Suppose (R, m) is a local Noetherian domain with quotient field K and m-adic completion Ȓ. It is well known that the fibers of the morphism Spec(Ȓ) ₒ Spec(R), i.e., the formal fibers of R, encode important information about the structure of R. Perhaps the most important condition in Grothendieck’s definition of R being excellent is that the formal fibers of R be geometrically regular. Indeed, a local Noetherian ring is excellent provided it is universally catenary and has geometrically regular formal fibers [G, (7.8.3), page 214]. But the structure of the formal fibers of R is often difficult to determine. We are interested here in bringing out the interrelatedness of properties of the generic formal fiber of R with the existence of certain local Noetherian domains C birationally dominating R and having C/mC is a finite R-module.

Let V be a real vector space. An arrangement of hyperplanes in V is a finite family of hyperplanes of V through the origin. We say that is essential if ∩H ∊H = {0}

Let G be a countably generated discrete group. A right-invariant mean μ on G is a bounded linear functional of the space L∞(G) of bounded functions on G having the property:

We say that G is amenable if it is equipped with a right-invariant mean. Finite groups, abelian groups, in fact, groups of subexponential growth are amenable. Solvable group are also amenable. Subgroups and quotients of amenable groups are amenable. On the other hand, free groups having two generators and over are non-amenable.

In this paper we are concerned with zero-sum two-player finite horizon games for stochastic partial differential equations (SPDE in short). The main aim is to formulate the principle of dynamic programming for the upper (or lower) value function and investigate the relationship between upper (or lower) value function and viscocity solution of min-max (or max-min) equation on Hilbert space.

Let X ⊆ Pr be a reduced, irreducible and non-degenerate projective variety over an algebraically closed field K of characteristic 0. Let reg(x) be the Castelnuovo-Mumford regularity of the sheaf of ideals associated to X.

Then it is an open problem—due to D. Eisenbud (see e.g. [E-Go])—whether

(0.1) reg(X) ≤ deg(x) - codim (x) + 1,

where deg(x) denotes the degree of X and codim(x) denotes the codimension of X. In many cases, this inequality has been proven to hold true.

Let Mn be an n-dimensional Riemannian manifold minimally immersed in the unit sphere Sn+p (1) of dimension n + p. When Mn is compact, Chern, do Carmo and Kobayashi [1] proved that if the square ‖h‖2 of length of the second fundamental form h in Mn is not more than , then either Mn is totally

geodesic, or Mn is the Veronese surface in S4 (1) or Mn is the Clifford torus .

In this paper, we generalize the results due to Chern, do Carmo and Kobayashi [1] to complete Riemannian manifolds.

The theory of newforms is very important and useful for arithmetical study of modular forms of integral weight. It is natural to try to extend this theory into the case of modular forms of half-integral weight Until now, several authors have attempted to find a theory of newforms of half-integral weight (cf. [She], [N], [K], [M-R-V], [She-W]). But complete results have not been obtained yet.