Related papers: The Calculus and Gauge Integrals, by Ralph Henstoc…
We give sufficient conditions on the exponent $p: \mathbb R^d\rightarrow [1,\infty)$ for the boundedness of the non-centered Gaussian maximal function on variable Lebesgue spaces $L^{p(\cdot)}(\mathbb R^d, \gamma_d)$, as well as of the new…
Complex analysis is a powerful tool to study classical integrable systems, statistical physics on the random lattice, random matrix theory, topological string theory,... All these topics share certain relations, called "loop equations" or…
In this paper, a generalised integral called the Laplace integral is defined on unbounded intervals, and some of its properties, including necessary and sufficient condition for differentiating under the integral sign, are discussed. It is…
One of the key elements of Feynman's formulation of non-relativistic quantum mechanics is a so-called Feynman path integral. It plays an important role in the theory, but it appears as a postulate based on intuition rather than a…
Gian-Carlo Rota suggested in one of his last articles the problem of developing a theory around the notion of integration algebras, complementary to the already existing theory of differential algebras. This idea was mainly motivated by…
This paper gives an overview of several key innovations in the 19th century which led to complex geometry in the 20th century. This includes the creation of the complex plane, the work of Abel on addition theorems for generalized elliptic…
Differential lambda-calculus was first introduced by Thomas Ehrhard and Laurent Regnier in 2003. Despite more than 15 years of history, little work has been done on a differential calculus with integration. In this paper, we shall propose a…
In this paper, we develop a novel approach to the Weingarten calculus by employing the notion of virtual isometries. Traditionally, Weingarten calculus provides explicit formulas for integrating polynomial functions over compact matrix…
This is the paper "Niels Henrik Abel and the birth of fractional calculus", Podlubny, I., Magin, R. L., Trymorush I., Fractional Calculus and Applied Analysis, vol.20, no.5, pp.1068-1075, 2017 (https://doi.org/10.1515/fca-2017-0057) with…
The theory of fractional calculus has developed in a number of directions over the years, including: the formulation of multiple different definitions of fractional differintegration; the extension of various properties of standard calculus…
A small and unsystematic selection of my favorite appearances of mathematicians and mathematics in German literature. It includes classic and romantic (Lessing, Goethe, Wezel, F. Schlegel, Kleist, Novalis, Grillparzer, Heine), modern…
During his brief life, the Polish mathematician and logician Adolf Lindenbaum (1904--1941) contributed to mathematical logic, among other things, by several significant achievements. Some results of Lindenbaum's, which bear his name, were…
The Arithmetic is interpreted in all the groups of Richard Thompson and Graham Higman, as well as in other groups of piecewise affine permutations of an interval which generalize the groups of Thompson and Higman. In particular, the…
Universality of eigenvalue spacings is one of the basic characteristics of random matrices. We give the precise meaning of universality and discuss the standard universality classes (sine, Airy, Bessel) and their appearance in unitary,…
The irreducible unitary representations of the Banach Lie group $U_0(\H)$ (which is the norm-closure of the inductive limit $\cup_k U(k)$) of unitary operators on a separable Hilbert space $\H$, which were found by Kirillov and Ol'shanskii,…
In this paper we introduce a new fractional integral that generalizes six existing fractional integrals, namely, Riemann-Liouville, Hadamard, Erd\'elyi-Kober, Katugampola, Weyl and Liouville fractional integrals in to one form. Such a…
This is an intuitive survey of extrinsic and intrinsic notions of convergence of manifolds complete with pictures of key examples and a discussion of the properties associated with each notion. We begin with a description of three extrinsic…
This is an introduction to linear algebra and group theory. We first review the linear algebra basics, namely the determinant, the diagonalization procedure and more, and with the determinant being constructed as it should, as a signed…
We attempt to reconstruct the irreducible unitary representations of the Banach Lie group $U_0(\H)$ of all unitary operators $U$ on a separable Hilbert space $\H$ for which $U-{\mathbb I}$ is compact, originally found by Kirillov and…
After the language of module and theirs morphisms, this short course presents matricial calculus and determinants in a commutative ring as appliction of ``remarquable identities'' in the ring of polynomials with integer coefficients with…