Breaking hyperbolicity for smooth symplectic toral diffeomorphisms

We study a smooth symplectic 2-parameter unfolding of an almost hyperbolic diffeomorphism on two-dimensional torus. This diffeomorphism has a fixed point of the type of the degenerate saddle. In the parameter plane there is a bifurcation curve corresponding to the transition from the degenerate saddle into a saddle and parabolic fixed point, separatrices of these latter points form a channel on the torus. We prove that a saddle period-2 point exists for all parameter values close to the co-dimension two point whose separatrices intersect transversely the boundary curves of the channel that implies the existence of a quadratic homoclinic tangency for this period-2 point which occurs along a sequence of parameter values accumulating at the co-dimension 2 point. This leads to the break of stable/unstable foliations existing for almost hyperbolic diffeomorphism. Using the results of Franks [1] on π ₁-diffeomorphisms, we discuss the possibility to get an invariant Cantor set of a positive measure being non-uniformly hyperbolic.     

On d-dimensional d-semimetrics and simplex-type inequalities for high-dimensional sine functions

We show that high-dimensional analogues of the sine function (more precisely, the $d$-dimensional polar sine and the $d$-th root of the $d$-dimensional hypersine) satisfy a simplex-type inequality in a real pre-Hilbert space $H$. Adopting the language of Deza and Rosenberg, we say that these $d$-dimensional sine functions are $d$-semimetrics. We also establish geometric identities for both the $d$-dimensional polar sine and the $d$-dimensional hypersine. We then show that when $d=1$ the underlying functional equation of the corresponding identity characterizes a generalized sine function. Finally, we show that the $d$-dimensional polar sine satisfies a relaxed simplex inequality of two controlling terms “with high probability”.     

Vibrations of shallow shells that are circular in plan and have local supports

The paper deals with vibrations of doubly curved shallow shells that are circular in plan and are reinforced by local rod-type supporting elements or cylindrical shells. The coefficients of the frequency equation are found by using numerical-analytical methods to solve boundary-value problems for fixed values of frequency. The natural frequencies and modes of vibration of a system composed of a shell and elastic supports are determined in the course of solving the problem. It is shown that it is possible to also account for reactive moments and shearing forces, in addition to the normal reactions of an elastic support. The potential of the approach which is developed is illustrated by the solution of specific problems. Special Design Office of the S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 3, pp. 69–75, March, 1999.     

Lattice Embeddings into the R.E. Degrees Preserving 0 and 1

We show that a finite distributive lattice can be embedded intothe r.e. degrees preserving least and greatest element if andonly if the lattice contains a join-irreducible noncappableelement.     

Study of reaction of sulfur halides with unsaturated. Compounds. 24. Reaction of sulfur monochloride with adamantylidene-adamantane

Conclusions In the reaction of adamantylideneadamantane with S₂Cl₂ at an equimolar ratio of the reagents, 2,2-epithio-2-(2-adamantyl)adamantane and 4-chloro-2,2-epithio-2-(2-adamantyl)-adamantane are formed, while with an excess of S₂Cl₂, the products are 4e-chloro- and 4e,4e-dichloroadamantylideneadamantane.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 3, pp. 617–620, March.Article 23, see [1].     

Finite element approximation of a model for phase separation of a multi-component alloy with a concentration-dependent mobility matrix

We consider a model for phase separation of a multi-componentalloy with a concentration-dependent mobility matrix and logarithmicfree energy. In particular we prove that there exists a uniquesolution for sufficiently smooth initial data. Further, we provean error bound for a fully practical piecewise linear finiteelement approximation in one and two space dimensions. Finallynumerical experiments with three components in one space dimensionare presented.     

Preparation of bromomethylene tetrathioprotadamantane and bromomethylchloro tetrathioadamantane

Conclusions When either bromine or ethylsulfenyl bromide is reacted with 1-methylene-3,5,7-trimethyl-2,4,6,8-tetrathioprotoadamantane an unsaturated bromide, which adds HCl in the presence of ZnCl₂, is formed as the result of bromination at the terminal methylene group. The hydrochlorination is accompanied by rearrangement of the 2,4,6,8-tetrathioprotoadamantane to the 2,4,6,8-tetrathioadamantane. Institute of Chemistry, Bashkir Branch of the Academy of Sciences of the USSR, Ufa. Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 10, pp. 2371–2373, October, 1976