Related papers: Euler and the Ordinary Differential Equations

Ordinary Differential Equations are derived for the adjoint Euler equations firstly using the method of characteristics in 2D. For this system of partial-differential equations, the characteristic curves appear to be the streamtraces and…

This paper looks at how ancient mathematicians (and especially the Pythagorean school) were faced by problems/paradoxes associated with the infinite which led them to juggle two systems of numbers: the discrete whole/rationals which were…

The angle defect, which is the standard way to measure curvature at the vertices of polyhedral surfaces, goes back at least as far as Descartes. Although the angle defect has been widely studied, there does not appear to be in the…

This paper is devoted to Poincar\'e's work in probability. Though the subject does not represent a large part of the mathematician's achievements, it provides significant insight into the evolution of Poincar\'e's thought on several…

This paper aims to provide physics teachers with tools to help deepen the understanding of the laws of electromagnetism. The fundamental contributions of our proposal are: a) to use quotes from mythical characters in the history of science…

A new pseudospectral calculation of collapsing Euler vortices \cite{HouLi06} has called into question the long-term conclusions of singular behavior described earlier in \cite{Kerr93,Kerr05}. This review is designed to: improve the…

In this paper, we derive some interesting symmetric properties for the geenralized Euler numbers and polynomials.

One of the important ways development takes place in mathematics is via a process of generalization. On the basis of a recent characterization of this process we propose a principle that generalizations of mathematical structures that are…

We describe the problems of quantum chemistry, the intuition behind classical heuristic methods used to solve them, a conjectured form of the classical complexity of quantum chemistry problems, and the subsequent opportunities for quantum…

Schanuel has pointed out that there are mathematically interesting categories whose relationship to the ring of integers is analogous to the relationship between the category of finite sets and the semi-ring of non-negative integers. Such…

I relate some coefficients encountered when computing the functional determinants on spheres to the central differentials of nothing. In doing this I use some historic works, in particular transcribing the elegant symbolic formalism of…

These lecture notes for the course APM 351 at the University of Toronto are aimed at mathematicians and physicists alike. It is not meant as an introductory course to PDEs, but rather gives an overview of how to view and solve differential…

This essay considers the special character of mathematical reasoning, and draws on observations from interactive theorem proving and the history of mathematics to clarify the nature of formal and informal mathematical language. It proposes…

This is the transcript of a lecture given at UMass-Lowell in which I compare and contrast the work of Godel and of Turing and my own work on incompleteness. I also discuss randomness in physics vs randomness in pure mathematics.

We prove necessary optimality conditions of Euler-Lagrange type for generalized problems of the calculus of variations on time scales with a Lagrangian depending not only on the independent variable, an unknown function and its delta…

These lectures notes give an introduction to the fast developing area of research dealing with perturbative descriptions of the gravitational instability in an expanding universe. I just sketch the outlines of some proofs, and many…

A basic example of shear flow was introduced by DiPerna and Majda to study the weak limit of oscillatory solutions of the Euler equations of incompressible ideal fluids. In particular, they proved by means of this example that weak limit of…

An explicit construction of closed, orientable, smooth, aspherical 4-manifolds with any odd Euler characteristic greater than 12 is presented. The manifolds constructed here are all Haken manifolds in the sense of B. Foozwell and H.…

Euler considers the following problem: A boat with a perfect rudder moves at constant speed across a stream flowing in straight fillets at assigned speeds. Assuming that the downstream velocity of the boat equals that of the river, how…

We consider the Euler approach to construction and to investigation of the superintegrable systems related to the addition theorems. As an example we reconstruct Drach systems and get some new two-dimensional superintegrable Stackel…