606 research outputs found
Computing Small Certificates of Inconsistency of Quadratic Fewnomial Systems
B{\'e}zout 's theorem states that dense generic systems of n multivariate
quadratic equations in n variables have 2 n solutions over algebraically closed
fields. When only a small subset M of monomials appear in the equations
(fewnomial systems), the number of solutions may decrease dramatically. We
focus in this work on subsets of quadratic monomials M such that generic
systems with support M do not admit any solution at all. For these systems,
Hilbert's Nullstellensatz ensures the existence of algebraic certificates of
inconsistency. However, up to our knowledge all known bounds on the sizes of
such certificates -including those which take into account the Newton polytopes
of the polynomials- are exponential in n. Our main results show that if the
inequality 2|M| -- 2n \sqrt 1 + 8{\nu} -- 1 holds for a quadratic
fewnomial system -- where {\nu} is the matching number of a graph associated
with M, and |M| is the cardinality of M -- then there exists generically a
certificate of inconsistency of linear size (measured as the number of
coefficients in the ground field K). Moreover this certificate can be computed
within a polynomial number of arithmetic operations. Next, we evaluate how
often this inequality holds, and we give evidence that the probability that the
inequality is satisfied depends strongly on the number of squares. More
precisely, we show that if M is picked uniformly at random among the subsets of
n + k + 1 quadratic monomials containing at least (n 1/2+)
squares, then the probability that the inequality holds tends to 1 as n grows.
Interestingly, this phenomenon is related with the matching number of random
graphs in the Erd{\"o}s-Renyi model. Finally, we provide experimental results
showing that certificates in inconsistency can be computed for systems with
more than 10000 variables and equations.Comment: ISSAC 2016, Jul 2016, Waterloo, Canada. Proceedings of ISSAC 201
Shape coexistence in neutron-deficient Kr isotopes: Constraints on the single-particle spectrum of self-consistent mean-field models from collective excitations
We discuss shape coexistence in the neutron-deficient Kr72-Kr78 isotopes in
the framework of configuration mixing calculations of particle-number and
angular-momentum projected axial mean-field states obtained from
self-consistent calculations with the Skyrme interaction SLy6 and a
density-dependent pairing interaction. While our calculation reproduces
qualitatively and quantitatively many of the global features of these nuclei,
such as coexistence of prolate and oblate shapes, their strong mixing at low
angular momentum, and the deformation of collective bands, the ordering of our
calculated low-lying levels is at variance with experiment. We analyse the role
of the single-particle spectrum of the underlying mean-field for the spectrum
of collective excitations.Comment: accepted for publication in Phys. Rev.
Unstaggered-staggered solitons in two-component discrete nonlinear Schr\"{o}dinger lattices
We present stable bright solitons built of coupled unstaggered and staggered
components in a symmetric system of two discrete nonlinear Schr\"{o}dinger
(DNLS) equations with the attractive self-phase-modulation (SPM) nonlinearity,
coupled by the repulsive cross-phase-modulation (XPM) interaction. These mixed
modes are of a "symbiotic" type, as each component in isolation may only carry
ordinary unstaggered solitons. The results are obtained in an analytical form,
using the variational and Thomas-Fermi approximations (VA and TFA), and the
generalized Vakhitov-Kolokolov (VK) criterion for the evaluation of the
stability. The analytical predictions are verified against numerical results.
Almost all the symbiotic solitons are predicted by the VA quite accurately, and
are stable. Close to a boundary of the existence region of the solitons (which
may feature several connected branches), there are broad solitons which are not
well approximated by the VA, and are unstable
Renormalization Group Theory for a Perturbed KdV Equation
We show that renormalization group(RG) theory can be used to give an analytic
description of the evolution of a perturbed KdV equation. The equations
describing the deformation of its shape as the effect of perturbation are RG
equations. The RG approach may be simpler than inverse scattering theory(IST)
and another approaches, because it dose not rely on any knowledge of IST and it
is very concise and easy to understand. To the best of our knowledge, this is
the first time that RG has been used in this way for the perturbed soliton
dynamics.Comment: 4 pages, no figure, revte
Indoor environment acceptability in the Continuing Care Retirement Community setting
Includes a sample letter and questionnaires.Department: Architecture
Second harmonic generation: Goursat problem on the semi-strip and explicit solutions
A rigorous and complete solution of the initial-boundary-value (Goursat)
problem for second harmonic generation (and its matrix analog) on the
semi-strip is given in terms of the Weyl functions. A wide class of the
explicit solutions and their Weyl functions is obtained also.Comment: 20 page
Completely integrable models of non-linear optics
The models of the non-linear optics in which solitons were appeared are
considered. These models are of paramount importance in studies of non-linear
wave phenomena. The classical examples of phenomena of this kind are the
self-focusing, self-induced transparency, and parametric interaction of three
waves. At the present time there are a number of the theories based on
completely integrable systems of equations, which are both generations of the
original known models and new ones. The modified Korteweg-de Vries equation,
the non- linear Schrodinger equation, the derivative non-linear Schrodinger
equation, Sine-Gordon equation, the reduced Maxwell-Bloch equation, Hirota
equation, the principal chiral field equations, and the equations of massive
Thirring model are gradually putting together a list of soliton equations,
which are usually to be found in non-linear optics theory.Comment: Latex, 17 pages, no figures, submitted to Pramana
Differential constraints for the Kaup -- Broer system as a reduction of the 1D Toda lattice
It is shown that some special reduction of infinite 1D Toda lattice gives
differential constraints compatible with the Kaup -- Broer system. A family of
the travelling wave solutions of the Kaup -- Broer system and its higher
version is constructed.Comment: LaTeX, uses IOP styl
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