The ISZ method (Interval-Symbol method with Zero rewriting) based on stabilization theory was proposed to reduce the amount of exact computations as much as possible but obtain the exact results by aid of floating-poi...
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The ISZ method (Interval-Symbol method with Zero rewriting) based on stabilization theory was proposed to reduce the amount of exact computations as much as possible but obtain the exact results by aid of floating-point computations. In this paper, we applied the ISZ method to Trager's algorithm which factors univariate polynomials over algebraic number fields. By Maple experiments, we show the efficiency of the ISZ method over the purely exact approach which uses exact computations throughout the execution of the algorithm. Furthermore, we propose a new method called the ISZ* method, which is similar to the ISZ method but beforehand excludes insufficient precisions of floating-point approximation by checking the correctness of the obtained supports. We confirmed that the ISZ* method is more *** than the ISZ method when the initially set precision is not sufficiently high.
In this paper, we present a novel variation of the coded matrix multiplication problem which we refer to as fully private grouped matrix multiplication (FPGMM). In FPGMM, a master wants to compute a group of matrix pr...
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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...
Designing high-dimensional chaotic maps with expected dynamic properties is an attractive but challenging task. The dynamic properties of a chaotic system can be reflected by the Lyapunov exponents (LEs). Using the in...
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Designing high-dimensional chaotic maps with expected dynamic properties is an attractive but challenging task. The dynamic properties of a chaotic system can be reflected by the Lyapunov exponents (LEs). Using the inherent relationship between the parameters of a chaotic map and its LEs, this paper proposes an n-dimensional polynomial chaotic system (nD-PCS) that can generate nD chaotic maps with any desired LEs. The nD-PCS is constructed from n parametric polynomials with arbitrary orders, and its parameter matrix is configured using the preliminaries in linear algebra. Theoretical analysis proves that the nD-PCS can produce high-dimensional chaotic maps with any desired LEs. To show the effects of the nD-PCS, two high-dimensional chaotic maps with hyperchaotic behaviors were generated. A microcontroller-based hardware platform was developed to implement the two chaotic maps, and the test results demonstrated the randomness properties of their chaotic signals. Performance evaluations indicate that the high-dimensional chaotic maps generated from nD-PCS have the desired LEs and more complicated dynamic behaviors compared with other high-dimensional chaotic maps. In addition, to demonstrate the applications of nD-PCS, we developed a chaos-based secure communication scheme. Simulation results show that nD-PCS has a stronger ability to resist channel noise than other high-dimensional chaotic maps.
The proceedings contain 21 papers. The special focus in this conference is on computeralgebra in Scientific Computing. The topics include: A General Method of Finding New Symplectic Schemes for Hamiltonian ...
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(纸本)9783031147876
The proceedings contain 21 papers. The special focus in this conference is on computeralgebra in Scientific Computing. The topics include: A General Method of Finding New Symplectic Schemes for Hamiltonian Mechanics;a Mechanical Method for Isolating Locally Optimal Points of Certain Radical Functions;subresultant Chains Using Bézout Matrices;application of Symbolic-Numerical Modeling Tools for Analysis of Gyroscopic Stabilization of Gyrostat Equilibria;computer Science for Continuous Data: Survey, Vision, theory, and Practice of a computer Analysis System;computational Aspects of Equivariant Hilbert Series of Canonical Rings for algebraic Curves;symbolic-Numeric Algorithm for Calculations in Geometric Collective Model of Atomic Nuclei;analyses and Implementations of Chordality-Preserving Top-Down Algorithms for Triangular Decomposition;accelerated Subdivision for Clustering Roots of polynomials Given by Evaluation Oracles;preface;survey on Generalizations of the Intermediate Value Theorem and applications;on Equilibrium Positions in the Problem of the Motion of a System of Two Bodies in a Uniform Gravity Field;an Interpolation Algorithm for Computing Dixon Resultants;distance Evaluation to the Set of Matrices with Multiple Eigenvalues;on Boundary Conditions Parametrized by Analytic Functions;computing the Integer Hull of Convex Polyhedral Sets;a Comparison of Algorithms for Proving Positivity of Linearly Recurrent Sequences;stability Analysis of Periodic Motion of the Swinging Atwood Machine;new Heuristic to Choose a Cylindrical algebraic Decomposition Variable Ordering Motivated by Complexity Analysis.
MSC Codes 16W99, 17B37, 16S10, 16T10, 16T30, 57R56The Rota-Baxter operator and the modified Rota-Baxter operator on various algebras are both important in mathematics and mathematical physics. The former is originated...
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The functional (de)composition of polynomials is a topic in pure and computeralgebra with many applications. The structure of decompositions of (suitably normalized) polynomials f = g circle h in F[x] over a field F ...
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The functional (de)composition of polynomials is a topic in pure and computeralgebra with many applications. The structure of decompositions of (suitably normalized) polynomials f = g circle h in F[x] over a field F is well understood in many cases, but less well when the degree of f is divisible by the positive characteristic p of F. This work investigates the decompositions of r-additive polynomials, where every exponent and also the field size is a power of r, which itself is a power of p. The decompositions of an r-additive polynomial f are intimately linked to the Frobenius-invariant subspaces of its root space V in the algebraic closure of F. We present an efficient algorithm to compute the rational Jordan form of the Frobenius automorphism on V. A formula of Fripertinger (2011) then counts the number of Frobenius-invariant subspaces of a given dimension and we derive the number of decompositions with prescribed degrees. (C) 2020 Elsevier Ltd. All rights reserved.
We prove upper bounds which are independent of the dimension of the ambient space, on the number of realizable zero-nonzero patterns as well as sign conditions (when the field of coefficients is ordered) of a finite s...
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In his contribution to the Proceedings of the 1998 AMS–IMS–SIAM Joint Summer Research Conference on q-Series, Combinatorics, and computeralgebra, Andrews considered a variant of Schur’s partition theorem, concerni...
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The additive monoid R+(x) is defined as the set of all nonnegative integer linear combinations of binomial polynomials ((x)(n)) for n is an element of Z(+). This paper is concerned with the inquiry into the structure ...
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The additive monoid R+(x) is defined as the set of all nonnegative integer linear combinations of binomial polynomials ((x)(n)) for n is an element of Z(+). This paper is concerned with the inquiry into the structure of R+(alpha) for complex numbers alpha. Particularly interesting is the case of algebraic awhich are not non-negative integers. This question is motivated by the study of functors between Deligne categories Rep(S-t)(and also Rep(GL(t))) for t is an element of C\Z(+). We prove that this object is a ring if and only if ais an algebraic number that is not a nonnegative integer. Furthermore, we show that all algebraic integers generated by alpha, i.e. all elements of O-Q(a), are also contained in this ring. We also give two explicit representations of R+(alpha) for both algebraic integers and general algebraic numbers alpha. One is in terms of inequalities for the valuations with respect to certain prime ideals and the other is in terms of explicitly constructed generators. We show how these results work in the context of the study of symmetric monoidal functors between Deligne categories in positive characteristic. Moreover, this leads to a particularly simple description of R+(alpha) for both quadratic algebraic numbers and roots of unity. (C) 2021 Elsevier Inc. All rights reserved.
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