Based on Gosper's algorithm for indeenite hypergeometric summation, Zeilberger's algorithm for proving binomial coeecient identities constitutes a recent breakthrough in symbolic computation. Mathematica implementations of these algorithms are described. Nontrivial examples are given in order to illustrate the usage of these packages which are available by… (More)
The problem of computing a closed form for sums of special functions arises in many parts of mathematics and computer science, especially in combinatorics and complexity analysis. Here we discuss two algorithms for indeenite summation of rational functions, due to Abramov and Paule. We describe some improvements and a parallel implementation on a… (More)
Indeenite summation essentially deals with the problem of inverting the diierence operator : f(X) ! f(X + 1) ? f(X). In the case of rational functions over a eld k we consider the following version of the problem given 2 k(X), determine ; 2 k(X) such that = +, where is as \small" as possible (in a suitable sense). In particular, we address the question what… (More)
Division of integers is called exact if the remainder is zero. We show that the high-order part and the low-order part of the exact quotient can be computed independently from each other. A sequential implementation of this algorithm is up to twice as fast as ordinary exact division and four times as fast as the general classical division algorithm if the… (More)
Twenty-three diarylcarbenium ions and 38 pi-systems (arenes, alkenes, allyl silanes and stannanes, silyl enol ethers, silyl ketene acetals, and enamines) have been defined as basis sets for establishing general reactivity scales for electrophiles and nucleophiles. The rate constants of 209 combinations of these benzhydrylium ions and pi-nucleophiles, 85 of… (More)
In the title compound, C17H18O3, the two non-spiro C atoms of the cyclo-propane ring bear a formyl and a phenyl substituent which are trans-oriented. In the crystal, mol-ecules are linked by weak C-H⋯O and C-H⋯π contacts resulting in a three-dimensional supra-molecular structure.
In the title compound, C23H25ClO4, the cyclo-hexane ring adopts a chair conformation with the 4-meth-oxy-phenyl substituent in an axial position and the chloro-(4-meth-oxy-phen-yl)methyl substituent in an equatorial position. The packing features inversion dimers formed by pairs of C-H⋯O contacts and strands along  and  established by further… (More)