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## Universality for random matrices and log-gases

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 04/12/2012

Relevância na Pesquisa

45.4%

Eugene Wigner's revolutionary vision predicted that the energy levels of
large complex quantum systems exhibit a universal behavior: the statistics of
energy gaps depend only on the basic symmetry type of the model.
Simplified models of Wigner's thesis have recently become mathematically
accessible. For mean field models represented by large random matrices with
independent entries, the celebrated Wigner-Dyson-Gaudin-Mehta (WDGM) conjecture
asserts that the local eigenvalue statistics are universal. For invariant
matrix models, the eigenvalue distributions are given by a log-gas with
potential $V$ and inverse temperature $\beta = 1, 2, 4$. corresponding to the
orthogonal, unitary and symplectic ensembles. For $\beta \not \in \{1, 2, 4\}$,
there is no natural random matrix ensemble behind this model, but the analogue
of the WDGM conjecture asserts that the local statistics are independent of
$V$.
In these lecture notes we review the recent solution to these conjectures for
both invariant and non-invariant ensembles. We will discuss two different
notions of universality in the sense of (i) local correlation functions and
(ii) gap distributions.
We will demonstrate that the local ergodicity of the Dyson Brownian motion is
the intrinsic mechanism behind the universality. In particular...

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## Beyond quantum microcanonical statistics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

35.42%

Descriptions of molecular systems usually refer to two distinct theoretical
frameworks. On the one hand the quantum pure state, i.e. the wavefunction, of
an isolated system which is determined to calculate molecular properties and to
consider the time evolution according to the unitary Schr\"odinger equation. On
the other hand a mixed state, i.e. a statistical density matrix, is the
standard formalism to account for thermal equilibrium, as postulated in the
microcanonical quantum statistics. In the present paper an alternative
treatment relying on a statistical analysis of the possible wavefunctions of an
isolated system is presented. In analogy with the classical ergodic theory, the
time evolution of the wavefunction determines the probability distribution in
the phase space pertaining to an isolated system. However, this alone cannot
account for a well defined thermodynamical description of the system in the
macroscopic limit, unless a suitable probability distribution for the quantum
constants of motion is introduced. We present a workable formalism assuring the
emergence of typical values of thermodynamic functions, such as the internal
energy and the entropy, in the large size limit of the system. This allows the
identification of macroscopic properties independently of the specific
realization of the quantum state. A description of material systems in
agreement with equilibrium thermodynamics is then derived without constraints
on the physical constituents and interactions of the system. Furthermore...

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## Bubble statistics and positioning in superhelically stressed DNA

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

45.38%

#Quantitative Biology - Biomolecules#Condensed Matter - Soft Condensed Matter#Physics - Biological Physics

We present a general framework to study the thermodynamic denaturation of
double-stranded DNA under superhelical stress. We report calculations of
position- and size-dependent opening probabilities for bubbles along the
sequence. Our results are obtained from transfer-matrix solutions of the
Zimm-Bragg model for unconstrained DNA and of a self-consistent linearization
of the Benham model for superhelical DNA. The numerical efficiency of our
method allows for the analysis of entire genomes and of random sequences of
corresponding length ($10^6-10^9$ base pairs). We show that, at physiological
conditions, opening in superhelical DNA is strongly cooperative with average
bubble sizes of $10^2-10^3$ base pairs (bp), and orders of magnitude higher
than in unconstrained DNA. In heterogeneous sequences, the average degree of
base-pair opening is self-averaging, while bubble localization and statistics
are dominated by sequence disorder. Compared to random sequences with identical
GC-content, genomic DNA has a significantly increased probability to open large
bubbles under superhelical stress. These bubbles are frequently located
directly upstream of transcription start sites.; Comment: to be appeared in Physical Review E

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## Noncommutative analysis, Multivariable spectral theory for operators in Hilbert space, Probability, and Unitary Representations

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

45.4%

#Mathematics - Functional Analysis#Primary 47L60, 46N30, 46N50, 42C15, 65R10, Secondary 46N20, 22E70,
31A15, 58J65, 81S25

Over the decades, Functional Analysis has been enriched and inspired on
account of demands from neighboring fields, within mathematics, harmonic
analysis (wavelets and signal processing), numerical analysis (finite element
methods, discretization), PDEs (diffusion equations, scattering theory),
representation theory; iterated function systems (fractals, Julia sets, chaotic
dynamical systems), ergodic theory, operator algebras, and many more. And
neighboring areas, probability/statistics (for example stochastic processes,
Ito and Malliavin calculus), physics (representation of Lie groups, quantum
field theory), and spectral theory for Schr\"odinger operators.
We have strived for a more accessible book, and yet aimed squarely at
applications; -- we have been serious about motivation: Rather than beginning
with the four big theorems in Functional Analysis, our point of departure is an
initial choice of topics from applications. And we have aimed for flexibility
of use; acknowledging that students and instructors will invariably have a host
of diverse goals in teaching beginning analysis courses. And students come to
the course with a varied background. Indeed, over the years we found that
students have come to the Functional Analysis sequence from other and different
areas of math...

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## Optimal measures and Markov transition kernels

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

45.34%

#Mathematics - Optimization and Control#Computer Science - Computational Complexity#Computer Science - Information Theory#Mathematical Physics#Mathematics - Functional Analysis#Statistics - Machine Learning

We study optimal solutions to an abstract optimization problem for measures,
which is a generalization of classical variational problems in information
theory and statistical physics. In the classical problems, information and
relative entropy are defined using the Kullback-Leibler divergence, and for
this reason optimal measures belong to a one-parameter exponential family.
Measures within such a family have the property of mutual absolute continuity.
Here we show that this property characterizes other families of optimal
positive measures if a functional representing information has a strictly
convex dual. Mutual absolute continuity of optimal probability measures allows
us to strictly separate deterministic and non-deterministic Markov transition
kernels, which play an important role in theories of decisions, estimation,
control, communication and computation. We show that deterministic transitions
are strictly sub-optimal, unless information resource with a strictly convex
dual is unconstrained. For illustration, we construct an example where, unlike
non-deterministic, any deterministic kernel either has negatively infinite
expected utility (unbounded expected error) or communicates infinite
information.; Comment: Replaced with a final and accepted draft; Journal of Global
Optimization...

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## On stochastic parameterizing manifolds: Pullback characterization and Non-Markovian reduced equations

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 14/10/2013

Relevância na Pesquisa

45.32%

A general approach to provide approximate parameterizations of the "small"
scales by the "large" ones, is developed for stochastic partial differential
equations driven by linear multiplicative noise. This is accomplished via the
concept of parameterizing manifolds (PMs) that are stochastic manifolds which
improve in mean square error the partial knowledge of the full SPDE solution
$u$ when compared to the projection of $u$ onto the resolved modes, for a given
realization of the noise.
Backward-forward systems are designed to give access to such PMs in practice.
The key idea consists of representing the modes with high wave numbers (as
parameterized by the sought PM) as a pullback limit depending on the
time-history of the modes with low wave numbers.
The resulting manifolds obtained by such a procedure are not subject to a
spectral gap condition such as encountered in the classical theory. Instead,
certain PMs can be determined under weaker non-resonance conditions.
Non-Markovian stochastic reduced systems are then derived based on such a PM
approach. Such reduced systems take the form of SDEs involving random
coefficients that convey memory effects via the history of the Wiener process,
and arise from the nonlinear interactions between the low modes...

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## Concentration of Measure Inequalities and Their Communication and Information-Theoretic Applications

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 10/10/2015

Relevância na Pesquisa

45.35%

During the last two decades, concentration of measure has been a subject of
various exciting developments in convex geometry, functional analysis,
statistical physics, high-dimensional statistics, probability theory,
information theory, communications and coding theory, computer science, and
learning theory. One common theme which emerges in these fields is
probabilistic stability: complicated, nonlinear functions of a large number of
independent or weakly dependent random variables often tend to concentrate
sharply around their expected values. Information theory plays a key role in
the derivation of concentration inequalities. Indeed, both the entropy method
and the approach based on transportation-cost inequalities are two major
information-theoretic paths toward proving concentration.
This brief survey is based on a recent monograph of the authors in the
Foundations and Trends in Communications and Information Theory (online
available at http://arxiv.org/pdf/1212.4663v8.pdf), and a tutorial given by the
authors at ISIT 2015. It introduces information theorists to three main
techniques for deriving concentration inequalities: the martingale method, the
entropy method, and the transportation-cost inequalities. Some applications in
information theory...

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## High Precision Measurements of $\theta_{\odot}$ in Solar and Reactor Neutrino Experiments

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

45.42%

#High Energy Physics - Phenomenology#Astrophysics#High Energy Physics - Experiment#Nuclear Experiment

We discuss the possibilities of high precision measurement of the solar
neutrino mixing angle $\theta_\odot \equiv \theta_{12}$ in solar and reactor
neutrino experiments. The improvements in the determination of
$\sin^2\theta_{12}$, which can be achieved with the expected increase of
statistics and reduction of systematic errors in the currently operating solar
and KamLAND experiments, are summarised. The potential of LowNu $\nu-e$ elastic
scattering experiment, designed to measure the $pp$ solar neutrino flux, for
high precision determination of $\sin^2\theta_{12}$, is investigated in detail.
The accuracy in the measurement of $\sin^2\theta_{12}$, which can be achieved
in a reactor experiment with a baseline $L \sim (50-70)$ km, corresponding to a
Survival Probability MINimum (SPMIN), is thoroughly studied. We include the
effect of the uncertainty in the value of $\sin^2\theta_{13}$ in the analyses.
A LowNu measurement of the $pp$ neutrino flux with a 1% error would allow to
determine $\sin^2\theta_{12}$ with an error of 14% (17%) at 3$\sigma$ from a
two-generation (three-generation) analysis. The same parameter
$\sin^2\theta_{12}$ can be measured with an uncertainty of 2% (6%) at 1$\sigma$
(3$\sigma$) in a reactor experiment with $L \sim60 $ km...

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## A New Class of Backward Stochastic Partial Differential Equations with Jumps and Applications

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 04/05/2011

Relevância na Pesquisa

45.31%

#Mathematics - Probability#Computer Science - Systems and Control#Mathematical Physics#Mathematics - Analysis of PDEs#Mathematics - Optimization and Control#Mathematics - Statistics Theory

We formulate a new class of stochastic partial differential equations
(SPDEs), named high-order vector backward SPDEs (B-SPDEs) with jumps, which
allow the high-order integral-partial differential operators into both drift
and diffusion coefficients. Under certain type of Lipschitz and linear growth
conditions, we develop a method to prove the existence and uniqueness of
adapted solution to these B-SPDEs with jumps. Comparing with the existing
discussions on conventional backward stochastic (ordinary) differential
equations (BSDEs), we need to handle the differentiability of adapted triplet
solution to the B-SPDEs with jumps, which is a subtle part in justifying our
main results due to the inconsistency of differential orders on two sides of
the B-SPDEs and the partial differential operator appeared in the diffusion
coefficient. In addition, we also address the issue about the B-SPDEs under
certain Markovian random environment and employ a B-SPDE with strongly
nonlinear partial differential operator in the drift coefficient to illustrate
the usage of our main results in finance.; Comment: 22 pagea, 1 figure

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## Second-order asymptotics for quantum hypothesis testing

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

35.42%

In the asymptotic theory of quantum hypothesis testing, the minimal error
probability of the first kind jumps sharply from zero to one when the error
exponent of the second kind passes by the point of the relative entropy of the
two states in an increasing way. This is well known as the direct part and
strong converse of quantum Stein's lemma. Here we look into the behavior of
this sudden change and have make it clear how the error of first kind grows
smoothly according to a lower order of the error exponent of the second kind,
and hence we obtain the second-order asymptotics for quantum hypothesis
testing. This actually implies quantum Stein's lemma as a special case.
Meanwhile, our analysis also yields tight bounds for the case of finite sample
size. These results have potential applications in quantum information theory.
Our method is elementary, based on basic linear algebra and probability theory.
It deals with the achievability part and the optimality part in a unified
fashion.; Comment: Published in at http://dx.doi.org/10.1214/13-AOS1185 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org)

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## Turbulence analysis of an experimental flux rope plasma

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 08/10/2013

Relevância na Pesquisa

45.35%

We have previously generated elongated Taylor double-helix flux rope plasmas
in the SSX MHD wind tunnel. These plasmas are remarkable in their rapid
relaxation (about one Alfv\'en time) and their description by simple analytical
Taylor force-free theory despite their high plasma beta and high internal flow
speeds. We report on the turbulent features observed in these plasmas including
frequency spectra, autocorrelation function, and probability distribution
functions of increments. We discuss here the possibility that the turbulence
facilitating access to the final state supports coherent structures and
intermittency revealed by non-Gaussian signatures in the statistics.
Comparisons to a Hall-MHD simulation of the SSX MHD wind tunnel show similarity
in several statistical measures.; Comment: 20 pages, 9 figures, submitted to Plasma Physics Controlled Fusion
for Special Issue on Flux Ropes

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## Unbounded Probability Theory and Its Applications

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

45.36%

The paper deals with the order statistics and empirical mathematical
expectation (which is also called the estimate of mathematical expectation in
the literature) in the case of infinitely increasing random variables. The
Kolmogorov concept which he used in the theory of complexity and the
relationship with thermodynamics which was pointed out already by Poincar\'e
are considered.
The mathematical expectation (generalizing the notion of arithmetical mean,
which is generally equal to infinity for any increasing sequence of random
variables) is compared with the notion of temperature in thermodynamics by
using an analog of nonstandard analysis.
The relationship with the Van-der-Waals law of corresponding states is shown.
Some applications of this concept in economics, in internet information
network, and self-teaching systems are considered.; Comment: 23 p. Latex, minor corrections

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## The theory of the double preparation: discerned and indiscerned particles

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 06/11/2013

Relevância na Pesquisa

45.36%

In this paper we propose a deterministic and realistic quantum mechanics
interpretation which may correspond to Louis de Broglie's "double solution
theory". Louis de Broglie considers two solutions to the Schr\"odinger
equation, a singular and physical wave u representing the particle (soliton
wave) and a regular wave representing probability (statistical wave). We return
to the idea of two solutions, but in the form of an interpretation of the wave
function based on two different preparations of the quantum system. We
demonstrate the necessity of this double interpretation when the particles are
subjected to a semi-classical field by studying the convergence of the
Schr\"odinger equation when the Planck constant tends to 0. For this
convergence, we reexamine not only the foundations of quantum mechanics but
also those of classical mechanics, and in particular two important paradox of
classical mechanics: the interpretation of the principle of least action and
the the Gibbs paradox. We find two very different convergences which depend on
the preparation of the quantum particles: particles called indiscerned
(prepared in the same way and whose initial density is regular, such as atomic
beams) and particles called discerned (whose density is singular...

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## Error analysis of free probability approximations to the density of states of disordered systems

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 27/02/2012

Relevância na Pesquisa

45.41%

#Condensed Matter - Disordered Systems and Neural Networks#Mathematics - Statistics Theory#Physics - Chemical Physics#Quantum Physics#46L54, 65C60, 97K70

Theoretical studies of localization, anomalous diffusion and ergodicity
breaking require solving the electronic structure of disordered systems. We use
free probability to approximate the ensemble- averaged density of states
without exact diagonalization. We present an error analysis that quantifies the
accuracy using a generalized moment expansion, allowing us to distinguish
between different approximations. We identify an approximation that is accurate
to the eighth moment across all noise strengths, and contrast this with the
perturbation theory and isotropic entanglement theory.; Comment: 5 pages, 3 figures, submitted to Phys. Rev. Lett

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## Critical level statistics at the Anderson transition in four-dimensional disordered systems

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 22/10/1998

Relevância na Pesquisa

45.36%

#Condensed Matter - Disordered Systems and Neural Networks#Condensed Matter - Mesoscale and Nanoscale Physics

The level spacing distribution is numerically calculated at the
disorder-induced metal--insulator transition for dimensionality d=4 by applying
the Lanczos diagonalisation. The critical level statistics are shown to deviate
stronger from the result of the random matrix theory compared to those of d=3
and to become closer to the Poisson limit of uncorrelated spectra. Using the
finite size scaling analysis for the probability distribution Q_n(E) of having
n levels in a given energy interval E we find the critical disorder W_c = 34.5
\pm 0.5, the correlation length exponent \nu = 1.1 \pm 0.2 and the critical
spectral compressibility k_c \approx 0.5.; Comment: 10 pages, LaTeX2e, 7 fig, invited talk at PILS (Percolation,
Interaction, Localization: Simulations of Transport in Disordered Systems)
Berlin, Germany 1998, to appear in Annalen der Physik

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## Concentration of Measure Inequalities in Information Theory, Communications and Coding (Second Edition)

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

45.37%

During the last two decades, concentration inequalities have been the subject
of exciting developments in various areas, including convex geometry,
functional analysis, statistical physics, high-dimensional statistics, pure and
applied probability theory, information theory, theoretical computer science,
and learning theory. This monograph focuses on some of the key modern
mathematical tools that are used for the derivation of concentration
inequalities, on their links to information theory, and on their various
applications to communications and coding. In addition to being a survey, this
monograph also includes various new recent results derived by the authors. The
first part of the monograph introduces classical concentration inequalities for
martingales, as well as some recent refinements and extensions. The power and
versatility of the martingale approach is exemplified in the context of codes
defined on graphs and iterative decoding algorithms, as well as codes for
wireless communication. The second part of the monograph introduces the entropy
method, an information-theoretic technique for deriving concentration
inequalities. The basic ingredients of the entropy method are discussed first
in the context of logarithmic Sobolev inequalities...

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## Orthogonal polynomial ensembles in probability theory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

45.41%

#Mathematics - Probability#15A52, 33C45, 60-02, 60C05, 60F05, 60K35, 82C22, 82C41 (Primary)
05E10, 15A90, 42C05 (Secondary)

We survey a number of models from physics, statistical mechanics, probability
theory and combinatorics, which are each described in terms of an orthogonal
polynomial ensemble. The most prominent example is apparently the Hermite
ensemble, the eigenvalue distribution of the Gaussian Unitary Ensemble (GUE),
and other well-known ensembles known in random matrix theory like the Laguerre
ensemble for the spectrum of Wishart matrices. In recent years, a number of
further interesting models were found to lead to orthogonal polynomial
ensembles, among which the corner growth model, directed last passage
percolation, the PNG droplet, non-colliding random processes, the length of the
longest increasing subsequence of a random permutation, and others. Much
attention has been paid to universal classes of asymptotic behaviors of these
models in the limit of large particle numbers, in particular the spacings
between the particles and the fluctuation behavior of the largest particle.
Computer simulations suggest that the connections go even farther and also
comprise the zeros of the Riemann zeta function. The existing proofs require a
substantial technical machinery and heavy tools from various parts of
mathematics, in particular complex analysis...

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## Moment-Based Spectral Analysis of Random Graphs with Given Expected Degrees

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 10/12/2015

Relevância na Pesquisa

35.42%

#Mathematics - Statistics Theory#Computer Science - Social and Information Networks#Mathematics - Probability#Physics - Physics and Society#Statistics - Applications#05C80, 60B20

In this paper, we analyze the limiting spectral distribution of the adjacency
matrix of a random graph ensemble, proposed by Chung and Lu, in which a given
expected degree sequence $\bar{w}_n^{^{T}} = (w^{(n)}_1,\ldots,w^{(n)}_n)$ is
prescribed on the ensemble. Let $\mathbf{a}_{i,j} =1$ if there is an edge
between the nodes $\{i,j\}$ and zero otherwise, and consider the normalized
random adjacency matrix of the graph ensemble: $\mathbf{A}_n$ $=$ $
[\mathbf{a}_{i,j}/\sqrt{n}]_{i,j=1}^{n}$. The empirical spectral distribution
of $\mathbf{A}_n$ denoted by $\mathbf{F}_n(\mathord{\cdot})$ is the empirical
measure putting a mass $1/n$ at each of the $n$ real eigenvalues of the
symmetric matrix $\mathbf{A}_n$. Under some technical conditions on the
expected degrees sequence, we show that with probability one,
$\mathbf{F}_n(\mathord{\cdot})$ converges weakly to a deterministic
distribution $F(\mathord{\cdot})$. Furthermore, we fully characterize this
distribution by providing explicit expressions for the moments of
$F(\mathord{\cdot})$

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## A versatile integral in physics and astronomy

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 23/09/2011

Relevância na Pesquisa

45.35%

#Astrophysics - Instrumentation and Methods for Astrophysics#Astrophysics - Solar and Stellar Astrophysics#Mathematical Physics

This paper deals with a general class of integrals, the particular cases of
which are connected to outstanding problems in astronomy and physics. Reaction
rate probability integrals in the theory of nuclear reaction rates, Kr\"atzel
integrals in applied analysis, inverse Gaussian distribution, generalized
type-1, type-2 and gamma families of distributions in statistical distribution
theory, Tsallis statistics and Beck-Cohen superstatistics in statistical
mechanics and the general pathway model are all shown to be connected to the
integral under consideration. Representations of the integral in terms of
generalized special functions such as Meijer's G-function and Fox's H-function
are also pointed out.; Comment: 11 pages, LaTeX

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## Spectral gaps in Wasserstein distances and the 2D stochastic Navier--Stokes equations

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

45.38%

#Mathematics - Probability#Mathematical Physics#Mathematics - Analysis of PDEs#Mathematics - Dynamical Systems#Mathematics - Spectral Theory#37A30, 37A25, 60H15 (Primary)

We develop a general method to prove the existence of spectral gaps for
Markov semigroups on Banach spaces. Unlike most previous work, the type of norm
we consider for this analysis is neither a weighted supremum norm nor an
${\L}^p$-type norm, but involves the derivative of the observable as well and
hence can be seen as a type of 1-Wasserstein distance. This turns out to be a
suitable approach for infinite-dimensional spaces where the usual Harris or
Doeblin conditions, which are geared toward total variation convergence, often
fail to hold. In the first part of this paper, we consider semigroups that have
uniform behavior which one can view as the analog of Doeblin's condition. We
then proceed to study situations where the behavior is not so uniform, but the
system has a suitable Lyapunov structure, leading to a type of Harris
condition. We finally show that the latter condition is satisfied by the
two-dimensional stochastic Navier--Stokes equations, even in situations where
the forcing is extremely degenerate. Using the convergence result, we show that
the stochastic Navier--Stokes equations' invariant measures depend continuously
on the viscosity and the structure of the forcing.; Comment: Published in at http://dx.doi.org/10.1214/08-AOP392 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org)

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