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# Euler's formula

About: Euler's formula is a research topic. Over the lifetime, 10484 publications have been published within this topic receiving 177356 citations.

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TL;DR: The algorithm can be used as a building block for solving other distributed graph problems, and can be slightly modified to run on a strongly-connected diagraph for generating the existent Euler trail or to report that no Euler trails exist.

Abstract: A new distributed Euler trail algorithm is proposed to run on an Euler diagraph G(V,E) where each node knows only its adjacent edges, converting it into a new state that each node knows how an existent Euler trail routes through its incoming and outgoing edges. The communication requires only 2middot;|E| one-bit messages. The algorithm can be used as a building block for solving other distributed graph problems, and can be slightly modified to run on a strongly-connected diagraph for generating the existent Euler trail or to report that no Euler trails exist.

13,828 citations

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Duke University

^{1}TL;DR: In this article, an introduction to vortex dynamics for incompressible fluid flows is given, along with vortex sheets, weak solutions and approximate-solution sequences for the Euler equation.

Abstract: Preface 1. An introduction to vortex dynamics for incompressible fluid flows 2. The vorticity-stream formulation of the Euler and the Navier-Stokes equations 3. Energy methods for the Euler and the Navier-Stokes equations 4. The particle-trajectory method for existence and uniqueness of solutions to the Euler equation 5. The search for singular solutions to the 3D Euler equations 6. Computational vortex methods 7. Simplified asympototic equations for slender vortex filaments 8. Weak solutions to the 2D Euler equations with initial vorticity in L 9. Introduction to vortex sheets, weak solutions and approximate-solution sequences for the Euler equation 10. Weak solutions and solution sequences in two dimensions 11. The 2D Euler equation: concentrations and weak solutions with vortex-sheet initial data 12. Reduced Hausdorff dimension, oscillations and measure-valued solutions of the Euler equations in two and three dimensions 13. The Vlasov-Poisson equations as an analogy to the Euler equations for the study of weak solutions Index.

1,863 citations

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TL;DR: In this article, a second-order extension of the Lagrangean method is proposed to integrate the equations of ideal compressible flow, which is based on the integral conservation laws and is dissipative, so that it can be used across shocks.

Abstract: A method of second-order accuracy is described for integrating the equations of ideal compressible flow. The method is based on the integral conservation laws and is dissipative, so that it can be used across shocks. The heart of the method is a one-dimensional Lagrangean scheme that may be regarded as a second-order sequel to Godunov's method. The second-order accuracy is achieved by taking the distributions of the state quantities inside a gas slab to be linear, rather than uniform as in Godunov's method. The Lagrangean results are remapped with least-squares accuracy onto the desired Euler grid in a separate step. Several monotonicity algorithms are applied to ensure positivity, monotonicity, and nonlinear stability. Higher dimensions are covered through time splitting. Numerical results for one-dimensional and two-dimensional flows are presented, demonstrating the efficiency of the method. The paper concludes with a summary of the results of the whole series “Towards the Ultimate Conservative Difference Scheme.”

1,837 citations

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1,760 citations

01 Jan 2015

TL;DR: In this article, the main concepts and techniques necessary for someone who wishes to carry out numerical experiments involving stochastic differential equations (SDEs) are described and compared. And the convergence of Euler-Maruyama and Milstein and Taylor approximate solutions are compared.

Abstract: This paper provides an introduction to the main concepts and techniques necessary for someone who wishes to carryout numerical experiments involving Stochastic Differential Equation (SDEs). As SDEs are frictionless generally and the solutions are continuous stochastic process that represent diffusive dynamic especially in finance, it is required of us to take into account random effects and influences in real world systems which are essential in the accurate description of such situations. We include a review of Stochastic Differential equations (SDE), Geometric Brownian Motion, Euler- Maruyama, Milstein and Taylor approximate which gives a clear picture of their graphical approximate and exact solution. We finally compared the convergence of Euler-Maruyama and Milstein

1,400 citations