We present new functional equations in weights 5, 6 and 7 use them for explicit depth reduction of multiple polylogarithms. These identities generalize the crucial identity $\mathbf{Q}_4$ from recent work Goncharov Rudenko that was used their proof weight 4 case Zagier's Polylogarithm Conjecture.

From any given Frobenius manifold one may construct a so-called ‘dual’ structure which, while not satisfying the full axioms of a Frobenius manifold, shares many of its essential features, such as the existence of a prepotential satisfying the WDVV equations of associativity. Jacobi group orbit spaces naturally carry the structures of a Frobenius manifold and hence there exists a dual prepotent...

We present an efficient divisible digital cash scheme which is unlinkable and does not require Trusted Third Party. The size of the coin is proportional to the size of the primes we use, i.e., hundreds of bytes. The computational and communication complexity of the protocol is proportional to a polynomial of the size of the primes and polylogarithm of the maximum number of pieces to which a coi...

We present a new derivation of the spectral gap of the totally asymmetric exclusion process on a half-filled ring of size L by using the Bethe Ansatz. We show that, in the large L limit, the Bethe equations reduce to a simple transcendental equation involving the polylogarithm, a classical special function. By solving that equation, the gap and the dynamical exponent are readily obtained. Our m...

Abstract. It is shown that a broad class of generalized Dirichlet series (including the polylogarithm, related to the Riemann zeta-function) can be presented as a class of solutions of the Fourier transformed spatially homogeneous linear Boltzmann equation with a special Maxwell-type collision kernel. The result is based on an explicit integral representation of solutions to the Cauchy problem ...

Following earlier work by Abel and others, Kummer gave functional equations for the polylogarithm function Lim(z) up to m = 5 in 1850, but no example for larger m was known until recently. We give the first genuine 2-variable functional equation for the 7–logarithm. We investigate and relate identities for the 3-logarithm given by Goncharov and Wojtkowiak and deduce a certain family of function...

Let E be a CM-elliptic curve over Q with good ordinary reduction at a prime p ≥ 5. The purpose of this paper is to construct the p-adic elliptic polylogarithm of E, following the method of A. Beı̆linson and A. Levin. Our main result is that the specializations of this object at torsion points give the special values of the one-variable p-adic L-function of the Grössencharakter associated to E.

Period doubling bifurcation leading to subharmonic oscillations are undesired phenomena in switching converters. In past studies, their prediction has been mainly tackled by explicitly deriving a discrete time model and then linearizing it in the vicinity of the operating point. However, the results obtained from such an approach cannot be applied for design purpose. Alternatively, in this pape...

In this paper, we begin by reviewing the calculus induced framework of [10]. there, extended Polylogarithm functions over a subalgebra noncommutative rational power series, recognizable finite state (multiplicity) automata alphabet \(X=\{x_0,x_1\}\). The stability under shuffle products relies on nuclearity target space [32]. We also concentrated algebraic and analytic aspects extension allowin...