A computer algebra user interface manifesto

Many computer algebra systems have more than 1000 built-in functions, making expertise difficult. Using mock dialog boxes, this article describes a proposed interactive general-purpose wizard for organizing optional transformations and allowing easy fine grain control over the form of the result even by amateurs. This wizard integrates ideas including: * flexible subexpression selection; * complete control over the ordering of variables and commutative operands, with well-chosen defaults; * interleaving the choice of successively less main variables with applicable function choices to provide detailed control without incurring a combinatorial number of applicable alternatives at any one level; * quick applicability tests to reduce the listing of inapplicable transformations; * using an organizing principle to order the alternatives in a helpful manner; * labeling quickly-computed alternatives in dialog boxes with a preview of their results, * using ellipsis elisions if necessary or helpful; * allowing the user to retreat from a sequence of choices to explore other branches of the tree of alternatives or to return quickly to branches already visited; * allowing the user to accumulate more than one of the alternative forms; * integrating direct manipulation into the wizard; and * supporting not only the usual input-result pair mode, but also the useful alternative derivational and in situ replacement modes in a unified window.Comment: 38 pages, 12 figures, to be published in Communications in Computer Algebr

The κ - μ shadowed fading model with arbitrary intercluster correlation

In this paper, we propose a generalization of the well-known κ-μ shadowed fading model. Based on the clustering of multipath waves as the baseline model, the novelty of this new distribution is the addition of an arbitrary correlation for the scattered components within each cluster. It also inherits the random fluctuation of the dominant component, which is assumed to be the same for all clusters. Thus, it unifies a wide variety of models: Rayleigh, Rician, Rician shadowed, Nakagami- m, κ-μ and κ-μ shadowed as well as multivariate Rayleigh, Rician and Rician shadowed. The main statistics of the newly proposed model, i.e. moment generating function, probability density function and cumulative density function, are given in terms of exponentials and powers, and some numerical results are provided in order to analyze the impact of the arbitrary intercluster correlation.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tec

Appell polynomials and their relatives II. Boolean theory

The Appell-type polynomial family corresponding to the simplest non-commutative derivative operator turns out to be connected with the Boolean probability theory, the simplest of the three universal non-commutative probability theories (the other two being free and tensor/classical probability). The basic properties of the Boolean Appell polynomials are described. In particular, their generating function turns out to have a resolvent-type form, just like the generating function for the free Sheffer polynomials. It follows that the Meixner (that is, Sheffer plus orthogonal) polynomial classes, in the Boolean and free theory, coincide. This is true even in the multivariate case. A number of applications of this fact are described, to the Belinschi-Nica and Bercovici-Pata maps, conditional freeness, and the Laha-Lukacs type characterization. A number of properties which hold for the Meixner class in the free and classical cases turn out to hold in general in the Boolean theory. Examples include the behavior of the Jacobi coefficients under convolution, the relationship between the Jacobi coefficients and cumulants, and an operator model for cumulants. Along the way, we obtain a multivariate version of the Stieltjes continued fraction expansion for the moment generating function of an arbitrary state with monic orthogonal polynomials

On hyperquadratic continued fractions in power series fields over a finite field

The first part of this note is a short introduction on continued fraction expansions for certain algebraic power series. In the last part, as an illustration, we present a family of algebraic continued fractions of degree 4, including a toy example considered about thirty years ago in a pioneer work in this area