Smooth approximation of singular perturbations

We consider a certain subclass of self-adjoint extensions of the symmetric operator

     

Spectral asymptotics for nonsmooth perturbations of the harmonic oscillator

We obtain the spectral asymptotics of a nonsmooth perturbation of the one-dimensional harmonic oscillator. We use the technique of perturbation theory which is based on an asymptotic presentation of the part of the kernel of the nonperturbed operator resolvent in some neighborhood of an eigenvalue.     

Random perturbations effects on the stability of Tension Leg Platforms in offshore engineering and of other large structures

We analyze the stability of Tension Leg Platform offshore structures, as well as of some large flexible space structures, under random perturbations of their main parameters. An asymptotic theory and numerical methods are used as tools to study moment stability, and numerical simulations of the Monte Carlo type are also conducted for both purposes, of checking the results of the asymptotic theory and to handle cases when such a theory cannot be applied.     

One-dimensional ventsel-freidlin theory and its analogue for singular perturbations of degenerate diffusions

We make some remarks about deriving the large deviations estimates for the Ventsel-Freidlin perturbed system and adapt these methods to derive similar results for singular perturbations of degenerate one-dimensional diffusions where β and ω are independent Brownian motions. This corresponds to a singular perturbation of the degenerate second-order operator     

On the classical limit of Bohmian mechanics for Hagedorn wave packets

We consider the classical limit of quantum mechanics in terms of Bohmian trajectories. For wave packets as defined by Hagedorn we show that the Bohmian trajectories converge to Newtonian trajectories in probability.     

Quantum non-Markovian Langevin equations and transport coefficients for an inverted oscillator

Based on the non-Markovian quantum Langevin equations, we obtain time-dependent transport coefficients for an inverted oscillator coupled linearly in the coordinate to a thermostat. We comparatively analyze the diffusion coefficients for harmonic and inverted oscillators and study the role of quantum statistical effects in the passage through a parabolic barrier. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 3, pp. 425–443, September, 2008.     

On the exact discretization of the classical harmonic oscillator equation

We discuss the exact discretization of the classical harmonic oscillator equation (including the inhomogeneous case and multidimensional generalizations) with a special stress on the energy integral. We present and suggest some numerical applications.     

Relative eigenvalue and singular value perturbations of scaled diagonally dominant matrices

The paper derives improved relative perturbation bounds for the eigenvalues of scaled diagonally dominant Hermitian matrices and new relative perturbation bounds for the singular values of symmetrically scaled diagonally dominant square matrices. The perturbation result for the singular values enlarges the class of well-behaved matrices for accurate computation of the singular values. AMS subject classification (2000)  65F15     

Stability criterion for small perturbations for a quasi-gasdynamic system of equations

The stability of small perturbations against a constant background is studied for a system of quasi-gasdynamic equations in an arbitrary number of space variables. It is established that, for a fixed adiabatic exponent γ, the stability is determined only by the background Mach number, and a necessary and sufficient condition for stability at any Mach number is $\gamma \leqslant \bar \gamma $ , where $\bar \gamma \approx 6.2479$ . The proof is based on a direct analysis of the corresponding complex characteristic numbers depending on several parameters. The multidimensional case is successfully reduced to the one-dimensional one. Then, the generalized Routh-Hurwitz criterion is applied in conjunction with analytical calculations based on Mathematica.     

Global attractors for singular perturbations of the Cahn-Hilliard equations

We consider the singular perturbations of two boundary value problems, concerning respectively the viscous and the nonviscous Cahn-Hilliard equations in one dimension of space. We show that the dynamical systems generated by these two problems admit global attractors in the phase space , and that these global attractors are at least upper-semicontinuous with respect to the vanishing of the perturbation parameter.     

A unified approach to the large deviations for small perturbations of random evolution equations

LetX ^ɛ = {X ^ɛ (t ; 0 ⩽t ⩽ 1 } (ɛ > 0) be the processes governed by the following stochastic differential equations:

Small perturbations and pairs of matrices that have the same immanent

We study the behaviour of the covering number of an element of a family of vectors under small perturbations. We apply this study to obtain results on the pairs of matrices that have the same immanent.     

Asymptotics of Sobolev embeddings and singular perturbations for the -Laplacian

We consider the best constant for the embedding of into where , . Here with a smooth, bounded domain in and a large positive number. It is proven by the validity of the expansion

as , where is a positive constant depending on and . The behavior of associated extremals, which satisfy an equation involving the -Laplacian operator, is also analyzed.

     

Dispersion and Strichartz estimates for some finite rank perturbations of the Laplace operator

We consider the dispersion properties in L^p spaces of Schrödinger hamiltonians with a large number of obstacles modelled by rank one perturbations. We obtain both for the dispersion an Strichartz estimates nonperturbative results with respect to the coupling constants.     

Stability and bifurcation for periodic perturbations of the equilibrium of an oscillator with infinite or infinitesimal oscillation frequency

We consider small perturbations periodic in time of an oscillator whose restoring force has a leading term with exponent 3 or 1/3. The first case corresponds to oscillations with infinitesimal frequency and the second case to oscillations with infinite frequency. The smallness of the perturbation is determined both by the smallness of the considered neighborhood of the equilibrium point and by a small nonnegative parameter ε. For ε=0, the stability of the equilibrium point is studied. For ε>0, we find conditions for an invariant two-dimensional torus to branch off with “soft” or “rigid” loss of stability with loss index 1/2. Translated fromMatematicheskie Zametki, Vol. 65, No. 3, pp. 323–335, March, 1999.     

On the classical descriptions of the quantum phenomena in the harmonic oscillator and in a charged particle under the coulomb force

In this work it is shown that the intrinsic phenomenon (the quantization of the energy) that appears in the first and simple systems studied initially by the quantum theory as the harmonic oscillator and the movement of a charged particle under the Coulomb force, can be obtained from the study of dissipative systems. In others words, we show that this phenomenon of the quantization of the energy of a particle which moves as an harmonic oscillator and which loses and wins energy can be obtained via a classical system of equations. The same also applies to the phenomena of the quantization of the energy of a charged particle which moves under the Coulomb force and which loses and wins energy.