Some Combinatorial Operators in Language Theory

Multitildes are regular operators that were introduced by Caron et al. in order to increase the number of Glushkov automata. In this paper, we study the family of the multitilde operators from an algebraic point of view using the notion of operad. This leads to a combinatorial description of already known results as well as new results on compositions, actions and enumerations.Comment: 21 page

CUR Low Rank Approximation of a Matrix at Sublinear Cost

Low rank approximation of a matrix (hereafter LRA) is a highly important area of Numerical Linear and Multilinear Algebra and Data Mining and Analysis. One can operate with LRA at sublinear cost, that is, by using much fewer memory cells and flops than an input matrix has entries, but no sublinear cost algorithm can compute accurate LRA of the worst case input matrices or even of the matrices of small families in our Appendix. Nevertheless we prove that Cross-Approximation celebrated algorithms and even more primitive sublinear cost algorithms output quite accurate LRA for a large subclass of the class of all matrices that admit LRA and in a sense for most of such matrices. Moreover, we accentuate the power of sublinear cost LRA by means of multiplicative pre-processing of an input matrix, and this also reveals a link between C-A algorithms and Randomized and Sketching LRA algorithms. Our tests are in good accordance with our formal study.Comment: 29 pages, 5 figures, 5 tables. arXiv admin note: text overlap with arXiv:1906.0492

The Dual JL Transforms and Superfast Matrix Algorithms

We call a matrix algorithm superfast (aka running at sublinear cost) if it involves much fewer flops and memory cells than the matrix has entries. Using such algorithms is highly desired or even imperative in computations for Big Data, which involve immense matrices and are quite typically reduced to solving linear least squares problem and/or computation of low rank approximation of an input matrix. The known algorithms for these problems are not superfast, but we prove that their certain superfast modifications output reasonable or even nearly optimal solutions for large input classes. We also propose, analyze, and test a novel superfast algorithm for iterative refinement of any crude but sufficiently close low rank approximation of a matrix. The results of our numerical tests are in good accordance with our formal study.Comment: 36.1 pages, 5 figures, and 1 table. arXiv admin note: text overlap with arXiv:1710.07946, arXiv:1906.0411

Summing Squares and Cubes of Integers

Recreational mathematics can provide students with opportunities to explore mathematics in meaningful ways. Elementary number theory is one area of mathematics that lends itself readily to recreational mathematics. In this article, the author provides two examples from elementary number theory with results that students might find surprising, and which may be used to motivate them to study additional topics from number theory

3D discrete rotations using hinge angles

International audienceIn this paper, we study 3D rotations on grid points computed by using only integers. For that purpose, we investigate the intersection between the 3D half-grid and the rotation plane. From this intersection, we define 3D hinge angles which determine a transit of a grid point from a voxel to its adjacent voxel during the rotation. Then, we give a method to sort all 3D hinge angles with integer computations. The study of 3D hinge angles allows us to design a 3D discrete rotation and to estimate the rotation between a pair of digital images in correspondence

BPS States and Automorphisms

The purpose of the present paper is twofold. In the first part, we provide an algebraic characterization of several families of $\nu= \frac{1}{2^n}$ $n\leq 5$ BPS states in M theory, at threshold and non-threshold, by an analysis of the BPS bound derived from the ${\cal N}=1$ D=11 SuperPoincar\'e algebra. We determine their BPS masses and their supersymmetry projection conditions, explicitly. In the second part, we develop an algebraic formulation to study the way BPS states transform under GL(32,\bR) transformations, the group of automorphisms of the corresponding SuperPoincar\'e algebra. We prove that all $\nu={1/2}$ non-threshold bound states are SO(32) related with $\nu={1/2}$ BPS states at threshold having the same mass. We provide further examples of this phenomena for less supersymmetric $\nu={1/4},{1/8}$ non-threshold bound states.Comment: 16 pages, RevTex, no figures, 3 tables. Published versio