Precise asymptotics in some strong limit theorems for multidimensionally indexed random variables

Consider Z₊^d (d2)—the positive d-dimensional lattice points with partial ordering , let {X_k,kZ₊^d} be i.i.d. random variables with mean 0, and set S_n=∑_knX_k, nZ₊^d. We establish precise asymptotics for ∑_n|n|^r/p−2P(|S_n||n|^1/p), and for

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, (0δ1) as 0, and for

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as .     

Refinement of Convergence Rates for Tail Probabilities

Let X₁, X₂,... be, i.i.d. random variables, and put . We find necessary and sufficient moment conditions for , where δ≥ 0 and q>0, and with a_n>0 and b_n is either or The series f(x) we deal with are classical series studied by Hsu and Robbins, Erdős, Spitzer, Baum and Katz, Davis, Lai, Gut, etc     

Precise Asymptotics in Spitzer's Law of Large Numbers

Let X, X ₁, X ₂,... be a sequence of i.i.d. random variables such that EX=0, assume the distribution of X is attracted to a stable distribution with exponent <1, and set S _n=X ₁+ ··· +X _n. We prove that

Dynamics of hyperbranched polymers and dendrimers: theoretical models

Chemical reactions depend in many ways on the dynamics of the underlying reactants, and an important aspect is the distance covered by the reactants before the reaction act occurs. Hence, even diffusion-limited reactions between point particles in confined geometries and in low-dimensional systems display decay forms which are very different from those obtained from simple chemical kinetics. Clearly, even more complex decay forms hold for macromolecules, given their internal degrees of freedom. Here we discuss how the dynamics of macromolecules in solution relates to their topological structure and focus on the motion of macromolecular segments (monomers) under the influence of external fields. After a general survey of the method of generalized Gaussian structures (GGS) we recall the wealth of forms which are observed, depending on the topology and on the microscopic dynamics involved. Paradigmatic are the findings for the class of hyperbranched macromolecules; to these belong the dendrimers. While the dendrimers do not show a typical scaling behavior, as found, say, for linear chains, the situation is different for particular classes of regular hyperbranched polymers which are fractal. We end by discussing their pattern of motion in the GGS-Rouse-Zimm picture.     

Synthesis of dihydroxy telechelic oligomers of β-butyrolactone catalyzed by titanium(IV)-alkoxides and their use as macrodiols in polyurethane chemistry

We report on a solvent-free approach for the synthesis of low molecular weight, α,ω-dihydroxy telechelic poly(β-butyrolactone). In the presence of Ti(IV) alkoxides, mixtures of β-butyrolactone and diols, like di- or triethylene glycol, were reacted in ratios between 4:1 and 10:1. The oligomerization proceeds at elevated temperatures (80–100°C). Different alkoxide substituents (R = Me, iPr, tBu) of the Ti(IV)(OR)₄ catalyst were investigated. The resulting oligomers were characterized by nuclear magnetic resonance (NMR), infra-red (IR), gel-permeation chromatography (GPC), titration, and matrix-assisted laser desorption-time-of-flight mass spectrometry (MALDI-ToF-MS) analysis. Aside from low molecular weight products, special effort was devoted to achieve high O-acyl cleavage selectivity and to circumvent the formation of unsaturated end-groups in order to form exclusively dihydroxy-telechelic oligomers. Optimized results in terms of selectivity and reaction rates were achieved at 100°C using catalyst loadings of 0.2 mol% with respect to the monomer. The molecular weights determined by GPC were in good accordance with the ratio of monomer to diol used, confirming successful oligomer formation. Polyurethanes prepared from crude macrodiols without any additional catalyst feature molecular weights up to 50,000 g/mol. The reported work serves as concept to utilize β-lactones for tailored polyol synthesis; the resulting products are suitable for polyurethane chemistry.     

Ueber eine Anwendung der Theorie der fastperiodischen Funktionen auf das Levi-Civitasche Problem der mittleren Bewegung

Ohne Zusammenfassung     

An extension of the Beckner’s type Poincaré inequality to convolution measures on abstract Wiener spaces

We generalize the Beckner’s type Poincaré inequality (Beckner, W. Proc. Amer. Math. Soc. (1989) 105:397–400) to a large class of probability measures on an abstract Wiener space of the form μ?ν, where μ is the reference Gaussian measure and ν is a probability measure satisfying a certain integrability condition. As the Beckner inequality interpolates between the Poincaré and logarithmic Sobolev inequalities, we utilize a family of products for functions which interpolates between the usual point-wise multiplication and the Wick product. Our approach is based on the positivity of a quadratic form involving Wick powers and integration with respect to those convolution measures. In addition, we prove that in the finite-dimensional case the class of densities of convolutions measures satisfies a point-wise covariance inequality.     

Synthesis and characterization of zinc oxide nanoparticles

The research is aimed at synthesis and characterization of nanoscaled zinc oxide particles and their application on linen fibrous supports, for thermal properties. To impart thermal activity to the fibrous nanocomposites, nanoparticles as well as fibrous nanocomposites were produced in different hydrothermal conditions of temperature (90 °C). To characterize the nanoparticles composition, their shape, size, and crystallinity, investigations technique, such as Fourier transformed infrared spectroscopy, scanning electron microscopy, and X-ray powder diffractometry were used. Differential scanning calorimetry analysis profiles were also revealed. The thermal treatment of linen fabrics with nanosized ZnO does not modify significantly their thermal stability.