The fractional polylogarithms, depending on a complex parameter α, are defined by a series which is analytic inside the unit disk. After an elementary conversion of the series into an integral presentation, we show that the fractional polylogarithms are multivalued analytic functions in the complex plane minus 0 and 1. For non-integer values of α, we prove the analytic continuation, compute the...

The Gauss hypergeometric functions 2 F 1 with arbitrary values of parameters are reduced to two functions with fixed values of parameters, which differ from the original ones by integers. It is shown that in the case of integer and/or half-integer values of parameters there are only three types of algebraically independent Gauss hyperge-ometric functions. The ε-expansion of functions of one of ...

Sharp bounds are obtained for expressions involving Zeta and related functions at a distance of one apart. Since Euler discovered in 1736 a closed form expression for the Zeta function at the even integers, a comparable expression for the odd integers has not been forthcoming. The current article derives sharp bounds for the Zeta, Lambda and Eta functions at a distance of one apart. The methods...

Many practical optimal control problems include discrete decisions. These may be either time–independent parameters or time–dependent control functions as gears or valves that can only take discrete values at any given time. While great progress has been achieved in the solution of optimization problems involving integer variables, in particular mixed–integer linear programs, as well as in cont...

I consider the expansion of transcendental functions in a small parameter around rational numbers. This includes in particular the expansion around half-integer values. I present algorithms which are suitable for an implementation within a symbolic computer algebra system. The method is an extension of the technique of nested sums. The algorithms allow in addition the evaluation of binomial sum...

In this paper we prove that, given s ≥ 0, and a Borel non zero measure μ in Rm, if for μ-almost every x ∈ Rm the limit lim ε→0 ∫ |x−y|>ε x −y |x −y|s+1 dμ(y) exists and 0 < lim supr→0 μ(B(x, r))/r s < ∞, then s in an integer. In particular, if E ⊂ Rm is a set with positive and bounded s-dimensional Hausdorff measure Hs and for Hs-almost every x ∈ E the limit

Résumé. Soit f ∈ Z[x] un polynôme de degré d ≥ 3 sans racines de multiplicité d ou (d− 1). Erdős a conjecturé que si f satisfait les conditions locales necessaires alors f(p) est sans facteurs puissances (d − 1) pour une infinité de nombres premiers p. On prouve cela pour toutes les fonctions f dont l’entropie est assez grande. On utilise dans la preuve un principe de répulsion pour les points ...

Low bitwidth integer arithmetic has been widely adopted in hardware implementations of deep neural network inference applications. However, despite the promised energy-efficiency improvements demanding edge applications, use low for training remains limited. Unlike inference, demands high dynamic range and numerical accuracy quality results, making low-bitwidth particularly challenging. To addr...

A polyhedron P has the integer decomposition property, if every integer vector in kP is the sum of k integer vectors in P . We explain that the projections of polyhedra defined by totally unimodular constraint matrices have the integer decomposition property, in order to deduce the same property for coflow polyhedra defined by Cameron and Edmonds. We then apply this result to the convex hull of...