The localization of energy in gauge field theories and in linear gravitation

It is shown how the energy-positivity criterion enables us to localize the energy in various field theories. For this purpose the role of surface integrals in a canonical formalism is investigated. The same techniques are applied to linearized gravity, where the mixed Cauchy-boundary value problem in a finite volume is analyzed. Unconstrained degrees of freedom and boundary data which have to be controlled are found. This paper is part of a program to analyze the possibility of localization of gravitational energy in complete General Relativity.     

Primitive lattice points in rational ellipses and related arithmetic functions

Assuming the Riemann Hypothesis to be true, an asymptotic with a sharp error term is established for the number of primitive lattice points inside a rational ellipseau ²+buv+cv ²x (a, b, c integers,b ²–4ac<0). A generalization of the result is given applying (as an example) to counting functions of Pythagorean triangles.     

Unions and Intersections of Families of Lp-Spaces

Let L^Ψ and E^Ψ be the ORLICZ space and the space of finite elements respectively, on a measure space (Ω, Σ, μ), and let T ? (0, ∞). It is proved that if inf {p: p ? T} ? T or sup {p: p ? T} ? T and μ is an infinite atomless measure, then there is no ORLICZ function Ψ such that: \documentclass{article}\pagestyle{empty}\begin{document}$ L^\varphi = Lin\mathop { \cup L^p }\limits_{p\varepsilon T} $\end{document} or \documentclass{article}\pagestyle{empty}\begin{document}$ E^\varphi = Lin\mathop { \cup L^p }\limits_{p\varepsilon T} $\end{document} and moreover, there is no ORLICZ function Ψ such that: \documentclass{article}\pagestyle{empty}\begin{document}$ L^\varphi = Lin\mathop { \cap L^p }\limits_{p\varepsilon T} $\end{document} or \documentclass{article}\pagestyle{empty}\begin{document}$ E^\varphi = Lin\mathop { \cap L^p }\limits_{p\varepsilon T} $\end{document}.     

Final results on μe-pair production in neutrino interactions in the SKAT freon bubble chamber experiment

We present the final results from the search for μe pairs produced in neutrino interactions using the freon filled bubble chamber SKAT. The rate of μ^? e ⁺ pairs to charged current events above the charm threshold is \(R_{\mu ^ - e^ + } = (4.8 \pm 1.1)10^{ - 3} \) . Assuming charm particle production to be the origin of the positron we calculate \(R_{\Lambda _c^ + } = (6.2 \pm 3.1)10^{ - 2} \) andR _D =(2.8±0.9)10^?2. We observe no considerable μ^? e ^? pair production above the background. In the regionE _v >3 GeV,p _μ,e >1.0 GeV/c andp _μ>p _e we find with a 90% confidence level the limit \(R_{\mu ^ - e^ - }< 1.7 10^{ - 4} \) .     

New evidence on the structure of potassium salts of 12-tungstophosphoric acid, KxH3-xPW12O40

Physicochemical properties and compositions of KxH(3-x)PW12O40 salts, where 2 < or = x < or = 3, have been investigated. It has been found that freshly prepared K2HPW12O40 salt (drying at 313 K) contains particles of heteropolyacid and particles of the neutral potassium salt, the sample being in 78.6% amorphous. On aging at room temperature, the heteropolyacid spreads to form a surface layer covering the neutral potassium salt particles K3PW12O40. Heat treatment of KxH(3-x)PW12O40 salts, where 2 < or = x < 3, from 313 K to higher temperatures induces the transformation of the heteropolyacid-covering K(3) core into a well-dispersed, amorphous surface layer. On further heating of the acidic potassium salts, the surface layer decomposes between 855 and 915 K with the formation of a PW8O26-type bronze as a new phase, the K3PW12O40 salt remaining unchanged. The latter starts to decompose at 1093 K, and in the case of all samples, the process is completed at about 1183 K. Rietveld structure refinement, XPS, and 31P NMR measurements of acidic potassium salts indicate that the core of these salts is always formed by the K3PW12O40 salt, which is covered by a heteropolyacid. Comparison of lattice parameters of the K3 salt and HPW leads to the conclusion that the layer is composed of partially or completely dehydrated heteropolyacid molecules. The coverage of the core by HPW in the K2 sample was estimated to be equal to one monolayer.     

New 2+2, 3+3 and 4+4 macrocycles derived from 1,2-diaminocyclohexane and 2,6-diformylpyridine

Two new Schiff base macrocycles - a 4+4 condensation product and a meso-type 2+2 condensation product - were obtained in a reaction of trans-1,2-diaminocyclohexane and 2,6-diformylpyridine. Reduction of these compounds led to the corresponding 4+4 and 2+2 macrocyclic amines. The macrocycles were characterised by NMR spectroscopy and electrospray mass spectrometry. The symmetry and stereochemistry of these macrocycles, as well as of new 3+3 and 4+4 diastereomers identified in solution, has been established. X-Ray structures of the 2+2 and 4+4 Schiff base macrocycles confirm the configurations determined on the basis of spectroscopic investigations. The crystal structures reveal that the centres of the square-shaped 4+4 macrocycles form channels as a result of columnar stacking.     

The synthesis and the solvent and substituent effect on the spectroscopic characteristic of 3-ethyl-2-(p-substitued styryl)benzothiazolium iodides

The photophysical and photochemical properties of p-substitued 2-styryl-ethylbenzothiazolium iodides, possessing different electron-withdrawing or electron-donating groups are described. The dyes were prepared by the condensation of 3-ethyl-2-methylbenzothiazole salts with p-substituted benzaldehydes. The synthesis of suitable substrates is presented as well. We describe here the absorption, emission spectra and the luminescence quantum yield of hemicyanine dyes (SH) measured in 11 different organic solvents of varying polarity. Molecular structure of the synthesized dyes was established by (1)H NMR, electronic absorption and fluorescence spectrometry. The spectral data confirmed that all the compounds exist in E-configuration of their styryl residues. The planar molecular conformation is typical for the compounds with five-membered side aromatic moieties (for example benzothiazole). The compounds possessing N-alkyl substituent in phenyl ring, in contrast to the compounds with other substituents, exhibit low fluorescence quantum yield in THF solution. This indicates that for N-alkyl derivatives the non-radiative processes are much more effective than the radiative ones. The electronic absorption and fluorescence emission spectra of tested dyes demonstrate high sensitivity to the nature of substituent introduced into the aromatic ring.     

Polymerization of acrylonitrile by catalytic systems consisting of triphenylphosphine and a lewis acid

Polymerization of acrylonitrile in the presence of systems that consisted of triphenylphosphine (PPh₃) and a Lewis acid R_mMX_n (ZnCl₂, Me₃Al, Et₃Al, Et₂AlCl, EtAlCl₂, AlCl₃) was studied. The systems that contained Me₃Al and Et₃Al (i.e., Lewis acid of moderate acidity) were the most efficient catalysts. Conductometric measurements carried out in the polymerization systems showed the presence of ions. The presence of phosphonium cation in the polyacrylonitrile chain formed by the PPh₃–R_mMX_n catalytic systems was determined by IR, ¹H-NMR, and ³¹P-NMR spectroscopy. The average molecular weight measurements and kinetic chain lengths of polyacrylonitrile formed within the reaction time in the presence of PPh₃–Et₃Al showed that transfer reactions occur. According to the results obtained, the polymerization reaction of acrylonitrile by PPh₃–R_mMX_n involved a zwitterion formed by the attack of PPh₃ on acrylonitrile complexed by Lewis acid [Ph₃P^⊕? CH₂? C^?H? C?N → MR_mX_n] and the anion [CH₂?C^?? C?N] formed by the proton abstraction from the monomer.