Caín y Abel

Argumentative Reasoning with ABEL

Most formal approaches to argumentative reasoning under uncertainty focus on the analysis of qualitative aspects. An exception is the framework of probabilistic argumentation systems. Its philosophy is to include both qualitative and quantitative aspects through a simple way of combining logic and probability theory. Probabilities are used to weigh arguments for and against particular hypothese...

Formal Development with ABEL

The Abel-Zeilberger Algorithm

We use both Abel’s lemma on summation by parts and Zeilberger’s algorithm to find recurrence relations for definite summations. The role of Abel’s lemma can be extended to the case of linear difference operators with polynomial coefficients. This approach can be used to verify and discover identities involving harmonic numbers and derangement numbers. As examples, we use the Abel-Zeilberger alg...

Computing the Abel map

A Riemann surface is associated with its Jacobian through the Abel map. Using a plane algebraic curve representation of the Riemann surface, we provide an algorithm for the numerical computation of the Abel map. Since our plane algebraic curves are of arbitrary degree and may have arbitrary singularities, the Abel map of any connected compact Riemann surface may be obtained in this way.

Diagonalizable Generalized Abel Integral

We find a large class of integral operators which allow a systematic and explicit diagonalization. These operators generalize Abel’s integral operator, for which such a diagonalization has recently been found.