Major substep in a lattice sieve algorithm which solves the Euclidean shortest vector problem (SVP) is the computation of sums and Euclidean norms of many vector pairs. Finding a solution to the SVP is the foundation of an attack against many lattice based crypto systems. We optimize the main subfunction of a sieve for the regular main processor and for the co-processor to speed up the algorith...

1 OPERATOR DECOMPOSITIONS In this lecture we will consider two basic decompositions of operators that will be used repeatedly in the course: the spectral decomposition and the singular-value decomposition. The spectral decomposition holds only for normal operators, while the singular-value decomposition holds for all operators. The singular-value decomposition will be particularly important in ...

TOPSIS is a multicriteria decision making technique based on the minimization of geometric distances that allows the ordering of compared alternatives in accordance with their distances from the ideal and anti-ideal solutions. The technique, that usually measures distances in the Euclidean norm, implicitly supposes that the contemplated attributes are independent. However, as this rarely occurs...

In this paper we initiate the study of tropical Voronoi diagrams. We start out with investigating bisectors finitely many points respect to arbitrary polyhedral norms. For more general scenario show that three are homeomorphic a non-empty open subset Euclidean space, provided certain degenerate cases excluded. Specializing our results then yields structural and algorithms for

We study approximation properties of weighted $\mathrm{L}^2$-orthogonal projectors onto spaces polynomials bounded degree in the Euclidean unit ball, where weight is reflection-invariant form $(1-\lVert x \rVert^2)^\alpha \prod_{i=1}^d \lvert x_i \rvert^{\gamma_i}$, $\alpha, \gamma_1, \dots, \gamma_d > -1$. Said are measured Dunkl-Sobolev-type norms which same $\mathrm{L}^2$ norm used to contro...

Abstract In this paper, we study the semiclassical behavior of distorted plane waves, on manifolds that are Euclidean near infinity or hyperbolic infinity, and non-positive curvature. Assuming there is a strip without resonances below real axis, show waves bounded in $L^2_{loc}$ independently $h$ they admit unique measure prove bounds their $L^p_{loc}$ norms.

In this paper, we establish a correlation between the bihyperbolic numbers set and semi-Euclidean space. There are three different norms on space that allow us to define hypersurfaces Hence, construct some topological structures these called norm e, s, t topologies. On other hand, introduce hyperbolic topologies set. Moreover, by using idempotent spectral representations of numbers, new

it is necessary to generate the automorphism group of a chemical graph in computer-aidedstructure elucidation. an euclidean graph associated with a molecule is defined by a weightedgraph with adjacency matrix m = [dij], where for i≠j, dij is the euclidean distance between thenuclei i and j. in this matrix dii can be taken as zero if all the nuclei are equivalent. otherwise,one may introduce dif...