On Asymptotic Properties of Maximal Tubes and Bands in a Neighborhood of an Isolated Singularity in Minkowski Space

We study the asymptotic behavior of maximal surfaces like bands and tubes in a neighborhood of an isolated singular point. In particular, we prove possibility of expansion of the radius vector of a two-dimensional surface in a power series with real-analytic coefficients in the time coordinate. We show also that the tangent rays at a singular point constitute a light-like surface. We prove an exact estimate for the existence time for multidimensional maximal tubes in terms of their asymptotic behavior at a singular point and describe completely the class of surfaces on which this estimate is attained.     

On Asymptotic Properties of Maximal Tubes and Bands in a Neighborhood of an Isolated Singularity in Minkowski Space

We study the asymptotic behavior of maximal surfaces like bands and tubes in a neighborhood of an isolated singular point. In particular, we prove possibility of expansion of the radius vector of a two-dimensional surface in a power series with real-analytic coefficients in the time coordinate. We show also that the tangent rays at a singular point constitute a light-like surface. We prove an exact estimate for the existence time for multidimensional maximal tubes in terms of their asymptotic behavior at a singular point and describe completely the class of surfaces on which this estimate is attained.     

Formal asymptotic solution of a singularly perturbed nonlinear optimal control problem

A system of equations that arises in a singularly perturbed optimal control problem is studied. We give conditions under which a formal asymptotic solution exists. This formal asymptotic solution consists of an outer expansion and left and right boundary-layer expansions. A feature of our procedure is that we do nota priori eliminate the control function from the problem. In particular, we construct a formal asymptotic expansion for the control directly. We apply our procedure to a Mayer-type problem. The paper concludes with a worked example.     

Behavior at infinity of a solution to a differential-difference equation

We obtain an asymptotic expansion for a solution to an mth order nonhomogeneous differential-difference equation of retarded or neutral type. Account is taken of the influence of the roots of the characteristic equation. The exact asymptotics of the remainder is established depending on the asymptotic properties of the free term of the equation.     

Behavior at infinity of a solution to a matrix differential-difference equation

We obtain an asymptotic expansion for a solution to a nonhomogeneous retarded- or neutraltype differential-difference equation. The case of unbounded delays is considered. The influence is accounted for the roots of the characteristic equation. We establish the exact asymptotics for the remainder depending on the asymptotic properties of the free matrix term of the equation.     

An asymptotic formula for the voltage potential in the case of a near-surface conductivity inclusion

We rigorously derive an asymptotic expansion of the steady-state voltage potentials in the presence of a conductivity inclusion of small volume that is close to a planar surface. This new formula is motivated by the practically important inverse problem of imaging a conductivity inclusion near a planar interface.     

Asymptotic expansion of the solution of a differential-difference equation of general form

We obtain an asymptotic expansion of the solution of an mth-order inhomogeneous differential-difference equation of general form. We establish an integral estimate with a submultiplicative weight for the remainder of the expansion depending on the existence of the corresponding submultiplicative moment of the right-hand side of the equation.     

On a singularly perturbed initial value problem in the case of a double root of the degenerate equation

We study the initial value problem of a singularly perturbed first order ordinary differential equation in case that the degenerate equation has a double root. We construct the formal asymptotic expansion of the solution such that the boundary layer functions decay exponentially. This requires a modification of the standard procedure. The asymptotic solution will be used to construct lower and upper solutions guaranteeing the existence of a unique solution and justifying its asymptotic expansion.     

Numerical approximation of vector-valued highly oscillatory integrals

We present a method for the efficient approximation of integrals with highly oscillatory vector-valued kernels, such as integrals involving Airy functions or Bessel functions. We construct a vector-valued version of the asymptotic expansion, which allows us to determine the asymptotic order of a Levin-type method. Levin-type methods are constructed using collocation, and choosing a basis based on the asymptotic expansion results in an approximation with significantly higher asymptotic order. AMS subject classification (2000)  65D30     

Asymptotic expansion of the solution to the cauchy problem for the volterra chain with a step-like initial condition

We describe the relationship between the solution of the Volterra chain that tends asymptotically to constants at infinity and the Whitham-modulated genus-1 Riemann surface. We construct the leading term of the large-time asymptotic expansion of the solution to the Cauchy problem for the Volterra chain under consideration. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 111, No. 3, pp. 335–344, June, 1997.     

Asymptotic expansion formulas for functionals of ε-Markov processes with a mixing property

The ε-Markov process is a general model of stochastic processes which includes nonlinear time series models, diffusion processes with jumps, and many point processes. With a view to applications to the higher-order statistical inference, we will consider a functional of the ε-Markov process admitting a stochastic expansion. Arbitrary order asymptotic expansion of the distribution will be presented under a strong mixing condition. Applying these results, the third order asymptotic expansion of theM-estimator for a general stochastic process will be derived. The Malliavin calculus plays an essential role in this article. We illustrate how to make the Malliavin operator in several concrete examples. We will also show that the thirdorder expansion formula (Sakamoto and Yoshida (1998, ISM Cooperative Research Report, No. 107, 53–60; 1999, unpublished)) of the maximum likelihood estimator for a diffusion process can be obtained as an example of our result.     

A systematization of the saddle point method. Application to the Airy and Hankel functions

The standard saddle point method of asymptotic expansions of integrals requires to show the existence of the steepest descent paths of the phase function and the computation of the coefficients of the expansion from a function implicitly defined by solving an inversion problem. This means that the method is not systematic because the steepest descent paths depend on the phase function on hand and there is not a general and explicit formula for the coefficients of the expansion (like in Watson's Lemma for example). We propose a more systematic variant of the method in which the computation of the steepest descent paths is trivial and almost universal: it only depends on the location and the order of the saddle points of the phase function. Moreover, this variant of the method generates an asymptotic expansion given in terms of a generalized (and universal) asymptotic sequence that avoids the computation of the standard coefficients, giving an explicit and systematic formula for the expansion that may be easily implemented on a symbolic manipulation program. As an illustrative example, the well-known asymptotic expansion of the Airy function is rederived almost trivially using this method. New asymptotic expansions of the Hankel function Hn(z) for large n and z are given as non-trivial examples.     

Asymptotic analysis of option pricing in a Markov modulated market

We address asymptotic analysis of option pricing in a regime switching market where the risk free interest rate, growth rate and the volatility of the stocks depend on a finite state Markov chain. We study two variations of the chain namely, when the chain is moving very fast compared to the underlying asset price and when it is moving very slow. Using quadratic hedging and asymptotic expansion, we derive corrections on the locally risk minimizing option price.     

Asymptotics of solutions of discontinuous singularly perturbed boundary-value problems

We construct an asymptotic expansion of a boundary-value problem for a singularly perturbed system of differential equations with the right-hand side discontinuous at certain surface. Odessa University, Odessa. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 6, pp. 861–864, June, 1999.     

Asymptotic expansion of a solution to a singular perturbation optimal control problem on an interval with integral constraint

A control problem for solutions of a boundary value problem for a second-order ordinary differential equation with a small parameter at the second derivative is considered on a closed interval. The control is scalar and subject to integral constraints. We construct a complete asymptotic expansion in powers of the small parameter in the Erdélyi sense.     

Analyticity of the resolvent for elastic waves in a perturbed isotropic half space

In this article the analytic and asymptotic properties of the resolvent for elastic waves in a three dimensional domain perturbed from the isotropic half space R ³₊ are studied. In this case, the asymptotic expansion of the resolvent at the origin has logarithmic terms. This property is totally different from the case of the whole 3‐dimensional space. The existence of the surface waves like the Rayleigh waves makes this difference. As an application of the asymptotic properties of the resolvent, the rate of the local energy decay estimates for the dynamical equations is obtained. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotic expansions for the distributions of functions of a correlation matrix

This paper deals with asymptotic expansions for the non-null distributions of certain test statistics concerning a correlation matrix in a multivariate normal distribution. For this purpose an asymptotic expansion is given for the distribution of a function of the sample correlation matrix. As special cases of the resulting expansion, asymptotic expansions for the distributions of the sample correlation coefficient, Fisher's z-transformation and arcsine transformation are also given.     

An asymptotic expansion of the solution to a system of integrodifferential equations with exact asymptotics for the remainder

We obtain an asymptotic expansion of the solution to a system of first order integrodifferential equations taking into account the influence of the roots of the characteristic equation. We establish exact asymptotics for the remainder in dependence on the asymptotic properties of original functions.     

Isomonodromy Approach to Boundary Value Problems for the Ernst Equation

We use the isomonodromy properties of theta-functional solutions of the Ernst equation and an asymptotic expansion in the spectral parameter to establish algebraic relations, enforced by the underlying Riemann surface, between the metric functions and their derivatives. These relations determine which classes of boundary value problems can be solved on a given surface. The situation on lower-genus Riemann surfaces is studied in detail.     

An asymptotic formula for the voltage potential in the case of a near-surface conductivity inclusion

We rigorously derive an asymptotic expansion of the steady-state voltage potentials in the presence of a conductivity inclusion of small volume that is close to a planar surface. This new formula is motivated by the practically important inverse problem of imaging a conductivity inclusion near a planar interface. Partly supported by ACI Jeunes Chercheurs (0693) from the Ministry of Education and Scientific Research, France. Partially supported by grant R02-2003-000-10012-0 from the Korea Science and Engineering Foundation. Received: December 2, 2003; revised: January 18, 2004