We give an elementary and rigorous proof of the Thomae type formula for the singular curves µ N = m j=1 (λ − λ 2j) N−1 m j=0 (λ − λ 2j+1). To derive the Thomae formula we use the traditional variational method which goes back to Riemann, Thomae and Fuchs. An important step of the proof is the use of the Szegö kernel computed explicitly in algebraic form for non-singular 1/N-periods. The proof i...

This paper is concerned with finding approximate solution for the singular integral equations. Relating the singular integrals to Cauchy principal-value integrals, we expand the kernel and the density function of singular integral equation by the sum of the chebyshev polynomials of the first, second, third and fourth kinds. Some numerical examples are presented to illustrate the accuracy and ef...

Loop-level scattering amplitudes for massless particles have singularities in regions where tree amplitudes are perfectly smooth. For example, a 2 → 4 gluon scattering process has a singularity in which each incoming gluon splits into a pair of gluons, followed by a pair of 2 → 2 collisions between the gluon pairs. This singularity mimics double parton scattering because it occurs when the tran...

We analyze in classical L(R)-spaces, n = 2 or n = 3, 1 < q < ∞, a singular integral operator arising from the linearization of a hydrodynamical problem with a rotating obstacle. The corresponding system of partial differential equations of second order involves an angular derivative which is not subordinate to the Laplacian. The main tools are Littlewood– Paley theory and a decomposition of the...

We derive and analyze two equivalent integral formulations for the time-harmonic electromagnetic scattering by a dielectric object. One is a volume integral equation (VIE) with a strongly singular kernel and the other one is a coupled surface-volume system of integral equations with weakly singular kernels. The analysis of the coupled system is based on standard Fredholm integral equations, and...

The present paper considers the operator pencil A ( λ ) = 0 + 1 , where ≠ are bounded linear mappings between complex Hilbert spaces and is neither one-to-one nor onto. Assuming that an isolated singularity of image closed, certain operators defined recursively starting from they shown to provide a characterization null space in principal part resolvent logarithmic residues at 0. relations with...

In this paper we deal with a nonlinear singular integral inequality which arises in the study of partial differential equations. The integral term is non local in time and space and the kernel involved is also singular in both the time and the space variable. The estimates we prove may be used to establish (global) existence and asymptotic behavior results for solutions of the corresponding pro...