On the possibility of gravitational collapse in the relativistic theory of gravity

In the example of dust matter, it is shown that a gravitational attractive force does not lead to the formation of “black holes” in the relativistic theory of gravity. It is proved that in the absence of matter, the gravitational field is also absent. Therefore, a vacuum is not a source of a gravitational field. The mechanism of energy production in the process of the accumulation of matter into massive objects is discussed.     

Relativistic theory of gravity and a Brans-Dicke scalar field

The natural generalization of the relativistic theory of gravity (RTG) by incorporating a Brans-Dicke scalar field is discussed. The equation for a scalar-tensor gravitational field in Minkowski space and the expression for the total energy-momentum metric tensor of a gravitational field and nongravitational matter is derived from the variational principle with a gravitational Lagrangian quadratic in the first derivatives of the scalar and tensor gravitational potentials. The two-parameter spherically symmetrical static solution for vacuum equations with a zero mass tensor graviton was obtained. This solution has a true singular Schwarzschild surface. In the case of a nonzero mass graviton, an approximate nonsingular solution for the beginning of the universe was obtained. It is noted that in the frame of the scalar-tensor generalization of RTG, a nonsingular homogeneous isotropic cosmology can be represented, not only by cyclic models, but also by models with an infinitely expanding universe and a simultaneously decreasing gravitational scalar.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 106, No. 2, pp. 325–332, February, 1996.     

On stability radii of infinite-dimensional time-varying discrete--time systems

This note introduces a stability radius for discrete–timelinear time–varying systems on Banach spaces under structuredtime–varying perturbations of multiple–output feedbacktype. Additive perturbations are considered which can be representedas a series of infinitely many perturbation terms. We derivecounterparts to some results established for time–varyingdifferentiable systems in [5, 7] and apply the results to periodicsystems.     

Rotations of a near-symmetrical satellite in an elliptical orbit with Mercury-type resonance

The motion of a satellite, i.e., a rigid body, about to the centre of mass under the action of the gravitational moments of a central Newtonian gravitational field in an elliptical orbit of arbitrary eccentricity is investigated. It is assumed that the satellite is almost dynamically symmetrical. Plane periodic motions for which the ratio of the average value of the absolute angular velocity of the satellite to the average motion of its centre of mass is equal to 3/2 (Mercury-type resonance) are examined. An analytic solution of the non-linear problem of the existence of such motions and their stability to plane perturbations is given. In the special case in which the central ellipsoid of inertia of the satellite is almost spherical, the stability to spatial perturbations is also examined, but only in a linear approximation. ©2008.     

Variation of Lyapunov exponents on a strange attractor

Summary We introduce the idea of local Lyapunov exponents which govern the way small perturbations to the orbit of a dynamical system grow or contract after afinite number of steps,L, along the orbit. The distributions of these exponents over the attractor is an invariant of the dynamical system; namely, they are independent of the orbit or initial conditions. They tell us the variation of predictability over the attractor. They allow the estimation of extreme excursions of perturbations to an orbit once we know the mean and moments about the mean of these distributions. We show that the variations about the mean of the Lyapunov exponents approach zero asL and argue from our numerical work on several chaotic systems that this approach is asL ^–v. In our examplesv 0.5–1.0. The exponents themselves approach the familiar Lyapunov spectrum in this same fashion.     

Existence and Uniqueness of a Solution to the Cauchy Problem for the Damped Boussinesq Equation

We consider the Cauchy problem for the damped Boussinesq equation governing long wave propagation in a viscous fluid of small depth. For the cases of one, two, and three space dimensions local in time existence and uniqueness of a solution is proved. We show that for discontinuous initial perturbations this solution is infinitely differentiable with respect to time t and space co-ordinates for t>0 on a bounded time interval.     

Definition and properties of characteristic frequencies of a linear equation

The definition of the characteristic frequencies of zeroes and changes of sign for solutions is given. It is equal to the upper medium (with respect to the time half-axis) of their number on the half-interval of length π. We also define the main frequencies for a linear homogeneous equation of order n. These main frequencies for an equation with constant coefficients coincide with the absolute values of the imaginary parts of the roots of the corresponding characteristic polynomial. It is proved that for the second-order equation the main frequencies are the same for all solutions and that they are stable with respect to uniformly small and infinitely small perturbations of the coefficients. For the third-order equation they can be different, and for any of the main frequencies an example of nonstability is given. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 25, pp. 249–294, 2005.     

Buffer phenomenon in systems close to two-dimensional Hamiltonian ones

Plane Hamiltonian systems perturbed by small time-periodic terms are considered. The conditions are established under which exponentially stable periodic solutions are accumulated infinitely in these systems as the perturbations tend to zero or, in other words, the buffer phenomenon occurs. It is shown that this phenomenon is typical for a wide range of classical mechanical problems described by equations of the pendulum type.     

Curvature phase transition inR 2 quantum gravity and induction of Einstein gravity

The phase transition with respect to the curvature in the effective potential ofR ² quantum gravity with matter is studied. The effective potential is calculated in the framework of the renormalization-group approach up to terms linear in the curvature. A universal expression is obtained for the induced gravitational and cosmological constants. The effective potential, and also the induced cosmological and gravitational constants depend on the relationships between the coupling constants of the original theory and on the gauge parameters. When the matter is represented by a single scalar field values fixed by asymptotic freedom are chosen for the coupling constants. There is no gauge dependence for the unified parametrization-and gauge-invariant effective action.Tomsk State Pedagogical Institute. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 90, No. 3, pp. 469–480, March, 1992.     

Stability and Sensitivity of Tridiagonal LU Factorization without Pivoting

In this paper the accuracy of LU factorization of tridiagonal matrices without pivoting is considered. Two types of componentwise condition numbers for the L and U factors of tridiadonal matrices are presented and compared. One type is a condition number with respect to small relative perturbations of each entry of the matrix. The other type is a condition number with respect to small componentwise perturbations of the kind appearing in the backward error analysis of the usual algorithm for the LU factorization. We show that both condition numbers are of similar magnitude. This means that the algorithm is componentwise forward stable, i.e., the forward errors are of similar magnitude to those produced by a componentwise backward stable method. Moreover the presented condition numbers can be computed in O(n) flops, which allows to estimate with low cost the forward errors. AMS subject classification (2000) 65F35, 65F50, 15A12, 15A23, 65G50.Received October 2003. Accepted August 2004. Communicated by Per Christian Hansen.Froilán M. Dopico: This research has been partially supported by the Ministerio de Ciencia y Tecnología of Spain through grants BFM2003-06335-C03-02 (M. I. Bueno) and BFM2000-0008 (F. M. Dopico).     

Viscous flow through corrugated tube by boundary element method

Summary The creeping flow of a Newtonian fluid through a sinusoidally-corrugated tube is solved by the Boundary Element Method. Agreement with another numerical method is noted. In addition, it is shown that previous perturbation theory is valid only when the corrugation amplitude is small (<0.3a) and the wavelength of the corrugation is large (>3a), wherea is the mean radius of the tube.

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Zusammenfassung Das Problem der schleichenden Bewegung eines Newton'schen Fluids durch ein Rohr mit sinusförmig gewellter Wand wird mit Hilfe der Boundary Element-Methode gelöst. Übereinstimmung mit einer anderen numerischen Methode wird festestellt. Zudem wird gezeigt, daß eine früher gefundene Störungstheorie nur gültig ist wenn die Wellenamplitude klein (<0.3a) und die Wellenlänge groß (>3a) ist (a=mittlerer Rohrradius).

     

Scattering of Dirac Particles by Electromagnetic Fields with Small Support in Two Dimensions and Effect from Scalar Potentials

We study the asymptotic behavior of scattering amplitudes for the scattering of Dirac particles in two dimensions when electromagnetic fields with small support shrink to point-like fields. The result is strongly affected by perturbations of scalar potentials and the asymptotic form changes discontinuously at half-integer fluxes of magnetic fields even for small perturbations. The analysis relies on the behavior at low energy of resolvents of magnetic Schrödinger operators with resonance at zero energy. The magnetic scattering of relativistic particles appears in the interaction of cosmic string with matter. We discuss this closely related subject as an application of the obtained results. Communicated by Bernard Helffersubmitted 05/05/03, accepted 31/07/03     

Traveling wave solutions of harmonic heat flow

We prove the existence of a traveling wave solution of the equation in an infinitely long cylinder of radius R, which connects two locally stable and axially symmetric steady states at x ₃ = ±∞. Here u is a director field with values in The traveling wave has a singular point on the cylinder axis. Letting R→ ∞ we obtain a traveling wave defined in all space.     

The multifractal analysis of Birkhoff averages for conformal repellers under random perturbations

We study the stability of multifractal structures for dynamical systems under small perturbations. For a repeller associated with an expanding C ^1+β -conformal topological mixing map, we show that the multifractal structure of Birkhoff averages is stable under small random perturbations.     

Stability of orbit spaces of endomorphisms

In this paper it is first proved that, for a hyperbolic set of aC ¹ (non-invertible) endomorphism of a compact manifold, the dynamical structure of its orbit space (inverse limit space) is stable underC ¹-small perturbations and is semi-stable underC ⁰-small perturbations. It is then proved that if an Axiom A endomorphism satisfies no-cycle condition then its orbit space is Θ-stable andR-stable underC ¹-small perturbations and is semi-Θ-stable and semi-R-stable underC ⁰-small perturbations. This research is supported by the National Natural Science Foundation of China     

Smooth approximations to solutions of perturbed Weyl and Dirac equations

Iterations of spherical mean values of the initial condition are used to approximate solutions of the Weyl and the Dirac equations in the Hilbert space [L ²(ℝ ⁿ )] ^N . In this work we smooth the potential and the initial condition, without restrictions on the increase rate as ∥x∥ → ∞. By iterating perturbations of such mean values, we prove apriori estimates for the approximate solutions of the perturbed Weyl and Dirac equations in the topology of uniform convergence of time and space derivatives on compact subsets in ℝ×ℝ ⁿ . Translated fromMatematicheskie Zametki, Vol. 60, No. 4, pp. 538–555, October, 1996. I thank Professor A. M. Chebotarev and Professor R. Quezada for their comments and help in the preparation of this paper. This research was partially supported by SNI and CONACyT-E, grant No 0233P-E9506.     

On existence and uniqueness of stationary points in semi-infinite optimization

The paper deals with semi-infinite optimization problems which are defined by finitely many equality constraints and infinitely many inequality constraints. We generalize the concept of strongly stable stationary points which was introduced by Kojima for finite problems; it refers to the local existence and uniqueness of a stationary point for each sufficiently small perturbed problem, where perturbations up to second order are allowed. Under the extended Mangasarian-Fromovitz constraint qualification we present equivalent conditions for the strong stability of a considered stationary point in terms of first and second derivatives of the involved functions. In particular, we discuss the case where the reduction approach is not satisfied. Received June 30, 1995 / Revised version received October 9, 1998? Published online June 11, 1999     

On the Signs of Fourier Coefficients of Cusp Forms

Let be a discrete subgroup of SL(2, ) with a fundamental region of finite hyperbolic volume. (Then, is a finitely generated Fuchsian group of the first kind.) Let 0}} {a(n)e^{2\pi i(n + {\kappa })z/{\lambda }} } ,{ }z \in \mathcal{H}.$$ " align="middle" border="0"> be a nontrivial cusp form, with multiplier system, with respect to . Responding to a question of Geoffrey Mason, the authors present simple proofs of the following two results, under natural restrictions upon . Theorem. If the coefficients a(n) are real for all n, then the sequence {a(n)} has infinitely many changes of sign. Theorem. Either the sequence {Re a(n)} has infinitely many sign changes or Re a(n) = 0 for all n. The same holds for the sequence {Im a(n)}.     

Solution of the Ulam Stability Problem for Euler–Lagrange Quadratic Mappings

In 1940 S. M. Ulam proposed at the University of Wisconsin theproblem: “Give conditions in order for a linear mapping near an approximately linear mapping to exist.” In 1968 S. U. Ulam proposed the moregeneral problem: “When is it true that by changing a little the hypotheses of a theorem one can still assert that the thesis of the theorem remains true or approximately true?” In 1978 P. M. Gruber proposed theUlam type problem: “Suppose a mathematical object satisfies a certain property approximately. Is it then possible to approximate this object by objects, satisfying the property exactly?” According to P. M. Gruber this kind of stability problems is of particular interest in probability theory and in the case of functional equations of different types. In 1982–1996 we solved the above Ulam problem, or equivalently the Ulam type problem for linear mappings and established analogous stability problems. In this paper we first introduce newquadratic weighted meansandfundamental functional equationsand then solve theUlam stability problemfornon-linear Euler–Lagrange quadratic mappingsQ:X → Y, satisfying a mean equation and functional equation[formula]for all 2-dimensional vectors (x₁, x₂) X², withXa normed linear space (Y a real complete normed linear space), and any fixed pair (a₁, a₂) of realsa_iand any fixed pair (m₁, m₂) of positive realsm_i(i = 1, 2), [formula]     

Pulsating magnetohydrodynamic flow in an annular channel

Pulsating laminar flow of a viscous incompressible electrically conducting fluid in an annular channel between two infinitely long circular cylinders under a radial impressed magnetic field is considered. The solutions of magnetohydrodynamic equations have been obtained on the assumption that the space between two cylinders is small compared to their mean radius. The solutions were also obtained on the assumption of small magnetic Reynolds number with special consideration of those for low and high frequencies.