New index transforms of the Lebedev–Skalskaya type

New index transforms, involving the real part of the modified Bessel function of the first kind as the kernel are considered. Mapping properties such as the boundedness and invertibility are investigated for these operators in the Lebesgue spaces. Inversion theorems are proved. As an interesting application, a solution of the initial value problem for the second-order partial differential equation, involving the Laplacian, is obtained. It is noted that the corresponding operators with the imaginary part of the modified Bessel function of the first kind lead to the familiar Kontorovich–Lebedev transform and its inverse.     

A Convolution Operator Related to the Generalized Mehler–Fock and Kontorovich–Lebedev Transforms

In this paper we study a generalization of an index integral involving the product of modified Bessel functions and associated Legendre functions. It is applied to a convolution construction associated with this integral, which is related to the classical Kontorovich–Lebedev and generalized Mehler–Fock transforms. Mapping properties and norm estimates in weighted L _p -spaces, 1 ≤ p ≤ 2, are investigated. An application to a class of convolution integral equations is considered. Necessary and sufficient conditions are found for the solvability of these equations in L ₂.     

On the dominated convergence theorem for the Kurzweil–Stieltjes integral

This paper is concerned with a version of the Lebesgue dominated convergence theorem (DCT) which has been stated for the Kurzweil–Stieltjes integral of real functions. Our objective in this work is to analyze the extension of this result to include vector functions with values in Banach spaces. We establish that the mentioned convergence theorem for the Kurzweil–Stieltjes integral can be formulated in weaker versions for reflexive and separable Banach spaces, and spaces having the Schur property, nonetheless it is not verified in the general case.     

Stability and Fourier–Mukai transforms on elliptic fibrations

We systematically develop Bridgeland's [7] and Bridgeland–Maciocia's [10] techniques for studying elliptic fibrations, and identify criteria that ensure 2-term complexes are mapped to torsion-free sheaves under a Fourier–Mukai transform. As an application, we construct an open immersion from a moduli of stable complexes to a moduli of Gieseker stable sheaves on elliptic threefolds. As another application, we give various 1–1 correspondences between fibrewise semistable torsion-free sheaves and codimension-1 sheaves on Weierstrass surfaces.     

On the Ostrowski’s inequality for Riemann-Stieltjes integral and applications

An Ostrowski type integral inequality for the Riemann-Stieltjes integral ∫_a^b ƒ (t) du (t), where ƒ is assumed to be of bounded variation on [a,b] andu is ofr-H- H?lder type on the same interval, is given. Applications to the approximation problem of the Riemann-Stieltjes integral in terms of Riemann-Stieltjes sums are also pointed out.     

On the efficiency of the Asmussen–Kroese-estimator and its application to stop-loss transforms

Recently, Asmussen and Kroese [6] proposed efficient stochastic estimators to approximate the probability that a random sum exceeds a certain threshold. This article aims to extend the analysis of some of these estimators: Several questions induced by [6] for the threshold setting are discussed. In particular, it is shown that Asmussen and Kroese provide the first heavy-tailed example with vanishing relative error. Furthermore, estimators of similar type for the approximation of moments of stop-loss transforms are given. Their asymptotic performance is analysed and a numerical illustration is attached at the end.     

A note on Clebsch–Gordan integral,Fourier–Legendre expansions and closed form for hypergeometric series

The Ramanujan Journal - Let $$p\ge 3$$ be a prime number. In this article, we study the canonical expansion of the primitive $$p^n$$ -th root of unity $$\zeta _{p^n}$$ in p-adic...     

Equidistribution for meromorphic transforms and the dd~c-method

In this paper we give an introduction to the notion of meromorphic transform. We describe some equidistribution problems and their solution, using the ddc-method. In particular, we give some statistical properties of the equilibrium measure for meromorphic maps on compact Kahler manifolds: K-mixing, exponential decay of correlations and central limit theorem.     

On a problem for the Navier–Stokes equations with the infinite Dirichlet integral

We study existence of global in time solutions to the Navier–Stokes equations in a two dimensional domain with an unbounded boundary. The problem is considered with slip boundary conditions involving nonzero friction. The main result shows a new L-bound on the vorticity. A key element of the proof is the maximum principle for a reformulation of the problem. Under some restrictions on the curvature of the boundary and the friction the result for large data (including flux) with the infinite Dirichlet integral is obtained.     

On a problem for the Navier–Stokes equations with the infinite Dirichlet integral

We study existence of global in time solutions to the Navier–Stokes equations in a two dimensional domain with an unbounded boundary. The problem is considered with slip boundary conditions involving nonzero friction. The main result shows a new L-bound on the vorticity. A key element of the proof is the maximum principle for a reformulation of the problem. Under some restrictions on the curvature of the boundary and the friction the result for large data (including flux) with the infinite Dirichlet integral is obtained.Received: October 31, 2002; revised: September 17, 2003