657 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
An axially-symmetric Newtonian Boson Star
A new solution to the coupled gravitational and scalar field equations for a
condensed boson field is found in Newtonian approximation. The solution is
axially symmetric, but not spherically symmetric. For N particles the mass of
the object is given by , to be compared with for the spherically symmetric case.Comment: 4 pages, figures available on reques
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
Multilinear Operators: The Natural Extension Of Hirota's Bilinear Formalism
We introduce multilinear operators, that generalize Hirota's bilinear
operator, based on the principle of gauge invariance of the functions.
We show that these operators can be constructed systematically using the
bilinear 's as building blocks. We concentrate in particular on the
trilinear case and study the possible integrability of equations with one
dependent variable. The 5th order equation of the Lax-hierarchy as well as
Satsuma's lowest-order gauge invariant equation are shown to have simple
trilinear expressions. The formalism can be extended to an arbitrary degree of
multilinearity.Comment: 9 pages in plain Te
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Systematic Multi-Domain Alzheimer's Risk Reduction Trial (SMARRT): Study Protocol.
This article describes the protocol for the Systematic Multi-domain Alzheimer's Risk Reduction Trial (SMARRT), a single-blind randomized pilot trial to test a personalized, pragmatic, multi-domain Alzheimer's disease (AD) risk reduction intervention in a US integrated healthcare delivery system. Study participants will be 200 higher-risk older adults (age 70-89 years with subjective cognitive complaints, low normal performance on cognitive screen, and ≥ two modifiable risk factors targeted by our intervention) who will be recruited from selected primary care clinics of Kaiser Permanente Washington, oversampling people with non-white race or Hispanic ethnicity. Study participants will be randomly assigned to a two-year Alzheimer's risk reduction intervention (SMARRT) or a Health Education (HE) control. Randomization will be stratified by clinic, race/ethnicity (non-Hispanic white versus non-white or Hispanic), and age (70-79, 80-89). Participants randomized to the SMARRT group will work with a behavioral coach and nurse to develop a personalized plan related to their risk factors (poorly controlled hypertension, diabetes with evidence of hyper or hypoglycemia, depressive symptoms, poor sleep quality, contraindicated medications, physical inactivity, low cognitive stimulation, social isolation, poor diet, smoking). Participants in the HE control group will be mailed general health education information about these risk factors for AD. The primary outcome is two-year cognitive change on a cognitive test composite score. Secondary outcomes include: 1) improvement in targeted risk factors, 2) individual cognitive domain composite scores, 3) physical performance, 4) functional ability, 5) quality of life, and 6) incidence of mild cognitive impairment, AD, and dementia. Primary and secondary outcomes will be assessed in both groups at baseline and 6, 12, 18, and 24 months
Gurevich-Zybin system
We present three different linearizable extensions of the Gurevich-Zybin
system. Their general solutions are found by reciprocal transformations. In
this paper we rewrite the Gurevich-Zybin system as a Monge-Ampere equation. By
application of reciprocal transformation this equation is linearized.
Infinitely many local Hamiltonian structures, local Lagrangian representations,
local conservation laws and local commuting flows are found. Moreover, all
commuting flows can be written as Monge-Ampere equations similar to the
Gurevich-Zybin system. The Gurevich-Zybin system describes the formation of a
large scale structures in the Universe. The second harmonic wave generation is
known in nonlinear optics. In this paper we prove that the Gurevich-Zybin
system is equivalent to a degenerate case of the second harmonic generation.
Thus, the Gurevich-Zybin system is recognized as a degenerate first negative
flow of two-component Harry Dym hierarchy up to two Miura type transformations.
A reciprocal transformation between the Gurevich-Zybin system and degenerate
case of the second harmonic generation system is found. A new solution for the
second harmonic generation is presented in implicit form.Comment: Corrected typos and misprint
Algebraic Bethe ansatz for a quantum integrable derivative nonlinear Schrodinger model
We find that the quantum monodromy matrix associated with a derivative
nonlinear Schrodinger (DNLS) model exhibits U(2) or U(1,1) symmetry depending
on the sign of the related coupling constant. By using a variant of quantum
inverse scattering method which is directly applicable to field theoretical
models, we derive all possible commutation relations among the operator valued
elements of such monodromy matrix. Thus, we obtain the commutation relation
between creation and annihilation operators of quasi-particles associated with
DNLS model and find out the -matrix for two-body scattering. We also observe
that, for some special values of the coupling constant, there exists an upper
bound on the number of quasi-particles which can form a soliton state for the
quantum DNLS model.Comment: 17 pages, Latex, minor typos corrected, to be published in Nucl.
Phys.
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
Shock waves in one-dimensional Heisenberg ferromagnets
We use SU(2) coherent state path integral formulation with the stationary
phase approximation to investigate, both analytically and numerically, the
existence of shock waves in the one- dimensional Heisenberg ferromagnets with
anisotropic exchange interaction. As a result we show the existence of shock
waves of two types,"bright" and "dark", which can be interpreted as moving
magnetic domains.Comment: 10 pages, with 3 ps figure
Novel multi-band quantum soliton states for a derivative nonlinear Schrodinger model
We show that localized N-body soliton states exist for a quantum integrable
derivative nonlinear Schrodinger model for several non-overlapping ranges
(called bands) of the coupling constant \eta. The number of such distinct bands
is given by Euler's \phi-function which appears in the context of number
theory. The ranges of \eta within each band can also be determined completely
using concepts from number theory such as Farey sequences and continued
fractions. We observe that N-body soliton states appearing within each band can
have both positive and negative momentum. Moreover, for all bands lying in the
region \eta > 0, soliton states with positive momentum have positive binding
energy (called bound states), while the states with negative momentum have
negative binding energy (anti-bound states).Comment: LaTeX, 20 pages including 2 figure
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