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

Senior Recital: Angela Anzai, piano

This recital is presented in partial fulfillment of requirements for the degree Bachelor of Music in Performance. Ms. Anzai studies piano with Soohyun Yun.https://digitalcommons.kennesaw.edu/musicprograms/1509/thumbnail.jp

血流数値解析に基づいたステントデザインに関する研究

要約のみTohoku University太田信課

水の活性化を伴う吸エルゴン反応のためのチタン酸カルシウム光触媒の設計と開発

京都大学新制・課程博士博士(人間・環境学)甲第23265号人博第980号新制||人||232(附属図書館)2020||人博||980(吉田南総合図書館)京都大学大学院人間・環境学研究科相関環境学専攻(主査)教授 吉田 寿雄, 教授 田部 勢津久, 教授 中村 敏浩学位規則第4条第1項該当Doctor of Human and Environmental StudiesKyoto UniversityDFA