An almost sure invariance principle for stationary ergodic sequences of Banach space valued random variables

Summary In this paper we establish an almost sure invariance principle with an error termo((t log logt)^1/2) (ast) for partial sums of stationary ergodic martingale difference sequences taking values in a real separable Banach space. As partial sums of weakly dependent random variables can often be well approximated by martingales, this result also leads to almost sure invariance principles for a wide class of stationary ergodic sequences such as ø-mixing and -mixing sequences and functionals of such sequences. Compared with previous related work for vector valued random variables (starting with an article by Kuelbs and Philipp [27]), the present approach leads to a unification of the theory (at least for stationary sequences), moment conditions required by earlier authors are relaxed (only second order weak moments are needed), and our proofs are easier in that we do not employ estimates of the rate of convergence in the central limit theorem but merely the central limit theorem itself.     

Otpically illuminated dielectric interfaces as high-speed far-infrared modulators

A new device for the gigahertz modulation of far-infrared radiation is analytically and numerically analyzed. It consists of a thin layer of a high-mobility, direct-bandgap semiconductor, such as GaAs, in which a high-density electron-hole plasma is rapidly created and destroyed, thereby rapidly changing the free-carrier reflectivity of the active layer. Illumination by a high-power, near-infrared laser diode array generates the plasma through intrinsic photoconduction. It is shown that this device acis primarily as an amplitude modulator, and that its efficiency increases sharply with increasing far-IR frequency, in contrast to a Schottky diode, which acts primarily as a phase modulator, and whose efficiency falls off sharply with far-IR frequency. The breakeven frequency lies at about 1.5 THz, depending slightly on the assumed device parameters. The relative advantage of the new device increases rapidly with increasing far-infrared frequency. At an operating frequency of 2.5 THz (119 m), for example, a 1 GHz modulation bandwidth may be achieved with a single-sideband conversion loss of only-21 db, versus a Schottky's loss of-39 db, assuming a laser diode power of 1 W, which is readily available from recently developed laser diode arrays.     

Studium und Anwendung eines neuen Komplexes von Kupfer mit Salicylsäure

Microchimica Acta - Eine empfindliche spektrophotometrische Methode zur Bestimmung von Kupfer mit Salicyls~iure wurde beschrieben (Molarextinktion ε = 2,4 · 9 105). Die nachweisbaren...     

3,7-Dioxaazelaamides as Ionophores for Lithium Ion Selective Liquid Membrane Electrodes

Lipophilic neutral carriers were synthesized which show Li⁺/Na⁺ selectivities of up to ca. 80 in highly lipophilic liquid membranes. The sensor membranes exhibit improved response times and increased lifetimes as compared to systems described earlier. They allow reliable measurements of Li⁺ in blood serum within the clinical concentration range. A 1:1 Li⁺/ionophore complex of one representative (N,N,N′,N′-tetracyclohexyl-5,5-dimethyl-3,7-dioxaazelaamide) has been prepared, and its structure was elucidated by X-ray analysis.     

The average a posteriori error of numerical methods

Summary The definition of the average error of numerical methods (by example of a quadrature formula to approximateS(f)= f d on a function classF) is difficult, because on many important setsF there is no natural probability measure in the sense of an equidistribution. We define the average a posteriori error of an approximation by an averaging process over the set of possible information, which is used by (in the example of a quadrature formula,N(F)={(f(a ₁), ...,f/fF} is the set of posible information). This approach has the practical advantage that the averaging process is related only to finite dimensional sets and uses only the usual Lebesgue measure. As an application of the theory I consider the numerical integration of functions of the classF={f:[0,1]/f(x)–f(y)||x–y|}. For arbitrary (fixed) knotsa _i we determine the optimal coefficientsc _i for the approximation and compute the resulting average error. The latter is minimal for the knots . (It is well known that the maximal error is minimal for the knotsa _i .) Then the adaptive methods for the same problem and methods for seeking the maximum of a Lipschitz function are considered. While adaptive methods are not better when considering the maximal error (this is valid for our examples as well as for many others) this is in general not the case with the average error.     

Solution- and bound-state conformational study of N,N',N"-triacetyl chitotriose and other analogous potential inhibitors of hevamine: application of trNOESY and STD NMR spectroscopy

The solution-state conformations of N,N',N"-triacetyl chitotriose (1) and other potential chitinase inhibitors 2-4 were studied using a combination of NMR spectroscopy (NOESY) and molecular mechanics calculations. Determination solely of the global energy minimum conformation was found to be insufficient for an agreement with the NMR results. An appropriate consistency between the NMR experimental data and theoretical calculations was only reached by assessing the structures as population-weighted average conformers based on Boltzmann distributions derived from the calculated relative energies. Analogies, but also particular differences, between the synthetic compounds 2-4 and the naturally-occurring N,N',N"-triacetyl chitotriose were found. Furthermore, the conformation of compounds 1 and 2 when bound to hevamine was also studied using transferred NOESY experiments and the binding process was found to impart a level of conformational restriction on the ligands. The preferred conformation as determined for 1 in the bound state to hevamine belonged to one of the conformational families found for the compound when free in solution, although full characterisation of the bound-state conformations was impeded due to severe signal overlap. Saturation transfer difference NMR experiments were also employed to analyse the binding epitopes of the bound compounds. We thus determined that it is mainly the acetyl amido groups of the trisaccharide and the heterocyclic moiety which are in close contact with hevamine.     

Reaktionen mit phosphororganischen Verbindungen, 48. Mitt.

Reaction of the vicinal diols of steroids1, 5, 7, 10, 13, and16 with TPP/DEAD yields both regio-and stereospecifically the oxosteroids2, 6, 8, 11, 14, and15 by displacement of an axial hydrogen and extrusion ofTPPO besides the cholest-4-en-6-ols9 and12 and the cyclic carbonate3. 16, 17-androstandiol16 gives only the cyclic carbonate17. The different structures of the carbohydrates withcis-diol arrangement19 and21 lead exclusively to cyclic carbonates20 and22 in moderate yields. Treatment of1 with TPP/DEAD/HN₃ affords 3-azido-2-hydroxycholestane4 in addition to the above mentioned2.

     

Diastereoselective intermolecular rhodium-catalyzed [4 + 2 + 2] carbocyclization reactions: computational and experimental evidence for the intermediacy of an alternative metallacycle intermediate

Intermolecular rhodium-catalyzed [m + n + o] reactions of 1,6-enynes and various pi-components (carbon monoxide, alkynes, 1,3-butadienes, etc.) provide an expeditious approach for the construction of polycyclic fragments that represent important synthons for target-directed synthesis. We present computational and experimental evidence for the existence of a previously undescribed reaction pathway for the rhodium-catalyzed [4 + 2 + 2] reaction involving a 1,6-enyne. This model clearly demonstrates the origin of the excellent diastereoselectivity in this type of reaction and the remarkable tolerance of both (E)- and (Z)-isomers within the 1,6-enyne, which is generally prone to competitive ene-cycloisomerization.     

Zur Kenntnis der Ortho-Strontiumphosphate

Zusammenfassung Die für den Einbau in die Calciumphosphate der Knochensubstanz in Frage kommenden Strontiumphosphate wurden auf ihre Existenzbedingungen untersucht. Oktacalciumphosphat tauscht in verd. Lösungen einige Prozente des Calciums gegen Strontium aus. Obwohl der Struktur nach apatitartige Strontiumphosphate zwischen Sr/P=1.33 und 1.67 erhältlich sind, läßt sich ein röntgenoptisch reines Strontiumoktaphosphat nicht isolieren, jedoch ein solches mit kleinem Carbonatgehalt und — leicht — die den Calciumsalzen analogen Strontiumphosphatcarbonate verschiedener Zusammensetzung. Im Hydrothermalbereich (375^o) sind je nach relativer Zusammensetzung des Strontiumphosphat-Systems SrHPO₄, Sr₂P₂O₇, Sr₂(PO₄)₂ und Sr₅(PO₄)₃OH stabil und gut kristallin erhältlich. Das Salz Sr₆H₃(PO₄)₅ läßt sich aus Lösungen von Strontiumphosphaten in Kohlensäure-Wasser, H₂S-Wasser und in einem Barbiturat-Puffersystem rein darstellen.

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Ortho strontium phosphates

Strontium phosphates which possibly are being incorporated into the calcium phosphates of bony tissue were investigated. Octacalcium phosphate in dilute solutions exchanges a few percent of calcium against strontium. While strontium phosphates with apatite-like structures, and with ratios of Sr/P between 1.33 and 1.67 can be prepared, no pure strontium octaphosphate, as a distinctive X-ray phase can be isolated. Only preparations with small amounts of carbonate, as well as strontium phosphate carbonates analogous to the corresponding calcium salts can be obtained. At hydrothermal conditions (375^o), the compounds SrHPO₄, Sr₂P₂O₇, Sr₃(PO₄)₂ and Sr₅(PO₄)₃OH are stable, and can be prepared in well cristallized form, depending on the relative composition of the strontium phosphate system. The salt Sr₆H₃(PO₄)₅ can be prepared in a pure state from solutions of strontium phosphates in aqueous solutions of carbonic acid, H₂S and barbiturate buffer systems.

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Mit 2 Abbildungen

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Herrn Prof. Dr. Dr. h. c.Hans Nowotny gewidmet.