General form of the deformation of the Poisson superbracket

All continuous formal deformations of the Poisson algebra realized on Grassmann-valued compactly supported smooth functions on ℝ²ⁿ with 2n ≥ 4 are found up to an equivalence transformation. We show that in the algebras considered, there exist additional deformations that di.er from the Moyal bracket. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 2, pp. 163–178, August, 2006.     

Dependence of the properties of polycaproamide obtained in the presence of benzhydrazide on the symmetry type of the unit cell

The structure and properties of polycaproamide obtained in the presence of benzhydrazide have been studied. They are shown to depend on the duration of application of pressure in the mold. Under tensile deformation a transition from hexagonal to monoclinic polycaproamide takes place in the neck.Lugansk Machine-Building Institute. Institute of the Chemistry of High-Molecular Compounds, As UkrSSR, Kiev. Translated from Mekhanika Polimerov, Vol. 4, No. 6, pp. 1125–1127, November–December, 1968.     

Effect of temperature on the nature of the tear surface of styrene polymers

The tear surfaces of polystyrene and SKS-85 butadiene-styrene copolymer (85% styrene) have been investigated at temperatures from –45 to +100°C and from –60 to +40°C, respectively. The fracture surface of these polymers changes not only on transition from the glassy to the high-elastic state, but also within the glassy state itself, changes being observed both in the relative extent of the individual zones contributing to the fracture surface and in the nature of those zones. Changes in the nature of the fracture surface associated with a slowing of the fracture process occur at 0 and 40°C in the case of polystyrene and at –10°C in the case of copolymer SKS-85 and are attributable to secondary transitions in the polymers.Moscow Technological Institute of the Meat and Dairy Industry. State Institute of Polymer Adhesives, Kirovakan. Translated from Mekhanika Polimerov, Vol. 5, No. 2, pp. 257–264, March–April, 1969.     

Theory of the biomechanism of fractures of the diaphysis of long tubular bones

A new theoretical basis is proposed for the biomechanism of fractures of the diaphysis of long tubular bones resulting from bending deformation. Bone damage is shown to result from the concentration of maximum tensile stresses on the tension side and shear stresses in the compression zone. The origin of fan-shaped cracks and oblique fractures, occurring individually or in combination, is explained in these terms.Altai State Medical Institute. Translated from Mekhanika Polimerov, No. 2, pp. 318–322, March–April, 1971.     

Optimal reconstruction of the solution of the wave equation from inaccurate initial data

In the present paper, we consider the problem of the optimal reconstruction of the solution of the wave equation from the approximate values of the Fourier coefficients of the function specifying the initial form of the string. For an operator defined on the weight space of vectors from l ₂, we present the solution of the more general problem of reconstruction from the approximate values of the coordinates of these vectors.     

Variation of the unit root along the Dwork family of Calabi–Yau varieties

We study the variation of the unit roots of members of the Dwork families of Calabi–Yau varieties over a finite field by the method of Dwork–Katz and also from the point of view of formal group laws. A p-adic analytic formula for the unit roots away from the Hasse locus is obtained. This work was supported in part by Professor N. Yui’s Discovery Grant from NSERC, Canada.     

On the best approximation of the differentiation operator on the half-line

We solve a problem of S. B. Stechkin concerning the best approximation in the metric of C to the operator of k-th order differentiation on certain classes of differentiable functions defined on the half-line, by linear operators whose norms from L₂ into C are bounded. We consider the analogous problem for linear differential operators with constant coefficients. The bibliography contains 10 items.Translated from Matematicheskie Zametki, Vol. 6, No. 5, pp. 573–582, November, 1969.     

The extreme value of the generalized distances of the individual points in the multivariate normal sample

1. Summary The extreme value of the generalized distances, from the origin, ofN individual points which may be correlated each other, in thep-variate normal sample is defined and discussed. It contains, as special cases, (i) the extreme deviate from the population mean or the sample mean, (ii) the extreme deviate from the control variate and (iii) the range defined by (2.10) or (2.11) below. The exact sampling distributional theory of this statistic is extremely difficult to find, even its moments. However, the method of obtaining the approximate upper 100α percentage points for the ordinary significance levelα is given. The lower percentage points can be obtained in the similar way if necessary. In connection with the evaluation of the approximate percentage points, the two-dimensional chi-square distribution is discussed and the asymptotic formulas for the joint distribution function of the two generalized distances are given in the special forms for the present aim. The extreme deviate from the sample mean will be explained in some detail and the tables of the approximate upper 5, 2.5 and 1% points are given. For the cases (ii) and (iii) mentioned above the details are omitted and will be discussed in the case of need.     

Effect of the sign of the shear stresses on the shear strength of reinforced plastics

The effect of the sign of the shear stresses on the shear strength has been investigated for a glass-reinforced plastic of the SVAM type with various ratios of the longitudinal and transverse reinforcement in shear in the plane of reinforcement in axes turned through 45° relative to the direction of reinforcement. More than 80 tubularspecimens were tested. The shear strengths T ₄₅ ⁺ and T ₄₅ ^– corresponding to shear stresses of different signs can be found from uniaxial tests in tension and compression in the direction of the reinforcement.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 2, pp. 262–268, March–April, 1971.     

Effect of the molecular weight of polystyrene on the viscosity of concentrated solutions

The viscosity of solutions of polystyrene of various molecular weights (from 1.04 · 10² to 3.8 · 10⁵) in a poor solvent (decalin) and a good solvent (ethylbenzene) has been measured at temperatures from 15 to 70°C over a broad range of shear stresses from 10² to 10⁶ dyne/cm². The nature of the solvent has a considerable influence on the critical molecular weight and the absolute value of the viscosities of the solutions over the entire range of molecular weights and on the form of the flow curves of decalin solutions of polystyrene as a function of temperature. The heat of activation of viscous flow increases with increase in molecular weight and shear stress on the interval 20–80°C. The results obtained are explained in terms of the effect of the molecular weight of the polymer, the nature of the solvent, stress and temperature on structure formation in the solution and on the orientation of the macromolecules and structures in the flow process.Ural Gor'kii State University, Sverdlovsk. Translated from Mekhanika Polimerov, No. 5, pp. 920–926, September–October, 1970.     

The role of the porous structure of the filler in the reinfo rcement of polymers

Composites consisting of polyethylene and porous fillers with various pore sizes (from 22 to 115 Å) and surface types have been investigated. It is shown that the polymer filling the pores takes part in the deformation processes. The modulus of elasticity of the filled polymers is directly proportional to the filler surface per gram polymer. It is proposed to characterize the activity of the fillers by reference to the slope of the straight lines E-S with the hydrodynamic effect excluded.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 2, pp. 253–258, March–April, 1973.     

A more precise definition of the method of determining the fatigue life of polymer materials from the reheat temperature

Conclusions 1. The rate of increase in the reheat temperature on the linear segment remains constant (for a periodic loading with higher stress levels) until a critical damage is sustained for the stress in question.2. The magnitude of T_c for a given stress is independent of the loading history, provided the critical damage has not been sustained for this stress.3. With decreasing stress T_c decreases slightly. A method for the accelerated determination of T_c from a single specimen for different stress levels has been developed.4. A method of predicting the fatigue life from the results of short-term tests, which take into account the variation of T_c due to stress level, has been developed.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 906–913, September–October, 1977.     

The categoricity of the group of all computable automorphisms of the rational numbers

We prove that there is a first-order sentence ϕ such that the group of all computable automorphisms of the ordering of the rational numbers is its only model among the groups that are embeddable in the group of all computable permutations. Supported by a Scheme 2 grant from the London Mathematical Society. __________ Translated from Algebra i Logika, Vol. 46, No. 5, pp. 649–662, September–October, 2007.     

The essence of the generalized Taylor theorem as the foundation of the homotopy analysis method

The generalized Taylor theorem is the foundation of the homotopy analysis method proposed by Liao. This theorem is interesting but hard to understand from the mathematical point of view. Especially, there is a key parameter h whose meaning is still unknown. In the paper, we derive the generalized Taylor theorem from a usual way, that is, we prove that the generalized Taylor expansion is equivalent to a different representation of the usual Taylor expansion at different points. Therefore the meaning of the auxiliary parameter h is clarified. These results give a reasonable explanation of the parameter h and uncover the essence of the generalized Taylor theorem from which we can deeply understand the homotopy analysis method. Through the detailed analysis of some examples, we compare the series solution at the different points with the generalized Taylor series solution obtained by the homotopy analysis method.     

On the problem of the finiteness of the number of the combinatorial types of infinite convex polyhedra with equiangular faces

An infinite (complete) convex polyhedron with equiangular faces, that is, such that all the angles of each of its faces are equal to one another, is called irreducible if the number of monogonal faces belonging to it cannot be decreased (by identifying their sides) while preserving the equiangularity of all of the other faces and the convexity of the polyhedron itself (a lack of conditional edges). Any infinite convex polyhedron with equiangular faces can be obtained from the corresponding irreducible one by pasting in the missing number of monogons. It is proved that the number of combinatorially different irreducible polyhedra is finite, not counting three infinite series of frusta of cones with finite or infinite bases and right prisms with infinite bases. It is also established that, without exception, all infinite convex polyhedra with equiangular faces and total curvature 2are the derivatives of closed convex polyhedra with equiangular faces. The proof is carried out with the help of A. D. Milka's method from Ross. Zh. Mat., 1988, 3A830. Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 75–83, 1992.     

The best approximation of the differentiation operator in the metric of Lp

For Stechkin's problem of the best approximation for the differentiation operator we indicate the necessary and sufficient conditions that E_n be finite. We study some properties of continuous linear operators V from L_p into L_q.Translated from Matematicheskie Zametki, Vol. 12, No. 5, pp. 531–538, November, 1972.     

Contribution to the theory of the elastic properties of amorphous-crystalline polymers

The elastic constants of an amorphous-crystalline polymer are calculated from the known values of the elastic constants of its amorphous and crystalline components.Moscow Lenin Pedagogical Institute, Problem Laboratory of Polymer Physics. Translated from Mekhanika Polimerov, Vol. 5, No. 2, pp. 213–218, March–April, 1969.     

Operator methods in the problem of estimating the asymptotics of the time of the first hit for the birth and death process

Estimates are established for the exponential asymptotics of the time of the first hit of a certain leveln for the birth and death process.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 8, pp. 1104–1108, August, 1993.     

The cyclicity of the period annulus of the cubic isochronous center

This paper is concerned with limit cycles which bifurcate from periodic orbits of the cubic isochronous center. It is proved that in this situation, the cyclicity of the period annulus under cubic perturbations is equal to four. Moreover, for each k?=?0,1, . . .,4, there are perturbations that give rise to exactly k limit cycles bifurcating from the period annulus.     

Investigation of the nonuniform distribution of the elastic and strength characteristics over the cross section of the diaphysis of the human tibia

The elastic, deformation, and strength properties of six different zones of the cross section of the diaphysis of the human tibia have been experimentally investigated. It is shown that when the compact bone tissue is stressed in tension all these properties differ significantly from zone to zone. The greatest values of the initial modulus of elasticity and the tensile strength correspond to the frontal-outer zones of the bone. The nonlinear stress-strain curves are analytically approximated. The secant and tangent moduli are shown to depend on the stress intensity.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 940–946, September–October, 1971.