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Abstract In this paper we study the multiplicative function ρ k , λ (n) that counts the number of solutions of the equation x 1 2 + ⋯ + x k 2 ≡ λ (mod n) in (Z / n Z) k. In particular we give closed explicit formulas for ρ k , λ (p s). This leads to an algorithm with an arithmetic complexity of constant order that improves previous work by Tóth [10] and completes the quadratic case considered by Li and Ouyang in [8]. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, an effective discrimination algorithm is presented to deal with equations arising from physical problems. The aim of the algorithm is to discriminate and derive the single traveling wave solutions of a large class of nonlinear evolution equations. Many examples are given to illustrate the algorithm. At the same time, some factorization technique are presented to construct the traveling wave solutions of nonlinear evolution equations, such as Camassa–Holm equation, Kolmogorov–Petrovskii–Piskunov equation, and so on. Then a direct constructive method called multi-auxiliary equations expansion method is described to derive the multi-solitary wave solutions of nonlinear evolution equations. Finally, a class of novel multi-solitary wave solutions of the -dimensional asymmetric version of the Nizhnik–Novikov–Veselov equation are given by three direct methods. The algorithm proposed in this paper can be steadily applied to some other nonlinear problems. [Copyright &y& Elsevier]

This paper is concerned with nonhomogeneous refinement equations of the form ϕ(x)=∑lower limit α∈Zs a(α)ϕ(Mx−α)+g(x), x∈Rs, where the vector of functions ϕ=(ϕ1,…,ϕr)T is unknown, g is a given vector of compactly supported functions on Rs, a is a finitely supported sequence of r×r matrices called the refinement mask, and M is an s×s integer matrix such that limn→∞M−n=0. Our approach will be to consider the convergence rates of the cascade algorithms associated with nonhomogeneous refinement equations mentioned above. The cascade algorithms associated with mask a, nonhomogeneous term g, and dilation matrix M generates a sequence ϕn, n=1,2,…, by the iterative process ϕn(x)=∑lower limit α∈Zs a(α)ϕn−1(Mx−α)+g(x), x∈Rs, from a starting vector of function ϕ0 in (Lp(Rs))r (0The aim of this paper is to give a characterization of the convergence rates of the cascade algorithms associated with a,g,ϕ0 and dilation matrix M in (Lp(Rs))r (0 in terms of the p-norm joint spectral radius of a finite collection of some linear operators determined by the sequence a and the set E restricted to a certain invariant subspace, where the set E is a complete set of representatives of the distinct cosets of the quotient group Zs/MZs containing 0. Some examples are provided to illustrate the method. [Copyright &y& Elsevier]

Abstract: Multiobjective discrete programming is a well-known family of optimization problems with a large spectrum of applications. The linear case has been tackled by many authors during the past few years. However, the polynomial case has not been studied in detail due to its theoretical and computational difficulties. This paper presents an algebraic approach for solving these problems. We propose a methodology based on transforming the polynomial optimization problem to the problem of solving one or more systems of polynomial equations and we use certain Gröbner bases to solve these systems. Different transformations give different methodologies that are theoretically stated and compared by some computational tests via the algorithms that they induce. [Copyright &y& Elsevier]

Abstract: A Reed–Solomon code of length can be list decoded using the well-known Guruswami–Sudan algorithm. By a result of the interpolation part in this algorithm can be done in complexity , where denotes the designed list size and the multiplicity parameter. The parameters and are sometimes considered to be constants in the complexity analysis, but for high rate Reed–Solomon codes, their values can be very large. In this paper we will combine ideas from and the concept of key equations to get an algorithm that has complexity . This compares favorably to the complexities of other known interpolation algorithms. [Copyright &y& Elsevier]

Abstract: In this paper we deal with the verification of safety properties of infinite-state systems modeled by term rewriting systems. An over-approximation of the set of reachable terms of a term rewriting system is obtained by automatically constructing a finite tree automaton. The construction is parameterized by a set of equations on terms, and we also show that the approximating automata recognize at most the set of -reachable terms. Finally, we present some experiments carried out with the implementation of our algorithm. In particular, we show how some approximations from the literature can be defined using equational approximations. [Copyright &y& Elsevier]

Abstract: A canal surface is an envelope of a one-parameter family of spheres. In this paper we present an efficient algorithm for computing the implicit equation of a canal surface generated by a rational family of spheres. By using Laguerre and Lie geometries, we relate the equation of the canal surface to the equation of a dual variety of a certain curve in 5-dimensional projective space. We define the -basis for arbitrary dimension and give a simple algorithm for its computation. This is then applied to the dual variety, which allows us to deduce the implicit equations of the dual variety, the canal surface and any offset to the canal surface. [Copyright &y& Elsevier]

Abstract: Triangular decompositions for systems of polynomial equations with variables, with exact coefficients, are well developed theoretically and in terms of implemented algorithms in computer algebra systems. However there is much less research concerning triangular decompositions for systems with approximate coefficients. In this paper we discuss the zero-dimensional case of systems having finitely many roots. Our methods depend on having approximations for all the roots, and these are provided by the homotopy continuation methods of Sommese, Verschelde and Wampler. We introduce approximate equiprojectable decompositions for such systems, which represent a generalization of the recently developed analogous concept for exact systems. We demonstrate experimentally the favorable computational features of this new approach, and give a statistical analysis of its error. [Copyright &y& Elsevier]