Optimal Liouville-type Theorems for Noncooperative Elliptic Schrödinger Systems and Applications

We study multi-component elliptic Schrödinger systems arising in nonlinear optics and Bose-Einstein condensation phenomena. We prove new Liouville-type nonexistence theorems, as well as a priori bounds, decay and singularity estimates. This is shown under an optimal Sobolev growth restriction on the nonlinearities, thus improving on recent results of Dancer et al. and of Tavares et al. These systems are of non-cooperative form and hence cannot be tackled by maximum principle methods such as moving planes. Instead we rely on a delicate combination of Rellich-Pohozaev type identities, Sobolev and interpolation inequalities on S ^n-1 and feedback and measure arguments. We also extend our results to a rather general class of gradient-type systems.     

General Bell-CHSH type and entropic inequalities based on quantum tomograms

Review of Bell-CHSH type and entropic inequalities in composite quantum correlated systems in the probability representation of states is presented. The upper bounds for some new Bell-CHSH type inequalities within the framework of classical probability theory and in quantum tomography are compared. Violation of Bell-CHSH type inequalities are shown explicitly using the method of averaging in tomographic picture of quantum states. Joint tomographic entropies of multiqubit systems are studied. Limitations on inequalities for tomographic entropies are obtained. A negative result of possible connection between the violation of entropic and Bell-CHSH type inequalities in multi-partite states is reported.     

Inequalities for higher order Ising spins and for continuum fluids

The recently derived Fortuin, Kasteleyn and Ginibre (FKG) inequalities for lattice gasses are investigated for higher order Ising spin systems and multi-component lattice gasses. Conditions are given for the validity of the FKG inequalities for higher order spin systems with Hamiltonians of the form used recently as models for various physical systems, e.g.He ³–He ⁴ mixtures. We also investigate various inequalities for binary lattice gases and show how these can be carried over to continuum systems.Supported in part by U.S.A.F.O.S.R. # F 44620-71-C-0013.N.S.F. Graduate Trainee.     

Some Parameterized Quantum Midpoint and Quantum Trapezoid Type Inequalities for Convex Functions with Applications

Quantum information theory, an interdisciplinary field that includes computer science, information theory, philosophy, cryptography, and entropy, has various applications for quantum calculus. Inequalities and entropy functions have a strong association with convex functions. In this study, we prove quantum midpoint type inequalities, quantum trapezoidal type inequalities, and the quantum Simpson’s type inequality for differentiable convex functions using a new parameterized q-integral equality. The newly formed inequalities are also proven to be generalizations of previously existing inequities. Finally, using the newly established inequalities, we present some applications for quadrature formulas.     

The random-walk representation of classical spin systems and correlation inequalities

We use the random-walk representation to prove the first few of a new family of correlation inequalities for ferromagnetic ?⁴ lattice models. These inequalities state that the finite partial sums of the propagator-resummed perturbation expansion for the 4-point function form an alternating set of rigorous upper and lower bounds for the exact 4-point function. Generalizations to 2n-point functions are also given. A simple construction of the continuum ? _d ⁴ quantum field theory (d<4), based on these inequalities, is described in a companion paper.     

Correlation inequalities and the thermodynamic limit for classical and quantum continuous systems

We use Ginibre's general formulation of Griffiths' inequalities to derive new correlation inequalities for two-component classical and quantum mechanical systems of distinguishable particles interacting via two body potentials of positive type. As a consequence we obtain existence of the thermodynamic limit of the thermodynamic and correlation functions in the grand canonical ensemble at arbitrary temperatures and chemical potentials. For a large class of systems we show that the limiting correlation functions are clustering. (In a subsequent article these results are extended to the correlation functions of two-component quantum mechanical gases with Bose-Einstein statistics). Finally, a general construction of the thermodynamic limit of the pressure for gases which are not H-stable, above collapse temperature, is presented.Research supported in part by the U.S. National Science Foundation under grant MPS 75-11864A Sloan Foundation Fellow     

Proof of absence of spooky action at a distance in quantum correlations

I prove that there is no spooky action at a distance and nonlocal state-reduction during measurements on quantum entangled systems. The prediction of quantum theory as well as experimental results are in conflict with the concept of nonlocal state-reduction, as conclusively shown here under very general assumptions. This has far-reaching implications in the interpretation of quantum mechanics in general, and demands a radical change in its present interpretation of measurements on entangled multiparticle systems. Motivated by these results we re-examine Bell’s theorem for correlations of entangled systems and find that the correlation function used by Bell fails to incorporate phase correlations at source. It is the use of such an unphysical correlation function, and not failure of locality, that leads to the Bell’s inequalities.     

Global synchronization for a class of dynamical complex networks

This paper is concerned with the problem of global synchronization for a class of dynamical complex networks composed of general Lur’e systems. Based on the absolute stability theory and the Kalman-Yakubovich-Popov (KYP) lemma, sufficient conditions are established to guarantee global synchronization of dynamical networks with complex topology, directed and weighted couplings. Several global synchronization criteria formulated in the form of linear matrix inequalities (LMIs) or frequency-domain inequalities are also proposed for undirected dynamical networks. In order to obtain global results, no linearization technique is involved through derivation of the synchronization criteria. Numerical examples are provided to demonstrate the effectiveness of the proposed results.     

Generic Bell inequalities for multipartite arbitrary dimensional systems

We present generic Bell inequalities for multipartite arbitrary dimensional systems. The inequalities that any local realistic theory must obey are violated by quantum mechanics for even dimensional systems. A large set of variants are shown to naturally emerge from the generic Bell inequalities. We discuss particular variants of Bell inequalities that are violated for all the systems including odd dimensional systems.     

Adaptive synchronization of a class of fractional-order complex-valued chaotic neural network with time-delay

This paper is concerned with the adaptive synchronization of fractional-order complex-valued chaotic neural networks (FOCVCNNs) with time-delay. The chaotic behaviors of a class of fractional-order complex-valued neural network are investigated. Meanwhile, based on the complex-valued inequalities of fractional-order derivatives and the stability theory of fractional-order complex-valued systems, a new adaptive controller and new complex-valued update laws are proposed to construct a synchronization control model for fractional-order complex-valued chaotic neural networks. Finally, the numerical simulation results are presented to illustrate the effectiveness of the developed synchronization scheme.     

Nonlocality in many-body quantum systems detected with two-body correlators

Contemporary understanding of correlations in quantum many-body systems and in quantum phase transitions is based to a large extent on the recent intensive studies of entanglement in many-body systems. In contrast, much less is known about the role of quantum nonlocality in these systems, mostly because the available multipartite Bell inequalities involve high-order correlations among many particles, which are hard to access theoretically, and even harder experimentally. Standard, “theorist- and experimentalist-friendly” many-body observables involve correlations among only few (one, two, rarely three...) particles. Typically, there is no multipartite Bell inequality for this scenario based on such low-order correlations. Recently, however, we have succeeded in constructing multipartite Bell inequalities that involve two- and one-body correlations only, and showed how they revealed the nonlocality in many-body systems relevant for nuclear and atomic physics [Tura et al., Science 344 (2014) 1256]. With the present contribution we continue our work on this problem. On the one hand, we present a detailed derivation of the above Bell inequalities, pertaining to permutation symmetry among the involved parties. On the other hand, we present a couple of new results concerning such Bell inequalities. First, we characterize their tightness. We then discuss maximal quantum violations of these inequalities in the general case, and their scaling with the number of parties. Moreover, we provide new classes of two-body Bell inequalities which reveal nonlocality of the Dicke states—ground states of physically relevant and experimentally realizable Hamiltonians. Finally, we shortly discuss various scenarios for nonlocality detection in mesoscopic systems of trapped ions or atoms, and by atoms trapped in the vicinity of designed nanostructures.     

Mesoscopic Higher Regularity and Subadditivity in Elliptic Homogenization

We introduce a new method for obtaining quantitative results in stochastic homogenization for linear elliptic equations in divergence form. Unlike previous works on the topic, our method does not use concentration inequalities (such as Poincaré or logarithmic Sobolev inequalities in the probability space) and relies instead on a higher (C^k, k ≥ 1) regularity theory for solutions of the heterogeneous equation, which is valid on length scales larger than a certain specified mesoscopic scale. This regularity theory, which is of independent interest, allows us to, in effect, localize the dependence of the solutions on the coefficients and thereby accelerate the rate of convergence of the expected energy of the cell problem by a bootstrap argument. The fluctuations of the energy are then tightly controlled using subadditivity. The convergence of the energy gives control of the scaling of the spatial averages of gradients and fluxes (that is, it quantifies the weak convergence of these quantities), which yields, by a new “multiscale” Poincaré inequality, quantitative estimates on the sublinearity of the corrector.     

Van der Waals interaction between spherical voids

The problem of calculating the non-retarded Van der Waals type of interaction between two spherical voids in an electron gas is presented from a new approach. The formulation is based on an integral equation for the self-induced density oscillations derived from a semi-classical treatment of the density-density response theory for inhomogeneous electron systems. The interaction energy between two identical voids is found to obey d^-6 law for large separation d between the voids, the law being determined by the dipolar plasma oscillations alone.     

New Bell-type inequalities forN > 4 necessary for existence of a hidden variable

The purpose of this paper is to extend Bell's inequalities to obtain some general necessary conditions for the existence of a joint probability distribution for any finite collection of Bell-type random variables. Our results show that forN > 4 many new elementary inequalities beyond those of Bell must be satisfied by any hidden variable theory.     

Calculation of diatomic van der Waals systems inert gas atom-halogen type inert gas ion in the ground state

Interatomic potentials are calculated for the systems inert gas ion in the ground state-inert gas atom Ne⁺, Xe⁺-Ne, Ar, Kr, Xe, Fr. The calculation is performed by the effective pseudopotential method using the new form of the polarization interaction potential obtained by calculating the most important polarization diagrams of perturbation theory in the Thomas-Fermi approximation. The quasimolecular states of these van der Waals systems are calculated to refine the available data; some data are obtained for the first time. Translated from ZhurnalStrukturnoi Khimii, Vol. 39, No. 4, pp. 591–595, July–August, 1998     

Stochastic synchronization of coupled neural networks with intermittent control

In this Letter, we study the exponential stochastic synchronization problem for coupled neural networks with stochastic noise perturbations. Based on Lyapunov stability theory, inequality techniques, the properties of Weiner process, and adding different intermittent controllers, several sufficient conditions are obtained to ensure exponential stochastic synchronization of coupled neural networks with or without coupling delays under stochastic perturbations. These stochastic synchronization criteria are expressed in terms of several lower-dimensional linear matrix inequalities (LMIs) and can be easily verified. Moreover, the results of this Letter are applicable to both directed and undirected weighted networks. A numerical example and its simulations are offered to show the effectiveness of our new results.     

The Saturation of Several Universal Inequalities in Information-Processing

In this paper, we characterize the saturation of four universal inequalities in quantum information theory, including a variant version of strong subadditivity inequality for von Neumann entropy, the coherent information inequality, the Holevo quantity, and average entropy inequalities. These results shed new light on quantum information inequalities.     

随机扰动下一般混沌系统的H∞同步

本文对随机扰动下的一般混沌系统进行了H_∞控制同步研究,其中扰动是布朗运动随机过程.基于随机李雅普诺夫稳定性理论、线性矩阵不等式、It公式以及H_∞控制方法,通过设置控制器,从理论上提出了驱动系统和随机扰动下的响应系统的H_∞渐近同步的新标准,这些标准形式简单且易于用Matlab实现.最后的数值模拟表明提出的理论结果的正确性和有效性. 关键词： 随机扰动 H_∞同步')" href="#">H_∞同步 线性矩阵不等式 It公式     

A Measure of Quantum Correlation Based on von Neumann Entropy and Positive Operator-Valued Measurement

Quantum correlations in composite quantum systems are at the origin of the most peculiar features of quantum mechanics such as the violation of Bells inequalities and non-locality. In quantum information theory, they are viewed as quantum resources used by quantum algorithms and communication protocols to outperform their classical analogs. In this paper, we define a new measure of quantum correlation based on von Neumann entropy and positive operator-valued measurement,which has clear physical meaning and we can prove that it satisfying many good property for a measure of quantumness.