THE EULER-POINCARÉ CHARACTERISTIC AND MIXED MULTIPLICITIES

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...

Efficient algorithms for computing the Euler-Poincaré characteristic of symmetric semi-algebraic sets

We give algorithms with polynomially bounded complexities (for fixed degrees) for computing the generalized Euler-Poincaré characteristic of semi-algebraic sets defined by symmetric polynomials. This is in contrast to the best complexity of the known algorithms for the same problem in the non-symmetric situation, which is singly exponential. This singly exponential complexity for the latter pro...

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.

The Euler Characteristic

I will describe a few basic properties of the Euler characteristic and then I use them to prove special case of a cute formula due to Bernstein-Khovanskii-Koushnirenko. 1. Basic properties of the Euler characteristic The Euler characteristic is a function χ which associates to each reasonable topological space X an integer χ(X). For us a reasonable space would be a space which admits a finite s...

Exponentiation and Euler Characteristic