Algorithms to Evaluate Multiple Sums for Loop Computations

We present algorithms to evaluate two types of multiple sums, which appear in higher-order loop computations. We consider expansions of a generalized hypergeometric-type sums, \sum_{n_1,...,n_N} [Gamma(a1.n+c1) Gamma(a2.n}+c2) ... Gamma(aM.n+cM)] / [Gamma(b1.n+d1) Gamma(b2.n+d2) ... Gamma(bM.n+dM)] x1^n1...xN^nN with $ai.n=\sum_{j=1}^N a_{ij}nj$, etc., in a small parameter epsilon around rational values of ci,di's. Type I sum corresponds to the case where, in the limit epsilon -> 0, the summand reduces to a rational function of nj's times x1^n1...xN^nN; ci,di's can depend on an external integer index. Type II sum is a double sum (N=2), where ci,di's are half-integers or integers as epsilon -> 0 and xi=1; we consider some specific cases where at most six Gamma functions remain in the limit epsilon -> 0. The algorithms enable evaluations of arbitrary expansion coefficients in epsilon in terms of Z-sums and multiple polylogarithms (generalized multiple zeta values). We also present applications of these algorithms. In particular, Type I sums can be used to generate a new class of relations among generalized multiple zeta values. We provide a Mathematica package, in which these algorithms are implemented.Comment: 30 pages, 2 figures; address of Mathematica package in Sec.6; version to appear in J.Math.Phy

Agonistic behavior of captive saltwater crocodile, crocodylus porosus in Kota Tinggi, Johor

Agonistic behavior in Crocodylus porosus is well known in the wild, but the available data regarding this behavior among the captive individuals especially in a farm setting is rather limited. Studying the aggressive behavior of C. porosus in captivity is important because the data obtained may contribute for conservation and the safety for handlers and visitors. Thus, this study focuses on C. porosus in captivity to describe systematically the agonistic behaviour of C. porosus in relation to feeding time, daytime or night and density per pool. This study was carried out for 35 days in two different ponds. The data was analysed using Pearson’s chi-square analysis to see the relationship between categorical factors. The study shows that C. porosus was more aggressive during daylight, feeding time and non-feeding time in breeding enclosure (Pond C, stock density =0.0369 crocodiles/m2) as compared to non-breeding pond (Pond B, stock density =0.3317 crocodiles/m2) where it is only aggressive during the nighttime. Pond C shows the higher domination in the value of aggression in feeding and non-feeding time where it is related to its function as breeding ground. Chi-square analysis shows that there is no significant difference between ponds (p=0.47, χ2= 2.541, df= 3), thus, there is no relationship between categorical factors. The aggressive behaviour of C. porosus is important for the farm management to evaluate the risk in future for the translocation process and conservation of C. porosus generally

Detecting Simultaneous Integer Relations for Several Real Vectors

An algorithm which either finds an nonzero integer vector ${\mathbf m}$ for given $t$ real $n$-dimensional vectors ${\mathbf x}_1,...,{\mathbf x}_t$ such that ${\mathbf x}_i^T{\mathbf m}=0$ or proves that no such integer vector with norm less than a given bound exists is presented in this paper. The cost of the algorithm is at most ${\mathcal O}(n^4 + n^3 \log \lambda(X))$ exact arithmetic operations in dimension $n$ and the least Euclidean norm $\lambda(X)$ of such integer vectors. It matches the best complexity upper bound known for this problem. Experimental data show that the algorithm is better than an already existing algorithm in the literature. In application, the algorithm is used to get a complete method for finding the minimal polynomial of an unknown complex algebraic number from its approximation, which runs even faster than the corresponding \emph{Maple} built-in function.Comment: 10 page

Parallel integer relation detection: techniques and applications

For guidance on citations see FAQs. c ○ [not recorded] Version: [not recorded] Link(s) to article on publisher’s website