The Euler-Poincaré Formula Through Contact Surfaces of Voxelized Objects

Multidimensional Euler – Poincaré equations 1

Given a Lagrangian L : J 1 P → R, with P = M × G → M, invariant under the natural action of G on J 1 P, we deduce the analog of the Euler–Poincaré equations. The geometry of the reduced variational problem as well as its link with the Noether Theorem and an example are also given.

Euler - Poincaré formulation of hybrid plasma models

Several different hybrid Vlasov-fluid systems are formulated as Euler-Poincaré systems and compared in the same framework. In particular, the discussion focuses on three major hybrid MHD models. These are the current-coupling scheme and two different variants of the pressurecoupling scheme. The Kelvin-Noether theorem is presented explicitly for each scheme, together with the Poincaré invariants...

Some Notes on the Poincaré-Bertrand Formula

Euler - Maclaurin Formula

a Bk({1− t}) k! f (t)dt where a and b are arbitrary real numbers with difference b − a being a positive integer number, Bn and bn are Bernoulli polynomials and numbers, respectively, and k is any positive integer. The condition we impose on the real function f is that it should have continuous k-th derivative. The symbol {x} for a real number x denotes the fractional part of x. Proof of this th...

Estimation of the number of alveolar capillaries by the Euler number (Euler-Poincaré characteristic).

The lung parenchyma provides a maximal surface area of blood-containing capillaries that are in close contact with a large surface area of the air-containing alveoli. Volume and surface area of capillaries are the classic stereological parameters to characterize the alveolar capillary network (ACN) and have provided essential structure-function information of the lung. When loss (rarefaction) o...