It is proved that every dissipative-repulsive periodic system admits a periodic solution, which is comparable with some well-known results due to Yoshizawa, Hale and Lopes, and Burton and Zhang for dissipative systems.

Lorentzian versions of classical Riemannian volume comparison theorems by Günther, Bishop and Bishop-Gromov, are stated for suitable natural subsets of general semi-Riemannian manifolds. The problem is more subtle in the Bishop-Gromov case, which is extensively discussed. For the general semi-Riemannian case, a local version of the Günther and Bishop theorems is given and applied.

Given an orientable Riemannian manifold, we consider the bundle of oriented orthonormal frames and the tangent sphere bundle over it, which admit natural Riemannian metrics defined by the Riemannian connection. We show that there is a natural homomorphism between the Lie algebras of fiber preserving Killing vector fields on these bundles. In particular, for any orientable Riemannian manifold of dimension two, we show that the homomorphism is extended to an isomorphism between these Lie algebras.

We study the magnetic Schrödinger operator $H$ on $R^n$, $n\geq3$. We assume that the electrical potential $V$ and the magnetic potential {\bf a} belong to a certain reverse Hölder class, including the case that $V$ is a non-negative polynomial and the components of {\bf a} are polynomials. We show some estimates for operators of Schrödinger type by using estimates of the fundamental solution for $H$. In particular, we show that the operator $\nabla^2(-\Delta+V)^{-1}$ is a Calderón-Zygmund operator.

The purpose of this paper is to give basic tools for the classification of nonsingular toric Fano verieties by means of the notions of primitive collections and primitive relations due to Batyrev. By using them we can easily deal with equivariant blow-ups and blow-downs, and get an easy criterion to determine whether a given nonsingular toric variety is a Fano variety or not. As applications of these results, we get a toric version of a theorem of Mori, and can classify, in principle, all nonsingular toric Fano varieties obtained from a given nonsingular toric Fano variety by finite successions of equivariant blow-ups and blow-downs through nonsingular toric Fano varieties. Especially, we get a new method for the classification of nonsingular toric Fano varieties of dimension at most four. These methods are extended to the case of Gorenstein toric Fano varieties endowed with natural resolutions of singularities. Especially, we easily get a new method for the classification of Gorenstein toric Fano surfaces.

Problems related to minimal maps are studied. In particular, we prove an existence result for the Dirichlet problem at infinity for minimal diffeomorphisms between the hyperbolic discs. We also give a representation formula for a minimal diffeomorphism between the hyperbolic discs by means of the generalized Gauss map of a complete maximal surface in the anti-de Sitter 3-space.

We shall give a saturation class for approximations by eigenfunction expansions of the Laplacian in an open domain in the Euclidean space.

We develop a general theory of martingale transform operators with operator-valued multiplying sequences. Applications are given to classical operators such as Doob's maximal function and the square function. Some geometric properties of the underlying Banach spaces are also considered.