1,721,074 research outputs found
A package on formal power series.
No full textFormal Laurent-Puiseux series of the form are important in many branches of mathematics. Whereas {\sc Mathematica} supports the calculation of truncated series with its {\tt Series} command, and the {\sc Mathematica} package {\tt SymbolicSum} that is shipped with {\sc Mathematica} version 2 is able to convert formal series of the type mentioned above in some instances to their corresponding generating functions, in six publications of the author we developed an algorithmic procedure to do these conversions that is implemented by the author, A.\ Rennoch and G.\ Stölting in the {\sc Mathematica} package {\tt PowerSeries}. The implementation enables the user to reproduce most of the results of the extensive bibliography on series of Hansen, E.\ R.: A table of series and products. Prentice-Hall, 1975. Moreover a subalgorithm of its own significance generates differential equations satisfied by the input function
REDUCE Package for the Indefinite and Definite Summation
No full textThis article describes the REDUCE package ZEILBERG implemented by Gregor Stölting and the author. The REDUCE package ZEILBERG is a careful implementation of the Gosper (The sum package contains also a partial implementation of the Gosper algorithm.) and Zeilberger algorithms for indefinite, and definite summation of hypergeometric terms, respectively. An expression is called a {\sl hypergeometric term} (or {\sl closed form}), if is a rational function with respect to . Typical hypergeometric terms are ratios of products of powers, factorials, function terms, binomial coefficients, and shifted factorials (Pochhammer symbols) that are integer-linear in their arguments
Orthogonal and multiple orthogonal polynomials, random matrices, and Painlevé equations
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Algorithms for the Indefinite and Definite Summation
No full textThe celebrated Zeilberger algorithm which finds holonomic recurrence equations for definite sums of hypergeometric terms is extended to certain nonhypergeometric terms. An expression is called hypergeometric term if both and are rational functions. Typical examples are ratios of products of exponentials, factorials, function terms, binomial coefficients, and Pochhammer symbols that are integer-linear with respect to and in their arguments. We consider the more general case of ratios of products of exponentials, factorials, function terms, binomial coefficients, and Pochhammer symbols that are rational-linear with respect to and in their arguments, and present an extended version of Zeilberger's algorithm for this case, using an extended version of Gosper's algorithm for indefinite summation. In a similar way the Wilf-Zeilberger method of rational function certification of integer-linear hypergeometric identities is extended to rational-linear hypergeometric identities. The given algorithms on definite summation apply to many cases in the literature to which neither the Zeilberger approach nor the Wilf-Zeilberger method is applicable. Examples of this type are given by theorems of Watson and Whipple, and a large list of identities (``Strange evaluations of hypergeometric series'') that were studied by Gessel and Stanton. It turns out that with our extended algorithms practically all hypergeometric identities in the literature can be verified. Finally we show how the algorithms can be used to generate new identities. REDUCE and MAPLE implementations of the given algorithms can be obtained from the author, many results of which are presented in the paper
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Formal Power Series.
No full text{\newcommand{\C}{{\rm {\mbox{C{\llap{{\vrule height1.52ex}\kern.4em}}}}}} \newcommand{\Z} {{\rm {\mbox{\protect\makebox[.2em][l]{\sf Z}\sf Z}}}} \newcommand{\Maple}{{\sc Maple}} Formal Laurent-Puiseux series of the form f(x)=\sum\limits_{k=k_0}^{\infty}a_{k}x^{k/n} \label{eq:formalLPS} with coefficients a_{k}\in\C\;(k\in\Z) are important in many branches of mathematics. \Maple\ supports the computation of {\em truncated\/} series with its {\tt series} command, and through the {\tt powerseries} package infinite series are available. In the latter case, the series is represented as a table of coefficients that have already been determined together with a function for computing additional coefficients. This is known as {\em lazy evaluation\/}. But these tools fail, if one is interested in an explicit formula for the coefficients . In this article we will describe the \Maple\ implementation of an algorithm presented in several papers of the second author which computes an {\em exact\/} formal power series of a given function. This procedure will enable the user to reproduce most of the results of the extensive bibliography on series. We will give an overview of the algorithm and then present some parts of it in more detail. This package is available through the \Maple-share library with the name {\tt FPS}. We flavor this procedure with the following example. %\begin{maple} \begin{verbatim}> FormalPowerSeries(sin(x), x=0);\end{verbatim} \begin{samepage} \begin{verbatim} infinity ----- k (2 k + 1) \ (-1) x ) ---------------- / (2 k + 1)! ----- k = 0 \end{verbatim} \end{samepage}
Variations on the Author
Full text link“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Spaces of Functions Satisfying Simple Differential Equations.
No full text{\newcommand{\N} {{\rm {\mbox{\protect\makebox[.15em][l]{I}N}}}} In several publications the first author published an algorithm for the conversion of analytic functions for which derivative rules are given into their representing power series at the origin and vice versa, implementations of which exist in {\sc Mathematica}, {\sc Maple} and {\sc Reduce}. One main part of this procedure is an algorithm to derive a homogeneous linear differential equation with polynomial coefficients for the given function. We call this type of ordinary differential equations {\sl simple}. Whereas the opposite question to find functions satisfying given differential equations is studied in great detail, our question to find differential equations that are satisfied by given functions seems to be rarely posed. In this paper we consider the family of functions satisfying a simple differential equation generated by the rational, the algebraic, and certain transcendental functions. It turns out that forms a linear space of transcendental functions. % with polynomial function coefficients. Further is closed under multiplication and under the composition with rational functions and rational powers. These results had been published by Stanley who had proved them by theoretical algebraic considerations. In contrast our treatment is purely algorithmically oriented. We present algorithms that generate simple differential equation for , , ( rational), and (), given simple differential equations for , and , and give a priori estimates for the order of the resulting differential equations. We show that all order estimates are sharp. After finishing this article we realized that in independent work Salvy and Zimmermann published similar algorithms. Our treatment gives a detailed description of those algorithms and their validity.
Appropriate Similarity Measures for Author Cocitation Analysis
Full text linkWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
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