768 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
Integration of a generalized H\'enon-Heiles Hamiltonian
The generalized H\'enon-Heiles Hamiltonian
with an additional
nonpolynomial term is known to be Liouville integrable for three
sets of values of . It has been previously integrated by genus
two theta functions only in one of these cases. Defining the separating
variables of the Hamilton-Jacobi equations, we succeed here, in the two other
cases, to integrate the equations of motion with hyperelliptic functions.Comment: LaTex 2e. To appear, Journal of Mathematical Physic
Quantized representation of some nonlinear integrable evolution equations on the soliton sector
The Hirota algorithm for solving several integrable nonlinear evolution
equations is suggestive of a simple quantized representation of these equations
and their soliton solutions over a Fock space of bosons or of fermions. The
classical nonlinear wave equation becomes a nonlinear equation for an operator.
The solution of this equation is constructed through the operator analog of the
Hirota transformation. The classical N-solitons solution is the expectation
value of the solution operator in an N-particle state in the Fock space.Comment: 12 page
Optical Bistability in Nonlinear Optical Coupler with Negative Index Channel
We discuss a novel kind of nonlinear coupler with one channel filled with a
negative index material (NIM). The opposite directionality of the phase
velocity and the energy flow in the NIM channel facilitates an effective
feedback mechanism that leads to optical bistability and gap soliton formation
Is there a Jordan geometry underlying quantum physics?
There have been several propositions for a geometric and essentially
non-linear formulation of quantum mechanics. From a purely mathematical point
of view, the point of view of Jordan algebra theory might give new strength to
such approaches: there is a ``Jordan geometry'' belonging to the Jordan part of
the algebra of observables, in the same way as Lie groups belong to the Lie
part. Both the Lie geometry and the Jordan geometry are well-adapted to
describe certain features of quantum theory. We concentrate here on the
mathematical description of the Jordan geometry and raise some questions
concerning possible relations with foundational issues of quantum theory.Comment: 30 page
Controlled Generation of Dark Solitons with Phase Imprinting
The generation of dark solitons in Bose-Einstein condensates with phase
imprinting is studied by mapping it into the classic problem of a damped driven
pendulum. We provide simple but powerful schemes of designing the phase imprint
for various desired outcomes. We derive a formula for the number of dark
solitons generated by a given phase step, and also obtain results which explain
experimental observations.Comment: 4pages, 4 figure
A Riemann-Hilbert Problem for an Energy Dependent Schr\"odinger Operator
\We consider an inverse scattering problem for Schr\"odinger operators with
energy dependent potentials. The inverse problem is formulated as a
Riemann-Hilbert problem on a Riemann surface. A vanishing lemma is proved for
two distinct symmetry classes. As an application we prove global existence
theorems for the two distinct systems of partial differential equations
for suitably restricted,
complementary classes of initial data
Perturbative analysis of wave interactions in nonlinear systems
This work proposes a new way for handling obstacles to asymptotic
integrability in perturbed nonlinear PDEs within the method of Normal Forms -
NF - for the case of multi-wave solutions. Instead of including the whole
obstacle in the NF, only its resonant part is included, and the remainder is
assigned to the homological equation. This leaves the NF intergable and its
solutons retain the character of the solutions of the unperturbed equation. We
exploit the freedom in the expansion to construct canonical obstacles which are
confined to te interaction region of the waves. Fo soliton solutions, e.g., in
the KdV equation, the interaction region is a finite domain around the origin;
the canonical obstacles then do not generate secular terms in the homological
equation. When the interaction region is infifnite, or semi-infinite, e.g., in
wave-front solutions of the Burgers equation, the obstacles may contain
resonant terms. The obstacles generate waves of a new type, which cannot be
written as functionals of the solutions of the NF. When an obstacle contributes
a resonant term to the NF, this leads to a non-standard update of th wave
velocity.Comment: 13 pages, including 6 figure
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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
Q-stars and charged q-stars
We present the formalism of q-stars with local or global U(1) symmetry. The
equations we formulate are solved numerically and provide the main features of
the soliton star. We study its behavior when the symmetry is local in contrast
to the global case. A general result is that the soliton remains stable and
does not decay into free particles and the electrostatic repulsion preserves it
from gravitational collapse. We also investigate the case of a q-star with
non-minimal energy-momentum tensor and find that the soliton is stable even in
some cases of collapse when the coupling to gravity is absent.Comment: Latex, 19pg, 12 figures. Accepted in Phys. Rev.
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