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...

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.

If the limit of this as t → 0 exists, then we say that h(x) is Abel summable with the limit as the sum. If h(x) is integrable on R, then h is Abel summable with the usual integral of h being the Abel sum. One can show that the function At(x) can be expressed as an average of Gaussians. This is a quite useful fact, and basically it is a way of saying that Gaussians are more concentrated. For ins...

In a recent paper, a new 3-parameter class of Abel type equations, so-called AIR, all of whose members can be mapped into Riccati equations, is shown. Most of the Abel equations with solution presented in the literature belong to the AIR class. Three canonical forms were shown to generate this class, according to the roots of a cubic. In this paper, a connection between those canonical forms an...

We identify contact transformations which linearize the given equations in the Riccati and Abel chains of nonlinear scalar and coupled ordinary differential equations to the same order. The identified contact transformations are not of Cole-Hopf type and are new to the literature. The linearization of Abel chain of equations is also demonstrated explicitly for the first time. The contact transf...

The mathematical model of many physical problems takes the form of the well-known Abel integral equation. This equation was applied by Niels Abel in 1823 to describe a sliding point mass in a vertical plane on a unknown curve under gravitational force. The point mass starts its motion without initial velocity from a point which has a vertical distance x from the lowest point of the curve [1]. U...

A solution of the Abel equation ẋ = A(t)x + B(t)x such that x(0) = x(1) is called a periodic orbit of the equation. Our main result proves that if there exist two real numbers a and b such that the function aA(t) + bB(t) is not identically zero, and does not change sign in [0, 1] then the Abel differential equation has at most one nonzero periodic orbit. Furthermore, when this periodic orbit ex...

The Abel identity is (x + y) = n ∑ i=0 ( n i ) x(x − iz)i−1(y + iz)n−i, where x, y and z are real numbers. In this paper we deduce several polynomials expansions, referred to as Abel-type identities, by using Foata’s method, and also show some of their applications.