1,868 research outputs found

    Linear complexity of sequences and multisequences

    Get PDF

    On Some Dynamical Systems in Finite Fields and Residue Rings

    Full text link
    We use character sums to confirm several recent conjectures of V. I. Arnold on the uniformity of distribution properties of a certain dynamical system in a finite field. On the other hand, we show that some conjectures are wrong. We also analyze several other conjectures of V. I. Arnold related to the orbit length of similar dynamical systems in residue rings and outline possible ways to prove them. We also show that some of them require further tuning

    Fast algorithm for border bases of Artinian Gorenstein algebras

    Get PDF
    Given a multi-index sequence σ\sigma, we present a new efficient algorithm to compute generators of the linear recurrence relations between the terms of σ\sigma. We transform this problem into an algebraic one, by identifying multi-index sequences, multivariate formal power series and linear functionals on the ring of multivariate polynomials. In this setting, the recurrence relations are the elements of the kerne lII\sigma of the Hankel operator $H$\sigma associated to σ\sigma. We describe the correspondence between multi-index sequences with a Hankel operator of finite rank and Artinian Gorenstein Algebras. We show how the algebraic structure of the Artinian Gorenstein algebra AA\sigmaassociatedtothesequence associated to the sequence \sigma yields the structure of the terms $\sigma\alphaforall for all α\alpha ∈\in N n.Thisstructureisexplicitlygivenbyaborderbasisof. This structure is explicitly given by a border basis of Aσ\sigma,whichispresentedasaquotientofthepolynomialring, which is presented as a quotient of the polynomial ring K[x 1 ,. .. , xn]bythekernel] by the kernel Iσ\sigmaoftheHankeloperator of the Hankel operator Hσ\sigma.Thealgorithmprovidesgeneratorsof. The algorithm provides generators of Iσ\sigmaconstitutingaborderbasis,pairwiseorthogonalbasesof constituting a border basis, pairwise orthogonal bases of Aσ\sigma$ and the tables of multiplication by the variables in these bases. It is an extension of Berlekamp-Massey-Sakata (BMS) algorithm, with improved complexity bounds. We present applications of the method to different problems such as the decomposition of functions into weighted sums of exponential functions, sparse interpolation, fast decoding of algebraic codes, computing the vanishing ideal of points, and tensor decomposition. Some benchmarks illustrate the practical behavior of the algorithm

    Tensor-Sparsity of Solutions to High-Dimensional Elliptic Partial Differential Equations

    Full text link
    A recurring theme in attempts to break the curse of dimensionality in the numerical approximations of solutions to high-dimensional partial differential equations (PDEs) is to employ some form of sparse tensor approximation. Unfortunately, there are only a few results that quantify the possible advantages of such an approach. This paper introduces a class Σn\Sigma_n of functions, which can be written as a sum of rank-one tensors using a total of at most nn parameters and then uses this notion of sparsity to prove a regularity theorem for certain high-dimensional elliptic PDEs. It is shown, among other results, that whenever the right-hand side ff of the elliptic PDE can be approximated with a certain rate O(n−r)\mathcal{O}(n^{-r}) in the norm of H−1{\mathrm H}^{-1} by elements of Σn\Sigma_n, then the solution uu can be approximated in H1{\mathrm H}^1 from Σn\Sigma_n to accuracy O(n−r′)\mathcal{O}(n^{-r'}) for any r′∈(0,r)r'\in (0,r). Since these results require knowledge of the eigenbasis of the elliptic operator considered, we propose a second "basis-free" model of tensor sparsity and prove a regularity theorem for this second sparsity model as well. We then proceed to address the important question of the extent such regularity theorems translate into results on computational complexity. It is shown how this second model can be used to derive computational algorithms with performance that breaks the curse of dimensionality on certain model high-dimensional elliptic PDEs with tensor-sparse data.Comment: 41 pages, 1 figur

    On the Degree Growth in Some Polynomial Dynamical Systems and Nonlinear Pseudorandom Number Generators

    Full text link
    In this paper we study a class of dynamical systems generated by iterations of multivariate polynomials and estimate the degreegrowth of these iterations. We use these estimates to bound exponential sums along the orbits of these dynamical systems and show that they admit much stronger estimates than in the general case and thus can be of use for pseudorandom number generation.Comment: Mathematics of Computation (to appear
    • …
    corecore