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From Kajihara's transformation formula to deformed MacDonald-Ruijsenaars and Noumi-Sano operators

1999年

作者： International Conference on Finite Fields: theory applications and Algorithms

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arXiv 2021年

作者： Hallnäs, Martin Langmann, Edwin Noumi, Masatoshi Rosengren, Hjalmar Department of Mathematical Sciences Chalmers University of Technology University of Gothenburg GothenburgSE-412 96 Sweden Department of Physics KTH Royal Institute of Technology StockholmSE-106 91 Sweden Department of Mathematics KTH Royal Institute of Technology StockholmSE-100 44 Sweden Department of Mathematics Kobe University Rokko Kobe657-8501 Japan

Kajihara obtained in 2004 a remarkable transformation formula connecting multiple basic hypergeometric series associated with A-type root systems of different ranks. By multiple principle specialisations of his formula, we deduce kernel identities for deformed Macdonald-Ruijsenaars (MR) and Noumi-Sano (NS) operators. The deformed MR operators were introduced by Sergeev and Veselov in the first order case and by Feigin and Silantyev in the higher order cases. As applications of our kernel identities, we prove that all of these operators pairwise commute and are simultaneously diagonalised by the super-Macdonald polynomials. We also provide an explicit description of the algebra generated by the deformed MR and/or NS operators by a Harish-Chandra type isomorphism and show that the deformed MR (NS) operators can be viewed as restrictions of inverse limits of ordinary MR (NS) operators. Copyright © 2021, The Authors. All rights reserved.

关键词： number theory

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Fibonacci多项式的置换性质

四川大学学报（自然科学版） 2020年第6期57卷 1047-1051页

作者：王智坚四川大学数学学院成都610064

置换多项式在代数学、组合学、数论、编码理论、密码学等领域中均有广泛而又重要的应用.本文主要研究Fibonacci多项式.通过计算其函数值的等幂和,本文得到了判定这些定义在有限域上的Fibonacci多项式为置换多项式的必要条件,解决了Ferna... 详细信息

置换多项式在代数学、组合学、数论、编码理论、密码学等领域中均有广泛而又重要的应用.本文主要研究Fibonacci多项式.通过计算其函数值的等幂和,本文得到了判定这些定义在有限域上的Fibonacci多项式为置换多项式的必要条件,解决了Fernando和Rashid提出的一个公开问题,从而推广了有关Fibonacci多项式置换性的已有结论.

关键词：置换性 Fibonacci多项式有限域编码理论

Efficiently decodable non-adaptive group testing

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Efficiently decodable non-adaptive group testing

21st Annual ACM-SIAM Symposium on Discrete Algorithms

作者： Indyk, Piotr Ngo, Hung Q. Rudra, Atri CSAIL MIT United States Department of Computer Science and Engineering University at Buffalo SUNY United States

ISBN: (纸本)9780898717013

We consider the following "efficiently decodable" non-adaptive group testing problem. There is an unknown string χ ∈ {0, 1} n with at most d ones in it. We are allowed to test any subset S ⊆ [n] of the indices. The answer to the test tells whether χi = 0 for all i ∈ S or not. The objective is to design as few tests as possible (say, t tests) such that χ can be identified as fast as possible (say, poly(t)-time). Efficiently decodable non-adaptive group testing has applications in many areas, including data stream algorithms and data forensics. A non-adaptive group testing strategy can be represented by a 1 x n matrix, which is the stacking of all the characteristic vectors of the tests. It is well-known that if this matrix is d-disjunct, then any test outcome corresponds uniquely to an unknown input string. Furthermore, we know how to construct d-disjunct matrices with t = O(d2 log n) efficiently. However, these matrices so far only allow for a "decoding" time of O(nt), which can be exponentially larger than poly(t) for relatively small values of d. This paper presents a randomness efficient construction of d-disjunct matrices with t = O(d2 log n) that can be decoded in time poly(d)·t log2 t + O(t2). To the best of our knowledge, this is the first result that achieves an efficient decoding time and matches the best known O(d2 log n) bound on the number of tests. We also derandomize the construction, which results in a polynomial time deterministic construction of such matrices when d = O(log n/ log log n). A crucial building block in our construction is the notion of (d, )-list disjunct matrices, which represent the more general "list group testing" problem whose goal is to output less than d + positions in χ, including all the (at most d) positions that have a one in them. List disjunct matrices turn out to be interesting objects in their own right and were also considered independently by [Cheraghchi, FCT 2009]. We present connections between list disjunct matrices

关键词： Matrix algebra

Generalized minimum distance functions

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arXiv 2017年

作者： GONZÁLEZ-SARABIA, MANUEL MARTÍNEZ-BERNAL, JOSÉ VILLARREAL, RAFAEL H. VIVARES, CARLOS E. Instituto Politécnico Nacional Departamento de Ciencias Básicas UPIITA Av. IPN No. 2580 Col. La Laguna Ticomán Gustavo A. Madero Ciudad de MéxicoC.P. 07340 Mexico Departamento de Matemáticas Centro de Investigación y de Estudios Avanzados del IPN Apartado Postal 14-740 Mexico City D.F.07000

MSC Codes 13P25, 14G50, 94B27, 11T71Using commutative algebra methods we study the generalized minimum distance function (gmd function) and the corresponding generalized footprint function of a graded ideal in a polynomial ring over a field. The number of solutions that a system of homogeneous polynomials has in any given finite set of projective points is expressed as the degree of a graded ideal. If X is a set of projective points over a finite field and I is its vanishing ideal, we show that the gmd function and the Vasconcelos function of I are equal to the r-th generalized Hamming weight of the corresponding Reed-Muller-type code CX(d) of degree d. We show that the generalized footprint function of I is a lower bound for the r-th generalized Hamming weight of CX(d). Then we present some applications to projective nested cartesian codes. To give applications of our lower bound to algebraic coding theory, we show an interesting integer inequality. Then we show an explicit formula and a combinatorial formula for the second generalized Hamming weight of an affine cartesian code. Copyright © 2017, The Authors. All rights reserved.

关键词： polynomials

Robustness Analysis of Scaled Resource Allocation Models Using the Imperial PEPA Compiler

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Robustness Analysis of Scaled Resource Allocation Models Usi...

International Symposium on Parallel and Distributed Computing

作者： William S. Sanders Srishti Srivastava Ioana Banicescu Computer Science & Engineering Mississippi State University MS Mississippi State USA Computer Science University of Southern Indiana Evansville USA

ISBN: (数字)9781728189468

ISBN: (纸本)9781728189475

The increase in scale provided by distributed computing systems has expanded scientific discovery and engineering solutions. Stochastic modeling with Performance Evaluation Process algebra (PEPA) has been used to evaluate the robustness of static resource allocations in parallel and distributed computing systems. These evaluations have previously been performed through the PEPA Plug-In for the Eclipse Integrated Development Environment and have been limited by factors that include: i) the size and complexity of the underlying, in-use PEPA model, ii) a small number of resource allocation models available for analysis, and iii) the human interaction necessary to configure the PEPA Eclipse Plug-In, thus limiting potential automation. As the size and complexity of the underlying PEPA models increases, the number of states to be evaluated for each model also greatly increases, leading to a case of state space explosion. In this work, we validate the Imperial PEPA Compiler (IPC) as a replacement for the PEPA Eclipse Plug-In for the robustness analysis of resource allocations. We make available an implementation of the IPC as a Singularity container, as part of a larger online repository of PEPA resources. We then develop and test a programmatic method for generating PEPA models for resource allocations. When combined with our IPC container, this method allows automated analysis of resource allocation models at scale. The use of the IPC allows the evaluation of larger models than it is possible when using the PEPA Eclipse Plug-In. Moreover, the increases in scale in both model size and number of models, support the development of improved makespan targets for robustness metrics, including those among applications subject to perturbations at runtime, as found in typical parallel and distributed computing environments.

关键词： Computational modeling Robustness Resource management Analytical models Complexity theory Runtime algebra

Sparse resultant based minimal solvers in computer vision and their connection with the action matrix

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arXiv 2023年

作者： Bhayani, Snehal Heikkilä, Janne Kukelova, Zuzana Center for Machine Vision and Signal Analysis Faculty of Information Technology and Electrical Engineering Oulu Finland Visual Recognition Group Department of Cybernetics Czech Technical University Prague Czech Republic

Many computer vision applications require robust and efficient estimation of camera geometry from a minimal number of input data measurements, i.e., solving minimal problems in a RANSAC framework. Minimal problems are usually formulated as complex systems of sparse polynomial equations. The systems usually are overdetermined and consist of polynomials with algebraically constrained coefficients. Most state-of-the-art efficient polynomial solvers are based on the action matrix method that has been automated and highly optimized in recent years. On the other hand, the alternative theory of sparse resultants and Newton polytopes has been less successful for generating efficient solvers, primarily because the polytopes do not respect the constraints on the coefficients. To tackle this challenge, in this paper, we propose a simple iterative scheme to test various subsets of the Newton polytopes and search for the most efficient solver. Moreover, we propose to use an extra polynomial with a special form to further improve the solver efficiency via a Schur complement computation. We show that for some camera geometry problems our extra polynomial-based method leads to smaller and more stable solvers than the state-of-the-art Gröbner basis-based solvers. The proposed method can be fully automated and incorporated into existing tools for automatic generation of efficient polynomial solvers. It provides a competitive alternative to popular Gröbner basis-based methods for minimal problems in computer vision. Additionally, we study the conditions under which the minimal solvers generated by the state-of-the-art action matrix-based methods and the proposed extra polynomial resultant-based method, are equivalent. Specifically we consider a step-by-step comparison between the approaches based on the action matrix and the sparse resultant, followed by a set of substitutions, which would lead to equivalent minimal solvers. © 2023, CC BY.

关键词： Matrix algebra

On A Novel Security Scheme for The Encryption and Decryption Of 2×2 Fuzzy Matrices with Rational Entries Based on The algebra of Neutrosophic Integers and El-Gamal Crypto-System

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Neutrosophic Sets and Systems 2023年 54卷 19-32页

作者： Abobala, Mohammad Allouf, Ali Tishreen University Department Of Mathematics Latakia Syria Tishreen University Faculty Of computer engineering and automatio Latakia Syria

The main goal behind mathematical cryptography is to keep messages and multimedia messages secret at a time when modern means of communication have spread and become very diverse. Fuzzy matrices as strong tools which was defined to deal with incomplete and uncertain data and many relationships in real life problems especially those which are related to images and graphs, may considered as important subjects for secret information and communication. The aim of this research paper is to present a new model and method for encrypting 2×2 fuzzy matrices using the basic concepts in neutrosophic number theory and El Gamal algorithm in cryptography, where we generalize El Gamal algorithm to become applicable to the ring of neutrosophic integer numbers that represents the studied fuzzy matrices. On the other hand, we study the applications of the novel algorithm to the encryption and decryption of some fuzzy relations represented in terms of fuzzy *** addition, we illustrate many examples to clarify the validity of the new algorithm. © 2023,Neutrosophic Sets and Systems. All Rights Reserved.

关键词： Cryptography

computer-aided analysis of high-dimensional Glass networks: periodicity, chaos, and bifurcations in a ring circuit

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arXiv 2024年

作者： Belgacem, Ismail Edwards, Roderick Farcot, Etienne University of Victoria Canada School of Mathematical Sciences University of Nottingham United Kingdom

Glass networks model systems of variables that interact via sharp switching. A body of theory has been developed over several decades that, in principle, allows rigorous proof of dynamical properties in high dimensions that is not normally feasible in nonlinear dynamical systems. Previous work has, however, used examples of dimension no higher than 6 to illustrate the methods. Here we show that the same tools can be applied in dimensions at least as high as 20. An important application of Glass networks is to a recently-proposed design of a True Random number Generator that is based on an intrinsically chaotic electronic circuit. In order for analysis to be meaningful for the application, the dimension must be at least 20. Bifurcation diagrams show what appear to be periodic and chaotic bands. Here we demonstrate that the analytic tools for Glass networks can be used to rigorously show where periodic orbits are lost, and the types of bifurcations that occur there. The main tools are linear algebra and the stability theory of Poincaré maps. All main steps can be automated, and we provide computer code. The methods reviewed here have the potential for many other applications involving sharply switching interactions, such as artificial neural networks. © 2024, CC BY-NC-SA.

关键词： Random number generation

Classifying the nonsingular intersection curve of two quadric surfaces

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Classifying the nonsingular intersection curve of two quadri...

Geometric Modeling and Processing

作者： Changhe Tu Wenping Wang Jiaye Wang Faculty of Computer Science and Technology Shandong University Jinan China Department of Computer Science and Information Systems University of Hong Kong Hong Kong China

We present new results on classifying the morphology of the nonsingular intersection curve of two quadrics by studying the roots of the characteristic equation, or the discriminant, of the pencil spanned by the two quadrics. The morphology of a nonsingular algebraic curve means the structural (or topological) information about the curve, such as the number of disjoint connected components of the curve in P/spl Ropf//sup 3/ (the 3D real projective space), and whether a particular component is a compact set in any affine realization of P/spl Ropf//sup 3/. For example, we show that two quadrics intersect along a nonsingular space quartic curve in P/spl Ropf//sup 3/ with one connected component if and only if their characteristic equation has two distinct real roots and a pair of complex conjugate roots. Since the number of the real roots of the characteristic equation can be counted robustly with exact arithmetic, our results can be used to obtain structural information reliably before computing the parameterization of the intersection curve; thus errors in the subsequent computation that is most likely done using floating point arithmetic will not lead to erroneous topological classification of the intersection curve. The key technique used to prove our results is to reduce two quadrics into simple forms using a projective transformation, a technique equivalent to the simultaneous block diagonalization of two real symmetric matrices, a topic that has been studied in matrix algebra.

关键词： Equations Robustness Surface morphology computer science Floating-point arithmetic Information systems Symmetric matrices Computational geometry Design automation computer graphics