem of choosing the correct radical when walking along the surface between supersingular elliptic curves. The techniques proposed in this paper provide substantial speed-ups.
ADVANCES IN CRYPTOLOGY- ASIACRYPT 2022, PT II
(2022)
Proceedings Paper
Mathematics
Jakub Byszewski, Gunther Cornelissen, Marc Houben, Lois van der Meijden
DYNAMICS: TOPOLOGY AND NUMBERS
(2020)
