Discrete Sturm-Liouville equations with point interaction

Scattering solutions and several properties of scattering function of a discrete Sturm-Liouville boundary value problem with point interaction (PBVP) are derived. Moreover, resolvent operator, continuous and discrete spectrum of this PBVP are investigated. An asymptotic equation is utilized to get the properties of eigenvalues. An example illustrating the main results is given.

On the approximation of vorticity fronts by the Burgers–Hilbert equation

This paper proves that the motion of small-slope vorticity fronts in the two-dimensional incompressible Euler equations is approximated on cubically nonlinear timescales by a Burgers–Hilbert equation derived by Biello and Hunter (2010) using formal asymptotic expansions. The proof uses a modified energy method to show that the contour dynamics equations for vorticity fronts in the Euler equations and the Burgers–Hilbert equation are both approximated by the same cubically nonlinear asymptotic equation. The contour dynamics equations for Euler vorticity fronts are also derived.

Asymptotic analysis of an anti-plane dynamic problem for a three-layered strongly inhomogeneous laminate

Anti-plane dynamic shear of a strongly inhomogeneous dynamic laminate with traction-free faces is analysed. Two types of contrast are considered, including those for composite structures with thick or thin stiff outer layers. In both cases, the value of the cut-off frequency corresponding to the lowest antisymmetric vibration mode tends to zero. For this mode, the shortened dispersion relations and the associated formulae for displacement and stresses are obtained. The latter motivate the choice of appropriate settings, supporting the limiting forms of the original anti-plane problem. The asymptotic equation derived for a three-layered plate with thick faces is valid over the whole low-frequency range, whereas the range of validity of its counterpart for another type of contrast is restricted to a narrow vicinity of the cut-off frequency.

The Newton Polynomial

This chapter focuses on the Newton polynomial based on assumption that K is a differential-valued field of H-type with asymptotic integration and small derivation. Here K is also assumed to be equipped with a monomial group and (Γ‎, ψ‎) is the asymptotic couple of K. Throughout, P is an element of K{Y}superscript Not Equal To. The chapter first revisits the dominant part of P before discussing the elementary properties of the Newton polynomial. It then presents results about the shape of the Newton polynomial and considers realizations of three cuts in the value group Γ‎ of K. It also describes eventual equalizers, along with further consequences of ω‎-freeness and λ‎-freeness, the asymptotic equation over K, and some special H-fields.

LONG MEMORY STOCHASTIC VOLATILITY IN OPTION PRICING

The aim of this paper is to present a stochastic model that accounts for the effects of a long-memory in volatility on option pricing. The starting point is the stochastic Black–Scholes equation involving volatility with long-range dependence. We define the stochastic option price as a sum of classical Black–Scholes price and random deviation describing the risk from the random volatility. By using the fact that the option price and random volatility change on different time scales, we derive the asymptotic equation for this deviation involving fractional Brownian motion. The solution to this equation allows us to find the pricing bands for options.