Some model theory for monotone quantifiers

A generalization of the existential and universal quantifier, the monotone quantifiers, are studied. It is shown that the model theory for monotone quantifiers behaves very much like classical model theory. Completeness theorems, definability theorems and preservation theorems are given. Ultraproducts, reduced products and Back and Forth arguments are studied.This work is part of the research program of the G.N.S.A.G. group of the Italian C.N.R.     

A central limit theorem for age- and density-dependent population processes

Consider a population consisting of one type of individual living in a fixed region with area A. In [8], we constructed a stochastic population model in which the death rate is affected by the age of the individual and the birth rate is affected by the population density P_A(t), i.e., the population size divided by the area A of the given region. In [8], we proposed a continuous deterministic model which in general is a nonlinear Volterra type integral equation and proved that under appropriate conditions the sequence P_A(t) would converge to the solution P(t) of our integral equation in the sense that

$\text{lim}\text{→∞}\text{P}\text{sup}\text{0?s?t}\text{|P}{}_{\mathrm{A}}\text{(s) ? P(s)|>ε}\text{=0}\text{for every}\text{ε > 0}$

.In this paper, we obtain a “central limit theorem” for the random element √A(P_A(t)?P(t)). We prove that under appropriate conditions √A(P_A(t)?P(t)) will converge to a Gaussian process. (See Theorem 3.4 for the explicit formula of this Gaussian process.)     

Crown compounds for anions. The nature of chemical bonds in the complexes of halide anions with cyclic trimeric perfluoro-o-phenylenemercury and some of its analogs

Geometries and electronic structures of the complexes of halide anions with cyclic trimerico-phenylenemercury, (o-C₆H₄Hg)₃, perfluoro-o-phenylenemercury, (o-C₆F₄Hg)₃, vinylenemercury, (C₂H₂Hg)₃, and perfluorovinylenemercury, (C₂F₂Hg)₃, were modelled by the MNDO method. Calculations were performed for [L-X]^– semisandwich complexes, [X-L-X]^2– bipyramidal complexes, and [L-X-L]^– sandwich complexes (where X=Hal,L is a mercury-containing macrocycle). Based on the results of calculations, we concluded that it was advantageous to describe the chemical bonding between halide anions and mercury-containing macrocycles in terms of generalized chemical bonds, which were successfully used for -complexes of transition metals. In the [L-X]^– semisandwich complexes, the halide anion and the metallacycle are involved in the formation of three generalized chemical bonds: one headlight-shaped -bond and two two-lobe -bonds. In the [X-L-X]^2– bipyramidal complexes, each halide anion forms three generalized chemical bonds with the macrocycle. It is possible because the macrocycleL has unoccupied molecular orbitals suitable for the formation of such bonds; these MOs consist mainly of the orbitals of mercury atoms directed toward both the upper and lower halogen atoms. In the [L-X-L]^– sandwich complexes, the halide anion cannot be bonded to each ringvia three bonds, and, hence, an unsymmetrical structure is formed, in which the rings are located at different distances from the central atom: the [L-X]^– semisandwich complex solvated by macrocycleL.Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 6, pp. 1035–1042, June, 1995.The authors are grateful to V. I. Faustov for valuable remarks.This work was supported by the Russian Foundation for Basic Research (Project No. 93-03-18342).     

Inorganic-Organic Hybrid Polymers from the Polymerisation of Methacrylate-Substituted Oxotantalum Clusters with Methylmethacrylate: A Thermomechanical and Spectroscopic Study

New inorganic-organic hybrid materials were prepared by free-radical polymerization of methyl methacrylate (MMA) with methacrylate-substituted oxotantalum cluster [Ta₄O₄(OEt)₈(OMc)₄] and their properties evaluated. The cluster was prepared by the reaction of the parent alkoxide with methacrylic acid. Samples of the hybrid materials were produced with Ta-cluster to methyl methacrylate in the ratios of 1:50 and 1:100 and were characterized by thermal and spectroscopic techniques. The glass transition temperatures of the hybrid materials are shifted to higher temperatures than pure PMMA as a result of cross-linking of the polymer by the oxotantalum clusters. The increase in T_g is also observed from the dynamic mechanical analysis (DMA). Evidence of crosslinking between the Ta-cluster and PMMA is obtained from infrared spectroscopic study. Surface studies performed by X-ray photoelectron spectroscopy (XPS) provide information about the atomic concentrations of the surface and indicate tantalum bonded to oxygen.     

Cross-linking of polyolefins: a study by thermoporosimetry with benzene derivatives as swelling solvents.

The solvents o-, m-, p-xylene, p-dichlorobenzene, 1,2,4-trichlorobenzene, and naphthalene were calibrated as condensates used in the thermoporosimetry technique. Exponential relationships were found connecting the pore radii R(p) (in nm) and the freezing-point depression of the swelling solvent deltaT (in degrees C) on the one hand and the apparent energy of crystallization W(a) (in J cm(-3)) and deltaT on the other hand: R(p) = t exp[-1/(c deltaT)]; W(a) = W0 exp(deltaT/f). Pore- or mesh-size distributions can be derived from differential scanning calorimetry results by using the following equation: dV(p)/dR(p) = k{[cY(T)deltaT2]/[W(a)R(p)]}. All the numerical parameters were determined. Polyethylene and polypropylene samples, cross-linked with high-energy electrons or gamma-rays, were submitted to thermoporosimetry study. Relative mesh-size distributions, which depend on the polymer/solvent pair, were calculated for these polyolefins with o-, m-, and p-xylene as solvent and were found to be in the same sequence as those of their degrees of swelling and the irradiation doses received.     

Net information measures for modified Yukawa and Hulthén potentials

The dimensional analyses of the position and momentum variances‐based quantum mechanical Heisenberg uncertainty measure, as well as the entropic information measures given by the Shannon information entropy sum and the product of Fisher information measures are carried out for two widely used nonrelativistic isotropic exponential‐cosine screened Coulomb potentials generated by multiplying the superpositions of (i) Yukawa‐like, ?Z(e^?μr/r), and (ii) Hulthén‐like, ?Zμ(1/(e^μr ? 1)), potentials by cos(bμr) followed by addition of the term a/r², where a and b ≥ 0, μ are the screening parameters and Z, in case of atoms, denotes the nuclear charge. Under the spherical symmetry, all the information measures considered are shown to be independent of the scaling of the set [μ, Z] at a fixed value of μ/Z, a, and b and the other parameters defining the superpositions of the potentials. Numerical results are presented, which support the validity of the scaling properties. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Lipschitzian inverse functions,directional derivatives,and applications inC 1,1 optimization

The paper shows that Thibault's limit sets allow an iff-characterization of local Lipschitzian invertibility in finite dimension. We consider these sets as directional derivatives and extend the calculus in a way that can be used to clarify whether critical points are strongly stable inC ^1,1 optimization problems.Many fruitful discussions with colleagues D. Klatte and K. Tammer as well as with H. Th. Jongen and F. Nozicka have influenced the present investigations in a very constructive manner. For the original papers concerning the sets f(x; u), the author is indebted to Prof. L. Thibault.     

Estimates of the tail of the stationary density function of certain nonlinear autoregressive processes

We consider the Markov chainX _n+1=T(X _n )+ _n , where { _n ;n1} is a ^d -valued random sequence of independent identically distributed random variables, and the functionT: ^{d d} is measurable and satisfies a suitable growth condition. Under certain conditions involvingT and the probability distribution of _n , we show that this Markov chain is ergodic. Moreover, we obtain sharp upper bounds for the tail of the corresponding stationary probability density function. In our proofs, we make use of the Leray-Schauder fixed-point theorem.     

Rates of convergence of α-stable random motions

The rate-of-convergence problem in the central limit theorem is considered for -stable distributions on the noncommutative group of motions in ^d. The method used here is based on the theory of probability metrics.     

An open optimization problem in statistical quality control

We ask the experts in global optimization if there is an efficient solution to an optimization problem in acceptance sampling: Here, one often has incomplete prior information about the quality of incoming lots. Given a cost model, a decision rule for the inspection of a lot may then be designed that minimizes the maximum loss compatible with the available information. The resulting minimax problem is sometimes hard to solve, as the loss functions may have several local maxima which vary in an unpredictable way with the parameters of the decision rule.