Acceleration of Series

The rate of convergence of infinite series can be accelerated b y a suitable splitting of each term into two parts and then combining the second part of the n-th term with the first part of the (n+1) -th term t get a new series and leaving the first part of the first term as an "orphan". Repeating this process an infinite number of times, the series will often approach zero, and we obtain the series of orphans, which may converge faster than the original series. H euristics for determining the splits are given. Various mathematical constants, originally defined as series having a term ratio which approaches 1, are accelerated into series having a term ratio less than 1. This is done with the constants of Euler and Catalan. The se ries for pi/4 = arctan 1 is transformed into a variety of series, among which is one having a term ration of 1/27 and another having a term ratio of 54/3125. A series for 1/pi is found which has a term ratio of 1/64 and each term of which is an integer divided by a powe r of 2, thus making it easy to evaluate the sum in binary arithmetic. We express zeta(3) in terms of pi-3 and a series having a term ra tio of 1/16. Various hypergeometric function identities are found, as well as a series for (arcsin y)-2 curiously related to a series f or y arcsin y. Convergence can also be accelerated for finite sums, as is shown for the harmonic numbers. The sum of the reciprocals of the Fibonacci numbers has been expressed as a series having the convergence rate of theta function. Finally, it is shown that a series whose n-th term ratio is (n+p)(n+q)/(n+r)(n+s), where p, q, r, s are integers, is equal to c + d pi-2, where c and d are rational

HAKMEM

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A Complete Invariant Generation Approach for P-solvable Loops

We present an algorithm for generating all polynomial invariants of P-solvable loops with assignments and nested conditionals. We prove termination of our algorithm. The proof relies on showing that the dimensions of the prime ideals from the minimal decomposition of the ideals generated at an iteration of our algorithm either remain the same or decrease at the next iteration of the algorithm. Our experimental results report that our method takes less iterations and/or time than other polynomial invariant generation techniques

Placing dignity at the centre of welfare policy

Fully packed loop models describe the statistics of closely packed nested polygons on the square lattice. Many exact results can be obtained for these models, even for finite geometries, using their close relationship to alternating-sign matrices and the solvable six-vertex and O(n=1) lattice models. Some results for the exact partition function of fully packed loop models on various finite geometries are briefly reviewed, as well as the well-known order-disorder bulk phase transition present in these models. A detailed study is presented of the distribution of boundary nests of polygons in fully packed loop models with mirror or rotational symmetry. The probability distribution function of such nests, as well as the average number of nests, are obtained analytically, albeit conjecturally. It is further shown that fully packed loop models undergo another phase transition as a function of the boundary nest fugacity. At criticality, we derive a scaling form for the nest distribution function which displays an unusual non-Gaussian cubic exponential behaviour.Comment: 30 pages, contributed chapte