Related papers: Generalized Riemann sums
This short note presents a peculiar generalization of the Riemann hypothesis, as the action of the permutation group on the elements of continued fractions. The problem is difficult to attack through traditional analytic techniques, and…
It is of course well known that the usual definitions of Riemann integration and Riemann integrals are equivalent to simpler definitions which can be expressed in terms of just one sequence of partitions, using dyadic intervals or dyadic…
The main purpose of the present article is to give some new Hilbert's sum type inequalities, which in special cases yield the classical Hilbert's inequalities. Our results provide some new estimates to these types of inequalities.
Using a concept of filter we propose one generalization of Riemann integral, that is integration with respect to filter. We study this problem, demonstrate different properties and phenomena of filter integration.
This year marks the hundredth anniversary of Einstein's 1915 landmark paper "Die Feldgleichungen der Gravitation" in which the field equations of general relativity were correctly formulated for the first time, thus rendering general…
Dedekind sums have applications in quite a number of fields of mathematics. Therefore, their distribution has found considerable interest. This article gives a survey of several aspects of the distribution of these sums. In particular, it…
The goals of this paper are first to describe and then to apply an ergodic-theoretic generalization of the Siegel integral formula from the geometry of numbers. The general formula will be seen to serve both as a guide and as a tool for…
A survey is given on mathematical structures which emerge in multi-loop Feynman diagrams. These are multiply nested sums, and, associated to them by an inverse Mellin transform, specific iterated integrals. Both classes lead to sets of…
Some of my previous publications were incomplete in the sense that non trivial zeros belonging to a particular type of fundamental domain have been inadvertently ignored. Due to this fact, I was brought to believe that computations done by…
We study discrete expressions of the form $$ T_n(g)=\sum_{i=1}^n a_i g(S_i), \qquad S_i=\sum_{j=1}^i a_j, $$ where $a_i>0$ and $\sum_{i=1}^n a_i=1$. If $g:[0,1]\to\mathbb{R}$ is a decreasing integrable function, we have $$ \sum_{i=1}^n a_i…
The Compositional Integral is defined, formally constructed, and discussed. A direct generalization of Riemann's construction of the integral; it is intended as an alternative way of looking at First Order Differential Equations. This brief…
We present a simplified integral of functions of several variables. Although less general than the Riemann integral, most functions of practical interest are still integrable. On the other hand, the basic integral theorems can be obtained…
Could elementary complex analysis, which covers the topics such as algebra of complex numbers, elementary complex functions, complex differentiation and integration, series expansions of complex functions, residues and singularities, and…
The main purpose of this article is to study higher power mean values of generalized quadratic Gauss sums using estimates for character sums, analytic method and algebraic geometric methods. In this article, we prove two conjectures which…
An open problem concerning Riemann sums, posed by O. Furdui, is considered.
The celebrated Riemann-Siegel formula compares the Riemann zeta function on the critical line with its partial sums, expressing the difference between them as an expansion in terms of decreasing powers of the imaginary variable $t$. Siegel…
The 75th anniversary of Turing's seminal paper and his centennial year anniversary occur in 2011 and 2012, respectively. It is natural to review and assess Turing's contributions in diverse fields in the light of new developments that his…
Our aim is to give a friendly introduction for students to systolic inequalities. We will stress the relationships between the classical formulation for Riemannian metrics and more recent developments related to symplectic measurements and…
A natural extension of Riemannian geometry to a much wider context is presented on the basis of the iterated differential form formalism developed in math.DG/0605113 and an application to general relativity is given.
The object of this paper is to generalize a theorem on the binomial coefficient [4] to the case in an arithmetic progression. We will also give a slightly stronger result than Langevin's [2].