Asymptotic expansions in the integral and local limit theorems in banach spaces with applications to ω-statistics

LetB be a real separable Banach space and letX,X ₁,X ₂,...∈B denote a sequence of independent identically distributed random variables taking values inB. DenoteS _n =n ^?1/2(X ₁+...X _n ). Let π:B→R be a polynomial. We consider (truncated) Edgeworth expansions and other asymptotic expansions for the distribution function of the r.v. π(S _n ) with uniform and nonuniform bounds for the remainder terms. Expansions for the density of π(S _n ) and its higher order derivatives are derived as well. As an application of the general results we get expansions in the integral and local limit theorems for ω-statistics $$\omega _n^p (q)\mathop { = n^{{p \mathord{\left/ {\vphantom {p 2}} \right. \kern-\nulldelimiterspace} 2}} }\limits^\Delta \smallint _{(0,1)} \{ F_n (x) - x\} ^p q(x)dx$$ and investigate smoothness properties of their distribution functions. Herep≥2 is an even number,q: [0, 1]→[0, ∞] is a measurable weight function, andF _n denotes the empirical distribution function. Roughly speaking, we show that in order to get an asymptotic expansion with remainder termO(n ^?α), α<p/2, for the distribution function of the ω-statistic, it is sufficient thatq is nontrivial, i.e., mes{t∈(0, 1):q(t)≠0}>0. Expansions of arbitrary length are available provided the weight functionq is absolutely continuous and positive on an nonempty subinterval of (0, 1). Similar results hold for the density of the distribution function and its derivatives providedq satisfies certain very mild smoothness condition and is bounded away from zero. The last condition is essential since the distribution function of the ω-statistic has no density whenq is vanishing on an nonempty subinterval of (0, 1).     

Spectral properties of random Schrödinger operators with unbounded potentials

We investigate spectral properties of random Schrödinger operators H = - + _n()(1 + |n|) acting onl ²(Z ^d), where _n are independent random variables uniformly distributed on [0, 1].Research partially supported by a Sloan Doctoral Dissertation Fellowship and NSERC under grant OGP-0007901Research partially supported by NSF grant DMS-9101716     

Sturmian functions for nonlocal interactions

The problem of calculation of Sturmian functions (positive energy Weinberg states) for nonlocal (exchange) interactions is considered. It is shown that the method of continued fractions proposed by Horáek and Sasakawa makes the calculation of Sturmian eigenfunctions and eigenvalues feasible even for complicated nonlocal interactions. As an example Sturmian functions and Sturmian eigenvalues for the low energy electron-hydrogen scattering in the static exchange approximation are calculated. In addition a very general proof of convergence of the method of continued fractions is presented.Dedicated to the memory of Professor Jozef Kvasnica.     

A model of solidification under microgravity conditions

The influence of the microgravity environment on solidification processes is discussed. A simple model of the solidification of a binary-alloy is presented with the chemical diffusion influenced by the gravitational field. Using the results of Mullins and Sekerka, we employ the linear theory of hydrodynamic stability to investigate the interfacial instability driving the pattern-forming processes in solidification. As a result, we estimate the characteristic size of the elements of the emerging pattern. We show that, in spite of good agreement of our result with the size of cellulae observed in experiments, the model cannot explain the changes in the patterns occurring in space environment. In conclusion we shortly discuss the possibility of adding realism to our simple model by including the effect of convection.     

Analytische Aspekte der Oxydation organischer Stickstoffverbindungen mit Chromsäure. XXVIII

Zusammenfassung Die schon früher beschriebene gasvolumetrische Methode zur N-N-Gruppenbestimmung durch Oxydationsaufschluß mit Chromsäure wurde zur Bestimmung des heterocyklischen Stickstoffes in Indazolinonderivaten mit Erfolg benützt und dadurch weiter die allgemeine Anwendbarkeit der Methode bestätigt. Enthält die Substanz gleichzeitig Amino- oder Nitrogruppen, können diese aus derselben Einwaage als Ammoniak bzw. Salpetersäure bestimmt werden.

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Analytical aspects of the oxidation of organic nitrogen compounds with chromic acid

Summary The gas volumetric method for determining N-N groups by oxidative fusion with chromic acid (which has already been described previously) was successfully employed to determine the heterocyclic nitrogen in indazolinone derivatives. The general applicability of the method was thereby confirmed. If the substance contains amino or nitro groups at the same time, these can be determined in the same sample as ammonia or nitric acid.

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XXVII. Mitteilung, siehe¹.     

Development of a radiochemical neutron activation analysis procedure for determination of rhenium in biological and environmental samples at ultratrace level

Summary Radiochemical neutron activation procedures using liquid-liquid extraction with tetraphenylarsonium chloride in chloroform from 1M HCl and solid extraction with ALIQUAT 336 incorporated in a polyacrylonitrile binding matrix from 0.1M HCl were developed for accurate determination of rhenium in biological and environmental samples at the sub-ng ^. g^-1 level. Concentrations of Re in the range of 0.1 to 2.4 ng ^. g^-1 were determined in several botanical reference materials (RM), while in a RM of road dust a value of ~10 ng ^. g^-1 was found. Significantly elevated values of Re, up to 90 ng ^. g^-1 were found in seaweed (brown algae). Results for Re in the brown algae Fucus vesiculosus in which elevated ⁹⁹Tc values had previously been determined suggested possible competition between Re and Tc in the accumulation process.     

Fourth-order diagrammatic MB-RSPT calculations of the interaction energy between two helium atoms

The efficiency of the MB-RSPT in the calculations of the correlation contributions to the interaction energies was investigated, using He₂ as a model Van der Waals system. The attention has been focused on the convergency of the perturbation expansion in the calculations of the interaction energy and on the analysis of the fourth-order terms of MB-RSPT. The rôle of the renormalization term in the correct long-range behaviour of the interaction potential has been emphasized.     

Reaction of N-phenylsulfonylbenzoxazolone with some amines

The reaction of N-phenylsulfonylbenzoxazolone with amines was investigated in order to study the mechanism of the reaction between aryloxazolones and amines. The sulfonyl group accelerates the reaction considerably. The structure of 2-(phenylsulfamido)phenyl esters of the corresponding N-substituted carbamic acids was assigned to the products on the basis of the IR spectra and alternative synthesis. The reaction with secondary aliphatic amines (dimethyl- and diethylamine)proceeds considerably more slowly and results in the formation of 2-(phenylsulfamido)phenol.Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 10, pp. 1333–1336, October, 1971.     

Determination of phosphorous in organic compounds and metal complexes by inductively-coupled plasma atomic emission spectrometry

Phosphorus is determined by inductively-coupled plasma atomic emission spectrometry at 213.618 nm. Different types of organic and inorganic phosphorus compounds and metal complexes were examined after direct dissolution in water, aqueous ethanol or acid, or after decomposition by oxygen flask combustion or Kjeldahl digestion; results were within the usual limit of precision and accuracy for such determinations. The effect of small (<20%) ethanol concentrations in the aqueous solution on the signals obtained from a low-power (1200 W) argon plasma is examined and discussed.     

About the ratio of the size of a maximum antichain to the size of a maximum level in finite partially ordered sets

LetP be a finite partially ordered set. The lengthl(x) of an elementx ofP is defined by the maximal number of elements, which lie in a chain withx at the top, reduced by one. Letw(P) (d(P)) be the maximal number of elements ofP which have the same length (which form an antichain). Further let . The numbers and as well as all partially ordered sets for which these maxima are attained are determined.