Factorization of Non-Commutative Polynomials. (arXiv:1002.3180v1 [cs.MS])

Factorization of Non-Commutative Polynomials. (arXiv:1002.3180v1 [cs.MS]): "

We describe an algorithm for the factorization of non-commutative polynomials
over a field. The first sketch of this algorithm appeared in an unpublished
manuscript (literally hand written notes) by James H. Davenport more than 20
years ago. This version of the algorithm contains some improvements with
respect to the original sketch. An improved version of the algorithm has been
fully implemented in the Axiom computer algebra system.

"

Equal--area method for scalar conservation laws. (arXiv:1002.3260v1 [math.NA])

Equal--area method for scalar conservation laws. (arXiv:1002.3260v1 [math.NA]): "

We study one-dimensional conservation law. We develop a simple numerical
method for computing the unique entropy admissible weak solution to the initial
problem. The method basis on the equal-area principle and gives the solution
for given time directly.

"

Max-relaxation iteration procedure for building of Barabanov norms: convergence and examples. (arXiv:1002.3251v1 [math.RA])

Max-relaxation iteration procedure for building of Barabanov norms: convergence and examples. (arXiv:1002.3251v1 [math.RA]): "

The problem of construction of Barabanov norms for analysis of properties of
the joint (generalized) spectral radius of matrix sets has been discussed in a
number of publications. In previous papers of the author the method of
Barabanov norms was the key instrument in disproving the Lagarias-Wang
Finiteness Conjecture. The related constructions were essentially based on the
study of the geometrical properties of the unit balls of some specific
Barabanov norms. In this context the situation when one fails to find among
current publications any detailed analysis of the geometrical properties of the
unit balls of Barabanov norms looks a bit paradoxical. Partially this is
explained by the fact that Barabanov norms are defined nonconstructively, by an
implicit procedure. So, even in simplest cases it is very difficult to
visualize the shape of their unit balls. The present work may be treated as the
first step to make up this deficiency. In the paper an iteration procedure is
considered that allows to build numerically Barabanov norms for the irreducible
matrix sets and simultaneously to compute the joint spectral radius of these
sets.

"

Creating materials with a desired refraction coefficient: numerical experiments. (arXiv:1002.3533v1 [math.NA])

Creating materials with a desired refraction coefficient: numerical experiments. (arXiv:1002.3533v1 [math.NA]): "

A recipe for creating materials with a desired refraction coefficient is
implemented numerically. The following assumptions are used: \bee
\zeta_m=h(x_m)/a^\kappa,\quad d=O(a^{(2-\kappa)/3}),\quad
M=O(1/a^{2-\kappa}),\quad \kappa\in(0,1), \eee where $\zeta_m$ and $x_m$ are
the boundary impedance and center of the $m$-th ball, respectively, $h(x)\in
C(D)$, Im$h(x)\leq 0$, $M$ is the number of small balls embedded in the cube
$D$, $a$ is the radius of the small balls and $d$ is the distance between the
neighboring balls.

An error estimate is given for the approximate solution of the many-body
scattering problem in the case of small scatterers. This result is used for the
estimate of the minimal number of small particles to be embedded in a given
domain $D$ in order to get a material whose refraction coefficient approximates
the desired one with the relative error not exceeding a desired small quantity.

"