Eigenvector p-Bases of Rings of Constants of Derivations

Let A be a UFD of characteristic p > 0, let 𝒵 be a set of some eigenvectors of a derivation of A. We prove, under some additional assumptions, a necessary and sufficient condition for 𝒵 to be a p-basis of the minimal ring of constants containing 𝒵. The main preparatory result is the unique decomposition theorem with respect to a factor from a given subalgebra containing A^p.     

Preservation of the existence of fixed and coincidence points under homotopy of mappings of ordered sets

In problems of topology and analysis, well-known theorem on the preservation by any continuous homotopy of the property of a mapping to have a fixed point and the property of a pair of mappings to have a coincidence point are extensively applied. Thus, for contraction mappings and some of their generalizations, Frigon’s results on the preservation of the property to have a fixed point by a homotopy of a special type are known. This paper presents theorems on the preservation by order homotopy of the property of a pair of mappings to have a coincidence point. As a corollary, conditions under which such a homotopy preserves the property of a mapping to have a fixed point are obtained.     

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.     

Linear problems of the stability of a type of rotation of a satellite about the centre of mass

The stability in the first approximation of the rotation of a satellite about a centre of mass is investigated. In the unperturbed motion the satellite performs, in absolute space, three rotations around the normal to the orbital plane in a time equal to two periods of rotation of its centre of mass in the orbit (Mercury-type rotation). Three cases of such rotations are considered: the rotations of a dynamically symmetrical satellite and a satellite, the central ellipsoid of inertia of which is close to a sphere, in an elliptic orbit of arbitrary eccentricity, and the rotation of a satellite with three different principal central moments of inertia in a circular orbit.     

Estimation of the number of components of finite mixtures of multivariate distributions

An estimator of the number of components of a finite mixture ofk-dimensional distributions is given on the basis of a one-dimensional independent random sample obtained by a transformation of ak-dimensional independent random sample. A consistency of the estimator is shown. Some simulation results are given in a case of finite mixtures of two-dimensional normal distributions.     

有资格限制的指派问题的求解方法

在实际的指派工作中，常会遇到某个人有没有资格去承担某项工作的问题，因此，本建立了有资格限制的指派问题的数学模型。在此数学模型中，将效益矩阵转化为判定矩阵，由此给出了判定此种指派问题是否有解的方法；在有解的情况下，进一步将效益矩阵转化为求解矩阵，从而将有资格限制的指派问题化为传统的指派问题来求解。最后给出了一个数值例子来说明这样的处理方法是有效的。     

Cohomology of categories and extensions of twisted diagrams of groups

A twisted diagram of groups assigns a group to every object of an indexing category and a homomorphism of groups to every morphism. However, it does not have to be completely functorial — it preserves composition only up to a compatible family of inner automorphisms. A. Haefliger defined a special case: the complex of groups. We prove that there exists a natural bijective correspondence between equivalence classes of epimorphisms of twisted diagrams of groups and elements of the second cohomology group of a certain small category. If this category is defined by a discrete group, then we obtain the well known classification of extensions of groups.     

Asymptotic Behavior of Solutions of Equations of Main Resonance

We investigate a system of two first-order differential equations that appears when averaging nonlinear systems over fast one-frequency oscillations. The main result is the asymptotic behavior of a two-parameter family of solutions with an infinitely growing amplitude. In addition, we find the asymptotic behavior of another two-parameter family of solutions with a bounded amplitude. In particular, these results provide the key to understanding autoresonance as the phenomenon of a considerable growth of forced nonlinear oscillations initiated by a small external pumping.     

Two-step generation of the equations of motion of closed-chains

This paper presents a two-step generation of the equations of motion of planar mechanisms using point and joint coordinates. First, the formulation replaces a rigid body by a dynamically equivalent constrained system of particles and uses Newton’s second law to study the motion of the particles without introducing any rotational coordinates. Then, the equations of motion are transformed to a reduced set in terms of selected relative joint variables using a velocity transformation matrix. For an open-chain, this process automatically eliminates all of the non-working constraint forces and leads to an efficient integration of the equations of motion. For a closed-chain, suitable joints should be cut and few cut-joints constraint equations are included. An example of a closed-chain is used to demonstrate the generality and efficiency of the proposed method.     

Choice of the best alternative in case of a continuous set of states of nature-application of fuzzy numbers

On the basis of a known application of an order weighted averaging operator to the decision making in the case of a discrete set of states of nature, a general approach to the case of a continuous set of states of nature is proposed. The general approach encompasses various types of attitudes of the decision maker, expressed in the form of fuzzy numbers.     

Estimates of rate of propagation of perturbations in quasilinear divergent degenerate parabolic equations of high order

A method is given for obtaining integral growth lemmas for solutions of boundary problems for a large class of quasilinear evolution equations. As a possible application a sharp estimate is obtained of the dependence of the support of a solution of a mixed problem and a Cauchy problem for a quasilinear divergent parabolic equation on time.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 10, pp. 1451–1456, October, 1992.     

Singularities of the structure of two-sided ideals of a domain of elementary divisors

We prove that, in a domain of elementary divisors, the intersection of all nontrivial two-sided ideals is equal to zero. We also show that a Bézout domain with finitely many two-sided ideals is a domain of elementary divisors if and only if it is a 2-simple Bézout domain.     

Studies of the dynamics of space motion of a class of parachute systems

A mathematical model of space motion of a mechanical system that consists of a load and a parachute is presented. With the help of the apparatus of the Lyapunov functions, a particular case of the model is studied when the load can be considered as a mass point that does not have resistance. A sufficient condition of stability is obtained. The results can be applied to the model/ing and construction of parachute systems.Translated from Dinamicheskie Sistemy, No. 9, pp. 99–102, 1990.     

Asymptotics of the spectrum of nonsmooth perturbations of differential operators of order 2m

On a finite closed interval, we obtain the asymptotics of the eigenvalues of a differential operator of order 2m perturbed by a differential operator of order 2m ? 2 given by a quasidifferential expression. We also consider the case of multiple eigenvalues.     

Bases of the identities of products of certain varieties of groups

The following theorem is proved: The product of any variety of two-step solvable groups and a variety having a finite basis of identity relations has a finite basis of identity relations.Translated from Matematicheskie Zametki, Vol. 5, No. 1, pp. 137–144, January, 1969.     

Existence of normal Hall subgroups by means of orders of products

Let G be a finite group, let π be a set of primes and let p be a prime. We characterize the existence of a normal Hall π‐subgroup in G in terms of the order of products of certain elements of G. This theorem generalizes a characterization of A. Moretó and the second author by using the orders of products of elements for those groups having a normal Sylow p‐subgroup 6 . As a consequence, we also give a π‐decomposability criterion for a finite group also by means of the orders of products.     

Comparisons of models for estimation of safe doses using measures of the heaviness of tail of a distribution

Summary Procedures to estimate a dose of which the incidence probability is very small (e.g. 10⁻⁶) have been developed to evaluate the safety of chemical compounds. To compare models for estimation of safe doses quantitatively, a measure of the heaviness of tail of a distribution and a measure of tail at the origin are introduced. These measures have a theoretical basis for the comparison of tail behavior between distributions. Using the two measures, a tail ordering is defined to present a criterion for the comparison of models and is discussed for the probit, the logit, the Weibull, the (generalized) multihit, the (generalized) multitarget and the multistage models. The multistage model is most conservative among them, while the probit model has the reverse property. The Weibull model is more conservative than the logit. The multihit and multitarget models are found to be more sensitive than the Weibull and the logit. The Institute of Statistical Mathematics National Institute of Environmental Health Sciences     

Construction of periodic solutions of integrodifferential equations of viscoelasticity

A theorem on the estimation of the periodic solutions of a linear integrodifferential equation and a theorem of existence and uniqueness of the periodic solution of a nonlinear integrodifferential equation are presented without proof.     

Properties of the solutions of a certain class of variational inequalities

In a curvilinear quadrangle one considers an elliptic operator with linear principal terms and discontinuous leading coefficients. One investigates the solution of a variational inequality with a constraint on the derivatives, tangent to the boundary and to the discontinuity lines of the coefficients. On certain parts of the boundary one imposes the first boundary condition and on others a condition on a directional derivative. One proves the existence of a solution with square summable second derivatives at each point of the subdomains where the leading coefficients are smooth.Translated from Problemy Matematicheskogo Analiza, No. 10, pp. 83–92, 1986.     

Complexity of additive computations of systems of linear forms

The additive computation of a system of linear forms can be represented by a sequence of square matrices Q₁,...,Q_T (Q_i equals the identity matrix increased or decreased by 1 in one entry). The complexity of the additive computation is the minimal number of matrices in such a representation. A connection between the additive complexity of a system with coefficient matrix A and of a system with coefficient matrix A^T is proved.