The problem of locating the discontinuities of a function by means of its truncated Fourier series, an interesting question in and of itself, arises naturally from an attempt to overcome the Gibbs phenomenon: the poor approximative properties of the Fourier partial sums of a discontinuous function. In [3], Cai et al. developed the idea already introduced in their previous papers and suggested a...

Chandra and XMM-Newton observations of many clusters reveal sharp discontinuities in the surface brightness, which, unlike shocks, have lower gas temperature on the X-ray brighter side of the discontinuity. For that reason these features are called “cold fronts”. It is believed that some cold fronts are formed when a subcluster merges with another cluster and the ram pressure of gas flowing out...

We study one-dimensional scattering for a decaying potential with rapid periodic oscillations and strong localized singularities. In particular, we consider the Schrödinger equation Hǫ ψ ≡ ` −∂ x + V0(x) + q (x, x/ǫ) ́ ψ = kψ for k ∈ R and ǫ ≪ 1. Here, q(·, y + 1) = q(·, y), has mean zero and |V0(x) + q(x, ·)| → 0 as |x| → ∞. The distorted plane waves of Hǫ are solutions of the form: eV ǫ±(x; k)...

We consider solving three-dimensional electromagnetic problems in parameter regimes where the quasi-static approximation applies and the permeability, permittivity, and conductivity may vary significantly. The difficulties encountered include handling solution discontinuities across interfaces and accelerating convergence of traditional iterative methods for the solution of the linear systems o...

A new multi-state Harten–Lax–van Leer (HLL) approximate Riemann solver for the ideal magnetohydrodynamic (MHD) equations is developed based on the assumption that the normal velocity is constant over the Riemann fan. This assumption is same as that used in the HLLC (‘‘C’’ denotes Contact) approximate Riemann solver for the Euler equations. From the assumption, it is naturally derived that the R...

The characteristic functions of many affine jump-diffusion models, such as Heston’s stochastic volatility model and all of its extensions, involve multivalued functions such as the complex logarithm. If we restrict the logarithm to its principal branch, as is done in most software packages, the characteristic function can become discontinuous, leading to completely wrong option prices if option...

In the present paper, we generalize the method suggested in an earlier paper by the author and overcome its main deficiency. First, we modify the well-known Prony method, which subsequently will be utilized for recovering exactly the locations of jump discontinuities and the associated jumps of a piecewise constant function by means of its Fourier coefficients with respect to any system of the ...

The solid-to-solid phase transitions that result from shock loading of certain materials, such as the graphite-to-diamond transition and the transition in iron, have long been subjects of a substantial theoretical and experimental literature. Recently a model for such transitions was introduced which, based on a CS condition CS and without use of fitting parameters, accounts quantitatively for ...

By investigating in detail discontinuities of the first kind of real-valued functions and the analysis of unordered sums, where the summands are given by values of a positive real-valued function, we develop a measure-theoretical framework which in particular allows us to describe rigorously the representation and meaning of sums of jumps of type ∑ 0<s≤t Φ ◦ |∆Xs|, where X : Ω×R+ −→ R is a stoc...

Discontinuity in viscosities is of interest in many applications. Classical adaptive numerical methods perform under the restricting assumption that the discontinuities of the viscosity are captured by the initial partition. This excludes applications where the jump of the viscosity takes place across curves, manifolds or at a priori unknown positions. We present a novel estimate measuring the ...