In a recent paper, the canonical forms of a new multi-parameter class of Abel differential equations, so-called AIR, all of whose members can be mapped into Riccati equations, were shown to be related to the differential equations for the hypergeometric 2F1, 1F1 and 0F1 functions. In this paper, a connection between the AIR canonical forms and the Heun General (GHE), Confluent (CHE) and Biconfl...

Reconstructing images from ill-posed inverse problems often utilizes total variation regularization in order to recover discontinuities in the data while also removing noise and other artifacts. Total variation regularization has been successful in recovering images for (noisy) Abel transformed data, where object boundaries and data support will lead to sharp edges in the reconstructed image. I...

In this paper, we consider the number of 2-protected nodes in random digital trees. For tries, Gaither, Homma, Sellke and Ward (2012) and Gaither and Ward (2013) derived asymptotic expansions for mean and variance. We re-derive their results (with some corrections) and show their conjectured central limit theorem. This will be done by recent unified approaches by Fuchs, Hwang and Zacharovas (20...

We prove a Tauberian theorem to recover moderate oscillation of a real sequence u = (u n) out of Abel limitability of the sequence (V (1) n (Δu)) and some additional condition on the general control modulo of oscillatory behavior of integer order of u = (u n).

We study integrals of the type ∫ t 0 φ(s)dLs, where φ is a positive locally bounded Borel function and Lt denotes the local time at level 0 of a Bessel process of dimension d, 0 < d < 2. 1991 Mathematics Subject Classifications : 60J65, 60J55, 45E10.

Today, different formalisms exist to solve reasoning problems under uncertainty. For most of the known formalisms, corresponding computer implementations are available. The problem is that each of the existing systems has its own user interface and an individual language to model the knowledge and the queries. This paper proposes ABEL, a new and general language to express uncertain knowledge a...

Abstract Our aim in this paper is presenting an attractive numerical approach giving accurate solution to the nonlinear fractional Abel differential equation based on a reproducing kernel algorithm with model endowed Caputo–Fabrizio derivative. By means of such approach, we utilize Gram–Schmidt orthogonalization process create orthonormal set bases that leads appropriate Hilbert space $\mathcal...

We consider the Abel Equation ρ′ = p(θ)ρ2 + q(θ)ρ3 (*) with p(θ), q(θ) polynomials in sin θ, cos θ. The center problem for this equation (which is closely related to the classical center problem for polynomial vector fields on the plane) is to find conditions on p and q under which all the solutions ρ(θ) of this equation are periodic, i.e. ρ(0) = ρ(2π) for all initial values ρ(0). We consider t...