熵函数法的数学理论

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Centralizers and entropy

We prove that for a large class of bidimensional real analytic diffeomorphisms the centralizer is trivial: they only commute with their own integer powers. In particular this property holds for an open and dense subset of those having positive topological entropy.Partially supported by JNICT, PBIC/C/CEN 1020/92.     

A relaxed primal-dual path-following algorithm for linear programming

In this paper, we provide an easily satisfied relaxation condition for the primaldual interior path-following algorithm to solve linear programming problems. It is shown that the relaxed algorithm preserves the property of polynomial-time convergence. The computational results obtained by implementing two versions of the relaxed algorithm with slight modifications clearly demonstrate the potential in reducing computational efforts.Partially supported by the North Carolina Supercomputing Center, the 1993 Cray Research Award, and a National Science Council Research Grant of the Republic of China.     

Estimate of exponential convergence rate in total variation by spectral gap

This note is devoted to study the exponential convergence rate in the total variation for reversible Markov processes by comparing it with the spectral gap. It is proved that in a quite general setup, with a suitable restriction on the initial distributions, the rate is bounded from below by the spectral gap. Furthermore, in the compact case or for birth-death processes or half-line diffusions, the rate is shown to be equal to the spectral gap. Research supported in part by NSFC, Qiu Shi Sci. & Tech. Found. DPFIHE, MCSEC and Univ., of Rome I, Italy     

Convergence rates for Markov chain order estimates using EDC criterion

The Efficient Determination Criterion (EDC) generalizes the AIC and BIC criteria and provides a class of consistent estimators for the order of a Markov chain with finite state space. In this note, we derive rates of convergence for the EDC estimates. *Partially supported by CNPq, CAPES/PROCAD, FAPDF/PRONEX, FINATEC and FUNPE/UnB. **Partially supported by CAPES.     

On spanning tree problems with multiple objectives

We investigate two versions of multiple objective minimum spanning tree problems defined on a network with vectorial weights. First, we want to minimize the maximum ofQ linear objective functions taken over the set of all spanning trees (max-linear spanning tree problem, ML-ST). Secondly, we look for efficient spanning trees (multi-criteria spanning tree problem, MC-ST).Problem ML-ST is shown to be NP-complete. An exact algorithm which is based on ranking is presented. The procedure can also be used as an approximation scheme. For solving the bicriterion MC-ST, which in the worst case may have an exponential number of efficient trees, a two-phase procedure is presented. Based on the computation of extremal efficient spanning trees we use neighbourhood search to determine a sequence of solutions with the property that the distance between two consecutive solutions is less than a given accuracy.Partially supported by Deutsche Forschungsgemeinschaft and HCº Contract no. ERBCHRXCT 930087.Partially supported by Alexander von Humboldt-Stiftung.     

Law of large numbers and central limit theorem for randomly forced PDE's

We consider a class of dissipative PDE's perturbed by an external random force. Under the condition that the distribution of perturbation is sufficiently non-degenerate, a strong law of large numbers (SLLN) and a central limit theorem (CLT) for solutions are established and the corresponding rates of convergence are estimated. It is also shown that the estimates obtained are close to being optimal. The proofs are based on the property of exponential mixing for the problem in question and some abstract SLLN and CLT for mixing-type Markov processes.     

Finitely generated groups of polynomial subgroup growth

We determine the structure of finitely generated residually finite groups in which the number of subgroups of each finite indexn is bounded by a fixed power ofn. To John Thompson, an inspiration to group theory, on his being awarded the Wolf Prize Partially supported by BSF and GIF grants. Partially supported by a BSF grant.     

Coxeter groups are almost convex

In [C] Cannon introduced the notion of almost convexity for the Cayley graph of a finitely generated group. In this paper, we observe that standard facts about Coxeter groups imply that the Cayley graph associated to any Coxeter system is almost convex.Partially supported by NSF grants DMS-8905378 and DMS-9007959.     

Almost block independence

Summary A stationary finite state process is almost block independent, if long blocks are almost independent in the ¯d-sense. We show here that the concept of almost block independence is equivalent to very weak Bernoulli and to finitely determined, hence that the class of almost block independent processes coincides with the class of stationary codings of i.i.d. processes.Research supported in part by NSF grant MP-575-08324     

Range conditions for the exponential Radon transform

A new approach to the range conditions for the exponential Radon transform is introduced. Partially supported by NSF EPSCOR Grant and Wesley Foundation Grant no. 9012019. Partially supported by NSF Grant 33-1807-231.     

Entropy of flows, revisited

We introduce a concept of measure-theoretic entropy for flows and study its invariance under measure-theoretic equivalences. Invariance properties of the corresponding topological entropy is studied too. We also answer a question posed by Bowen-Walters in [3] concerning the equality between the topological entropy of the time-one map of an expansive flow and the time-one map of its symbolic suspension.Partially supported by FAPESP-Brasil, Grant #96/11671-6.Partially supported by CNPq-Brasil, Grant #300557/89-2.     

Geometric ergodicity of affine processes on cones

For affine processes on finite-dimensional cones, we give criteria for geometric ergodicity — that is exponentially fast convergence to a unique stationary distribution. Ergodic results include both the existence of exponential moments of the limiting distribution, where we exploit the crucial affine property, and finite moments, where we invoke the polynomial property of affine semigroups. Furthermore, we elaborate sufficient conditions for aperiodicity and irreducibility. Our results are applicable to Wishart processes with jumps on the positive semidefinite matrices, continuous-time branching processes with immigration in high dimensions, and classical term-structure models for credit and interest rate risk.     

Exponential Mixing of the 3D Stochastic Navier-Stokes Equations Driven by Mildly Degenerate Noises

We prove the strong Feller property and exponential mixing for 3D stochastic Navier-Stokes equation driven by mildly degenerate noises (i.e. all but finitely many Fourier modes being forced) via a Kolmogorov equation approach.     

Closedness Properties of Internal Relations III: Pointed Protomodular Categories

A pointed variety of universal algebras is protomodular in the sense of D. Bourn, if and only if it is classically ideal determined in the sense of A. Ursini (this result is due to D. Bourn and G. Janelidze). We prove a characterization theorem for pointed protomodular categories, which is a (pointed) categorical version of Ursini’s characterization theorem for classically ideal determined varieties, involving classically 0-regular algebras. A suitable simplification of the property of a pair of relations, which is used to define a classically 0-regular algebra, yields a new closedness property of a single binary relation – we show that a finitely complete pointed category is protomodular if and only if every binary internal relation R→A ² in it has this closedness property. Partially supported by South African National Research Foundation, and Georgian National Science Foundation (GNSF/ST06/3-004).     

On the entropy method and exponential convergence to equilibrium for a recombination–drift–diffusion system with self-consistent potential

We consider a Shockley–Read–Hall recombination–drift–diffusion model coupled to Poisson’s equation and subject to boundary conditions, which imply conservation of the total charge. As main result, we derive an explicit functional inequality between relative entropy and entropy production rate, which implies exponential convergence to equilibrium with explicit constant and rate. We report that the key entropy–entropy production inequality ought rather not to be formulated on the states space of the parabolic–elliptic system, but on the reduced states space of the associated nonlocal drift–diffusion problem, where the Poisson equation is replaced by the corresponding Green function.     

How often is 84(g−1) achieved?

For any finitely generated group Γ, the asymptotics of the set of orders of finite quotient groups of Γ are determined by the minimum dimension of a complex linear group containing an infinite quotient of Γ. We give a proof and an application to the asymptotic behavior of the set of integersg for which the Hurwitz bound is sharp. Partially supported by NSF Grant DMS-97-27553.     

Coding Markov chains from the past

It is shown that a mixing Markov chain is a unilateral or one-sided factor of every ergodic process of equal or greater entropy. This extends the work of Sinai, who showed that the result holds for independent processes, and the work of Ornstein and Weiss, who showed that the result holds for mixing Markov chains in which all transition probabilities are positive. The proof exploits the Rothstein-Burton joinings-space formulation of Ornstein’s isomorphism theory, and uses a random coding argument. Partially supported by an NSF Graduate Fellowship, an NSF Postdoctoral Fellowship, and NSF Grant # DMS 84-03182 during the writing of this article.     

Sporadic randomness,Maxwell's Demon and the Poincaré recurrence times

In the case of fully chaotic systems, the distribution of the Poincaré recurrence times is an exponential whose decay rate is the Kolmogorov–Sinai (KS) entropy. We address the discussion of the same problem, the connection between dynamics and thermodynamics, in the case of sporadic randomness, using the Manneville map as a prototype of this class of processes. We explore the possibility of relating the distribution of Poincaré recurrence times to “thermodynamics”, in the sense of the KS entropy, also in the case of an inverse power-law. This is the dynamic property that Zaslavsky [Physics Today 52 (8) (1999) 39] finds to be responsible for a striking deviation from ordinary statistical mechanics under the form of Maxwell's Demon effect. We show that this way of establishing a connection between thermodynamics and dynamics is valid only in the case of strong chaos, where both the sensitivity to initial conditions and the distribution of the Poincaré recurrence times are exponential. In the case of sporadic randomness, resulting at long times in the Lévy diffusion processes, the sensitivity to initial conditions is initially a power-law, but it becomes exponential again in the long-time scale, whereas the distribution of Poincaré recurrence times keeps, or gets, its inverse power-law nature forever, including the long-time scale where the sensitivity to initial condition becomes exponential. We show that a non-extensive version of thermodynamics would imply the Maxwell's Demon effect to be determined by memory, and thus to be temporary, in conflict with the dynamic approach to Lévy statistics. The adoption of heuristic arguments indicates that this effect is possible, as a form of genuine equilibrium, after completion of the process of memory erasure.     

Pointwise ergodic theorems for the symmetric exclusion process

Summary Almost sure convergence theorems are proved for Cesaro averages of continous functions in the case of the symmetric exclsion processes in dimension d≧3. For the occupation time of a single site the same result is proved in all dimensions. Partially supported by CNPq