A nonclassical skew-selfadjoint system of two linear differential equations is considered, which depends rationally on the spectral parameter. Systems of this type are related to the sine-Gordon equation. We introduce the notion of W_p-functions (Weyl functions) which are defined in a neighborhood of the poles. The main results are theorems on the existence and uniqueness of the Weyl functions, on the uniqueness of the solutions of the inverse problem, and on explicit solutions for the direct and the inverse problem.

We prove that a two-sided sub-Gaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the Green kernel admits a uniform polynomial decay.

The purpose of this paper is to present a short elementary proof of a theorem due to G. Faltings and G. Laumon, which says that the global nilpotent cone is a Lagrangian substack in the cotangent bundle of the moduli space of G-bundles on a complex compact curve. This result plays a crucial role in the geometric Langlands program (see [BD]) since it insures that the $\mathscr{D}$-modules on the moduli space of G-bundles whose characteristic variety is contained in the global nilpotent cone are automatically holonomic and, in particular, have finite length.

We present sharp estimates for the degree and the height of the polynomials in the Nullstellensatz over the integer ring ℤ. The result improves previous work of P. Philippon, C. Berenstein and A. Yger, and T. Krick and L. M. Pardo.

We also present degree and height estimates of intrinsic type, which depend mainly on the degree and the height of the input polynomial system. As an application we derive an effective arithmetic Nullstellensatz for sparse polynomial systems.

The proof of these results relies heavily on the notion of local height of an affine variety defined over a number field. We introduce this notion and study its basic properties.

We prove a positivity property for the cup product in the T-equivariant cohomology of the flag variety. This was conjectured by D. Peterson and has as a consequence a conjecture of S. Billey. The result for the flag variety follows from a more general result about algebraic varieties with an action of a solvable linear algebraic group such that the unipotent radical acts with finitely many orbits. The methods are those used by S. Kumar and M. Nori.