2,045 research outputs found
Improved method for finding optimal formulae for bilinear maps in a finite field
In 2012, Barbulescu, Detrey, Estibals and Zimmermann proposed a new framework
to exhaustively search for optimal formulae for evaluating bilinear maps, such
as Strassen or Karatsuba formulae. The main contribution of this work is a new
criterion to aggressively prune useless branches in the exhaustive search, thus
leading to the computation of new optimal formulae, in particular for the short
product modulo X 5 and the circulant product modulo (X 5 -- 1). Moreover , we
are able to prove that there is essentially only one optimal decomposition of
the product of 3 x 2 by 2 x 3 matrices up to the action of some group of
automorphisms
Multihomogeneous resultant formulae by means of complexes
We provide conditions and algorithmic tools so as to classify and construct
the smallest possible determinantal formulae for multihomogeneous resultants
arising from Weyman complexes associated to line bundles in products of
projective spaces. We also examine the smallest Sylvester-type matrices,
generically of full rank, which yield a multiple of the resultant. We
characterize the systems that admit a purely B\'ezout-type matrix and show a
bijection of such matrices with the permutations of the variable groups. We
conclude with examples showing the hybrid matrices that may be encountered, and
illustrations of our Maple implementation. Our approach makes heavy use of the
combinatorics of multihomogeneous systems, inspired by and generalizing results
by Sturmfels-Zelevinsky, and Weyman-Zelevinsky.Comment: 30 pages. To appear: Journal of Symbolic Computatio
Solitons on tori and soliton crystals
Necessary conditions for a soliton on a torus to be a
soliton crystal, that is, a spatially periodic array of topological solitons in
stable equilibrium, are derived. The stress tensor of the soliton must be
orthogonal to \ee, the space of parallel symmetric bilinear forms on ,
and, further, a certain symmetric bilinear form on \ee, called the hessian,
must be positive. It is shown that, for baby Skyrme models, the first condition
actually implies the second. It is also shown that, for any choice of period
lattice , there is a baby Skyrme model which supports a soliton
crystal of periodicity . For the three-dimensional Skyrme model, it is
shown that any soliton solution on a cubic lattice which satisfies a virial
constraint and is equivariant with respect to (a subgroup of) the lattice
symmetries automatically satisfies both tests. This verifies in particular that
the celebrated Skyrme crystal of Castillejo {\it et al.}, and Kugler and
Shtrikman, passes both tests.Comment: 24 pages, revised version to be published. Added an existence proof
for baby Skyrmions of arbitrary degree on a general two-torus for a model
with general potential. Otherwise, minor improvement
Noncommutativity and theta-locality
In this paper, we introduce the condition of theta-locality which can be used
as a substitute for microcausality in quantum field theory on noncommutative
spacetime. This condition is closely related to the asymptotic commutativity
which was previously used in nonlocal QFT. Heuristically, it means that the
commutator of observables behaves at large spacelike separation like
, where is the noncommutativity parameter. The
rigorous formulation given in the paper implies averaging fields with suitable
test functions. We define a test function space which most closely corresponds
to the Moyal star product and prove that this space is a topological algebra
under the star product. As an example, we consider the simplest normal ordered
monomial and show that it obeys the theta-locality condition.Comment: LaTeX, 17 pages, no figures; minor changes to agree with published
versio
Recherche automatique de formules pour calculer des formes bilinéaires
National audienceThis talk will focus on the bilinear rank problem: given a bilinear map (e.g., the product of polynomials, of finite-field elements, or of matrices), what is the smallest number of multiplications over the coefficient ring required to evaluate this function?For instance, Karatsuba's method allows one to compute the product of two linear polynomials using only three multiplications instead of four. In this talk, we give a formalization of the bilinear rank problem, which is known to be NP-hard, and propose a generic algorithm to efficiently compute exact solutions, thus proving the optimality of (or even improving) known complexity bounds from the literature.Dans cet exposĂ©, nous nous intĂ©ressons au problĂšme du rang bilinĂ©aire : Ă©tant donnĂ©e une application bilinĂ©aire (par exemple, le calcul dâun produit de polynĂŽmes, dâĂ©lĂ©ments dâun corps fini, ou encore de matrices), quel est le nombre minimal de multiplications sur le corps de base nĂ©cessaires pour Ă©valuer cette application ?Ainsi, par exemple, la mĂ©thode de Karatsuba permet de calculer le produit de deux polynĂŽmes linĂ©aires en seulement trois multiplications au lieu de quatre. Nous donnons dans cet exposĂ© une formalisation du problĂšme du rang bilinĂ©aire, connu pour ĂȘtre NP-dur, et proposons un algorithme gĂ©nĂ©ral permettant de calculer efficacement des solutions exactes, qui nous permettent ainsi de prouver lâoptimalitĂ© de, voire mĂȘme dâamĂ©liorer certaines bornes de la littĂ©rature
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