637 research outputs found
Solving word equations modulo partial commutations
AbstractIt is shown that it is decidable whether an equation over a free partially commutative monoid has a solution. We give a proof of this result using normal forms. Our method is a direct reduction of a trace equation system to a word equation system with regular constraints. Hereby we use the extension of Makanin's theorem on the decidability of word equations to word equations with regular constraints, which is due to Schulz
Solving parametric systems of polynomial equations over the reals through Hermite matrices
We design a new algorithm for solving parametric systems having finitely many
complex solutions for generic values of the parameters. More precisely, let with and
, be the algebraic set
defined by and be the projection . Under the
assumptions that admits finitely many complex roots for generic values of
and that the ideal generated by is radical, we solve the following
problem. On input , we compute semi-algebraic formulas defining
semi-algebraic subsets of the -space such that
is dense in and the number of real points in
is invariant when varies over each .
This algorithm exploits properties of some well chosen monomial bases in the
algebra where is the ideal generated by in
and the specialization property of the so-called Hermite
matrices. This allows us to obtain compact representations of the sets by
means of semi-algebraic formulas encoding the signature of a symmetric matrix.
When satisfies extra genericity assumptions, we derive complexity bounds on
the number of arithmetic operations in and the degree of the
output polynomials. Let be the maximal degree of the 's and , we prove that, on a generic , one can compute
those semi-algebraic formulas with operations in and that the polynomials involved
have degree bounded by .
We report on practical experiments which illustrate the efficiency of our
algorithm on generic systems and systems from applications. It allows us to
solve problems which are out of reach of the state-of-the-art
Robust isogeometric preconditioners for the Stokes system based on the Fast Diagonalization method
In this paper we propose a new class of preconditioners for the isogeometric
discretization of the Stokes system. Their application involves the solution of
a Sylvester-like equation, which can be done efficiently thanks to the Fast
Diagonalization method. These preconditioners are robust with respect to both
the spline degree and mesh size. By incorporating information on the geometry
parametrization and equation coefficients, we maintain efficiency on
non-trivial computational domains and for variable kinematic viscosity. In our
numerical tests we compare to a standard approach, showing that the overall
iterative solver based on our preconditioners is significantly faster.Comment: 31 pages, 4 figure
Contextual partial commutations
We consider the monoid T with the presentation which is "close" to trace monoids. We prove two different types of results. First, we give a combinatorial description of the lexicographically minimum and maximum representatives of their congruence classes in the free monoid {a; b}* and solve the classical equations, such as commutation and conjugacy in T. Then we study the closure properties of the two subfamilies of the rational subsets of T whose lexicographically minimum and maximum cross-sections respectively, are rational in {a; b}*. © 2010 Discrete Mathematics and Theoretical Computer Science
Some symmetry classifications of hyperbolic vector evolution equations
Motivated by recent work on integrable flows of curves and 1+1 dimensional
sigma models, several O(N)-invariant classes of hyperbolic equations for an -component vector are considered. In each
class we find all scaling-homogeneous equations admitting a higher symmetry of
least possible scaling weight. Sigma model interpretations of these equations
are presented.Comment: Revision of published version, incorporating errata on geometric
aspects of the sigma model interpretations in the case of homogeneous space
Towards Mixed Gr{\"o}bner Basis Algorithms: the Multihomogeneous and Sparse Case
One of the biggest open problems in computational algebra is the design of
efficient algorithms for Gr{\"o}bner basis computations that take into account
the sparsity of the input polynomials. We can perform such computations in the
case of unmixed polynomial systems, that is systems with polynomials having the
same support, using the approach of Faug{\`e}re, Spaenlehauer, and Svartz
[ISSAC'14]. We present two algorithms for sparse Gr{\"o}bner bases computations
for mixed systems. The first one computes with mixed sparse systems and
exploits the supports of the polynomials. Under regularity assumptions, it
performs no reductions to zero. For mixed, square, and 0-dimensional
multihomogeneous polynomial systems, we present a dedicated, and potentially
more efficient, algorithm that exploits different algebraic properties that
performs no reduction to zero. We give an explicit bound for the maximal degree
appearing in the computations
An Explicit Construction of Casimir Operators and Eigenvalues : I
We give a general method to construct a complete set of linearly independent
Casimir operators of a Lie algebra with rank N. For a Casimir operator of
degree p, this will be provided by an explicit calculation of its symmetric
coefficients . It is seen that these coefficients can be
descibed by some rational polinomials of rank N. These polinomials are also
multilinear in Cartan sub-algebra indices taking values from the set . The crucial point here is that for each degree one needs, in
general, more than one polinomials. This in fact is related with an observation
that the whole set of symmetric coefficients is
decomposed into sum subsets which are in one to one correspondence with these
polinomials. We call these subsets clusters and introduce some indicators with
which we specify different clusters. These indicators determine all the
clusters whatever the numerical values of coefficients
are. For any degree p, the number of clusters is independent of rank N. This
hence allows us to generalize our results to any value of rank N.
To specify the general framework explicit constructions of 4th and 5th order
Casimir operators of Lie algebras are studied and all the polinomials
which specify the numerical value of their coefficients are given explicitly.Comment: 14 pages, no figures, revised version, to appear in Jour.Math.Phy
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