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.     

Some exact solutions of the equations of the mechanics of nonhomogeneous media

We consider problems of the torsion of nonhomogeneous prismatic rods and the oscillation of plats of variable thickness operating in shear. We obtain an inequality characterizing the dependence of the maximum tangential stress on the shear modulus. An estimate from below is obtained for the torsional moment. An optimization problem for plates of variable thickness for a specified class of thicknesses is reduced to a problem on characteristic values for the Helmholtz equation.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 25, pp. 87–90, 1987.     

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 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.     

Problem of determining the permittivity in the stationary system of Maxwell equations

The stationary system of Maxwell equations for a unmagnetized nonconducting medium is considered. For this system, the problem of determining the permittivity ε from given electric or magnetic fields is studied. It is assumed that the electromagnetic field is induced by a plane wave coming from infinity in the direction ν. It is also assumed that the permittivity is different from a given positive constant ε₀ only inside a compact domain Ω ? R ³ with a smooth boundary S. To find ε inside Ω, the solution of the corresponding direct problem for the system of electrodynamic equations on the shadow portion of the boundary of Ω is specified for all frequencies starting at some fixed ω₀ and for all ν. The high-frequency asymptotics of the solution to the direct problem is studied. It is shown that the information specified makes it possible to reduce the original problem to the well-known inverse kinematic problem of determining the refraction coefficient inside Ω from the traveling times of an electromagnetic wave. This leads to a uniqueness theorem for the solution of the problem under consideration and opens up the opportunity of its constructive solution.     

Effect of inhomogeneity of the medium on the directivity of the simplest sources of elastic waves

The initial data for the first correction term of the ray expansion of a longitudinal wave excited by a point center of expansion are found by the boundary-layer method. The same is done for a transverse wave from a center of rotation.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 140, pp. 77–87, 1984.     

On the validity of the Ginzburg-Landau equation

Summary The famous Ginzburg-Landau equation describes nonlinear amplitude modulations of a wave perturbation of a basic pattern when a control parameterR lies in the unstable regionO(ε ²) away from the critical valueR _c for which the system loses stability. Hereε>0 is a small parameter. G-L's equation is found for a general class of nonlinear evolution problems including several classical problems from hydrodynamics and other fields of physics and chemistry. Up to now, the rigorous derivation of G-L's equation for general situations is not yet completed. This was only demonstrated for special types of solutions (steady, time periodic) or for special problems (the Swift-Hohenberg equation). Here a mathematically rigorous proof of the validity of G-L's equation is given for a general situation of one space variable and a quadratic nonlinearity. Validity is meant in the following sense. For each given initial condition in a suitable Banach space there exists a unique bounded solution of the initial value problem for G-L's equation on a finite interval of theO(1/ε²)-long time scale intrinsic to the modulation. For such a finite time interval of the intrinsic modulation time scale on which the initial value problem for G-L's equation has a bounded solution, the initial value problem for the original evolution equation with corresponding initial conditions, has a unique solutionO(ε²) — close to the approximation induced by the solution of G-L's equation. This property guarantees that, for rather general initial conditions on the intrinsic modulation time scale, the behavior of solutions of G-L's equation is really inherited from solutions of the original problem, and the other way around: to a solution of G-L's equation corresponds a nearby exact solution with a relatively small error.     

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.     

Freshwater inflow requirements for the protection of the critical habitat and the drinking water sources in the Yangtze River Estuary,China

Freshwater inflow requirements (FIRs for short), which considered the requirements for protection of drinking water sources as well as the first-grade state protection wildlife (Acipenser sinensis) in larval periods, were analyzed in this paper for the Yangtze River Estuary, China. Based on the different levels of salinity objectives and the relationship between salinity and the freshwater inflows, the FIRs for the Yangtze River Estuary were determined. The estuary FIRs were determined based on the habitat ecosystem health from April to November with minimum and medium levels, from March to December with high level; and on the requirement of protection of drinking water sources in other months of the year, accordingly. Combined the salinity objectives of drinking water sources and critical habitat in the Yangtze River Estuary, the FIRs for the estuary are calculated to be 938.2 × 10⁹, 729.4 × 10⁹ and 615.5 × 10⁹ m³ in the whole year with different levels, which is equal to 100.8%, 78.4% and 66.2% of the average annual river discharge for the Yangtze River Estuary, respectively. Annual river discharges can satisfy the medium and minimum levels of FIRs for the estuary. However, the temporal variation of the actual runoff has distinct difference from the FIRs for the estuary in critical periods (May, July and August) for the habitat ecosystem, 5% of the FIRs for the estuary should be maintained from December to February for protection of drinking water sources.     

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.     

The problem of recovering the permittivity coefficient from the modulus of the scattered electromagnetic field

Under consideration is the stationary system of equations of electrodynamics relating to a nonmagnetic nonconducting medium. We study the problem of recovering the permittivity coefficient ε from given vectors of electric or magnetic intensities of the electromagnetic field. It is assumed that the field is generated by a point impulsive dipole located at some point y. It is also assumed that the permittivity differs from a given constant ε₀ only inside some compact domain Ω ? R³ with smooth boundary S. To recover ε inside Ω, we use the information on a solution to the corresponding direct problem for the system of equations of electrodynamics on the whole boundary of Ω for all frequencies from some fixed frequency ω ₀ on and for all y ∈ S. The asymptotics of a solution to the direct problem for large frequencies is studied and it is demonstrated that this information allows us to reduce the initial problem to the well-known inverse kinematic problem of recovering the refraction index inside Ω with given travel times of electromagnetic waves between two arbitrary points on the boundary of Ω. This allows us to state uniqueness theorem for solutions to the problem in question and opens up a way of its constructive solution.     

Iterative method of determining the relaxation-time spectrum from the components of the complex shear modulus

The determination of the relaxation-time spectrum from the results of measurements carried out under conditions of steady-state shear vibrations involves the solution of a Fredholm integral equation of the first kind, and constitutes an example of incorrectly formulated problems. Regularizing conditions are presented for the solution of the problem, together with one of a number of possible algorithms for constructing an iterative procedure in order to determine the relaxation-time spectrum from both components of the complex shear modulus at the same time. An example of the determination of such a spectrum is presented, together with an analysis of the relative errors.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 6, pp. 1069–1076, November–December, 1975.     

Aftermath of the Chernobyl Catastrophe from the Point of View of the Security Concept

The paper deals with uncertainty relations for time and energy operators, and the aftermath of the Chernobyl catastrophe is considered as an example. The mathematical approach developed by Holevo is analyzed, which allows us to assign the corresponding observables to non-self-adjoint operators and to establish uncertainty relations for nonstandard canonical conjugate pairs.

Relations for calculating the minimal time interval in which the energy jump can be discovered are given. Based on the intensity parameter introduced by the author, which is related to a special statistics called Gentile statistics and to the polylogarithm function, properties of stable chemical elements, such as time fluctuations and the jump of specific energy in the transition from the Bose—Einstein distribution to the Fermi—Dirac distribution, are mathematically described with regard to experimental data. The obtained data are arranged in a table for 255 stable chemical elements.

The mathematical approach developed by the author of the present paper allows one to describe the “antipode” (in a certain sense) of the standard thermodynamics, i.e., the thermodynamics of nuclear matter. This field of nuclear physics is very important for the study of properties of radioactive elements and, accordingly, from the standpoint of ensuring nuclear safety.

     

Collapse Rate of Solutions of the Cauchy Problem for the Nonlinear Schrödinger Equation

We prove that solutions of the Cauchy problem for the nonlinear Schrödinger equation with certain initial data collapse in a finite time, whose exact value we estimate from above. We obtain an estimate from below for the solution collapse rate in certain norms.

     

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.     

Influence of the inhomogeneity of porous media on the instantaneous compactification of the support of solution of the filtration problem

We study the phenomenon of instantaneous compactification and the initial behavior of the support of solution of the filtration equation for inhomogeneous porous media. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 8, pp. 1035–1044, August, 2006.     

Accelerating the convergence of the method of alternating projections

The powerful von Neumann-Halperin method of alternating projections (MAP) is an algorithm for determining the best approximation to any given point in a Hilbert space from the intersection of a finite number of subspaces. It achieves this by reducing the problem to an iterative scheme which involves only computing best approximations from the individual subspaces which make up the intersection. The main practical drawback of this algorithm, at least for some applications, is that the method is slowly convergent. In this paper, we consider a general class of iterative methods which includes the MAP as a special case. For such methods, we study an ``accelerated' version of this algorithm that was considered earlier by Gubin, Polyak, and Raik (1967) and by Gearhart and Koshy (1989). We show that the accelerated algorithm converges faster than the MAP in the case of two subspaces, but is, in general, not faster than the MAP for more than two subspaces! However, for a ``symmetric' version of the MAP, the accelerated algorithm always converges faster for any number of subspaces. Our proof seems to require the use of the Spectral Theorem for selfadjoint mappings.

     

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.     

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.     

On the operator representation of the Green's functions of the dynamic theory of elasticity

Starting from the operator notation for the dyadic Green's function for elastic displacement of the nonstationary theory of elasticity, we propose a method of factoring the mutually orthogonal components of the Green's function in the form of tensor products of operators acting respectively on the coordinates of the points of observation and the source. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 78–80.