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.     

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.     

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.     

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。     

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.     

Geometric construction of generators of CoHA of doubled quiver

Let Q be the double of a quiver. According to Efimov, Kontsevich and Soibelman, the cohomological Hall algebra (CoHA) associated with Q is a free super-commutative algebra. In this short note, we confirm a conjecture of Hausel, which gives a geometric realisation of the generators of the CoHA.     

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.     

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.     

Admissibility of estimators of the probability of unobserved outcomes

The problem of estimating the probability of unobserved outcomes or, as it is sometimes called, the conditional probability of a new species, is studied. Good's estimator, which is essentially the same as Robbins' estimator, namely the number of singleton species observed divided by the sample size, is studied from a decision theory point of view. The results obtained are as follows: (1) When the total number of different species is assumed bounded by some known number, Good's and Robbins' estimators are inadmissible for squared error loss. (2) If the number of different species can be infinite, Good's and Robbins' estimators are admissible for squared error loss. (3) Whereas Robbins' estimator is a UMVUE for theunconditional probability of a new species obtained in one extra sample point, Robbins' estimator is not a uniformly minimum mean squared error unbiased estimator of the conditional probability of a new species. This answers a question raised by Robbins. (4) It is shown that for Robbins' model and squared error loss, there are admissible Bayes estimators which do not depend only on a minimal sufficient statistic. A discussion of interpretations and significance of the results is offered. Research supported by NSF Grant DMS-88-22622.     

Analysis of the properties of solutions of the relativistic theory of gravity in the vicinity of a singular sphere

The properties of the static, spherically symmetric metric tensor of the relativistic theory of gravity are analyzed in the vicinity of a singular sphere. It is shown that a massive particle with a nongeodesic radial motion may reach this sphere and remain there at rest. Based on this property, it is inferred that a sphere formed by massive particles can serve as a source of singular metrics in the relativistic theory of gravity. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 111, No. 1, pp. 144–148, April, 1997.     

Stability of Periodic Points of Diffeomorphisms of Multidimensional Space

We study the diffeomorphism of a multidimensional space into itself with a hyperbolic fixed point at the origin and a nontransversal homoclinic point. From the works of Sh. Newhouse, B.F. Ivanov, L.P. Shilnikov, and other authors, it follows that there is a method of tangency for the stable and unstable manifold such that the neighborhood of a nontransversal homoclinic point can contain an infinite set of stable periodic points, but at least one of the characteristic exponents of those points tends to zero as the period increases. In this paper, we study diffeomorphisms such that the method of tangency for the stable and unstable manifold differs from the case studied in the works of the abovementioned authors. This paper continues previous works of the author, where diffeomorphisms are studied such that their Jacobi matrices at the origin have only real eigenvalues. In those previous works, we find conditions such that the neighborhood of a nontransversal homoclinic point of the studied diffeomorphism contains an infinite set of stable periodic points with characteristic exponents separated from zero. In the present paper, it is assumed that the Jacobi matrix of the original diffeomorphism at the origin has real eigenvalues and several pairs of complex conjugate eigenvalues. Under this assumption, we find conditions guaranteeing that a neighborhood of a nontransversal homoclinic point contains an infinite set of stable periodic points with characteristic exponents separated from zero.     

Estimates of probabilities of large deviations in problems of estimation of rated effects. I

Let there be given a velocity field described by some function that depends both on time and a point of a phase space. It is assumed that the velocity field is subject to small random perturbations that are, in the general case, generalized derivatives of a pre-Gaussian process. On the basis of observations of trajetories of the motion of the system in such a random environment, we would like to recover the given velocity field. We obtain a nuclear estimate of the velocity vector. Deviations of the estimate from the estimated quantity are controlled by means of exponential S. N. Berstein inequalities.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 1, pp. 27–35, January, 1991.     

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     

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.     

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.     

Arithmetic of a group of points of a formal group

For a formal group of finite height over a non-ramified extension of a ring of p-adic integers we construct a system of generators of a formal module that is the generalization of the canonic Shafarevich basis and the system of Henniart generators for Lubin — Tate groups.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademiya Nauk SSSR, Vol. 191, pp. 9–23, 1991.     

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.     

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.