Rates of convergence for continued fractions of irrational numbers

The dynamics of the Gauss Map suggests a way to compare the convergence to a real number ζ ε(0,l) of a continued fraction and the divergence of the orbit of ζ Of particular interest is the comparison of the rate of convergence to ζ of its simple continued fraction and the rate of divergence by the Gauss Map of the orbit of ζ for all irrational numbers in (0,l). We state and prove sharp inequalities for the convergence of the sequence of rational convergents of an irrational number ζ. We show that the product of the rate of convergence of the continued fraction of ζ and the rate of divergence by the Gauss Map of the orbit of ζ equals 1.     

Identification and orbit determination of a tethered satellite system

Orbital motion of a tethered satellite system, composed of two satellites and an inextensible tether, is considered by using a perturbed two-body model. This approach is adopted so that the determination of the orbit of one of the satellites can be attempted without using observations of the motion of the other satellite in the system. The identification of the tethered condition of the system using observations of only one of the satellites in the tethered satellite system is considered. The characteristics of the `tether perturbed' motion of the observed satellite are investigated. Estimation of the state of the system using near perfect data is also illustrated. Observations of one satellite provide the entire state of the system and a parameter involving the ratio of the masses of satellites and the tether length.     

On solving equations of algebraic sum of equal powers

It is well known that a system of equations of sum of equal powers can be converted to an algebraic equation of higher degree via Newton's identities. This is the Viete-Newton theorem. This work reports the generalizations of the Viete-Newton theorem to a system of equations of algebraic sum of equal powers. By exploiting some facts from algebra and combinatorics, it is shown that a system of equations of algebraic sum of equal powers can be converted in a closed form to two algebraic equations, whose degree sum equals the number of unknowns of the system of equations of algebraic sum of equal powers.     

Scale changes and shared information in bargaining: An experimental study

This paper reports one of a series of experiments designed to test aspects of various game-theoretic models of bargaining. The results of this experiment consolidate those of previous experiments, which are first reviewed. The principal new result of this experiment is the observation, under conditions of partial information, of systematic violations of the axiom of independence of equivalent utility representations. The adequacy of various theories of bargaining to describe the observed data is also tested and discussed.     

The thermodynamically consistent equations of a thermoelastic saturated porous medium

A phenomenological model of a porous medium saturated with fluid is considered with in the framework of the hypothesis of interpenetrating continua. Assuming that there are no phase transitions, that the contribution of pulsations to the stress tensor and kinetic energy is small, and the components of the medium are in thermal equilibrium, mass, momentum and energy equations and a law of conservation of compatibility of the deformations and velocities are formulated. Using a representation of the force of interaction of the components in the form of the sum of equilibrium and dissipative components, a new form of inequality is obtained for the rate of entropy production. A definition of a thermoelastic saturated porous medium is given. The symmetry group of such a medium is considered as a set of two groups, corresponding to the symmetry of the skeleton and the fluid. It is shown that, in the class of thermoelastic porous media with an arbitrary type of symmetry of the skeleton, the saturating fluid can only be an ideal fluid, while the thermodynamic potentials and the porosity, stresses and entropies determined by them do not depend on the temperature gradient and the relative fluid velocity. It is found that the condition of incompressibility of only one of the components of the medium leads to the elimination of the porosity from the governing relations, rather than to kinematic limitations. The limitations imposed on the governing relations by the principle of thermodynamic consistency and the requirement of independence of the choice of the frame of reference are investigated. A form of the governing relations, necessary and sufficient to satisfy these principles, is obtained. It is shown that the Biot equations are one of the forms of thermodynamically consistent governing relations. A thermodynamic validation of the effective-stress tensor is given.     

On Dynamical Properties of a One-Parameter Family of Transformations Arising in Averaging of Semigroups

To describe the dynamics of quantum systems with degenerate symmetric but not self-adjoint Hamiltonian, we consider the Naimark extension of the Hamiltonian to a self-adjoint operator in an extended Hilbert space. We relate to the symmetric Hamiltonian a one-parameter family of averaged dynamical transformations of the set of quantum states obtained from a unitary group of transformations of the extended Hilbert space by using a conditional expected value to an algebra of bounded operators acting in the original space. We establish the absence of the semigroup property and injectivity of the family of averaged dynamical transformations. We obtain a representation of trajectories of the averaged family of dynamical transformations by maximum points of functionals on the space of mappings of the time interval into the set of quantum states.     

Algorithm of Exact Methods Obtaining of the Solution and Dynamical Flexibilities of Subsystem of the Mechatronic system

In this paper the application of analysis of transverse vibrating subsystem of mechatronic systems by means of the exact were the main purposes of work to solve the task of assignment of frequency-modal analysis and characteristics of mechatronic system. At first the problem of analysis in the form of the one differential equation of motion of mechanical subsystem or of the set of state equations of considered mechatronic model of object has been formulated and solved. Classic method to solve this problem have been used. The considered transverse vibrating mechanical subsystems of mechatronic system are a continuous beams with constant cross-section, with free ends and/or clamped on one end. A ring transducer, which is the integral part of mechatronic system, extorted by harmonic force or voltage excitation, is assumed to be perfectly bonded to the beam surface. Parameters of the transducer have important influence on values of natural frequencies and on form of characteristics of considered mechatronic system. The poles of dynamical characteristic calculated by mathematical exact method and the Galerkin's method have approximately different or the same values; it is depended on the combination of boundary conditions of elementary beam. The results of the calculations were not only presented in mathematical form but also as a transients of examined dynamical characteristic which are function of frequency of assumed excitation. Practical implications of this work is to present the introduction to synthesis of considered class of mechanical and/or mechatronic beam-systems with a constant changeable cross-section. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Prediction of the deformation properties of a polymeric composite with a granular-fibrous filler

Conclusion A variant of calculation of the characteristics of the deformation properties of a hybrid composite containing a complex disperse filler in the form of granular particles and short fibers was proposed. The effect of aggregation of the granular filler, the statistical distributions of the fibers by lengths and orientation in the material, and the anisotropy of the fibers are taken into consideration in the calculation. The statistical distribution of the orientation of the fibers is given by a function proportional to the distance from the center to the surface of a triaxial ellipsoid in the corresponding direction. The uniform random distribution of the fibers in bulk and in the plane is a special case of this distribution. The results of the analysis of the effect of the parameters of the statistical distributions of the length and orientation of short fibers on the elasticity characteristics of a composite are reported. The dependence of the creep of the composite on the ratio of the concentration of the components of the complex filler was determined, and the efficiency of partial replacement of a granular filler by a short-fiber filler to inhibit creep of the composite was demonstrated. The possibilities of predicting the long-term creep were experimentally confirmed on the example of LDPE filled with ground limestone and short glass fibers.Presented at the Sixth All-Union Conference on the Mechanics of Polymer and Composite Materials (Riga, November, 1986).Translated from Mekhanika Kompozitnykh Materialov, No. 5, pp. 898–909, September–October, 1987.     

Coupled damping vibrations of composite structures

This paper summarizes the results of a series of papers on developing methods for deter-mining elastic-dissipative characteristics of polymer composite materials (PCM) and predicting the corresponding dynamical responses from structures. For a prismatic anisotropic bar which is arbitrarily oriented with respect to the axes of elastic symmetry of an orthotropic plate, a mathematical model and a method for solving the problem of damping bending-torsional vibrations are developed. The interaction between the vibration modes of a composite bar is examined. Basic provisions of an iterative method for determining elastic and dissipative characteristics of PCM are formulated. Mathematical models of damping vibrations of fibrous composite thin-walled bars and plates are developed, and a two-stage method for solving the resulting complex eigenvalue problems is put for-ward. The effect of the composition and the reinforcement structure on the eigenfrequencies and the coefficients of mechanical losses of the structures in question is discussed. Controlability of the values of resonance frequencies and the coefficients of the mechanical losses due to variation of the degree of anisotropy of fiber materials and the degree of inhomogeneity of the structure over thickness has been demonstrated. The widely used method of potential energy of eigenforms is shown as being capable of providing correct values of the coefficients of mechanical losses only up to η = 0.02–0.03, resulting in significant errors for higher dissipative characteristics of composite structures. For the method presented, examples of practical implementation in developing vibration absorptive composite structures are given.     

On the Growth of Components of Meromorphic Solutions of Systems of Complex Differential Equations

This paper investigates the problem of the growth of the components of meromorphic solutions of a class of a system of complex algebraic differential equations, and generalized some of N. Toda＇s results concerning the growth of differential equations to the case of systems of differential equations. The paper considers the existence of admissible solutions of the system of differential equations.     

Nonlinear Electromagnetic Delay of Electromagnetic Signals Propagating in the Magnetic Meridian Plane of Pulsars and Magnetars

We analyze the propagation of electromagnetic waves in the magnetic meridian planes of neutron stars with a strong magnetic field in the framework of the parameterized post-Maxwellian electrodynamics of the vacuum. The origin of these electromagnetic waves is the curvature emission of X-rays and gamma rays from high-energy electrons in the vicinity of the magnetic poles of neutron stars. We show that in the case of a slowly varying intensity of X-ray and gamma-ray emission, the delay of the slow normal mode of electromagnetic waves relative to the fast mode results in a shift of the time dependence of the intensity of the detected radiation with one polarization relative to that of the radiation with the orthogonal polarization. In the case of single X-ray or gamma-ray pulses, the delay effect results in the polarization of the detected pulse varying during the pulse length, the leading edge of all pulses being polarized normally to the magnetic equator plane of the neutron star. We note that the modern level of the experimental technique, in principle, allows observing the manifestations of the delay effect for signals of different polarizations.     

On nonoscillating solutions of Emden-Fowler-type equations

The asymptotic properties of nonoscillating solutions of Emden-Fowler-type equations of arbitrary order are considered. The paper contains the results of the study of the asymptotic properties of solutions with integer-valued asymptotics as well as of solutions arising from the rapid decrease of the coefficient of the equation. To analyze the asymptotic behavior of solutions of the equations, methods of power geometry are used.     

Continuity of Entropy for Bimodal Maps

We characterize the continuity of the topological entropy ofbimodal maps of the interval and of the circle in terms of thebehaviour of the iterates of the turning points and of the valueof the topological entropy of the map under consideration. Inthe case of bimodal circle maps of degree one we also studythe continuity of the entropy in terms of their rotation intervals.     

Some laws of the elastic properties of oriented fibrous polymers and fibers

The author considers the results of calculations of the limiting values of the elastic properties (modulus of elasticity and load-extension diagram) of the main types of chemical fibers, using a model with "ideal orientation" of the molecules and the derived laws of deformation of polymer chains. A method is proposed for calculating the elastic properties of "ideally oriented" polymers from the velocity of propagation of an elastic deformation pulse and the effective density of the "skeletons" of the polymer chains. Values of the moduli of elasticity of the amorphous regions of the structure of oriented polymers are calculated. The calculated results are compared with experimental data on the elastic properties of fibers.Mekhanika Polimerov, Vol. 2, No. 1, pp. 34–42, 1966Paper read at the XIV All-Union Conference on High-Molecular Compounds, Oriented State.     

Riemannian Submersions with Discrete Spectrum

We prove some estimates on the spectrum of the Laplacian of the total space of a Riemannian submersion in terms of the spectrum of the Laplacian of the base and the geometry of the fibers. When the fibers of the submersions are compact and minimal, we prove that the spectrum of the Laplacian of the total space is discrete if and only if the spectrum of the Laplacian of the base is discrete. When the fibers are not minimal, we prove a discreteness criterion for the total space in terms of the relative growth of the mean curvature of the fibers and the mean curvature of the geodesic spheres in the base. We discuss in particular the case of warped products.     

On the “genetic memory” of chaotic attractor of the barotropic ocean model

The structure of attractor of barotropic ocean model is studied in this paper. Theorems of the existence of the attractor for the finite dimensional approximation of this model are proved as well as its convergence to the attractor of the model itself. Some properties of stationary solutions of this model and their stability are discussed.The structure of the attractor is partially explained by the sequence of bifurcations the system is subjected to by variations of leading parameters. The principal feature of the studied system is the existence of two “almost invariant” basins of chaotic attractor with very rare transitions between them. This is related to the rise of a couple of non-symmetric stable stationary solutions in the model with symmetric forcing.The “memory” of chaos appears also in the presence of maxima in the spectrum of energy. These maxima correspond either to the principal frequency of the limit cycle which arose in the Hopf bifurcation, or to the frequencies of the Feigenbaum phenomenon.