Related papers: The Calculus and Gauge Integrals, by Ralph Henstoc…
We provide an historical account of equivalent conditions for the Riemann Hypothesis arising from the work of Ramanujan and, later, Guy Robin on generalized highly composite numbers. The first part of the paper is on the mathematical…
We present a set of lectures on topics of advanced calculus in one real and complex variable with several new results and proofs on the subject, specially with detailed proof-always missing in the literature - of the Cissoti explicitly…
Rescaled Mellin-type transforms of the exponential functional of the Bourgade-Kuan-Rodgers statistic of Riemann zeroes are conjecturally related to the distribution of the total mass of the limit lognormal stochastic measure of…
This book gives an introduction to fundamental aspects of generalized Riemannian, complex, and K\"ahler geometry. This leads to an extension of the classical Einstein-Hilbert action, which yields natural extensions of Einstein and…
We study a generalization of graded Hecke algebras introduced by Drinfeld in 1986, in which the role of a finite group G is played by a reductive algebraic group. This includes continuous generalizations of symplectic reflection algebras…
We provide a general construction of integral TQFTs over a general commutative ring, $\mathbf{k}$, starting from a finite Hopf algebra over $\mathbf{k}$ which is Frobenius and double balanced. These TQFTs specialize to the Hennings…
Heisenberg's uncertainty principle was originally posed for the limit of the accuracy of simultaneous measurement of non-commuting observables as stating that canonically conjugate observables can be measured simultaneously only with the…
We survey the main ideas in the early history of the subjects on which Riemann worked and that led to some of his most important discoveries. The subjects discussed include the theory of functions of a complex variable, elliptic and Abelian…
The primary aim of this chapter is, commemorating the 150th anniversary of Riemann's death, to explain how the idea of {\it Riemann sum} is linked to other branches of mathematics. The materials I treat are more or less classical and…
We prove that the Heisenberg Riesz transform is $L_2$--unbounded on a family of intrinsic Lipschitz graphs in the first Heisenberg group $\mathbb{H}$. We construct this family by combining a method from \cite{NY2} with a stopping time…
Historically the fractional calculus concept works an extended idea based on the question asked by Guillaume de L'H\^opital to Gottfried Wilhelm Leibniz in 1695 about the notation ${d^nf}/{dx^n}$ for the derivative operator "What if…
The present work is inspired by three interrelated themes: Weingarten calculus for integration over unitary groups, monotone Hurwitz numbers which enumerate certain factorisations of permutations into transpositions, and Jucys-Murphy…
It is well-known that the Lebesgue integral generalises the Riemann integral. However, as is also well-known but less frequently well-explained, this generalisation alone is not the reason why the Lebesgue integral is important and needs to…
Notable results on the special values of $L$-functions of Siegel modular forms were obtained by J. Sturm in the case when the degree $n$ is even and the weight $k$ is an integer. In this paper we extend this method to half-integer weights…
In the 1960's, Mandelstam proposed a new approach to gauge theories and gravity based on loops. The program for gauge theories was completed for Yang--Mills theories by Gambini and Trias in the 1980's. Gauge theories could be understood as…
We study the existence of Riemann-Stieltjes integrals of bounded functions against a given integrator. We are also concerned with the possibility of computing the resulting integrals by means of related Riemann integrals. In particular, we…
The Riemann Hypothesis, originally proposed by the eminent mathematician Bernard Riemann in 1859, remains one of the most profound challenges in number theory. It posits that all non-trivial zeros of the Riemann zeta function {\zeta}(s) are…
With the theory of general relativity, Einstein abolished the interpretation of gravitation as a force and associated it to the curvature of spacetime. Tensorial calculus and differential geometry are the mathematical resources necessary to…
The paper suggests a slightly more rigorous justification to Wang et al.'s work from 2007, and introduces the Slanted Line Integral.
This text grew out of notes I have used in teaching a one quarter course on integration at the advanced undergraduate level. My intent is to introduce the Lebesgue integral in a quick, and hopefully painless, way and then go on to…