Multiple recurrence and nilsequences

Aiming at a simultaneous extension of Khintchine(X,X,m,T)(X,\mathcal{X},\mu,T) and a set A ? XA\in\mathcal{X} of positive measure, the set of integers n such that A T^2nA T^knA)(A)^k+1-\mu(A{\cap} T^{n}A{\cap} T^{2n}A{\cap} \ldots{\cap} T^{kn}A)>\mu(A)^{k+1}-\epsilon is syndetic. The size of this set, surprisingly enough, depends on the length (k+1) of the arithmetic progression under consideration. In an ergodic system, for k=2 and k=3, this set is syndetic, while for kòf(x)f(Tⁿx)f(T²ⁿx)? f(T^knx)  dm(x)\int{f(x)f(T^{n}x)f(T^{2n}x){\ldots} f(T^{kn}x) \,d\mu(x)} , where k and n are positive integers and f is a bounded measurable function. We also derive combinatorial consequences of these results, for example showing that for a set of integers E with upper Banach density d^*(E)>0 and for all {n ? \mathbbZ\colon d^*(E?(E+n)?(E+2n)?(E+3n)) > d^*(E)⁴-e}\big\{n\in\mathbb{Z}{\colon} d^*\big(E\cap(E+n)\cap(E+2n)\cap(E+3n)\big) > d^*(E)^4-\epsilon\big\}     

Origin of organic gaseous products formed in the thermal decomposition of formates

The gaseous products of the thermal decomposition of formates have been investigated and compared with the products of formaldehyde and synthesis gas transformation under analogous conditions in the presence of the solids formed in the thermal decomposition of particular formates. It was found that formaldehyde transformation leads to organic compounds identical to those obtained in the thermal decomposition of the respective formates, while synthesis gas does not react under such conditions. This fact substantiates the hypothesis that formaldehyde is a precursor of the organic compounds identified in the thermal decomposition of formates. The nature of the organic componds obtained in the thermal decomposition of formates indicates that the formaldehyde formed in the initial stages is then transformed in the Cannizzaro or Tishchenko reactions.

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Zusammenfassung Die gasförmigen Produkte der thermischen Zersetzung von Formiaten wurden untersucht und mit den Produkten verglichen, die bei der Umwandlung von Formaldehyd und Synthesegas unter analogen Bedingungen in Gegenwart von bei der thermischen Zersetzung von speziellen Formiaten gebildeten Feststoffen entstehen. Formaldehydumwandlungen führen zu organischen Verbindungen, die identisch mit denen sind, die bei der thermischen Zersetzung des betreffenden Formiats gebildet werden, während Synthesegas unter solchen Bedingungen nicht reagiert. Dies erhärtet die Hypothese, daß Formaldehyd eine Vorstufe der bei der thermischen Zersetzung von Formiaten identifizierten organischen Verbindungen ist. Die Natur der bei der thermischen Zersetzung von Formiaten auftretenden organischen Verbindungen weist darauf hin, daß das in den ersten Reaktionsschritten gebildete Formaldehyd nach der Cannizzaro-oder Tishchenko-Reaktion weiterreagiert.

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Preparation and Characterisation of Nano-Structured WO<Subscript>3</Subscript>-TiO<Subscript>2</Subscript> Layers for Photoelectrochromic Devices

Nano-structured WO₃-TiO₂ layers were prepared by the sol-gel route. To obtain transparent, porous and crack free layers up to 0.8 μ m with a single dipping cycle a templating strategy was used. As a template three-dimensionally network based on organically modified silane was introduced to the WO₃ and TiO₂ sols. The WO₃ layers were dip-coated onto the conductive glass substrate (TCO) and the TiO₂ layers on the top of the WO₃ layer. The morphology and the structure of the layers were determined by Scanning Electron Microscopy (SEM), High Resolution Transmission Electron Microscopy (HR-TEM), Energy Dispersive X-Ray Spectroscopy (EDXS), Auger and Infrared spectroscopy. SEM image of the WO₃-TiO₂ layer confirmed the nano-porosity of the layers and give the size of the particles of about 10 nm for TiO₂ and 30 nm for WO₃ layer. Further analysis indicated that the titanium sol penetrates the WO₃ layer. Particles in the WO₃ layer consist of a crystalline monoclinic WO₃ core surrounded by a 5–10 nm amorphous phase consisting of WO₃, TiO₂ and SiO₂. The WO₃-TiO₂ layers were used to assemble all solid state photoelectrochromic (PE) devices. Under 1 sun irradiation (1000 W/m²) the visible transmittance of the PE device changes from 62% to 1.6%. The colouring and bleaching processes last about 10 minutes.     

The motion of a compressible viscous fluid around rotating body

We consider the motion of compressible viscous fluids around a rotating rigid obstacle when the velocity at infinity is non zero and parallel to the axis of rotation. We prove the existence of weak solution.     

Enantio- and Diastereoselective Oxidation of N-Alkylimines Using Chiral α-Bromonitriles and Hydrogen Peroxide System

Chiral α-bromonitriles were prepared with good chemical and optical yields starting from natural α-amino acids by dehydrating the corresponding α-bromoamides with thionyl chloride. The combined system α-bromonitriles/hydrogen peroxide was examined for the enantio- and diastereoselective oxidation of N-alkylimines in basic media at room temperature. The oxidation of N-tertiobutylarylimines leads to optically active oxaziridines with moderate enantiomeric excess. However, the oxidation of (S)-1-phenylethylarylimines affords the corresponding oxaziridines with good diasteromeric excess up to 97/3 as proved by gaseous-phase chromatography.     

Low-temperature versus oxygen plasma treatment of water-based TiO2 paste for dye-sensitized solar cells

High-temperature treatment steps in fabrication process of dye sensitized solar cell (DSSC) significantly contribute to the manufacturing costs and limit the use of temperature sensitive substrates. Therefore our aim was to develop a simple method for the preparation of water-based TiO₂ paste. The paste is based on peroxotitanic acid (PTA) sol–gel matrix and commercial TiO₂ nanoparticles (P25). Two fabrication processes to decompose/transform the PTA matrix in the printed TiO₂ layer are explored and combined: annealing at temperatures up to 250 °C and/or oxygen plasma treatment. The results show that the PTA matrix in the paste converts to anatase phase and to some extent also attaches to the TiO₂ nanoparticles P25 acting as an interconnecting network within TiO₂ layer. The transformation of the PTA matrix occurs around 250 °C, but in the presence of TiO₂ nanoparticles P25 it starts already at 120 °C. In addition the results reveal that the crystallization is achievable also solely with the oxygen plasma treatment. The efficiency of the TiO₂ layers, exposed to different post-deposition treatments, is evaluated in DSSCs. The results show that oxygen plasma treatment of the TiO₂ layers could efficiently replace temperature curing at 250 °C. Within this study the DSSCs with the efficiency up to 4.2 % measured under standard test conditions (1,000 W/m², AM1.5, 25 °C) were realized.     

On the quintuple product identity

In this note we present a new proof of the quintuple product identity which is based on our study of order theta functions with characteristics and the identities they satisfy. In this context the quintuple product identity is another example of an identity which when phrased in terms of theta functions, rather than infinite products and sums, has a simpler form and is much less mysterious.

     

Influence of the cation on the formation of free hydrogen and formaldehyde in the thermal decomposition of formates

Three stages have been distinguished in the thermal decomposition of formates: It was found that in the first stage carbon dioxide is evolved, leaving structural elements of the respective cation hydride in the reaction system: \(HCO_2^ - \to H^ - + CO_2 \) The second stage depends on the character of the bond in the intermediately formed hydride: The third stage is connected with formaldehyde transformation into different organic compounds, depending on the character of the metal oxide formed in the thermal decomposition of the formate. This explains the formation of free hydrogen and organic compounds identified as thermal decomposition products of various formates.