4,086 research outputs found
Angular Symmetry Breaking Induced by Electromagnetic Field
It is well known that velocities does not commute in presence of an
electromagnetic field. This property implies that angular algebra symmetries,
such as the sO(3) and Lorentz algebra symmetries, are broken. To restore these
angular symmetries we show the necessity of adding the Poincare momentum M to
the simple angular momentum L. These restorations performed succesively in a
flat space and in a curved space lead in each cases to the generation of a
Dirac magnetic monopole. In the particular case of the Lorentz algebra we
consider an application of our theory to the gravitoelectromagnetism. In this
last case we establish a qualitative relation giving the mass spectrum for
dyons.Comment: 19 page
Inverse spectral positivity for surfaces
Let be a complete non-compact Riemannian surface. We consider
operators of the form , where is the non-negative
Laplacian, the Gaussian curvature, a locally integrable function, and
a positive real number. Assuming that the positive part of is
integrable, we address the question "What conclusions on and can
one draw from the fact that the operator is non-negative ?"
As a consequence of our main result, we get a new proof of Huber's theorem and
Cohn-Vossen's inequality, and we improve earlier results in the particular
cases in which is non-positive and or
The weak Pleijel theorem with geometric control
Let , be a bounded open set, and denote
by , the eigenvalues of the Dirichlet Laplacian
arranged in nondecreasing order, with multiplicities. The weak form of
Pleijel's theorem states that the number of eigenvalues ,
for which there exists an associated eigenfunction with precisely nodal
domains (Courant-sharp eigenvalues), is finite. The purpose of this note is to
determine an upper bound for Courant-sharp eigenvalues, expressed in terms of
simple geometric invariants of . We will see that this is connected
with one of the favorite problems considered by Y. Safarov.Comment: Revised Oct. 12, 2016. To appear in Journal of Spectral Theory 6
(2016
A. Stern's analysis of the nodal sets of some families of spherical harmonics revisited
In this paper, we revisit the analyses of Antonie Stern (1925) and Hans Lewy
(1977) devoted to the construction of spherical harmonics with two or three
nodal domains. Our method yields sharp quantitative results and a better
understanding of the occurrence of bifurcations in the families of nodal
sets.This paper is a natural continuation of our critical reading of A. Stern's
results for Dirichlet eigenfunctions in the square, see arXiv:14026054.Comment: Accepted for publication in "Monatshefte f{\"u}r Mathematik
Coupling times with ambiguities for particle systems and applications to context-dependent DNA substitution models
We define a notion of coupling time with ambiguities for interacting particle
systems, and show how this can be used to prove ergodicity and to bound the
convergence time to equilibrium and the decay of correlations at equilibrium. A
motivation is to provide simple conditions which ensure that perturbed particle
systems share some properties of the underlying unperturbed system. We apply
these results to context-dependent substitution models recently introduced by
molecular biologists as descriptions of DNA evolution processes. These models
take into account the influence of the neighboring bases on the substitution
probabilities at a site of the DNA sequence, as opposed to most usual
substitution models which assume that sites evolve independently of each other.Comment: 33 page
Coupling from the past times with ambiguities and perturbations of interacting particle systems
We discuss coupling from the past techniques (CFTP) for perturbations of
interacting particle systems on the d-dimensional integer lattice, with a
finite set of states, within the framework of the graphical construction of the
dynamics based on Poisson processes. We first develop general results for what
we call CFTP times with ambiguities. These are analogous to classical coupling
(from the past) times, except that the coupling property holds only provided
that some ambiguities concerning the stochastic evolution of the system are
resolved. If these ambiguities are rare enough on average, CFTP times with
ambiguities can be used to build actual CFTP times, whose properties can be
controlled in terms of those of the original CFTP time with ambiguities. We
then prove a general perturbation result, which can be stated informally as
follows. Start with an interacting particle system possessing a CFTP time whose
definition involves the exploration of an exponentially integrable number of
points in the graphical construction, and which satisfies the positive rates
property. Then consider a perturbation obtained by adding new transitions to
the original dynamics. Our result states that, provided that the perturbation
is small enough (in the sense of small enough rates), the perturbed interacting
particle system too possesses a CFTP time (with nice properties such as an
exponentially decaying tail). The proof consists in defining a CFTP time with
ambiguities for the perturbed dynamics, from the CFTP time for the unperturbed
dynamics. Finally, we discuss examples of particle systems to which this result
can be applied. Concrete examples include a class of neighbor-dependent
nucleotide substitution model, and variations of the classical voter model,
illustrating the ability of our approach to go beyond the case of weakly
interacting particle systems.Comment: This paper is an extended and revised version of an earlier
manuscript available as arXiv:0712.0072, where the results were limited to
perturbations of RN+YpR nucleotide substitution model
Fluctuations of the front in a one-dimensional model for the spread of an infection
We study the following microscopic model of infection or epidemic reaction:
red and blue particles perform independent nearest-neighbor continuous-time
symmetric random walks on the integer lattice with jump rates
for red particles and for blue particles, the interaction rule
being that blue particles turn red upon contact with a red particle. The
initial condition consists of i.i.d. Poisson particle numbers at each site,
with particles at the left of the origin being red, while particles at the
right of the origin are blue. We are interested in the dynamics of the front,
defined as the rightmost position of a red particle. For the case ,
Kesten and Sidoravicius established that the front moves ballistically, and
more precisely that it satisfies a law of large numbers. Their proof is based
on a multi-scale renormalization technique, combined with approximate
sub-additivity arguments. In this paper, we build a renewal structure for the
front propagation process, and as a corollary we obtain a central limit theorem
for the front when . Moreover, this result can be extended to the case
where , up to modifying the dynamics so that blue particles turn red
upon contact with a site that has previously been occupied by a red particle.
Our approach extends the renewal structure approach developed by Comets,
Quastel and Ram\'{{\i}}rez for the so-called frog model, which corresponds to
the case.Comment: Published at http://dx.doi.org/10.1214/15-AOP1034 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
On Courant's nodal domain property for linear combinations of eigenfunctions, Part I
According to Courant's theorem, an eigenfunction as\-sociated with the -th
eigenvalue has at most nodal domains. A footnote in the book
of Courant and Hilbert, states that the same assertion is true for any linear
combination of eigenfunctions associated with eigenvalues less than or equal to
. We call this assertion the \emph{Extended Courant
Property}.\smallskipIn this paper, we propose simple and explicit examples for
which the extended Courant property is false: convex domains in
(hypercube and equilateral triangle), domains with cracks in , on
the round sphere , and on a flat torus .Comment: To appear in Documenta Mathematica.Modifications with respect to
version 4: Introduction rewritten. To shorten the paper two sections (Section
7, Numerical simulations and Section 8, Conjectures) have been removed and
will be published elsewhere. Related to the paper arXiv:1803.00449v2. Small
overlap with arXiv:1803.00449v1 which will be modified accordingl
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