Entropy production by block variable summation and central limit theorems

We prove a strict lower bound on the entropy produced when independent random variables are summed and rescaled. Using this, we develop an approach to central limit theorems from a dynamical point of view in which the entropy is a Lyapunov functional governing approach to the Gaussian limit. This dynamical approach naturally extends to cover dependent variables, and leads to new results in pure probability theory as well as in statistical mechanics. It also provides a unified framework within which many previous results are easily derived.Partially supported by an NSF postdoctoral fellowshipPartially supported by NSF grant DMS-8905772; A. P. Sloan Fellow in Mathematics     

Entropy,limit theorems,and variational principles for disordered lattice systems

We study infinite volume limits and Gibbs states of disordered lattice systems with bounded and continuous potentials. Our main tools are a generalization of relative entropy for random reference measures and a large deviation theory for nonstationary independent processes. We find that many familiar results of invariant potentials, such as large deviation theorems, variational principles, and equivalence of ensembles, continue to hold for disordered models, with suitably modified statements.     

On the failure of subadditivity of the Wigner–Yanase entropy

It was recently shown by Hansen that the Wigner–Yanase entropy is, for general states of quantum systems, not subadditive with respect to decomposition into two subsystems, although this property is known to hold for pure states. We investigate the question whether the weaker property of subadditivity for pure states with respect to decomposition into more than two subsystems holds. This property would have interesting applications in quantum chemistry. We show, however, that it does not hold in general, and provide a counterexample. Work partially supported by U.S. National Science Foundation grant PHY-0353181 and by an Alfred P. Sloan Fellowship. This paper may be reproduced, in its entirety, for non-commerical purposes.     

On Convexity of Generalized Wigner–Yanase–Dyson Information

The convexity of the Wigner–Yanase–Dyson information, as first proved by Lieb, is a deep and fundamental result because it leads to the strong subadditivity of quantum entropy. The Wigner–Yanase–Dyson information is a particular kind of quantum Fisher information with important applications in quantum estimation theory. But unlike the quantum entropy, which is the unique natural quantum extension of the classical Shannon entropy, there are many different variants of quantum Fisher information, and it is desirable to investigate their convexity. This article is devoted to studying the convexity of a direct generalization of the Wigner–Yanase–Dyson information. Some sufficient conditions are obtained, and some necessary conditions are illustrated. In a particular case, a surprising necessary and sufficient condition is obtained. Our results reveal the intricacy and subtlety of the convexity issue for general quantum Fisher information.      

Lie-admissible structures on Witt type algebras

In this paper we study Lie-admissible structures on Witt type algebras. Witt type algebras are Γ$\Gamma$-graded Lie algebras (where Γ$\Gamma$ is an abelian group) which generalize the Witt algebra. We give all third power-associative and flexible Lie-admissible structures on these algebras. In particular we generalize some results on the Witt algebra. After describing the second scalar cohomology group of Witt type algebras, we investigate third power-associative and flexible Lie-admissible structures on the central extension of some Witt type algebras. Finally we study a left-symmetric structure induced by a symplectic form for some Witt type algebras.     

Constructive inverse function theorems

An explicit method for finding every coefficient of the reversed series of a power series in one variable is presented. We also show how implicit functions on the plane may be solved for one of the variables. Our approach to the Lagrange inversion formula is based on factorization properties of partitions of integers and distributions of distinguishable objects. These results and techniques may have many applications in numerical analysis, combinatorics, and soliton theory.     

Quantized universal enveloping superalgebra of type Q and a super-extension of the Hecke algebra

The quantized universal enveloping algebra U _q(q(n)) of the strange Lie superalgebra q(n) and a super-analogue HC _q (N) of the Hecke algebra H _q (N) are constructed. These objects are in a duality similar to the known duality between U _q (gl(n)) and H _q (N).     

q-Deformed orthogonal and pseudo-orthogonal algebras and their representations

Deformed orthogonal and pseudo-orthogonal Lie algebras are constructed which differ from deformations of Lie algebras in terms of Cartan subalgebra and root vectors and which make it possible to construct representations by operators acting according to Gel'fand-Tsetlin-type formulas. Unitary representations of the q-deformed algebras U _q (so _n,1) are found.     

Hamilton-Jacobi formulation of Kolmogorov-Sinai entropy for classical and quantum dynamics

A Hamilton-Jacobi formulation of the Lyapunov spectrum and Kolmogorov-Sinai (KS) entropy is developed. It is numerically efficient and reveals a close relation between the KS invariant and the classical action. This formulation is extended to the quantum domain using the Madelung-Bohm orbits associated with the Schroedinger equation. The resulting quantum KS invariant for a given orbit equals the mean decay rate of the probability density along the orbit, while its ensemble average measures the mean growth rate of configuration-space information for the quantum system.     

Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions

We prove a functional central limit theorem for additive functionals of stationary reversible ergodic Markov chains under virtually no assumptions other than the necessary ones. We use these results to study the asymptotic behavior of a tagged particle in an infinite particle system performing simple excluded random walk.Supported by NSF Grant MCS-8301364, ONR Contract N00014-81-K-0012 and a Fellowship from John S. Guggenheim Memorial Foundation     

Quasi-classical limit of BKP hierarchy andW-infinity symmetries

Previous results on quasi-classical limit of the KP and Toda hierarchies are now extended to the BKP hierarchy. Basic tools such as the Lax representation, the Baker-Akhiezer function and the tau function are reformulated so as to fit into the analysis of quasi-classical limit. Two subalgebrasW ₁ ^B ₊ andw ₁ ^B ₊ of theW-infinity algebrasW _{1 +} andw _{1 +} are introduced as fundamental Lie algebras of the BKP hierarchy and its quasi-classical limit, the dispersionless BKP hierarchy. The quantumW-infinity algebraW ₁ ^B ₊ emerges in symmetries of the BKP hierarchy. In quasi-classical limit, theseW ₁ ^B ₊ symmetries are shown to be contracted intow ₁ ^B ₊ symmetries of the dispersionless BKP hierarchy.     

Quasi-classical limit of Toda hierarchy andW-infinity symmetries

Previous results on quasi-classical limit of the KP hierarchy and itsW-infinity symmetries are extended to the Toda hierarchy. The Planck constant now emerges as the spacing unit of difference operators in the Lax formalism. Basic notions, such as dressing operators, Baker-Akhiezer functions, and tau function, are redefined.W _{1 +} symmetries of the Toda hierarchy are realized by suitable rescaling of the Date-Jimbo-Kashiara-Miwa vertex operators. These symmetries are contracted tow _{1 +} symmetries of the dispersionless hierarchy through their action on the tau function.     

Characterizations of the quantum Witt algebra

The q-analogues of some concepts in the theory of nonassociative algebras are introduced and two characterizations are given for the quantum Witt algebra.     

On the relation between the divergence of trajectories and the Kolmogorov-Sinai entropy

Automorphisms of the torus are considered and the Kolmogorov-Sinai entropy explicitly computed via rate of growth of volumes. The usual rate of growth of exponentiating trajectories is shown to give only a lower bound to K-S entropy.     

Entropy,the central limit theorem and the algebra of the canonical commutation relation

The central limit theorem of Cushen and Hudson is reformulated on the algebra of the CCR. Namely, for a gauge invariant state , the weighted convolutions _n of the central limit tend to the quasi-free reduction _Q of pointwise. It is proved that if the initial relative entropy S(, _Q ) is finite, then S( _n , _Q ) goes to 0 and so _n – _Q 0. No restriction on the dimension of the test function space is made.     

Lotka-Volterra-type equations and their explicit integration

In this Letter, we give an explicit integration of some two-dimensionalised Lotka-Volterra-type equations associated with simple Lie algebras possessing a representation without branching.     

M2Ge17(M=Ni、Mn)及其阴离子的稳定性和磁性研究

采用密度泛函理论（density functional theory,DFT）中的广义梯度近似（generalized gradient approximation,GGA）对Ni2Ge17,Mn2Ge17,[Ni2Ge17]4-和[Mn2Ge17]2-三种同分异构体的几何结构、电子结构和磁性进行了研究。通过对三种同分异构体的稳定性分析,得出D4d结构的[Ni2Ge17]4-最为稳定。由磁性的分析得到,Ni2Ge17和[Ni2Ge17]4-的D2d、D4d、D4h结构的磁矩均为零,体现为非磁性。对于Mn2Ge17和[Mn2Ge17]2-的D4d结构出现了自旋分裂能级,分别带有9μB 和8μB的磁矩。其中,Mn原子与Ge原子之间存在弱的铁磁相互作用。对于D4h结构的Mn2Ge17,带有7μB的磁矩,Mn原子和Ge原子之间既存在弱的铁磁相互作用又存在弱的反铁磁相互作用,而Mn2Ge17和[Mn2Ge17]2-的D2d结构以及[Mn2Ge17]2-的D4h结构的磁矩为零。     

The quantum super-Yangian and Casimir operators of Uq(gl(M |N))

TheZ ₂ graded Yangian Yq(gl(M |N)) associated with the Perk-SchultzR matrix is introduced. Its structural properties, the central algebra in particular, are studied. AZ ₂-graded associative algebra epimorphism Yq(gl(M |N)) Uq (gl(M |N)) is obtained in explicit form. Images of central elements of the quantum super-Yangian under this epimorphism yield the Casimir operators of the quantum supergroup Uq(gl(M |N)) constructed in an earlier publication.