Quasipoint spectral measures in the theory of dynamical systems of conflict

A notion of spectral measure with quasipoint spectrum is introduced within the framework of the dynamic picture of interacting physical systems. It is shown that, in the case of conflict interaction with point measures, only quasipoint singularly continuous measures can be transformed into measures with pure point spectrum.     

Point Spectrum of Singularly Perturbed Self-Adjoint Operators

We study the inverse spectral problem for the point spectrum of singularly perturbed self-adjoint operators. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 5, pp. 654–658, May, 2005.     

A singular approach to a class of impulsive differential equation

In this paper, a singular approach to study the solutions of an impulsive differential equation from a qualitative and quantitative point of view is proposed. In the approach, a suitable singular perturbation term is introduced and a singularly perturbed system with infinite initial values is defined, in which, the reduced problem of the singularly perturbed system is exactly the impulsive differential equation under consideration. Then the boundary layer function method is applied to construct the uniformly valid asymptotic solutions to the singularly perturbed system. Based on the continuous asymptotic solution, the discontinuous solutions of the impulsive differential equation are described and approximated. An example, namely, a classical Lotka-Volterra prey-predator model with one pulse is carried out to illustrate the main results.     

Asymptotic normality of fluctuations of the procedure of stochastic approximation with diffusive perturbation in a Markov medium

We consider the asymptotic normality of a continuous procedure of stochastic approximation in the case where the regression function contains a singularly perturbed term depending on the external medium described by a uniformly ergodic Markov process. Within the framework of the scheme of diffusion approximation, we formulate sufficient conditions for asymptotic normality in terms of the existence of a Lyapunov function for the corresponding averaged equation. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 12, pp. 1686–1692, December, 2006.     

Consequences of Pure Point Diffraction Spectra for Multiset Substitution Systems

   Abstract. There is a growing body of results in the theory of discrete point sets and tiling systems giving conditions under which such systems are pure point diffractive. Here we look at the opposite direction: what can we infer about a discrete point set or tiling, defined through a primitive substitution system, given that it is pure point diffractive? Our basic objects are Delone multisets and tilings, which are self-replicating under a primitive substitution system of affine mappings with a common expansive map Q . Our first result gives a partial answer to a question of Lagarias and Wang: we characterize repetitive substitution Delone multisets that can be represented by substitution tilings using a concept of ``legal cluster.' This allows us to move freely between both types of objects. Our main result is that for lattice substitution multiset systems (in arbitrary dimensions), being a regular model set is not only sufficient for having pure point spectrum—a known fact—but is also necessary. This completes a circle of equivalences relating pure point dynamical and diffraction spectra, modular coincidence, and model sets for lattice substitution systems begun by the first two authors of this paper.     

Consequences of Pure Point Diffraction Spectra for Multiset Substitution Systems

Abstract. There is a growing body of results in the theory of discrete point sets and tiling systems giving conditions under which such systems are pure point diffractive. Here we look at the opposite direction: what can we infer about a discrete point set or tiling, defined through a primitive substitution system, given that it is pure point diffractive? Our basic objects are Delone multisets and tilings, which are self-replicating under a primitive substitution system of affine mappings with a common expansive map Q . Our first result gives a partial answer to a question of Lagarias and Wang: we characterize repetitive substitution Delone multisets that can be represented by substitution tilings using a concept of ``legal cluster.' This allows us to move freely between both types of objects. Our main result is that for lattice substitution multiset systems (in arbitrary dimensions), being a regular model set is not only sufficient for having pure point spectrum—a known fact—but is also necessary. This completes a circle of equivalences relating pure point dynamical and diffraction spectra, modular coincidence, and model sets for lattice substitution systems begun by the first two authors of this paper.     

On the structure of solutions of a system of singularly perturbed differential equations with nondiagonalizable limit operator

We construct uniform asymptotics of a solution of a heterogeneous system of singularly perturbed differential equations in the case of nondiagonalizable limit operator. We consider the case where the spectrum of the limit operator contains an unstable element at the point x = 0. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 7, pp. 867–876, July, 1998.     

Connections to fixed points and Sil’nikov saddle-focus homoclinic orbits in singularly perturbed systems

We consider a singularly perturbed system depending on two parameters with two (possibly the same) normally hyperbolic center manifolds. We assume that the unperturbed system has an orbit that connects a hyperbolic fixed point on one center manifold to a hyperbolic fixed point on the other. Then we prove some old and new results concerning the persistence of these connecting orbits and apply the results to find examples of systems in dimensions greater than three that possess Sil’nikov saddle-focus homoclinic orbits. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 1, pp. 28–55, January, 2008.     

A UNIFORM NUMERICAL METHOD FOR QUASILINEAR SINGULAR PERTURBATION PROBLEM WITH A TURNING POINT

A nonlinear difference scheme is given for solving a quasilinear siagularly perturbed two-point boundary value problem with a turning point. The method uses non-equidistant discretization meshes. The solution of the scheme is shown to be first order accurate in the discrete L^∞ norm, uniformly in the perturbation parameter.     

The exit problem for small random perturbations of dynamical systems with a hyperbolic fixed point

We consider the Markov diffusion process ξ^∈(t), transforming when ɛ=0 into the solution of an ordinary differential equation with a turning point ℴ of the hyperbolic type. The asymptotic behevior as ɛ→0 of the exit time, of its expectation of the probability distribution of exit points for the process ξ^∈(t) is studied. These indicate also the asymptotic behavior of solutions of the corresponding singularly perturbed elliptic boundary value problems.     

Coercive singular perturbations

Summary We consider general boundary value problems with small parameter ɛ in the operator and boundary conditions. Both the perturbed and reduced operators are supposed to be elliptic. We point outnecessary andsufficient conditions of Shapiro-Lopatinsky type for the singularly perturbed problem to be coercive, i.e. for a two-sided a priori estimate to hold for its solutions uniformly with respect to ɛ. Entrata in Redazione il 6 luglio 1977.     

Multifrequency oscillations of singularly perturbed systems

We consider a system of differential equations that consists of two parts, a regularly perturbed and a singularly perturbed one. We assume that the matrix of the linear part of the regularly perturbed system has pure imaginary eigenvalues, while the matrix of the singularly perturbed part is hyperbolic; i.e., all of its eigenvalues have nonzero real parts.     

Boosted Simon‐Wolff Spectral Criterion and Resonant Delocalization

Discussed here are criteria for the existence of continuous components in the spectra of operators with random potential. First, the essential condition for the Simon‐Wolff criterion is shown to be measurable at infinity. By implication, for the i.i.d. case and more generally potentials with the K‐property, the criterion is boosted by a zero‐one law. The boosted criterion, combined with tunneling estimates, is then applied for sufficiency conditions for the presence of continuous spectrum for random Schrödinger operators. The general proof strategy that this yields is modeled on the resonant delocalization arguments by which continuous spectrum in the presence of disorder was previously established for random operators on tree graphs. In another application of the Simon‐Wolff rank‐one analysis we prove the almost sure simplicity of the pure point spectrum for operators with random potentials of conditionally continuous distribution.© 2015 Wiley Periodicals, Inc.     

Non-autonomous semilinear evolution equations with almost sectorial operators

Inspired by the theory of semigroups of growth α, we construct an evolution process of growth α. The abstract theory is applied to study semilinear singular non-autonomous parabolic problems. We prove that, under natural assumptions, a reasonable concept of solution can be given to such semilinear singularly non-autonomous problems. Applications are considered to non-autonomous parabolic problems in space of H?lder continuous functions and to a parabolic problem in a domain with a one dimensional handle. Marcelo J. D. Nascimento: Research partially supported by FAPESP # 02/11855-2, Brazil.     

Asymptotic solutions of the Cauchy problem for the singularly perturbed Korteweg-de Vries equation with variable coefficients

We propose an algorithm for the construction of an asymptotic solution of the Cauchy problem for the singularly perturbed Korteweg-de Vries equation with variable coefficients and prove a theorem on the estimation of its precision. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 1, pp. 122–132, January, 2007.     

Singularly perturbed and nonlocal modulation equations for systems with interacting instability mechanisms

Summary Two related systems of coupled modulation equations are studied and compared in this paper. The modulation equations are derived for a certain class of basic systems which are subject to two distinct, interacting, destabilising mechanisms. We assume that, near criticality, the ratio of the widths of the unstable wavenumber-intervals of the two (weakly) unstable modes is small—as, for instance, can be the case in double-layer convection. Based on these assumptions we first derive a singularly perturbed modulation equation and then a modulation equation with a nonlocal term. The reduction of the singularly perturbed system to the nonlocal system can be interpreted as a limit in which the width of the smallest unstable interval vanishes. We study and compare the behaviour of the stationary solutions of both systems. It is found that spatially periodic stationary solutions of the nonlocal system exist under the same conditions as spatially periodic stationary solutions of the singularly perturbed system. Moreover, these solutions can be interpreted as representing the same quasi-periodic patterns in the underlying basic system. Thus, the ‘Landau reduction’ to the nonlocal system has no significant influence on the stationary quasi-periodic patterns. However, a large variety of intricate heteroclinic and homoclinic connections is found for the singularly perturbed system. These orbits all correspond to so-called ‘localised structures’ in the underlying system: They connect simple periodic patterns atx → ± ∞. None of these patterns can be described by the nonlocal system. So, one may conclude that the reduction to the nonlocal system destroys a rich and important set of patterns.     

Paths leading to the Nash set for nonsmooth games

Maschler, Owen and Peleg (1988) constructed a dynamic system for modelling a possible negotiation process for players facing a smooth n-person pure bargaining game, and showed that all paths of this system lead to the Nash point. They also considered the non-convex case, and found in this case that the limiting points of solutions of the dynamic system belong to the Nash set. Here we extend the model to i) general convex pure bargaining games, and to ii) games generated by “divide the cake” problems. In each of these cases we construct a dynamic system consisting of a differential inclusion (generalizing the Maschler-Owen-Peleg system of differential equations), prove existence of solutions, and show that the solutions converge to the Nash point (or Nash set). The main technical point is proving existence, as the system is neither convex valued nor continuous. The intuition underlying the dynamics is the same as (in the convex case) or analogous to (in the division game) that of Maschler, Owen, and Peleg. Received August 1997/Final version May 1998     

Spectral methods for some singularly perturbed third order ordinary differential equations

Spectral methods with interface point are presented to deal with some singularly perturbed third order boundary value problems of reaction-diffusion and convection-diffusion types. First, linear equations are considered and then non-linear equations. To solve non-linear equations, Newton’s method of quasi-linearization is applied. The problem is reduced to two systems of ordinary differential equations. And, then, each system is solved using spectral collocation methods. Our numerical experiments show that the proposed methods are produce highly accurate solutions in little computer time when compared with the other methods available in the literature.      

Asymptotic integration of systems of integro-differential equations with degeneracies

We construct asymptotic solutions of singularly perturbed homogeneous and heterogeneous systems of integro-differential Fredholm-type equations with degenerate matrix at the derivative. Translated from Ukrainskii Matematicheskii Zhumal, Vol. 51, No. 2, pp. 170–180, February, 1999.     

On the existence of almost-pure-strategy Nash equilibrium in <Emphasis Type="Italic">n</Emphasis>-person finite games

This paper gives wide characterization of n-person non-coalitional games with finite players’ strategy spaces and payoff functions having some concavity or convexity properties. The characterization is done in terms of the existence of two-point-strategy Nash equilibria, that is equilibria consisting only of mixed strategies with supports being one or two-point sets of players’ pure strategy spaces. The structure of such simple equilibria is discussed in different cases. The results obtained in the paper can be seen as a discrete counterpart of Glicksberg’s theorem and other known results about the existence of pure (or “almost pure”) Nash equilibria in continuous concave (convex) games with compact convex spaces of players’ pure strategies.