Weighted Completion of Galois Groups and Galois Actions on the Fundamental Group of P^1 - {0,1,infinity}

This is a revision of the paper that was previously entitled "Weighted Completion of Galois Groups and Some Conjectures of Deligne". Fix a prime number \l. We prove a conjecture stated by Ihara, which he attributes to Deligne, about the action of the absolute Galois group on the pro-\l completion of the fundamental group of the thrice punctured projective line. Similar techniques are also used to prove part of a conjecture of Goncharov, also about the action of the absolute Galois group on the fundamental group of the thrice punctured projective line. The main technical tool is the weighted completion of a profinite group with respect to a reductive representation (and other appropriate data). This theory is developed in this paper.Comment: 41 pages, amslate

Variants of Mersenne Twister Suitable for Graphic Processors

This paper proposes a type of pseudorandom number generator, Mersenne Twister for Graphic Processor (MTGP), for efficient generation on graphic processessing units (GPUs). MTGP supports large state sizes such as 11213 bits, and uses the high parallelism of GPUs in computing many steps of the recursion in parallel. The second proposal is a parameter-set generator for MTGP, named MTGP Dynamic Creator (MTGPDC). MT- GPDC creates up to 2^32 distinct parameter sets which generate sequences with high-dimensional uniformity. This facility is suitable for a large grid of GPUs where each GPU requires separate random number streams. MTGP is based on linear recursion over the two-element field, and has better high-dimensional equidistribution than the Mersenne Twister pseudorandom number generator.Comment: 23 pages, 6 figure

Universal Mixed Elliptic Motives

In this paper we construct a Q-linear tannakian category MEM_1 of universal mixed elliptic motives over the moduli space M_{1,1} of elliptic curves. It contains MTM, the category of mixed Tate motives unramified over the integers. Each object of MEM_1 is an object of MTM endowed with an action of SL_2(Z) that is compatible with its structure. Universal mixed elliptic motives can be thought of as motivic local systems over M_{1,1} whose fiber over the tangential base point d/dq at the cusp is a mixed Tate motive. The basic structure of the tannakian fundamental group of MEM is determined and the lowest order terms of all relations are found (using computations of Francis Brown), including the arithmetic relations, which describe the "infinitesimal Galois action". We use the presentation to give a new and more conceptual proof of the Ihara-Takao congruences.Comment: 93 pages: many small improvements; should be final versio

Checking the Quality of Approximation of pp-values in Statistical Tests for Random Number Generators by Using a Three-Level Test

Statistical tests of pseudorandom number generators (PRNGs) are applicable to any type of random number generators and are indispensable for evaluation. While several practical packages for statistical tests of randomness exist, they may suffer from a lack of reliability: for some tests, the amount of approximation error can be deemed significant. Reducing this error by finding a better approximation is necessary, but it generally requires an enormous amount of effort. In this paper, we introduce an experimental method for revealing defects in statistical tests by using a three-level test proposed by Okutomi and Nakamura. In particular, we investigate the NIST test suite and the test batteries in TestU01, which are widely used statistical packages. Furthermore, we show the efficiency of several modifications for some tests.Comment: 18 page

Galois actions on fundamental groups of curves and the cycle C−C−C-C^-

Suppose that C is a smooth, projective, geometrically connected curve of genus g > 2 defined over a number field K. Suppose that x is a K-rational point of C. Denote the Lie algebra of the unipotent completion (over Q_ell) of the fundamental group of the corresponding complex analytic curve by p(C,x). This is acted on by both G_K (the Galois group of K) and Gamma, the mapping class group of the pointed complex curve. In this paper we show that the algebraic cycle C_x-C_x^- in the jacobian of C controls the size of the image of G_K in Aut p(C,x). More precisely, we give necessary and sufficient conditions, in terms of two Galois cohomology classes determined by this cycle, for the Zariski closure of the image of G_K in Aut p(C,x) to contain the image of the mapping class group. We also prove an equivalent version for the pro-ell fundamental group, an unpointed version, and a Galois analogue of the Harris-Pulte Theorem.Comment: Latex2e, 35 page

On the fast computation of the weight enumerator polynomial and the tt value of digital nets over finite abelian groups

In this paper we introduce digital nets over finite abelian groups which contain digital nets over finite fields and certain rings as a special case. We prove a MacWilliams type identity for such digital nets. This identity can be used to compute the strict $t$-value of a digital net over finite abelian groups. If the digital net has $N$ points in the $s$ dimensional unit cube $[0,1]^s$, then the $t$-value can be computed in $\mathcal{O}(N s \log N)$ operations and the weight enumerator polynomial can be computed in $\mathcal{O}(N s (\log N)^2)$ operations, where operations mean arithmetic of integers. By precomputing some values the number of operations of computing the weight enumerator polynomial can be reduced further

Tannakian Fundamental Groups Associated to Galois Groups

This should be the final version of this paper. Numberous minor improvements have been made to the manuscript, one argument has been corrected, and an appendix has been added.Comment: LaTeX2e, 26 page

Finsler-Geometrical Approach to the Studying of Nonlinear Dynamical Systems

A two dimensional Finsler space associated with the differential equation $y''=Y_3 y'^3+Y_2 y'^2+Y_1 y'+Y_0$ is characterized by a tensor equation and called the Douglas space. An application to the Lorenz nonlinear dynamical equation is discussed from the standpoint of Finsler geometry.Comment: 22 pages, Latex; Reports of Math. Phys.(1998

Approximation of integration over finite groups, difference sets and association schemes

Let $G$ be a finite group and $f:G \to {\mathbb C}$ be a function. For a non-empty finite subset $Y\subset G$, let $I_Y(f)$ denote the average of $f$ over $Y$. Then, $I_G(f)$ is the average of $f$ over $G$. Using the decomposition of $f$ into irreducible components of ${\mathbb C}^G$ as a representation of $G\times G$, we define non-negative real numbers $V(f)$ and $D(Y)$, each depending only on $f$, $Y$, respectively, such that an inequality of the form $|I_G(f)-I_Y(f)|\leq V(f)\cdot D(Y)$ holds. We give a lower bound of $D(Y)$ depending only on $\#Y$ and $\#G$. We show that the lower bound is achieved if and only if $\#\{(x,y)\in Y^2 \mid x^{-1}y \in [a]\}/\#[a]$ is independent of the choice of the conjugacy class $[a]\subset G$ for $a \neq 1$. We call such a $Y\subset G$ as a pre-difference set in $G$, since the condition is satisfied if $Y$ is a difference set. If $G$ is abelian, the condition is equivalent to that $Y$ is a difference set. We found a non-trivial pre-difference set in the dihedral group of order 16, where no non-trivial difference set exists. The pre-difference sets in non-abelian groups of order 16 are classified. A generalization to commutative association schemes is also given.Comment: 20 page

A Computable Figure of Merit for Quasi-Monte Carlo Point Sets

Let $\mathcal{P} \subset [0,1)^S$ be a finite point set of cardinality $N$ in an $S$-dimensional cube, and let $f:[0,1)^S \to \mathbb{R}$ be an integrable function. A QMC integration of $f$ by $\mathcal{P}$ is the average of values of $f$ at each point in $\mathcal{P}$, which approximates the integration of $f$ over the cube. Assume that $\mathcal{P}$ is constructed from an $\mathbb{F}2$-vector space P\subset (\F2^n)^S by means of a digital net with $n$-digit precision. As an $n$-digit discretized version of Josef Dick's method, we introduce Walsh figure of merit (WAFOM) $\textnormal{WF}(P)$ of $P$, which satisfies a Koksma-Hlawka type inequality, namely, QMC integration error is bounded by $C_{S,n}||f||_n \textnormal{WF}(P)$ under $n$-smoothness of $f$, where $C_{S,n}$ is a constant depending only on $S,n$. We show a Fourier inversion formula for $\textnormal{WF}(P)$ which is computable in $O(n SN)$ steps. This effectiveness enables us a random search for $P$ with small value of $\textnormal{WF}(P)$, which would be difficult for other figures of merit such as discrepancy. From an analogy to coding theory, we expect that random search may find better point sets than mathematical constructions. In fact, a na\"{i}ve search finds point sets $P$ with small $\textnormal{WF}(P)$. In experiments, we show better performance of these point sets in QMC integration than widely used QMC rules. We show some experimental evidence on the effectiveness of our point sets to even non-smooth integrands appearing in finance