Estimates of stability of characterization of additive types of distributions

An estimate of stability of characterization of distribution types is obtained for the case of additive types. Under some conditions, the estimate has the order ε^1/3L(ε), where L(ε) is a slowly varying function. Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, Russia, 1996, Part I.     

Characterization of types of asymptotic discernibility of families of hypotheses

We give a characterization of the types of asymptotic discernibility of families of hypotheses in the case of hypothetical measures that are not, in general, mutually absolutely continuous. The case when the logarithm of the likelihood ratio admits an asymptotic expansion of the type of an expansion with local asymptotic normality is examined in detail. Examples are studied.Translated fromTeoriya Sluchainykh Protsessov, Vol. 15, pp. 64–71, 1987.     

Singularities of Solutions of One Class of Equations of Continuum Mechanics

We analyze one class of families of integral equations and describe the dependence of the singularities of solutions of integral equations on the dimensions of the families of kernels of equations. On the basis of these results, we propose procedures for the construction of approximate solutions for a small parameter.     

Variant of statistical interpretation of the formation of viscoelastic deformations of polymers

A variant of a statistical probability interpretation of elementary micromechanisms of formation of viscoelastic deformations of polymers is examined with the aid of the concept of dissymmetric media and the dissymmetry coefficient. A new rheological relationship is obtained, a particular case of which is the known equation of a typical linear viscoelastic body. A method is given for determining the relaxation spectrum for different intensity functions figuring in the rheological relationships. The results of the calculation are compared with the experimental data, exemplified by the creep of polyamide P-68.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, Vol. 9, No. 3, pp. 392–396, May–June, 1973.     

Study of the blow-up of solutions of systems of equations of hydrodynamic type

Mathematical Notes - We study the initial boundary-value problem for three-dimensional systems of equations of pseudoparabolic type. The system is similar to the Oskolkov system, but differs from...     

Preservation of equivalence of derivations under reduction of depth of formulas

In this paper we consider derivations in the (&, )-fragment of the intuitionistic propositional calculus. As is known, replacement of any occurrence of a formula [F] in a sequent S by an occurrence of the formula [p], where p is a new propositional variable, with the simultaneous addition to the antecedent of the formula F p or p F depending on the sign of the occurrence of F in S, leaves the derivability unchanged. We give a proof of the fact that the natural extension of this transformation to derivations preserves the relation of equivalence of derivations, i.e., transformed derivations are equivalent if and only if the originals are equivalent. (Derivations are considered equivalent if certain of their normal forms coincide, or, what is the same, if their deductive terms coincide.) It is proved that by the iteration of this transformation, each derivation of an arbitrary sequent S can be transformed into a derivation of a sequent S, depending only on S, whose succedent is a variable, and in the antecedent there occur only formulas of the form a,a & b, a b,,(a b) c, a & b c, a (b & c), wherea, b, c are variables. Here if S is balanced, then S is also balanced. (A sequent is called balanced if each variable occurs in it no more than twice.) The familiar correspondence between certain concepts of the theory of categories and concepts of the theory of proofs allows one to assert that there has been constructed a univalent functor, mapping a free Cartesian closed category into itself.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 88, pp. 197–207, 1979.     

一类完全正则半群簇子簇格的性质

本文研究了完全正则半群簇的子簇格［Ｖ＋∩ＰＶ，Ｖ＋∩ＰＶ］的某些格运算性质，我们证明了簇Ｖ＋∩ＰＶ可分解为Ｖ与Ｖ＋∩ＰＶ的并；对任意完全正则半群簇Ｗ，有Ｗ∩（Ｖ∨Ｖ＋∩ＰＶ）＝（Ｗ∩Ｖ）∨（Ｗ∩Ｖ＋∩ＰＶ）．特别地，我们得到了等式Ｖ＋∩ＰＶ＝Ｖ成立的若干条件．     

The use of splitting of the equations of motion in the numerical solution of problems of the dynamics of elastic systems

Linear elastic systems with a finite number of degrees of freedom, the initial equations of motion of which are constructed using the finite element method or other discretization methods, are considered. Since, in applied dynamics problems, the motions are usually investigated in a frequency range with an upper bound, the degrees of freedom of the initial system of equations are split into dynamic and quasi-dynamic degrees. Finally, the initial system of equations is split into a small number of differential equations for the dynamic degrees of freedom and into a system of algebraic equations for determining the quasi-static displacements, represented in the form of a matrix series. The number of terms of the series taken into account depends on the accuracy required.     

Axiomatizability of reducts of algebras of relations

In this paper, we prove that any subreduct of the class of representable relation algebras whose similarity type includes intersection, relation composition and converse is a non-finitely axiomatizable quasivariety and that its equational theory is not finitely based. We show the same result for subreducts of the class of representable cylindric algebras of dimension at least three whose similarity types include intersection and cylindrifications. A similar result is proved for subreducts of the class of representable sequential algebras. Received October 7, 1998; accepted in final form September 10, 1999.     

Limit of Iterates of Cascade Addition of Matrices

With the cascade connection of electrical multiport networks as a model, the cascade sum of two positive semi-definite matrices has been considered. The existence of the limit of iterates of cascade addition of a single matrix is proved and various characterizations for the limit are presented.     

Birational aspects of the geometry of Varieties of Sums of Powers

Varieties of Sums of Powers describe the additive decompositions of a homogeneous polynomial into powers of linear forms. The study of these varieties dates back to Sylvester and Hilbert, but only few of them, for special degrees and number of variables, are concretely identified. In this paper we aim to understand a general birational behavior of VSP. To do this we birationally embed these varieties into Grassmannians and prove the rational connectedness of many VSP in arbitrary degrees and number of variables.     

Determination of asymptotic solutions of problems of nonlinear theories of thermoviscoelasticity

It is shown that the asymptotic solution of a problem of the nonlinear theory of thermoviscoelasticity, if it exists, can be found directly from the solution of the asymptotic boundary-value problem without completely solving the starting problem.M. V. Lomonosov Moscow State University. Translated from Mekhanika Polimerov, No. 3, pp. 395–400, May–June, 1976.     

Emptiness of set of points of lower semicontinuity of Lyapunov exponents

We consider parametric families of differential systems with coefficients that are bounded and continuous on the half-line and uniformly in time continuously depend on a real parameter. For each Lyapunov exponent, we construct a family such that the Lyapunov exponent of its systems treated as a function of the parameter is not a lower semicontinuous function for any value of the parameter.     

Invariant properties of systems of equations of the mechanics of inhomogeneous fluids

The invariant properties of the fundamental system of equations of the mechanics of inhomogeneous fluids and its widely used approximations are analysed by the methods of continuous group theory. Simplifying assumptions significantly alter the invariant properties of the systems of equations. Both extending and contracting the admissible symmetry group can occur and provide evidence of the violation of the conditions of equivalence between the original and derived systems. For example, the use of the incompressibility and Boussinesq approximations leads to extension of the concept of an inertial reference frame to all frames moving relative to one another with arbitrary linear acceleration.     

Properties of Neighborhoods of Attractors of Dynamical Systems

Mathematical Notes - The paper deals with the study of dynamical systems admitting compact attractors, which describe the behavior of trajectories of the system at infinity and play an important...     

Regularity of minimizers of non-isotropic integrals of the calculus of variations

The regularity of the minimizers of a special type of non-isotropic variational minimization problem is studied. The particularity of the potential of energy is that it has different growth rate with respect to different parts of the derivatives of the function. In particular, the model treated in this paper can be described as $$\Phi (Du) = |\partial _1 u|^2 + |\partial _2 u|^2 + |\partial _3 u|^2 + |\partial _3 u - |^p .$$ By using a result of P.Marcellini (cf. [4]) and perturbation method, it is proved that the minimizer of the Dirichlet boundary value problem is a function of W _loc ^{1, ∞} .     

Effect of the properties of the material of the buckling of a plate

In [2] it was shown with reference to the cylindrical bending of a plate that the Kirchhoff-Love hypothesis is asympototically accurate for determining the critical load irrespective of the properties of the material. However, the question of the dependence of the critical load on the material properties was left unexplored. This question is now numerically investigated. Curves characterizing the variation of the critical load with relative shear modulus are presented.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, Vol. 5, No. 2, pp. 355–357. March–April, 1969.     

Construction of normal bases of rings of integral elements of certain fields of algebraic functions

This article is devoted to the construction of normal bases of rings of integral elements of certain fields of algebraic functions. These fields are algebraic extensions of the initial fields, given by appropriate algebraic equations. The extensions are given only by the monodromy group. The device of the matrix Riemann boundary-value problem on a Riemann surface is used.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 7, pp. 1004–1007, July, 1990.     

Lengths of periods of continued fractions of square roots of integers

Empirical study of the period’s length T of the continued fractions of $\sqrt{Q}$ (for growing integers Q) shows several strange asymptotical results, for instance, $T\leq C\sqrt{Q}\ln{Q}$ . These results show important differences between the statistics of the elements of the continued fractions of random real numbers and of square roots of random integers.     

非线性方程分枝解理论在线性闭环系统极点摄动量估计中的应用

本文利用非线性方程小分枝解的理论,研究多变量定常线性控制系统的状态矩阵与控制矩阵受到摄动时,闭环系统的极点所受到的摄动量的估计问题.