Euler Equations on Borel Subalgebras of Semisimple Lie Algebras

Abstract

We study the level spacing distribution p(s) in the spectrum of random networks. According to our numerical results, the shape of p(s) in the Erdős-Rényi (E-R) random graph is determined by the average degree k and p(s) undergoes a dramatic change when k is varied around the critical point of the percolation transition, k = 1. When k 1, the p(s) is described by the statistics of the Gaussian orthogonal ensemble (GOE), one of the major statistical ensembles in Random Matrix Theory, whereas at k = 1 it follows the Poisson level spacing distribution. Closely above the critical point, p(s) can be described in terms of an intermediate distribution between Poisson and the GOE, the Brodydistribution. Furthermore, below the critical point p(s) can be given with the help of the regularized Gamma-function. Motivated by these results, we analyse the behaviour of p(s) in real networks such as the internet, a word association network and a protein-protein interaction network as well. When the giant component of these networks is destroyed in a node deletion process simulating the networks subjected to intentional attack, their level spacing distribution undergoes a similar transition to that of the E-R graph.

Complete involutive algebras of functions on cotangent bundles of homogeneous spaces

“…Trofimov and Thimm devised a method for constructing functions in involution on a Lie algebra G by using chains of subalgebras [35,36]. Suppose we are given a chain of connected compact subgroups H = G 0 ⊂ G 1 ⊂ G 2 ⊂ .…”

“…Then the Lie-Poisson bracket of p 1 • π i and p 2 • π j vanishes identically on G [35,36]. With respect to (24) we have orthogonal decomposition of V :…”

The Algebra of Hamiltonian Systems

“…The following construction was first described by Vergne [1972] in the context of geometric quantization (construction of polarizations) and by Trofimov [1979] in the context of integrable systems. The following construction was first described by Vergne [1972] in the context of geometric quantization (construction of polarizations) and by Trofimov [1979] in the context of integrable systems.…”

Интегрируемые Гамильтоновы Системы На Алгебрах Ли Малой Размерности

“…В связи с этим естественно поставить вопрос В более поздних работах А. В. Браилова, А. В. Болсинова, В. В. Трофимо-ва, Т. А. Певцовой, О. И. Богоявленского, А. А. Архангельского, С. Сярова, Ле Нгок Тьеуена, А. М. Переломова, А. Г. Реймана, А. В. Беляева, М. Вернь, В. А. Гинзбурга полные коммутативные наборы полиномов были построены на различных сериях алгебр Ли: вещественных нильпотентных, вполне разреши-мых, борелевских подалгебрах полупростых алгебр Ли, многих полупрямых суммах и т.д. (см., например, [7]- [10]). Полный список соответствующих ре-зультатов приведен в [1].…”