Generalized binomial and negative binomial distributions of order<Emphasis Type="Italic">k</Emphasis> by the<Emphasis Type="Italic">l</Emphasis>-overlapping enumeration scheme

In this paper, we investigate the exact distribution of the waiting time for ther-th ℓ-overlapping occurrence of success-runs of a specified length in a sequence of two state Markov dependent trials. The probability generating functions are derived explicitly, and as asymptotic results, relationships of a negative binomial distribution of orderk and an extended Poisson distribution of orderk are discussed. We provide further insights into the run-related problems from the viewpoint of the ℓ-overlapping enumeration scheme. We also study the exact distribution of the number of ℓ-overlapping occurrences of success-runs in a fixed number of trials and derive the probability generating functions. The present work extends several properties of distributions of orderk and leads us a new type of geneses of the discrete distributions.     

Level of nodes in increasing trees revisited

Simply generated families of trees are described by the equation T(z) = ϕ(T(z)) for their generating function. If a tree has n nodes, we say that it is increasing if each node has a label ∈ { 1,…,n}, no label occurs twice, and whenever we proceed from the root to a leaf, the labels are increasing. This leads to the concept of simple families of increasing trees. Three such families are especially important: recursive trees, heap ordered trees, and binary increasing trees. They belong to the subclass of very simple families of increasing trees, which can be characterized in 3 different ways. This paper contains results about these families as well as about polynomial families (the function ϕ(u) is just a polynomial). The random variable of interest is the level of the node (labelled) j, in random trees of size n ≥ j. For very simple families, this is independent of n, and the limiting distribution is Gaussian. For polynomial families, we can prove this as well for j,n → ∞ such that n − j is fixed. Additional results are also given. These results follow from the study of certain trivariate generating functions and Hwang's quasi power theorem. They unify and extend earlier results by Devroye, Mahmoud, and others. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

The application of spectral distribution of product of two random matrices in the factor analysis

In the factor analysis model with large cross-section and time-series dimensions,we pro- pose a new method to estimate the number of factors.Specially if the idiosyncratic terms satisfy a linear time series model,the estimators of the parameters can be obtained in the time series model. The theoretical properties of the estimators are also explored.A simulation study and an empirical analysis are conducted.     

Sequential nonlinear estimation with nonaugmented priors

How can the basic compatibility of theory and observations be investigated for nonlinear processes without requiring stochastic characterizations for residual error terms? The present paper proposes a flexible least-cost approach. For each possible estimatex for the sequence of process states, letc _D (x) andx _M(x) denote the costs incurred for deviations away from the prior dynamic specifications and prior measurement specifications, respectively. Define the cost-efficiency frontier to be the greatest lower bound for the set of all possible cost pairs [c _D (x),c _M(x)], conditional on the given observations. State sequence estimatesx that attain the cost-efficiency frontier indicate the possible ways that the actual process could have developed over time in a manner minimally incompatible with the prior dynamic and measurement specifications. An algorithm is developed for the exact sequential updating of the cost-efficient state sequence estimates as the duration of the process increases and additional observations are obtained.     

LIMITING BEHAVIOR OF UNIFORM RECURSIVE TREES

The authors consider the limiting behavior of various branches in a uniform recursive tree with size growing to infinity.The limiting distribution ofζ_(n,m),the number of branches with size m in a uniform recursive tree of order n,converges weakly to a Poisson distribution with parameter 1/m with convergence of all moments.The size of any large branch tends to infinity almost surely.     

Limiting Lagrangians: A primal approach

In this paper, we consider a convex program with either a finite or an infinite number of constraints and its formal Lagrangian dual. We show that either the primal program satisfies a general condition which implies there is no duality gap or that there is a nonzero vectord with the following properties: First, wheneverd is added to the objective function, where is a positive number not greater than one, the resulting program satisfies the general sufficient condition cited above for no duality gap. Second, the optimal value of this perturbed program is attained and tends to the optimal value of the original program as tends to zero. Third, the optimal solutions of the perturbed programs form a minimizing sequence of the original program. As a consequence of the above, we derive the limiting Lagrangian theory of Borwein, Duffin, and Jeroslow.The authors are indebted to an unknown referee who suggested the very short and elegant proofs of Lemma 2.3 and Theorem 2.3.This work was completed while the first author was a member of the College of Management, Georgia Institute of Technology, Atlanta, Georgia.     

Molecular structure and nanosecond optical limiting property of a typical silver cubane-like cluster containing tetraselenotungstate

A typical silver cubane-like heteroselenometallic cluster, (μ₃-WSe₄)Ag₃(PPh₃)₃Cl·0.5SePPh₃, was synthesized from reaction of [Et₄N]₂[WSe₄] and Ag(PPh₃)₂(NO₃) in CH₂Cl₂ solution and structurally characterized by X-ray crystallography. The compound crystallizes in the trigonal space group R3 with a=16.258(4), b=22.80(3) Å, V=5217(7) ų, and Z=3. The structure contains a strongly distorted cubane-like {WAg₃Se₃Cl} core. The coordination geometries of the W center and each Ag atom are tetrahedral. The mean W-Ag separation is 3.010(3) Å. Optical nonlinearity of this cubane-like cluster was studied. Large optical limiting effect with threshold of 0.45 J/cm² was observed with the laser pulses of 7 ns at 532 nm.     

Limiting Partial Molar Volumes of Electrolytes in 2-Methyl-2-Butanol <Emphasis Type="Bold">+</Emphasis> Water Mixtures at 298.15 K

Partial molar volumes at infinite dilution, V⁰₂, of alkali–metal halides (LiCl, NaCl KCl RbCl CsCl, NaBr, KBr, KI), tetra-n-alkylammonium bromides, R₄NBr (R=Me, Et, n-Pr, n-Bu, n-Pen), NaBPh₄, and Ph₄PCl have been determined in binary solvent mixtures of water with 2-methyl-2-butanol covering the water-rich region and the alcohol-rich region at 298.15 K. V⁰₂ for alkali–metal halides show relatively little dependence on the solvent composition. However, in the case of hydrophobic electrolytes the observed effects are more pronounced. A good linear dependence between V⁰₂(R₄NBr) and the molecular weight of the tetra-n-alkylammonium cation is found. Limiting single-ion volumes have been obtained using the assumption that V⁰(Ph₄P⁺)–V⁰(BPh^–₄)=2.0 cm³-mol^–1. The trends in the single-ion volumes are discussed in both solvent regions.     

Transport of glycine in systems with cation-exchange membranes in hydrochloric acid solutions: Effect of a heterogeneous protonation reaction

The electrical mass transfer of cations in electromembrane systems (EMS) with an MK-40 cation-exchange membrane and glycine in aqueous solutions of hydrochloric acid is studied using the method of a rotating membrane disk. Limiting current densities and limiting steps of the transport of cations in such systems are determined. Shown is the possibility of an increase in the electrical mass transfer of glycine as a consequence of the occurrence of a heterogeneous reaction of protonation of its zwitterions. The effect of the membrane surface state on the kinetic regularities of transport of cations in EMS with glycine in solutions of hydrochloric acid is exposed.     

Volumetric properties of H2O, D2O, and methanol in acetonitrile at 278.15—318.15 K

The densities of H₂O, D₂O, and MeOH solutions in acetonitrile with the solute concentrations up to 0.07 molar fractions at 278.15, 288.15, 298.15, 308.15, and 318.15 K were measured using vibrating-tube densimetry with an error 8·10^–6 g cm^–3. The limiting partial molar volumes for the H/D isotopomers of water and IaII in acetonitrile (V^– ₂ ) and the isotope effects in V^– ₂ and in excess molar volumes of acetonitrile—water mixtures were calculated. Molecules of H₂O, D₂O, and IaII form associates with acetonitrile molecules via hydrogen bonds. The associates have the packing volumes close to those in the individual solute. The water and methanol molecules were assumed to be incorporated into the acetonitrile structure without substantial changes in the latter. However, this process results in some compression of the system with a simultaneous increase in its expansibility.