On classical analogues of free entropy dimension

We define a classical probability analogue of Voiculescu's free entropy dimension that we shall call the classical probability entropy dimension of a probability measure on Rn. We show that the classical probability entropy dimension of a measure is related with diverse other notions of dimension. First, it can be viewed as a kind of fractal dimension. Second, if one extends Bochner's inequalities to a measure by requiring that microstates around this measure asymptotically satisfy the classical Bochner's inequalities, then we show that the classical probability entropy dimension controls the rate of increase of optimal constants in Bochner's inequality for a measure regularized by convolution with the Gaussian law as the regularization is removed. We introduce a free analogue of the Bochner inequality and study the related free entropy dimension quantity. We show that it is greater or equal to the non-microstates free entropy dimension.     

Multiple nontrivial solutions for nonlinear periodic problems with the p-Laplacian

We consider a nonlinear periodic problem driven by the scalar p-Laplacian with a nonsmooth potential (hemivariational inequality). Using the degree theory for multivalued perturbations of (S)₊-operators and the spectrum of a class of weighted eigenvalue problems for the scalar p-Laplacian, we prove the existence of at least three distinct nontrivial solutions, two of which have constant sign.     

Discrete-time GI/Geo/1 queue with multiple working vacations

Consider the discrete time GI/Geo/1 queue with working vacations under EAS and LAS schemes. The server takes the original work at the lower rate rather than completely stopping during the vacation period. Using the matrix-geometric solution method, we obtain the steady-state distribution of the number of customers in the system and present the stochastic decomposition property of the queue length. Furthermore, we find and verify the closed property of conditional probability for negative binomial distributions. Using such property, we obtain the specific expression for the steady-state distribution of the waiting time and explain its two conditional stochastic decomposition structures. Finally, two special models are presented.      

Algorithmic analysis of theMAP/PH/1 retrial queue

In this paper the distribution of the maximum number of customers in a retrial orbit for a single server queue with Markovian arrival process and phase type services is studied. Efficient algorithm for computing the probability distribution and some interesting numerical examples are presented.     

Markovian network processes: Congestion-dependent routing and processing

A Markovian network process describes the movement of discrete units among a set of nodes that process the units. There is considerable knowledge of such networks, often called queueing networks, in which the nodes operate independently and the routes of the units are independent. The focus of this study, in contrast, is on networks with dependent nodes and routings. Examples of dependencies are parallel processing across several nodes, blocking of transitions because of capacity constraints on nodes, alternate routing of units to avoid congestion, and accelerating or decelerating the processing rate at a node depending on downstream congestion. We introduce a general network process representing the numbers of units at the nodes and derive its equilibrium distribution. This distribution takes the form of a product of functions of vectors in which the arguments of the functions satisfy an interchangeability property. This new type of distribution may apply to other multi-variate processes as well. A basic idea in our approach is a linking of certain micro-level balance properties of the network routing to the processing rates at the nodes. The link is via routing-balance partitions of nodes that are inherent in any network. A byproduct of this approach is a general characterization of blocking of transitions without the restriction that the process is reversible, which had been a standard assumption. We also give necessary and sufficient conditions under which a unit moving in the network sees a time average for the unmoved units (called the MUSTA property). Finally, we discuss when certain flows between nodes in an open network are Poisson processes.This research was sponsored in part by Air Force Office of Scientific Research contract 84-0367.     

How likely is Polya's drunkard to stay in x⩾ y⩾z?

In his celebrated paper, Polya has considered the random walk in the three-dimensional (cubic) lattice and showed that the probability of return to the origin is less than 1. Subsequent authors have shown that the probability is %34.053.... Here we consider the same random walk, with the restriction that the drunkard is only allowed to stay inxyz. It is shown that his probability of returning to the originand staying in the allowed region is %6.4844....     

Corrigendum to “EuBaFe2O5+w: Valence mixing and charge ordering are two separate cooperative phenomena” [J. Solid State Chem. 180 (2007) 148-157]

An error in the expression for the Fe²⁺, Fe³⁺ valence-mixing probability in RBaFe₂O_5+w is acknowledged and correct formula is derived. The new formula slightly improves the least-squares fit to the experimental concentrations of the Mössbauer component Fe^2.5+ as a function of the oxygen-nonstoichiometry parameter w for R=Eu.     

A subjective Bayesian approach to the theory of queues II — Inference and information in M/M/1 queues

This is a sequel to Part I of A Subjective Bayesian Approach to the Theory of Queues. The focus here is on inference and a use of Shannon's measure of information for assessing the amount of information conveyed by the various types of data from queues. The notation and terminology used here is established in Part I.     

Tetraarylester der μ-Imidodiphosphorsäure und ihre Thioderivate — Strukturuntersuchungen

Tetraarylesters of μ-Imido-Diphosphoric Acid and its Thio Derivatives — Structure Investigations New O,O′,O″,O?-tetratolyl- and ditolyl-diphenylesters of the μ-imido-diphosphoric acid and its mono and dithio derivatives were synthesized, compared with the corresponding tetraphenylesters and investigated by ¹H, ¹³C, and ³¹P NMR spectroscopy and X-ray crystal structure analysis. Structures of the O,O′,O″,O?-tetrakis-(2-methyl-phenyl)-μ-imidodiphosphate, 1b , as well as of the corresponding ortho-, meta- and para-tolylesters of the μ-imido-monothiodiphosphoric acid ( 2a , 2b , 2c ) were determined. All the compounds form dimers via N? H…?O hydrogen bonds in the crystal as well as in nonpolar solvents. The distances around the phosphorus atoms rise with decreasing electronegativity of the phosphorus substituents. Signs of the ²J_{P? N? P} coupling constants were determined by ¹³C{¹H, ³¹P} triple resonance experiments for some compounds. These constants become more negative owing to substitution of a phosphoryl by a thiophosphoryl group.     

Molecular mechanics study of zirconocenium catalyzed isospecific polymerization of propylene

Stereocontrol energy (ΔE⁰) is investigated as a measure of enantioselectivity of ansa-zircoocenium catalyst in propylene polymerization; it was calculated with MM2 (molecular mechanics) force field using π complex (°C) and transition state (TS) geometries obtained by ab initio molecular orbital methods. Both rac-ethylenebis (1-η⁵-indenyl) - ( 1 ) and rac-ethylenebis (1-η⁵-4,5,7,8-tetrahydroindenyl) ( 2 ) zirconocenium species are isospecific in either the π-complexes or the transition states. The stereoselectivity is greater if there is α-agostic interaction; it is lowered in the cases of β and γ agostic interactions. The ¹³C-NMR steric pentad distribution indicates the poly(propylene) to be much less stereoregular than that predicted by ΔE⁰. Following the occurrence of a regiochemical insertion error, the addition of another monomer via any mode is prohibitively unfavorable. The catalyst suffers loss of stereospecificity as temperature of polymerization increases. Insertion via transition states involving different agostic interactions could be one explanation for the observed loss. © 1995 John Wiley & Sons, Inc.