Related papers: Euler and the Ordinary Differential Equations
Partial differential equations are central objects in the mathematical modeling of natural and social sciences (sound propagation, heat diffusion, thermodynamics, electromagnetism, elasticity, fluid dynamics, quantum mechanics, population…
In their account of theory change in logic, Aberdein and Read distinguish 'glorious' from 'inglorious' revolutions--only the former preserves all 'the key components of a theory' [1]. A widespread view, expressed in these terms, is that…
In a piece published in 1981, H. M. Edwards touts the benefits of reading the masters. A quarter-century later, Edwards takes seriously his own advice by publishing an encomium on Euler's Institutiones (1755). While we agree with Edwards…
This paper reconstructs the derivations underlying the kinematical part of Einstein's 1905 special relativity paper, emphasizing their operational clarity and minimalist use of mathematics. Einstein employed modest tools-algebraic…
This is a short survey article written for the Erd\H{o}s centennial conference in Budapest in 2013. The main two topics covered are Szemer\'edi's theorem and its ramifications, and the Erd\H{o}s discrepancy problem. There is an emphasis on…
Einstein identified singularities in spacetimes, such as at the Schwarzschild radius, where later relativists only find a coordinate system assigning multiple values to a single spacetime event. These differing judgments derive from…
The Eulerian numbers form a triangular array with many interesting properties. The numbers arise from various combinatorial and probabilistic interpretations, and have been studied in a variety of mathematical contexts. In this article we…
Euler systems are certain compatible families of cohomology classes, which play a key role in studying the arithmetic of Galois representations. We briefly survey the known Euler systems, and recall a standard conjecture of Perrin-Riou…
The concept of a fluid algebra was introduced by Sullivan over a decade ago as an algebraic construct which contains everything necessary in order to write down a form of the Euler equation, as an ODE whose solutions have invariant…
In this article we describe special type of mathematical problems that may help develop teaching methods that motivate students to explore patterns, formulate conjectures and find solutions without only memorizing formulas and procedures.…
In the "6D treatment of Special Relativity" proposed by Igor A. Urusovskii one deals with universal light-like motion of matter pre-elements in the extended (3+3) space and with their regular rotation in the additional 3-space. On the other…
A recent essay [1] reminds us of how richly Boltzmann deserves to be admiringly commemorated for the originality of his ideas on the occasion of his 150th birthday. Without any doubt, the scientific community owes Boltzmann a great debt of…
Metaheuristic algorithms are methods devised to efficiently solve computationally challenging optimization problems. Researchers have taken inspiration from various natural and physical processes alike to formulate meta-heuristics that have…
Mathematical objects are generally abstract and not very approachable. Illustrations and interactive visualizations help both students and professionals to comprehend mathematical material and to work with it. This approach lends itself…
We solve a problem concerning Euler's paper "Variae considerationes circa series hypergeometricas" (\cite{E661}) suggested by G. Faber in the preface to Volume 16,2 of the first series of Euler's Opera Omnia by methods proven by Euler on…
We analyze the behavior of the Euler method for delay differential equations under nonstandard assumptions on the right-hand-side function f, when evaluations of f are corrupted by informational noise. We provide theoretical upper bounds on…
This is the English translation of Leonhard Euler's Latin paper "De solidis quorum superficiem in planum explicare licet". Euler explains several methods to obtain equations for developable surfaces. Therefore, this paper might be…
E565 in the Enestrom index. Translated from the Latin original, "De plurimis quantitatibus transcendentibus quas nullo modo per formulas integrales exprimere licet" (1775). Euler does not prove any results in this paper. It seems to me like…
We discuss umbral calculus as a method of systematically discretizing linear differential equations while preserving their point symmetries as well as generalized symmetries. The method is then applied to the Schr\"{o}dinger equation in…
Euler gives an asymptotic approximation for the function f(x) and recognizes that he is trying to interpolate the factorial function introduced in E19 "De progressionibus transcendentibus seu quarum termini generales algebraice dari…