Exponential Asymptotics

A temperature field in a thin moving sheet heated with a laser beam is calculated. The power density in the beam is distributed according the Gauss function. Cooling effects caused by the free or forced convection in ambient gas is taken into account. The problem becomes two dimensional by averaging the temperature field over the sheet thickness. Twodimensional integral Fourier transformation on space coordinates is applied to solve the problem. It gives an analytical solution of the problem in the Fourier space. The inverse Fourier transformation is fulfilled and the solution is represented via an integral having exponential asymptotes at infinity.

In this work, we obtained an approximate bound state solution to Schrodinger with Hulthen plus exponential Coulombic potential with centrifugal potential barrier using parametric Nikiforov-Uvarov method. We obtained both the eigen energy and the wave functions to non-relativistic wave equations. We implement Matlab algorithm to obtained the numerical bound state energies for various values of adjustable screening parameter at various quantum state.. The developed potential reduces to Hulthen potential and the numerical bound state energy conform to that of existing literature.

A ray-tracing procedure is derived for the new exponential asymptotically bounded ͑EAB͒ velocity model introduced in Part I of this paper. The model inherits the properties of a medium with linear-velocity variation in depth in the shallow zone and of a medium with constant velocity in the deep zone. Two types of rays departing from the source point on the earth's surface exist in this model, depending on the takeoff angles. The rays of the first kind are symmetric arcs that return up to the earth's surface and have a limited maximum depth of propagation. The rays of the second kind propagate down to infinite depth. In the shallow region, they are curved lines, but at large depth they become asymptotically straight. The form of the ray is governed by the takeoff angle at the source point, where a critical angle splits the two kinds of rays. This critical angle depends only on the ratio between the velocity at the source point and the asymptotic velocity of the EAB model. We derive the formulae required to calculate the two kinds of rays and solve the inverse problem of two-point ray tracing. Finally, we construct the 2D-and 3D-isochron surfaces for a finite offset.

We establish the existence and robustness of layered, time-periodic solutions to a reaction-diffusion equation in a bounded domain in R n , when the diffusion coefficient is sufficiently small and the reaction term is periodic in time and bistable in the state variable. Our results suggest that these patterned, oscillatory solutions are stable and locally unique. The location of the internal layers is characterized through a periodic traveling wave problem for a related one-dimensional reaction-diffusion equation. This one-dimensional problem is of independent interest and for this we establish the existence and uniqueness of a heteroclinic solution which, in constant-velocity moving coodinates, is periodic in time. Furthermore, we prove that the manifold of translates of this solution is globally exponentially asymptotically stable.

A systematic method is presented to provide various equivalent solution formulas for exact solutions to the sine-Gordon equation. Such solutions are analytic in the spatial variable x and the temporal variable t, and they are exponentially asymptotic to integer multiples of 2π as x → ±∞. The solution formulas are expressed explicitly in terms of a real triplet of constant matrices. The method presented is generalizable to other integrable evolution equations where the inverse scattering transform is applied via the use of a Marchenko integral equation. By expressing the kernel of that Marchenko equation as a matrix exponential in terms of the matrix triplet and by exploiting the separability of that kernel, an exact solution formula to the Marchenko equation is derived, yielding various equivalent exact solution formulas for the sine-Gordon equation.

A systematic method is presented to provide various equivalent solution formulas for exact solutions to the sine-Gordon equation. Such solutions are analytic in the spatial variable x and the temporal variable t, and they are exponentially asymptotic to integer multiples of 2π as x → ±∞. The solution formulas are expressed explicitly in terms of a real triplet of constant matrices. The method presented is generalizable to other integrable evolution equations where the inverse scattering transform is applied via the use of a Marchenko integral equation. By expressing the kernel of that Marchenko equation as a matrix exponential in terms of the matrix triplet and by exploiting the separability of that kernel, an exact solution formula to the Marchenko equation is derived, yielding various equivalent exact solution formulas for the sine-Gordon equation.

We introduce a directed analog of the local chromatic number defined by Erdős et al. [Discrete Math. 59 (1986) 21-34] and show that it provides an upper bound for the Sperner capacity of a directed graph. Applications and variants of this result are presented. In particular, we find a special orientation of an odd cycle and show that it achieves the maximum of Sperner capacity among the differently oriented versions of the cycle. We show that apart from this orientation, for all the others an odd cycle has the same Sperner capacity as a single edge graph. We also show that the (undirected) local chromatic number is bounded from below by the fractional chromatic number while for power graphs the two invariants have the same exponential asymptotics (under the co-normal product on which the definition of Sperner capacity is based). We strengthen our bound on Sperner capacity by introducing a fractional relaxation of our directed variant of the local chromatic number.

Wireless networking is a method by which homes, telecommunications networks and enterprise (business) installations avoid the costly process of introducing cables into a building, or as a connection between various equipment locations. For such a reasons this technology has become popular. Though it is familiar, its wireless channel is vulnerable to the eavesdroppers during message delivery (security is the major problem). In the previous cases this problem was solved by cryptographic methods such as RSA public key cryptosystem. But due to expensive key distribution and improvement in decoding technology, the message transmitted is said to be unsecured. The problem can be overcome by using artificial noise generation. This paper investigates and studies how to deliver the message securely in the wireless network using artificial noise generation concept.

An extension of the classical Cramér-Lundberg approximation for ruin probabilities to a model of nonlinearly perturbed risk processes is presented. We introduce correction terms for the Cramér-Lundberg and diffusion type approximations, which provide the right asymptotic behaviour of relative errors in a perturbed model. The dependence of these correction terms on relations between the rate of perturbation and the speed of growth of an initial capital is investigated. Various types of perturbations of risk processes are discussed. The results are based on a new type of exponential asymptotics for perturbed renewal equations. is used to model the business of an insurance company. In (1.1) the positive constant c is the gross premium rate, N(t) is a Poisson process (with parameter λ) counting the number of claims on the company in the time-interval ଝ

The problem of source localization has assumed increased importance in recent years. In this paper, we formulate a continuous time adaptive localization algorithm, that permits a mobile agent to estimate the location of a stationary source, using only the measured distance from the source. The algorithm is shown to be exponentially asymptotically stable, under a persistent excitation condition that has an appealing interpretation. We quantify the fact that exponential asymptotic stability endows the algorithm with the ability to track slow, bounded but potentially persistent and nontrivial drift in the source.

We consider Glauber dynamics of classical spin systems of Ising type in the limit when the temperature tends to zero in finite volume. We show that information on the structure of the most profound minima and the connecting saddle points of the Hamiltonian can be translated into sharp estimates on the distribution of the times of metastable transitions between such minima as well as the low lying spectrum of the generator. In contrast with earlier results on such problems, where only the asymptotics of the exponential rates is obtained, we compute the precise pre-factors up to multiplicative errors that tend to 1 as T ↓ 0. As an example we treat the nearest neighbor Ising model on the 2 and 3 dimensional square lattice. Our results improve considerably earlier estimates obtained by Neves-Schonmann [NS] and Ben Arous-Cerf [BC] and Alonso-Cerf [AC]. Our results employ the methods introduced by Bovier, Eckhoff, Gayrard, and Klein in [BEGK1,BEGK2].

In this paper we continue the investigation of the spectral theory and exponential asymptotics of primarily discrete-time Markov processes, following Kontoyiannis and Meyn . We introduce a new family of nonlinear Lyapunov drift criteria, which characterize distinct subclasses of geometrically ergodic Markov processes in terms of simple inequalities for the nonlinear generator. We concentrate primarily on the class of multiplicatively regular Markov processes, which are characterized via simple conditions similar to (but weaker than) those of Donsker-Varadhan. For any such process Φ = {Φ(t)} with transition kernel P on a general state space X, the following are obtained.

We study stationary solutions of the Cahn-Hilliard equation in a bounded smooth domain which have an interior spherical interface (bubbles). We show that a large class of interior points (the "nondegenerate peak" points) have the following property: there exists such a solution whose bubble center lies close to a given nondegenerate peak point. Our construction uses among others the Liapunov-Schmidt

The problem of steady subcritical free surface flow past a submerged inclined step is considered. The asymptotic limit of small Froude number is treated, with particular emphasis on the effect that changing the angle of the step face has on the surface waves. As demonstrated by , the divergence of a power series expansion in powers of the square of the Froude number is caused by singularities in the analytic continuation of the free surface; for an inclined step, these singularities may correspond to either the corners or stagnation points of the step, or both, depending on the angle of incline. Stokes lines emanate from these singularities, and exponentially small waves are switched on at the point the Stokes lines intersect with the free surface. Our results suggest that for a certain range of step angles, two wavetrains are switched on, but the exponentially subdominant one is switched on first, leading to an intermediate wavetrain not previously noted. We extend these ideas to the problem of flow over a submerged bump or trench, again with inclined sides. This time there may be two, three or four active Stokes lines, depending on the inclination angles. We demonstrate how to construct a base topography such that wave contributions from separate Stokes lines are of equal magnitude but opposite phase, thus cancelling out. Our asymptotic results are complemented by numerical solutions to the fully nonlinear equations.

We establish the existence and robustness of layered, time-periodic solutions to a reaction-diffusion equation in a bounded domain in R n , when the diffusion coefficient is sufficiently small and the reaction term is periodic in time and bistable in the state variable. Our results suggest that these patterned, oscillatory solutions are stable and locally unique. The location of the internal layers is characterized through a periodic traveling wave problem for a related one-dimensional reaction-diffusion equation. This one-dimensional problem is of independent interest and for this we establish the existence and uniqueness of a heteroclinic solution which, in constant-velocity moving coodinates, is periodic in time. Furthermore, we prove that the manifold of translates of this solution is globally exponentially asymptotically stable.

The problem of source localization has assumed increased importance in recent years. In this paper, we formulate a continuous time adaptive localization algorithm, that permits a mobile agent to estimate the location of a stationary source, using only the measured distance from the source. The algorithm is shown to be exponentially asymptotically stable, under a persistent excitation condition that has an appealing interpretation. We quantify the fact that exponential asymptotic stability endows the algorithm with the ability to track slow, bounded but potentially persistent and nontrivial drift in the source.

In this article, we study the following second-order system of ordinary differential equations with dissipation u + cu + dAu + kH u = P t u∈ n t ∈ where c, d, and k are positive constants, H n → n is a locally Lipschitz function, and P R → n is a continuous and bounded function. A is a n × n matrix whose eigenvalues are positive. Under these conditions, we prove that for some values of c, d, and k this system has a bounded solution which is exponentially asymptotically stable. Moreover; if P t is almost periodic, then this bounded solution is also almost periodic. These results are applied to the spatial discretization of very well-known second-order partial differential equations.

We consider Glauber dynamics of classical spin systems of Ising type in the limit when the temperature tends to zero in finite volume. We show that information on the structure of the most profound minima and the connecting saddle points of the Hamiltonian can be translated into sharp estimates on the distribution of the times of metastable transitions between such minima as well as the low lying spectrum of the generator. In contrast with earlier results on such problems, where only the asymptotics of the exponential rates is obtained, we compute the precise pre-factors up to multiplicative errors that tend to 1 as $T\downarrow 0$. As an example we treat the nearest neighbor Ising model on the 2 and 3 dimensional square lattice. Our results improve considerably earlier estimates obtained by Neves-Schonmann and Ben Arous-Cerf. Our results employ the methods introduced by Bovier, Eckhoff, Gayrard, and Klein.

Localised structures appear in a wide variety of systems, arising from a pinning mechanism due to the presence of a small-scale pattern or an imposed grid. When there is a separation of lengthscales, the width of the pinning region is exponentially small and beyond the reach of standard asymptotic methods. We show how this behaviour can be obtained using a variational method, for two systems. In the case of the quadratic-cubic Swift-Hohenberg equation, this gives results that are in agreement with recent work using exponential asymptotics. Secondly, the method is applied to a discrete system with cubic-quintic nonlinearity, giving results that agree well with numerical simulations.

An extension of the classical Cramér-Lundberg approximation for ruin probabilities to a model of nonlinearly perturbed risk processes is presented. We introduce correction terms for the Cramér-Lundberg and diffusion type approximations, which provide the right asymptotic behaviour of relative errors in a perturbed model. The dependence of these correction terms on relations between the rate of perturbation and the speed of growth of an initial capital is investigated. Various types of perturbations of risk processes are discussed. The results are based on a new type of exponential asymptotics for perturbed renewal equations. is used to model the business of an insurance company. In (1.1) the positive constant c is the gross premium rate, N(t) is a Poisson process (with parameter λ) counting the number of claims on the company in the time-interval ଝ

In this paper we continue the investigation of the spectral theory and exponential asymptotics of primarily discrete-time Markov processes, following Kontoyiannis and Meyn . We introduce a new family of nonlinear Lyapunov drift criteria, which characterize distinct subclasses of geometrically ergodic Markov processes in terms of simple inequalities for the nonlinear generator. We concentrate primarily on the class of multiplicatively regular Markov processes, which are characterized via simple conditions similar to (but weaker than) those of Donsker-Varadhan. For any such process Φ = {Φ(t)} with transition kernel P on a general state space X, the following are obtained.

El campo de las funciones especiales ha tenido un gran desarrollo en las últimas décadas dado que son muchos los fenómenos que se pueden estudiar mediante el uso de las mismas como, por ejemplo, procesos estocásticos relacionados, investigación de operaciones, teoría cuántica, ecuaciones funcionales, vibración de placas, conducción del calor, elasticidad, y radiación. En este trabajo se considera una ampliación de las teorías presentadas por M. Dotsenko en 1991, quien introdujo la generalización de la función hipergeométrica de Gauss, denotada por 2R1τ (z), y estableció su representación en serie e integral. Es importante notar que en 1999 Nina Virchenko y luego, en el 2003, Leda Galué consideraron esta función, introduciendo un conjunto de fórmulas de recurrencia y de diferenciación. En este trabajo se establecen algunas representaciones simples para la función 2R1τ (a, b; c; τ; z), las cuales serán muy útiles en futuras investigaciones puesto que permiten simplificar cálculos en el momento de solucionar problemas que involucren esta función.

MSC: 33D15, 33D90, 33D60, 34M03,  62E15