The dynamics of the dilute electrons can be modeled by the fundamental Vlasov-PoissonBoltzmann system when the electrons interact with themselves through collisions in the selfconsistent electric field. In this paper, it is shown that any smooth initial perturbation of a given global Maxwellian leads to a unique global-in-time classical solution when either the mean free path is small or the ba...

Conjecture 1 (Langlands). To each reductive group G over a number field K, each automorphic (complex) representation π of G, and each finite dimensional representation r of the (complex) group G, there is defined an automorphic L-function L(s, π, r), which enjoys an analytic continuation and functional equation generalizing the Riemann zeta function πΓ(s/2)ζ(s) (or Artin’s L-function L(s, σ), w...

The rational solutions with as denominators powers of to the elliptic diophantine equation y x x are determined An idea of Yuri Bilu is applied which avoids Thue and Thue Mahler equations and deduces four term S unit equations with special properties that are solved by linear forms in real and p adic logarithms Introduction In a recent paper SW my colleague R J Stroeker and I determined the com...

Aims. The so-called Limber equation is widely used in the literature to relate the projected angular clustering of galaxies to the spatial clustering of galaxies in an approximate way. This note gives estimates of where the regime of applicability of Limber's equation stops. Methods. This paper revisits Limber's equation, summarises its underlying assumptions, and compares its predictions to th...

The modular vector field plays an important role in the theory of Poisson manifolds and is intimately connected with the Poisson cohomology of the space. In this paper we investigate its significance in the theory of integrable systems. We illustrate in detail the case of the Toda lattice both in Flaschka and natural coordinates.

Given a classical r-matrix on a Poisson algebra, we show how to construct a natural family of compatible Poisson structures for the Hamiltonian formulation of Lax equations. Examples for which our formalism applies include the Benny hierachy, the dispersionless Toda lattice hierachy, the dispersionless KP and modified KP hierachies, the dispersionless Dym hierachy etc.

Using concepts of noncommutative probability we show that the Löwner’s evolution equation can be viewed as providing a map from paths of measures to paths of probability measures. We show that the fixed point of the Löwner map is the convolution semigroup of the semicircle law in the chordal case, and its multiplicative analogue in the radial case. We further show that the Löwner evolution “spr...