Related papers: Euler integration over definable functions

In this paper we present a new family of identities for Euler sums and integrals of polylogarithms by using the methods of generating function and integral representations of series. Then we apply it to obtain the closed forms of all…

In the context of the complex-analytic structure within the unit disk centered at the origin of the complex plane, that was presented in a previous paper, we show that a certain class of non-integrable real functions can be represented…

The primary purpose of this paper is to investigate the question of invertibility of the sum of operators. The setting is bounded and unbounded linear operators. Some interesting examples and consequences are given. As an illustrative…

We comment on recent results in the field of information based complexity, which state (in a number of different settings), that approximation of infinitely differentiable functions is intractable and suffers from the curse of…

While there is a well-established notion of what a computable ordinal is, the question which functions on the countable ordinals ought to be computable has received less attention so far. We propose a notion of computability on the space of…

We give explicit evaluations of the linear and non-linear Euler sums of hyperharmonic numbers $h_{n}^{\left( r\right) }$ with reciprocal binomial coefficients. These evaluations enable us to extend closed form formula of Euler sums of…

We believe that Euler constant is not just the "renormalized" value of the Riemann zeta function in 1. In a sense that we shall clarify it is in fact the normal and natural value of zeta of 1. In this paper we first propose a limit…

We focus on measurability and integrability for set valued functions in non-necessarily separable Fr\'echet spaces. We prove some properties concerning the equivalence between different classes of measurable multifunctions. We also provide…

The class of functions from the integers to the integers computable in polynomial time has been characterized recently using discrete ordinary differential equations (ODE), also known as finite differences. In the framework of ordinary…

We establish a surprising correspondence between groups definable in o-minimal structures and linear algebraic groups, in the nilpotent case. It turns out that in the o-minimal context, like for finite groups, nilpotency is equivalent to…

Consider a Banach space valued measurable function $f$ and an operator $u$ from the space where {$f$} takes values. If $f $ is Pettis integrable, a classical result due to J. Diestel shows that composing it with $u$ gives a Bochner…

We obtain a characterization of the weighted inequalities for the Riesz transforms on weighted local Morrey spaces. The condition is sufficient for the boundedness on the same spaces of all Calder\'on-Zygmund operators suitably defined on…

Operator monotone functions, introduced by Lowner in 1934, are an important class of real-valued functions. They arise naturally in matrix and operator theory and have various applications in other branches of mathematics and related…

We define a Weil-\'etale complex with compact support for duals (in the sense of the Bloch dualizing cycles complex $\mathbb{Z}^c$) of a large class of $\mathbb{Z}$-constructible sheaves on an integral $1$-dimensional proper arithmetic…

Inference in expressive probabilistic models is generally intractable, which makes them difficult to learn and limits their applicability. Sum-product networks are a class of deep models where, surprisingly, inference remains tractable even…

We describe a category of undirected graphs which comes equipped with a faithful functor into the category of (colored) modular operads. The associated singular functor from modular operads to presheaves is fully faithful, and its essential…

We show here how, using Euler's integration method and an associated function bounding the error in function of time, one can generate structures closely surrounding the invariant tori of dynamical systems. Such structures are constructed…

We extend Zeilberger's approach to special function identities to cases that are not holonomic. The method of creative telescoping is thus applied to definite sums or integrals involving Stirling or Bernoulli numbers, incomplete Gamma…

In this paper, we show that there is a one-to-one correspondence between operator monotone functions on the nonnegative reals and finite Borel measures on the unit interval. This correspondence appears as an integral representation of…

An explicit vertex operator algebra construction is given of a class of irreducible modules for toroidal Lie algebras.