Asymptotic Behavior of the Density in a Parabolic SPDE

Consider the density of the solution X(t, x) of a stochastic heat equation with small noise at a fixed t[0, T], x[0, 1]. In this paper we study the asymptotics of this density as the noise vanishes. A kind of Taylor expansion in powers of the noise parameter is obtained. The coefficients and the residue of the expansion are explicitly calculated. In order to obtain this result some type of exponential estimates of tail probabilities of the difference between the approximating process and the limit one is proved. Also a suitable iterative local integration by parts formula is developed.     

Symbolic Computation of Newton Sum Rules for the Zeros of Polynomial Eigenfunctions of Linear Differential Operators

A symbolic algorithm based on the generalized Lucas polynomials of first kind is used in order to compute the Newton sum rules for the zeros of polynomial eigenfunctions of linear differential operators with polynomial coefficients.     

Interrelationship of the Torsional and Rotational States of Asymmetric Nonrigid Molecules

The incorrectness of solution of a torsional-rotational equation for asymmetric nonrigid molecules in the system of internal axes is shown; this incorrectness is caused by the broken interrelationship between the torsional quantum number and the parity of the rotational quantum number K.     

Formation of Metallophthalocyanine Trimers in Liquid Solutions

For solutions of the complexes of phthalocyanine (PhC) with magnesium (Mg-PhC) and of tetrachloromethyl-PhC with zinc (Zn-PhCCh) in mixtures of glycerin with dimethyl formamide (DMF) and for Zn-PhCCh in toluene and DMF it has been found that the electron absorption spectra substantially depend on the substance concentration. It is shown that these changes are attributable to the association of the molecules of pigments. The number of molecules in the associates and the equilibrium constants for this association are determined. The formation of the trimers of Mg-PhC in binary solutions and the dimers of Zn-PhCCh in toluene, DNF, and binary solutions has been established. It has been found that on association the Q absorption bands of the pigments shift to shorter wavelengths as compared to the bands of the monomers. For the Mg-PhC trimers, the shift is = 1410 cm^–1, while for the Zn-PhCCh dimers it is 990 cm^–1 in toluene, 930 cm^–1 in DMF, and 925 cm^–1 in the mixture of DMF with glycerin. It is concluded that the structure of the trimers and dimers is of the type of a stack in which molecules are arranged in parallel one above the other at a distance of 3.5 to 4.0 Å between the planes of the molecules. In association, the fluorescence intensity of the pigments decreases substantially (the decrease corresponds to the decrease in the concentration of the monomer molecules of phthalocyanines), with the fluorescence spectra remaining unchanged. Consequently, the associates of the pigments do not display pronounced fluorescence.     

Effect of Deuteration on Quasiline Electronic‐Vibrational Spectra of Phthalocyanine

Quasiline electronicvibrational spectra of fluorescence and absorption (excitation of fluorescence in selective recording) of the molecules of phthalocyanine deuterated around the periphery of benzene rings (H₂Phcd ₁₆) and the center of the macrocycle (D₂Phc) are obtained. The vibrational frequencies of the ground state are almost insensitive to this deuteration (except for vibrations with the participation of angular deformations). In excitation spectra, changes in deuteration are more pronounced due to the effects of nonadiabatic vibronic interaction of the vibrational sublevels of the S ₁ state and of the purely electronic level S ₂.     

Optical Reflection Spectrum of the Human Tympanic Membrane

The results of an investigation of the reflection spectrum of the human tympanic membrane in the visible and near IR ranges ( = 400–900 nm) are reported.     

Continuation method for -sublinear mappings

Let be a real Banach space partially ordered by a closed convex cone with nonempty interior . We study the continuation method for the monotone operator which satisfies

for all , , where . Thompson's metric is among the main tools we are using.     

Adaptive Smoothing Method,Deterministically Computable Generalized Jacobians,and the Newton Method

In this note, we show that a well-known integral method, which was used by Mayne and Polak to compute an -subgradient, can be exploited to compute deterministically an element of the plenary hull of the Clarke generalized Jacobian of a locally Lipschitz mapping regardless of its structure. In particular, we show that, when a locally Lipschitz mapping is piecewise smooth, we are able to compute deterministically an element of the Clarke generalized Jacobian by the adaptive smoothing method. Consequently, we show that the Newton method based on the plenary hull of the Clarke generalized Jacobian can be implemented in a deterministic way for solving Lipschitz nonsmooth equations.     

Dependence Properties of Multivariate Mixture Distributions and Their Applications

Consider a multivariate mixture model where the random variables X ₁, ..., X _n given (₁, ..., _n ), are conditionally independent. Conditions are obtained under which different kinds of positive dependence hold among X _i 's. The results obtained are applied to a variety of problems including the concomitants of order statistics and of record values; and to frailty models.     

Generalized Steiner Systems GS(2, 4, v, 2) with v a Prime Power ≡ 7 (mod 12)

Generalized Steiner systems GS(2, 4, v, g) were first introduced by Etzion and were used to construct optimal constant weight codes over an alphabet of size g + 1 with minimum Hamming distance 5, in which each codeword has length v and weight 4. Etzion conjectured that the necessary conditions v 1 (mod 3) and v ; 7 are also sufficient for the existence of a GS(2,4,v,2). Except for the example of a GS(2,4,10,2) and some recursive constructions given by Etzion, nothing else is known about this conjecture. In this paper, Weil's theorem on character sum estimates is used to show that the conjecture is true for any prime power v 7 (mod 12) except v = 7, for which there does not exist a GS(2,4,7,2).