Página 1 dos resultados de 164 itens digitais encontrados em 0.201 segundos

Universality for random matrices and log-gases

Erdos, Laszlo
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...

Beyond quantum microcanonical statistics

Fresch, Barbara; Moro, Giorgio J.
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...

Bubble statistics and positioning in superhelically stressed DNA

Jost, Daniel; Zubair, Asif; Everaers, Ralf
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
45.38%
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

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

Jorgensen, Palle; Tian, Feng
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
45.4%
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...

Optimal measures and Markov transition kernels

Belavkin, Roman V.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
45.34%
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...

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

Chekroun, Mickael D.; Liu, Honghu; Wang, Shouhong
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...

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

Raginsky, Maxim; Sason, Igal
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...

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

Bandyopadhyay, Abhijit; Choubey, Sandhya; Goswami, Srubabati; Petcov, S. T.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
45.42%
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...

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

Dai, Wanyang
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 04/05/2011
Relevância na Pesquisa
45.31%
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

Second-order asymptotics for quantum hypothesis testing

Li, Ke
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)

Turbulence analysis of an experimental flux rope plasma

Schaffner, D. A.; Lukin, V. S.; Wan, A.; Brown, M. R.
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

Unbounded Probability Theory and Its Applications

Maslov, V. P.; Maslova, T. V.
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

The theory of the double preparation: discerned and indiscerned particles

Gondran, Michel; Gondran, Alexandre
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...

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

Chen, Jiahao; Hontz, Eric; Moix, Jeremy; Welborn, Matthew; Van Voorhis, Troy; Suárez, Alberto; Movassagh, Ramis; Edelman, Alan
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 27/02/2012
Relevância na Pesquisa
45.41%
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

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

Zharekeshev, I. Kh.; Kramer, B.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 22/10/1998
Relevância na Pesquisa
45.36%
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

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

Raginsky, Maxim; Sason, Igal
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...

Orthogonal polynomial ensembles in probability theory

Koenig, Wolfgang
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
45.41%
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...

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

Preciado, Victor M.; Rahimian, M. Amin
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 10/12/2015
Relevância na Pesquisa
35.42%
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 $F(\mathord{\cdot})$

A versatile integral in physics and astronomy

Mathai, A. M.; Haubold, H. J.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 23/09/2011
Relevância na Pesquisa
45.35%
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

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

Hairer, Martin; Mattingly, Jonathan C.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
45.38%
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)