Ray-Singer type theorem for the refined analytic torsion

We show that the refined analytic torsion is a holomorphic section of the determinant line bundle over the space of complex representations of the fundamental group of a closed oriented odd-dimensional manifold. Further, we calculate the ratio of the refined analytic torsion and the Farber-Turaev combinatorial torsion. As an application, we establish a formula relating the eta-invariant and the phase of the Farber-Turaev torsion, which extends a theorem of Farber and earlier results of ours. This formula allows to study the spectral flow using methods of combinatorial topology.     

Filled Julia Sets with Empty Interior Are Computable

We show that if a polynomial filled Julia set has empty interior, then it is computable.     

Kirwan-Novikov inequalities on a manifold with boundary

We extend the Novikov Morse-type inequalities for closed 1-forms in 2 directions. First, we consider manifolds with boundary. Second, we allow a very degenerate structure of the critical set of the form, assuming only that the form is non-degenerated in the sense of Kirwan. In particular, we obtain a generalization of a result of Floer about the usual Morse inequalities on a manifold with boundary. We also obtain an equivariant version of our inequalities.

Our proof is based on an application of the Witten deformation technique. The main novelty here is that we consider the neighborhood of the critical set as a manifold with a cylindrical end. This leads to a considerable simplification of the local analysis. In particular, we obtain a new analytic proof of the Morse-Bott inequalities on a closed manifold.

     

Exponential boundedness of solutions for impulsive delay differential equations

For an impulsive delay differential equation

exponent estimates of solutions have been obtained.     

On the Discrete Hardy Inequality

Necessary and sufficient conditions for the boundedness of thediscrete Hardy operator of the form , from to when 0 < q < 1 <p , is given.     

Linear operators generated by stochastic processes

Let X_t(0 t < ) be a homogeneous stochastically continuous stochastic process with independent increments; (, A, P) be the corresponding probability space; p, q1 be fixed numbers. Necessary and sufficient conditions are found for a stochastic integral defined on finite-valued functions to be extendable to a bounded operator from L_p (0, ) to L_q() (p q).Translated from Teoriya Sluchainykh Protsessov, No. 16, pp. 7–13, 1988.     

Sulphur-oxygen versus carbon-oxygen bond fission in the solvolysis of benzyl sulphenates

In contrast to p-anisyl trichloromethanesulphenate 1, which readily undergoes ethanolysis at room temperature with carbon-oxygen bond fission, the ethanolysis of the corresponding 2-nitrobenzenesulphenate 2 proceeds at a similar rate only at 100°, and involves sulphur-oxygen bond cleavage. While the solvolysis of 1 showed first-order kinetics, the solvolysis of 2 was second-order (first-order with respect to ester and to added base). The solvolysis rate of 2 decreases on going from 100% to 80% ethanol and by using pyridine instead of acetate as base, consistent with an S_N2 type mechanism involving nucleophilic displacement at sulphur by the base or lyate ion. The rate of solvolysis of 1 is greatly enhanced in polar solvents and correlates satisfactorily with the ionization of p-methoxyneophyl tosylate. An ionization mechanism to some ion pair species is suggested for the solvolysis of 1.     

Electron depleted bis(methylene)cyclobutenes: sulfinyl and sulfonyl substitution

Double [2,3] sigmatropic rearrangements of bis(propargyl sulfenates) to bis(allenic sulfoxides) and of bis(propargyl sulfinates) to bis(allenic sulfones) are shown to be a convenient and effective method for the preparation of conjugated diallene systems bearing two electron withdrawing trihalomethyl sulfoxide or sulfone substituents either on C-1 and C-6, or on C-3 and C-4. Such substituents are further shown to facilitate cyclization to bis(methylene)cyclobutenes, and to stabilize the latter. The electron withdrawing group substitution on the exocyclic methylene extremities proved more effective than similar substitution on the endocyclic double bond.     

Reducing waste contamination from animal-processing plants by anaerobic thermophilic fermentation and by flesh fly digestion

There is currently no market in Israel for the large amounts of waste from fish- and poultry-processing plants. Therefore, this waste is incinerated, as part of the measures to prevent the spread of pathogens. Anaerobic methanogenic thermophilic fermentation (AMTF) of wastes from the cattle-slaughtering industry was examined previously, as an effective system to treat pathogenic bacteria, and in this article, we discuss a combined method of digestion by thermophilic anaerobic bacteria and by flesh flies, as a means of waste treatment. The AMTF process was applied to the wastes on a laboratory scale, and digestion by rearing of flesh fly (Phaenicia sericata) and housefly (Musca domestica) larvae on the untreated raw material was done on a small scale and showed remarkable weight conversion to larvae. The yield from degradation of poultry waste by flesh fly was 22.47% (SD = 3.89) and that from fish waste degradation was 35.34% (SD = 12.42), which is significantly higher than that from rearing houseflies on a regular rearing medium. Bacterial contents before and after thermophilic anaerobic digestion, as well as the changes in the chemical composition of the components during the rearing of larvae, were also examined.     

The rearrangement and solvolysis of furfuryl arenesulfinates

Furfuryl benzenesulfinate, furfuryl p-toluenesulfinate and 5-nitrofurfuryl benzenesulfinate were synthesized, and their behaviour under various conditions was investigated. The first two esters were found to undergo readily rearrangement to sulfone. In nonhydroxylic solvents, a mixture of furfuryl aryl sulfone and 2-methyl-3-furyl aryl sulfone is obtained. The ratio between the two sulfones changes with the polarity of the solvent. In hydroxylic solvents, only rearrangement to the furfuryl aryl sulfone takes place, and this is accompanied by solvolysis of the ester. A kinetic study of the reaction in ethanol and aqueous ethanol solvents indicated an ionization mechanism. It is suggested that under these conditions the sulfone is formed by recombination of ion pairs. A kinetic study of the rearrangement under nonsolvolytic conditions was also performed in order to obtain the effect of the solvent and the effect of added salts on the rate of rearrangement. This study has shown that the rate of rearrangement is sensitive to the ionizing power of the solvent. The addition of perchlorate was found to have a stronger effect on the formation of the furfuryl sulfone than on the 2-methyl-3-furyl sulfone. In this case an ionic mechanism is also suggested, and the two sulfones may arise by recombination from two different species of ion pairs.