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.     

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.     

Apparent Molar Volume, Heat Capacity, and Conductance of Lithium Bis(trifluoromethylsulfone)imide in Glymes and Other Aprotic Solvents

Lithium bis(trifluoromethylsulfone)imide (LiTFSI) is a promising electrolyte for high-energy lithium batteries due to its high solubility in most solvents and electrochemical stability. To characterize this electrolyte in solution, its conductance and apparent molar volume and heat capacity were measured over a wide range of concentration in glymes, tetraethylsulfamide (TESA), acetonitrile, -butyrolactone, and propylene carbonate at 25°C and were compared with those of LiClO₄ in the same solvents. The glymes or n(ethylene glycol) dimethyl ethers (nEGDME), which have the chemical structure CH₃–O–(CH₂–CH₂–O) _n –CH₃ for n = 1 to 4, are particularly interesting since they are electrochemically stable, have a good redox window, and are analogs of the polyethylene oxides used in polymer-electrolyte batteries. TESA is a good plasticizer for polymer-electrolyte batteries. Whenever required, the following properties of the pure solvents were measured: compressibilities, expansibilities, temperature and pressure dependences of the dielectric constant, acceptor number, and donor number. These data were used in particular to calculate the limiting Debye-Hückel parameters for volumes and heat capacities. The infinite dilution properties of LiTFSI are quite similar to those of other lithium salts. At low concentrations, LiTFSI is strongly associated in the glymes and moderately associated in TESA. At intermediate concentrations, the thermodynamic data suggests that a stable solvate of LiTFSI in EGDME exists in the solution state. At high concentrations, the thermodynamic properties of the two lithium salts approach those of the molten salts. These salts have a reasonably high specific conductivity in most of the solvents. This suggests that the conductance of ions at high concentration in solvents of low dielectric constant is due to a charge transfer process rather than to the migration of free ions.     

Cathode of a fuel cell with a solid polymer electrolyte: Calculating overall currents in the presence of a gas-diffusion layer

Mathematical apparatus, which makes it possible to perform calculations of the current-voltage characteristics of cathodes of fuel cells with a solid polymer electrolyte in conditions where there are present extraneous diffusion restrictions is proposed. In so doing, the partial pressure of oxygen and the absolute pressure of gas in the gas chamber may assume any values. First of all presented are the results of calculations of the current-voltage characteristics intrinsic to active layers of the air and oxygen cathodes, which are performed under the assumption that the extraneous diffusion restrictions are absent altogether. Thereafter, in the same conditions (at the same parameters that characterize the active layer of a cathode), obtained are results of a calculation of the current-voltage characteristics inherent in the air and oxygen cathodes in the presence of extraneous diffusion restrictions. Afterward there is performed an analysis of the way a gas-diffusion layer restricts the process of generation of current in a cathode and of what measures should be taken in order for the extraneous diffusion restrictions to become less significant.