In this chapter, we discuss some recently obtained asymptotic expansions related to problems in numerical analysis and approximation theory. • We present a generalization of the Euler–Maclaurin (E–M) expansion for the trapezoidal rule approximation of finite-range integrals R b a f ðxÞdx, when f(x) is allowed to have arbitrary algebraic–logarithmic endpoint singularities. We also discuss effect...

We present an algorithm for the computation of logarithmic `-class groups of number fields. Our principal motivation is the effective determination of the `-rank of the wild kernel in the K-theory of number fields.

In this paper, we present a computational method for solving boundary integral equations with loga-rithmic singular kernels which occur as reformulations of a boundary value problem for the Laplacian equation. Themethod is based on the use of the Galerkin method with CAS wavelets constructed on the unit interval as basis.This approach utilizes the non-uniform Gauss-Legendre quadrature rule for ...

In this paper we studied the double scaling limit of a random unitary matrix ensemble near a singular point where a new cut is emerging from the support of the equilibrium measure. We obtained the asymptotic of the correlation kernel by using the Riemann-Hilbert approach. We have shown that the kernel near the critical point is given by the correlation kernel of a random unitary matrix ensemble...

This paper proposes numerical methods for solving hybrid weakly singular integro-differential equations of the second kind. The terms in these equations are in the following order: derivative term of a state, integro-differential term of a state with a weakly singular kernel, a state, integral term of a state with a smooth kernel, and force. The original class of weakly singular integro-differe...

We present an algorithm for the computation of logarithmic `-class groups of number fields. Our principal motivation is the effective determination of the `-rank of the wild kernel in the K-theory of number fields.

For quasilinear integro-diierential equations of the form u t ?aA(u) = f, where a is a scalar singular integral kernel that behaves like t ? , 1 2 < 1 and A is a second order quasilinear elliptic operator in divergence form, solutions are found for which A(u) is integrable over space and time.

Abstract: In this article we have elaborated the numerical schemes of reduction methods for approximate solution of system of singular integro-differential equations when the kernel has a weak singularity. The equations are defined on the arbitrary smooth closed contour of complex plane. We suggest the numerical schemes of the reduction method over the system of Faber-Laurent polynomials for th...

We show that a resummation model for the evolution kernel at small x creates a bridge between the weak and strong couplings. The resummation model embodies DGLAP and BFKL anomalous dimensions at leading logarithmic orders, as well as a kinematical constraint on the real emission part of the kernel. In the case of pure gluodynamics the strong coupling limit of the Pomeron intercept is consistent...

We reduce here end-point estimates for one singular operator (namely for dyadic square function) to Monge–Ampère equations with drift. The spaces are weighted spaces, and therefore the domain, where we solve our PDE is non-convex. If we are in the end-point situation our goal is either to find a logarithmic blow-up of the norm estimate from below, or to prove that there is upper estimate of the...