Related papers: Lecture notes about a simpler approach to Riemann …
We present a simple yet rigorous theory of integration that is based on two axioms rather than on a construction involving Riemann sums. With several examples we demonstrate how to set up integrals in applications of calculus without using…
We present a replacement for traditional Riemann integrals in undergraduate calculus, which supplements naive precalculus and at the same time opens a way to more sophisticated theories such as Lebesgue integration.
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…
Over the last year significant progress was made in the understanding of the computation of Feynman integrals using differential equations. These lectures give a review of these developments, while not assuming any prior knowledge of the…
We present a notion of primitive which corresponds exactly with the Riemann integral. We obtain a characterization of the integrability in the sense of Riemann which produces a Fundamental Theorem of Calculus without special assumptions. We…
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 the purpose of this article to outline a course that can be given to engineers looking for an understandable mathematical description of the foundations of distribution theory and the necessary functional analytic methods. Arguably,…
In this paper, we consider a discrete version of iterated integrals by the naive (equally divided) Riemann sum. In particular, basic three formulas for usual iterated integrals are discritized. Moreover, we proved cyclic sum formulas for…
Riemannian metric learning is an emerging field in machine learning, unlocking new ways to encode complex data structures beyond traditional distance metric learning. While classical approaches rely on global distances in Euclidean space,…
The period is a classical complex analytic invariant for a compact Riemann surface defined by integration of differential 1-forms. It has a strong relationship with the complex structure of the surface. In this chapter, we review another…
We introduce a simple procedure to integrate differential forms with arbitrary holomorphic poles on Riemann surfaces. It gives rise to an intrinsic regularization of such singular integrals in terms of the underlying conformal geometry.…
These brief lecture notes are intended mainly for undergraduate students in engineering or physics or mathematics who have met or will soon be meeting the Dirac delta function and some other objects related to it. These students might have…
This article presents a construction of the concept of stochastic integration in Riemannian manifolds from a purely functional-analytic point of view. We show that there are infinitely many such integrals, and that any two of them are…
These lecture notes provide a self-contained introduction to Euler integrals, which are frequently encountered in applications. In particle physics, they arise as Feynman integrals or string amplitudes. Our four selected topics demonstrate…
This book is a short introduction into dyadic analysis with applications to classical weighted norm inequalities.
These are the class notes of lectures given by Ralph Henstock at the New University of Ulster in 1970-71. The notes deal with the Riemann-complete integral (also known as the generalized Riemann integral, the gauge integral, and the…
The standard definition of integration of differential forms is based on local coordinates and partitions of unity. This definition is mostly a formality and not used used in explicit computations or approximation schemes. We present a…
We study a single-valued integration pairing between differential forms and dual differential forms which subsumes some classical constructions in mathematics and physics. It can be interpreted as a $p$-adic period pairing at the infinite…
This course on Feynman integrals starts from the basics, requiring only knowledge from special relativity and undergraduate mathematics. Topics from quantum field theory and advanced mathematics are introduced as they are needed. The course…