4,880 research outputs found
Reducing the size and number of linear programs in a dynamic Gr\"obner basis algorithm
The dynamic algorithm to compute a Gr\"obner basis is nearly twenty years
old, yet it seems to have arrived stillborn; aside from two initial
publications, there have been no published followups. One reason for this may
be that, at first glance, the added overhead seems to outweigh the benefit; the
algorithm must solve many linear programs with many linear constraints. This
paper describes two methods of reducing the cost substantially, answering the
problem effectively.Comment: 11 figures, of which half are algorithms; submitted to journal for
refereeing, December 201
On the pre-metric foundations of wave mechanics I: massless waves
The mechanics of wave motion in a medium are founded in conservation laws for
the physical quantities that the waves carry, combined with the constitutive
laws of the medium, and define Lorentzian structures only in degenerate cases
of the dispersion laws that follow from the field equations. It is suggested
that the transition from wave motion to point motion is best factored into an
intermediate step of extended matter motion, which then makes the
dimension-codimension duality of waves and trajectories a natural consequence
of the bicharacteristic (geodesic) foliation associated with the dispersion
law. This process is illustrated in the conventional case of quadratic
dispersion, as well as quartic ones, which include the Heisenberg-Euler
dispersion law. It is suggested that the contributions to geodesic motion from
the non-quadratic nature of a dispersion law might represent another source of
quantum fluctuations about classical extremals, in addition to the diffraction
effects that are left out by the geometrical optics approximation.Comment: 25 pages, 1 figur
Laguerre and Meixner symmetric functions, and infinite-dimensional diffusion processes
The Laguerre symmetric functions introduced in the note are indexed by
arbitrary partitions and depend on two continuous parameters. The top degree
homogeneous component of every Laguerre symmetric function coincides with the
Schur function with the same index. Thus, the Laguerre symmetric functions form
a two-parameter family of inhomogeneous bases in the algebra of symmetric
functions. These new symmetric functions are obtained from the N-variate
symmetric polynomials of the same name by a procedure of analytic continuation.
The Laguerre symmetric functions are eigenvectors of a second order
differential operator, which depends on the same two parameters and serves as
the infinitesimal generator of an infinite-dimensional diffusion process X(t).
The process X(t) admits approximation by some jump processes related to one
more new family of symmetric functions, the Meixner symmetric functions. In
equilibrium, the process X(t) can be interpreted as a time-dependent point
process on the punctured real line R\{0}, and the point configurations may be
interpreted as doubly infinite collections of particles of two opposite charges
with log-gas-type interaction. The dynamical correlation functions of the
equilibrium process have determinantal form: they are given by minors of the
so-called extended Whittaker kernel, introduced earlier in a paper by Borodin
and the author.Comment: LaTex, 26 p
Laguerre and Meixner orthogonal bases in the algebra of symmetric functions
Analogs of Laguerre and Meixner orthogonal polynomials in the algebra of
symmetric functions are studied. This is a detailed exposition of part of the
results announced in arXiv:1009.2037. The work is motivated by a connection
with a model of infinite-dimensional Markov dynamics.Comment: Latex, 52p
Sparse Gr\"obner Bases: the Unmixed Case
Toric (or sparse) elimination theory is a framework developped during the
last decades to exploit monomial structures in systems of Laurent polynomials.
Roughly speaking, this amounts to computing in a \emph{semigroup algebra},
\emph{i.e.} an algebra generated by a subset of Laurent monomials. In order to
solve symbolically sparse systems, we introduce \emph{sparse Gr\"obner bases},
an analog of classical Gr\"obner bases for semigroup algebras, and we propose
sparse variants of the and FGLM algorithms to compute them. Our prototype
"proof-of-concept" implementation shows large speed-ups (more than 100 for some
examples) compared to optimized (classical) Gr\"obner bases software. Moreover,
in the case where the generating subset of monomials corresponds to the points
with integer coordinates in a normal lattice polytope and under regularity assumptions, we prove complexity bounds which depend
on the combinatorial properties of . These bounds yield new
estimates on the complexity of solving -dim systems where all polynomials
share the same Newton polytope (\emph{unmixed case}). For instance, we
generalize the bound on the maximal degree in a Gr\"obner
basis of a -dim. bilinear system with blocks of variables of sizes
to the multilinear case: . We also propose
a variant of Fr\"oberg's conjecture which allows us to estimate the complexity
of solving overdetermined sparse systems.Comment: 20 pages, Corollary 6.1 has been corrected, ISSAC 2014, Kobe : Japan
(2014
Inhomogeneous extreme forms
G.F. Voronoi (1868-1908) wrote two memoirs in which he describes two
reduction theories for lattices, well-suited for sphere packing and covering
problems. In his first memoir a characterization of locally most economic
packings is given, but a corresponding result for coverings has been missing.
In this paper we bridge the two classical memoirs.
By looking at the covering problem from a different perspective, we discover
the missing analogue. Instead of trying to find lattices giving economical
coverings we consider lattices giving, at least locally, very uneconomical
ones. We classify local covering maxima up to dimension 6 and prove their
existence in all dimensions beyond.
New phenomena arise: Many highly symmetric lattices turn out to give
uneconomical coverings; the covering density function is not a topological
Morse function. Both phenomena are in sharp contrast to the packing problem.Comment: 22 pages, revision based on suggestions by referee, accepted in
Annales de l'Institut Fourie
Probing spatial homogeneity with LTB models: a detailed discussion
Do current observational data confirm the assumptions of the cosmological
principle, or is there statistical evidence for deviations from spatial
homogeneity on large scales? To address these questions, we developed a
flexible framework based on spherically symmetric, but radially inhomogeneous
Lemaitre-Tolman-Bondi (LTB) models with synchronous Big Bang. We expanded the
(local) matter density profile in terms of flexible interpolation schemes and
orthonormal polynomials. A Monte Carlo technique in combination with recent
observational data was used to systematically vary the shape of these profiles.
In the first part of this article, we reconsider giant LTB voids without dark
energy to investigate whether extremely fine-tuned mass profiles can reconcile
these models with current data. While the local Hubble rate and supernovae can
easily be fitted without dark energy, however, model-independent constraints
from the Planck 2013 data require an unrealistically low local Hubble rate,
which is strongly inconsistent with the observed value; this result agrees well
with previous studies. In the second part, we explain why it seems natural to
extend our framework by a non-zero cosmological constant, which then allows us
to perform general tests of the cosmological principle. Moreover, these
extended models facilitate explorating whether fluctuations in the local matter
density profile might potentially alleviate the tension between local and
global measurements of the Hubble rate, as derived from Cepheid-calibrated type
Ia supernovae and CMB experiments, respectively. We show that current data
provide no evidence for deviations from spatial homogeneity on large scales.
More accurate constraints are required to ultimately confirm the validity of
the cosmological principle, however.Comment: 18 pages, 12 figures, 2 tables; accepted for publication in A&
Anisotropic multiple scattering in diffuse media
The multiple scattering of scalar waves in diffusive media is investigated by
means of the radiative transfer equation. This approach amounts to a
resummation of the ladder diagrams of the Born series; it does not rely on the
diffusion approximation. Quantitative predictions are obtained, concerning
various observables pertaining to optically thick slabs, such as the mean
angle-resolved reflected and transmitted intensities, and the shape of the
enhanced backscattering cone. Special emphasis is put on the dependence of
these quantities on the anisotropy of the cross-section of the individual
scatterers, and on the internal reflections due to the optical index mismatch
at the boundaries of the sample. The regime of very anisotropic scattering,
where the transport mean free path is much larger than the scattering
mean free path , is studied in full detail. For the first time the
relevant Schwarzschild-Milne equation is solved exactly in the absence of
internal reflections, and asymptotically in the regime of a large index
mismatch. An unexpected outcome concerns the angular width of the enhanced
backscattering cone, which is predicted to scale as
, in contrast with the generally
accepted law, derived within the diffusion approximation.Comment: 53 pages TEX, including 2 tables. The 4 figures are sent at reques
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