We define motivic multiple polylogarithms and prove the double shuffle relations for them. We use this to study the motivic fundamental group π 1 (Gm −μN ), where μN is the group of all N -th roots of unity, and relate the structure of π 1 (Gm − μN ) to the geometry and topology of modular varieties Xm(N) := Γ1(m;N)\GLm(R)/R ∗ + ·Om for m = 1, 2, 3, 4, .... We get new results about the action o...

We give an elementary short proof for a well-known theorem of Guibas and Odlyzko stating that the sets of periods of words are independent of the alphabet size. As a consequence of our constructing proof, we give a linear time algorithm which, given a word, computes a binary one with the same periods. We give also a very short proof for the famous Fine and Wilf's periodicity lemma.

The periods of the three–form on a Calabi–Yau manifold are found as solutions of the Picard–Fuchs equations; however, the toric varietal method leads to a generalized hy-pergeometric system of equations, first introduced by Gelfand, Kapranov and Zelevinski, which has more solutions than just the periods. This same extended set of equations can be derived from symmetry considerations. Semi-perio...

Earlier papers (Allen and Hayes 1985,1986) described a compact axiomatic theory which provided a formal basis for temporal reasoning. This theory makes a sharp distinction between time points and periods of time. We show here, by considering possible models of the theory, that it can be slightly extended to give a better fit to intuition. In particular, the extended theory is preserved under ch...

The study of approximately periodic strings is relevant to diverse applications such as molecular biology data compression and computer assisted music analysis Here we study di erent forms of ap proximate periodicity under a variety of distance rules We consider three related problems for two of which we derive polynomial time algorithms we then show that the third problem is NP complete

This chapter reviews sensitive periods in human brain development based on the literature on children raised in institutions. Sensitive experiences occur when experiences are uniquely influential for the development of neural circuitry. Because in humans, we make inferences about sensitive periods from evaluations of complex behaviors, we underestimate the occurrence of sensitive periods at the...

During the past 20 years, basic science has shown that there are different critical periods for different visual functions during the development of the visual system. Visual functions processed at higher anatomical levels within the system have a later critical period than functions processed at lower levels. This general principle suggests that treatments for amblyopia should be followed in a...

1.0 This paper, which is a sequel to [6], is a step towards a geometric version of the Howe correspondence (an analogue of the theta-lifting in the framework of the geometric Langlands program). We consider only the (unramified) dual reductive pair (H = GO2m, G = GSp2n) over a smooth projective connected curve X. Let BunG (resp., BunH) denote the stack of Gtorsors (resp., H-torsors) on X. Using...

We consider multi-loop integrals in dimensional regularisation and the corresponding Laurent series. We study the integral in the Euclidean region and where all ratios of invariants and masses have rational values. We prove that in this case all coefficients of the Laurent series are periods.

We give an algorithm to compute the periods of smooth projective hypersurfaces of any dimension. This is an improvement over existing algorithms which could only compute the periods of plane curves. Our algorithm reduces the evaluation of period integrals to an initial value problem for ordinary differential equations of Picard–Fuchs type. In this way, the periods can be computed to extreme-pre...