367 research outputs found
Asymptotics of the Farey Fraction Spin Chain Free Energy at the Critical Point
We consider the Farey fraction spin chain in an external field . Using
ideas from dynamical systems and functional analysis, we show that the free
energy in the vicinity of the second-order phase transition is given,
exactly, by
Here is a reduced
temperature, so that the deviation from the critical point is scaled by the
Lyapunov exponent of the Gauss map, . It follows that
determines the amplitude of both the specific heat and susceptibility
singularities. To our knowledge, there is only one other microscopically
defined interacting model for which the free energy near a phase transition is
known as a function of two variables.
Our results confirm what was found previously with a cluster approximation,
and show that a clustering mechanism is in fact responsible for the transition.
However, the results disagree in part with a renormalisation group treatment
Similarity transformations for Nonlinear Schrodinger Equations with time varying coefficients: Exact results
In this paper we use a similarity transformation connecting some families of
Nonlinear Schrodinger equations with time-varying coefficients with the
autonomous cubic nonlinear Schrodinger equation. This transformation allows one
to apply all known results for that equation to the non-autonomous case with
the additional dynamics introduced by the transformation itself. In particular,
using stationary solutions of the autonomous nonlinear Schrodinger equation we
can construct exact breathing solutions to multidimensional non-autonomous
nonlinear Schrodinger equations. An application is given in which we explicitly
construct time dependent coefficients leading to solutions displaying weak
collapse in three-dimensional scenarios. Our results can find physical
applicability in mean field models of Bose-Einstein condensates and in the
field of dispersion-managed optical systems
Interpolation and harmonic majorants in big Hardy-Orlicz spaces
Free interpolation in Hardy spaces is caracterized by the well-known Carleson
condition. The result extends to Hardy-Orlicz spaces contained in the scale of
classical Hardy spaces , . For the Smirnov and the Nevanlinna
classes, interpolating sequences have been characterized in a recent paper in
terms of the existence of harmonic majorants (quasi-bounded in the case of the
Smirnov class). Since the Smirnov class can be regarded as the union over all
Hardy-Orlicz spaces associated with a so-called strongly convex function, it is
natural to ask how the condition changes from the Carleson condition in
classical Hardy spaces to harmonic majorants in the Smirnov class. The aim of
this paper is to narrow down this gap from the Smirnov class to ``big''
Hardy-Orlicz spaces. More precisely, we characterize interpolating sequences
for a class of Hardy-Orlicz spaces that carry an algebraic structure and that
are strictly bigger than . It turns out that the
interpolating sequences are again characterized by the existence of
quasi-bounded majorants, but now the weights of the majorants have to be in
suitable Orlicz spaces. The existence of harmonic majorants in such Orlicz
spaces will also be discussed in the general situation. We finish the paper
with an example of a separated Blaschke sequence that is interpolating for
certain Hardy-Orlicz spaces without being interpolating for slightly smaller
ones.Comment: 19 pages, 2 figure
Tverberg-type theorems for intersecting by rays
In this paper we consider some results on intersection between rays and a
given family of convex, compact sets. These results are similar to the center
point theorem, and Tverberg's theorem on partitions of a point set
Regularization of Linear Ill-posed Problems by the Augmented Lagrangian Method and Variational Inequalities
We study the application of the Augmented Lagrangian Method to the solution
of linear ill-posed problems. Previously, linear convergence rates with respect
to the Bregman distance have been derived under the classical assumption of a
standard source condition. Using the method of variational inequalities, we
extend these results in this paper to convergence rates of lower order, both
for the case of an a priori parameter choice and an a posteriori choice based
on Morozov's discrepancy principle. In addition, our approach allows the
derivation of convergence rates with respect to distance measures different
from the Bregman distance. As a particular application, we consider sparsity
promoting regularization, where we derive a range of convergence rates with
respect to the norm under the assumption of restricted injectivity in
conjunction with generalized source conditions of H\"older type
Blow-up collocation solutions of nonlinear homogeneous Volterra integral equations
In this paper, collocation methods are used for detecting blow-up solutions
of nonlinear homogeneous Volterra-Hammerstein integral equations. To do this,
we introduce the concept of "blow-up collocation solution" and analyze
numerically some blow-up time estimates using collocation methods in particular
examples where previous results about existence and uniqueness can be applied.
Finally, we discuss the relationships between necessary conditions for blow-up
of collocation solutions and exact solutions.Comment: 22 pages, 5 figures. New version: We have made some notation changes
in order to make emphasis in the fact that we use variable stepsizes, for
preventing misunderstandings (a referee misunderstood it). arXiv admin note:
text overlap with arXiv:1112.464
From Sturm-Liouville problems to fractional and anomalous diffusions
Some fractional and anomalous diffusions are driven by equations involving
fractional derivatives in both time and space. Such diffusions are processes
with randomly varying times. In representing the solutions to those diffusions,
the explicit laws of certain stable processes turn out to be fundamental. This
paper directs one's efforts towards the explicit representation of solutions to
fractional and anomalous diffusions related to Sturm-Liouville problems of
fractional order associated to fractional power function spaces. Furthermore,
we study a new version of the Bochner's subordination rule and we establish
some connections between subordination and space-fractional operatorComment: Accepted by Stochastic Processess and Their Application
On Robustness of Discrete Time Optimal Filters
A new result on stability of an optimal nonlinear filter for a Markov chain with respect to small perturbations on every step is established. An exponential recurrence of the signal is assumed
Functional Integration Approach to Hysteresis
A general formulation of scalar hysteresis is proposed. This formulation is
based on two steps. First, a generating function g(x) is associated with an
individual system, and a hysteresis evolution operator is defined by an
appropriate envelope construction applied to g(x), inspired by the overdamped
dynamics of systems evolving in multistable free energy landscapes. Second, the
average hysteresis response of an ensemble of such systems is expressed as a
functional integral over the space G of all admissible generating functions,
under the assumption that an appropriate measure m has been introduced in G.
The consequences of the formulation are analyzed in detail in the case where
the measure m is generated by a continuous, Markovian stochastic process. The
calculation of the hysteresis properties of the ensemble is reduced to the
solution of the level-crossing problem for the stochastic process. In
particular, it is shown that, when the process is translationally invariant
(homogeneous), the ensuing hysteresis properties can be exactly described by
the Preisach model of hysteresis, and the associated Preisach distribution is
expressed in closed analytic form in terms of the drift and diffusion
parameters of the Markovian process. Possible applications of the formulation
are suggested, concerning the interpretation of magnetic hysteresis due to
domain wall motion in quenched-in disorder, and the interpretation of critical
state models of superconducting hysteresis.Comment: 36 pages, 9 figures, to be published on Phys. Rev.
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