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.     

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.     

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.     

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.     

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.     

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.     

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。     

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.     

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.     

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.     

Application of the method of direct cutting-out to the solution of the problem of longitudinal shear of a wedge with thin heterogeneities of arbitrary orientation

The method of direct cutting-out consists of modeling of a finite body, in particular, with thin heterogeneities, using a much simpler problem for a bounded or a partially bounded body with thin heterogeneities located in the same manner and the presence of additional cracks or absolutely rigid inclusions of fairy large length, which are modeled by the boundary conditions of a bounded body. The method is tested on the problems of antiplane deformation of a symmetrically loaded crack in a wedge with free faces and an absolutely rigid inclusion placed with some tension in a wedge with restrained faces. For an elastic inclusion, we construct generalized conditions of interaction, which enable us to unify the procedure of giving different boundary conditions in the case of using the method of direct cutting-out.     

Definition of boundary values of thickness of chrome coverings of piston rings

An approach to the definition of adhesion and cohesion stresses that appear in a piston ring after chroming and its mounting on the piston, its performance in a motor, and corresponding conditions of the critical thicknesses of the covering on the basis of a representation of the ring as a two-layer beam is described.Translated from Dinamicheskie Sistemy, No. 9, pp. 72–76, 1990.     

Some properties of c-covers of a pair of Lie algebras

In [H. Safa and H. Arabyani, On c-nilpotent multiplier and c-covers of a pair of Lie algebras, Commun. Algebra 45(10) (2017), 4429–4434], we characterized the structure of the c-nilpotent multiplier of a pair of Lie algebras in terms of its c-covering pairs and discussed some results on the existence of c-covers of a pair of Lie algebras. In the present paper, it is shown under some conditions that a relative c-central extension of a pair of Lie algebras is a homomorphic image of a c-covering pair. Moreover, we prove that a c-cover of a pair of finite dimensional Lie algebras, under some assumptions, has a unique domain up to isomorphism and also that every perfect pair of Lie algebras admits at least one c-cover. Finally, we discuss a result concerning the isoclinism of c-covering pairs.     

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.     

Boundedness of the series of absolute values of blocks of sine series

The series of absolute values of blocks of a sine series whose coefficients satisfy a certain weakened monotonicity condition are considered. It is shown that to ensure the bounded convergence of such a series, one can take the same blocks as for the Fourier series of functions of bounded variation.     

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.     

Volatility of volatility of financial markets

We present empirical evidence for considering volatility of Eurodollar futures as a stochastic process, requiring a generalization of the standard Black-Scholes (BS) model which treats volatility as a constant. We use a previous development of a statistical mechanics of financial markets (SMFM) to model these issues.     

Geometry of the Space of Phylogenetic Trees

We consider a continuous space which models the set of all phylogenetic trees having a fixed set of leaves. This space has a natural metric of nonpositive curvature, giving a way of measuring distance between phylogenetic trees and providing some procedures for averaging or combining several trees whose leaves are identical. This geometry also shows which trees appear within a fixed distance of a given tree and enables construction of convex hulls of a set of trees. This geometric model of tree space provides a setting in which questions that have been posed by biologists and statisticians over the last decade can be approached in a systematic fashion. For example, it provides a justification for disregarding portions of a collection of trees that agree, thus simplifying the space in which comparisons are to be made.