Related papers: Divergent series: past, present, future
A working mathematician's summary of many results on the derived category, perverse sheaves, and vanishing cycles. This is the August 2025 version, with a completely revised section on vanishing cycles.
The aim of this text is to provide a linguistically accessible, but comprehensive introduction into a variety of topics in dynamical systems and its applications. Whilst preliminary knowledge of dynamical systems is useful, it is not…
This primer is intended as an introduction to differential forms, a central object in modern mathematical physics, for scientists and engineers.
This note gives a few rapidly convergent series representations of the sums of divisors functions. These series have various applications such as exact evaluations of some power series, computing estimates and proving the existence results…
This expository article written for the Notices of the American Mathematical Society provides an overview of transcendental functions arising as solutions of the discrete Painlev\'e equations, for which the developments of the last two…
Treating divergent series properly has been an ongoing issue in mathematics. However, many of the problems in divergent series stem from the fact that divergent series were discovered prior to having a number system which could handle them.…
A brief summary of recent developments in mathematical diffraction theory is given. Particular emphasis is placed on systems with aperiodic order and continuous spectral components. We restrict ourselves to some key results and refer to the…
This essay considers ways that recent uses of computers in mathematics challenge contemporary views on the nature of mathematical understanding. It also puts these challenges in a historical perspective and offers speculation as to a…
This paper provides some reflections on the field of mathematical software on the occasion of John Rice's 65th birthday. I describe some of the common themes of research in this field and recall some significant events in its evolution.…
Starting with a novel definition of divided differences, this essay derives and discusses the basic properties of, and facts about, (univariate) divided differences.
In this paper I review the theoretical progresses in studying the diffractive DIS in the colour--dipole approach.
The purpose of this article is to provide an exposition of domains of convergence of power series of several complex variables without recourse to relatively advanced notions of convexity.
The main objective of the present paper is to solve the several long standing divergence problems related to the logarithmic means of Fourier series in context of general orthonormal systems.
Euler gives a long introduction, giving all the arguments for and against the use of divergent series in calculus and then gives his own definition of the sum of a diverging series. Then in the second half of this paper he evaluates the the…
This is the translation of Leonhard Euler's paper "De Seriebus divergentibus" written in Latin into English. Leonhard Euler defines and discusses divergent series. He is especially interested in the example $1!-2!+3!-\text{etc.}$ and uses…
We prove that several results in different areas of number theory such as the divergent series, summation of arithmetic functions, uniform distribution modulo one and summation over prime numbers which are currently considered to be…
Using diffusion priors to solve inverse problems in imaging have significantly matured over the years. In this chapter, we review the various different approaches that were proposed over the years. We categorize the approaches into the more…
Differentiable physics provides a new approach for modeling and understanding the physical systems by pairing the new technology of differentiable programming with classical numerical methods for physical simulation. We survey the rapidly…
The aim of this review, based on a series of four lectures held at the 22nd "Saalburg" Summer School (2016), is to cover selected topics in the theory of perturbation series and their summation. The first part is devoted to strategies for…
Basic principles of mathematical modeling are reviewed in this book, with the focus on physics and its practical applications, and examples of selected mathematical methods are presented. Most of the models have been imported from physics…