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).     

HPLC fractionation and structural dynamics of humic acids

Different soil extracted humic acids as well as a commercial humic acid sodium salt were fractionated by HPLC. An almost complete recovery could be achieved for the dissolved material. All humic samples show a typical chromatogram of at least five fractions. The separation of humic substances is influenced by an altered hydrophobicity due to changes in the tertiary structure. Three fractions were further investigated by Diffuse Reflectance Infrared Fourier Transform (DRIFT) spectroscopy and by HPLC/HPLC separation. DRIFT spectroscopy provided data on the primary structure, whereas HPLC/HPLC results gave insight into the secondary and tertiary structure and their changes. Received: 18 June 1996 / Revised: 9 October 1996 / Accepted: 12 October 1996     

Versuche zur Herstellung eines «Bis(triasterans)» und Synthese des heterocyclischen (Tricyclo[4.4.1.01,6]undeca-3,8-dien-11,11-dimethyl)sulfits

Tentative Synthesis of ‘Bis(triasterane)’ and Synthesis of the Heterocyclic (Tricyclo[4.4.1.0^1,6]undeca-3,8-diene-11,11-dimethyl)sulfite The synthesis of the bis(triasterane) ( 1 ) has been tried; the reaction of ‘isotetraline’ (1,4,5,8-Tetrahydronaphthalene; 2 ) with diazomalonate yielded the tricyclic systems 5 and 6 , and not 4 . Hydrolysis of 5 gave the monocarboxylic acid 7 , and not the dicarboxylic acid 9 . The latter could be obtained from the dibromoderivative 8 , but 9 couldn't be converted to the acyl chloride 10 . The reduction of 9 with LiAlH₄ yielded the crystalline diol 11 , which was cyclized with SOCl₂ to the heterocycle 12 . The spectral data of the new compounds 5, 6, 7, 11 and 12 are reported and discussed.     

Photoelectron spectroscopy of isolated multiply negatively charged oligonucleotides

Ultraviolet photoelectron spectroscopy in an ion beam was used to investigate the electronic properties of isolated DNA oligonucleotides [dA(5)-4H](4-) and [dT(5)-4H](4-), carrying four excess negative charges. We find the fourth adiabatic electron affinity to be slightly negative for [dA(5)-4H](4-), while it is positive for [dT(5)-4H](4-). This implies a significant influence of the base composition on energetics, which is in turn relevant for analytic applications and also for charge transport properties.     

4-Aryldihydropyridines,a New Class of Highly Active Calcium Antagonists

The aryldihydropyridines first prepared by Hantzsch almost 100 years ago have recently been found to be highly effective calcium antagonists with suitable pharmacological profiles. An illustrative example is dimethyl-l,4-dihydro-2,6-dimethyl-4-(o-nitrophenyl)pyridine-3,5-dicarboxylate (Nifedipine) which is already employed therapeutically. This substance lowers the frequency of attack of angina pectoris and reduces blood pressure. The discovery of the therapeutic activity of this class of substances initiated renewed investigation of the Hantzsch condensation and the synthesis of numerous 4-aryldihydropyridines and related compounds. Qualitative and quantitative structure/activity relationships of these substances can be deduced from their biological data.     

SYNTHESIS OF POSITIVELY CHARGED PHTHALOCYANINES and THEIR ACTIVITY IN THE PHOTODYNAMIC THERAPY OF CANCER CELLS

Positively charged zinc containing or metal free phthalocyanines 6a-c and 7a-c were prepared via a three step procedure starting from 4-nitrophthalonitrile. The phthalocyanines contain alkyl chains of different length in order to influence the hydrophilic vs lipophilic character of the compounds. The partition between a hydrophilic (water) and lipophilic (octanol-1) phase was determined, and the photoredox activities were investigated. Initial results on the photodynamic activity of these compounds were compared with those of Dougherty's Photofrin II on different malignant and non-malignant cell lines (XP 29MAmal, CX1, HeLa, S180 and NO17). Positively charged phthalocyanines in vitro showed a higher photodynamic activity than Photofrin II.     

Theoretical and experimental study of the adsorption on radioactive gases on continuous flow columns

A bimolecular reaction model was used to describe the adsorption process in continuous flow columns filled with solid adsorbents. The analytical solution of the model for low gas concentrations and a cascade-type numerical method for higher gas concentrations were developed. An air flow apparatus using activated carbon as adsorbent and methyl-iodide labelled with¹²⁵I as adsorbate was constructed for measuring breakthrough- and accumulation curves.     

Über das Nicotellin und die Synthese eines neuen Terpyridyls

Zusammenfassung Das Nicotellin enthält wahrscheinlich nicht, wie die meisten anderen Tabakalkaloide, 2 N-Atome in der Molekel, sondern hat die Formel C₁₅H₁₁N₃. Es werden gewichtige Gründe dafür beigebracht, es als 3,2;6,3-Terpyridyl (I) aufzufassen. Diese Hypothese wird durch Synthese des damit isomeren 3,2;4,3-Terpyridyls (V) und Vergleich der Infrarotspektren der beiden Terpyridyle mit denen von 4 bekannten Bipyridylen gestützt. Betrachtungen über eine mögliche Synthese des Nicotellins in der Pflanze führen zu Versuchen, diese Synthese in vitro nachzuahmen; diese Versuche geben aber eine andere Verbindung, wahrscheinlich das 1,5-Di--pyridyl-pentadien-(1,4)-on-(3).Herrn Prof. Dr.L. Ebert zum 60. Geburtstage in aufrichtiger Verehrung gewidment.     

Vierring-synthesen durch [3 + 1]-cycloaddition . Synthese von 3-imino-azetidinen aus azomethin-yliden und isonitrilen

Azomethine ylide 3 , generated by thermolysis of 3,4-diphenyl-Δ¹-1,2,3-triazoline-5,5-dimethyldicarboxylate 1 , can be trapped by isonitriles to give 3-imino-1,4-diphenylazetidine-2,2-dimethyldicarboxylates 5 . The IR, ¹H-NMR, ¹³C-NMR and mass spectral data of the new compounds are discussed.     

Reversible storage of molecular hydrogen by sorption into multilayered TiO2 nanotubes

The sorption of hydrogen between the layers of the multilayered wall of nanotubular TiO2 was studied in the temperature range of -195 to 200 degrees C and at pressures of 0 to 6 bar. Hydrogen can intercalate between layers in the walls of TiO2 nanotubes forming host-guest compounds TiO2 x xH2, where x < or = 1.5 and decreases at higher temperatures. The rate of hydrogen incorporation increases with temperature and the characteristic time for hydrogen sorption in TiO2 nanotubes is several hours at 100 degrees C. The rate of intercalate formation is limited by the diffusion of molecular hydrogen inside the multilayered walls of the TiO2 nanotube. 1H NMR-MAS and XRD data confirm the incorporation of hydrogen between the layers in the walls of TiO2 nanotubes. The nature and possible applications of the observed intercalates are considered.