Let M be an n-dimensional manifold immersed in an m-dimensional euclidean space Em and let ∇ and ∇̃ be the covariant differentiations of M and Em, respectively. Let X and Y be two tangent vector fields on M. Then the second fundamental form h is given by

(1.1) ∇̃XY = ∇XY + h(X,Y).

This paper is devoted to the study of the asymptotic distribution of eigenvalues of the Laplace operator with zero boundary conditions in a quasi-bounded domain contained in Euclidean space R2.

This paper is one of a series relating the symmetry groups of the principal linear partial differential equations of mathematical physics and the coordinate systems in which variables separate for these equations. In particular, we mention [1] and paper [2] which is a survey of and introduction to the series. Here we apply group-theoretic methods to study the separable coordinate systems for the Helmholtz equation.

We shall be concerned here with cusps on the boundaries of Teichmüller spaces.

In [3] Cameron and Storvick introduced a very general operator-valued function space “integral”. In [3-5, 8, 9, 11, 13-20] the existence of this integral as an operator from L2 to L2 was established for certain functions. Recently the existence of the integral as an operator from L1 to L∞, has been studied [6, 7, 21]. In this paper we study the integral as an operator from Lp to Lp′, where 1 < p ≤ 2. The resulting theorems extend the theory substantially and indicate relationships between the L2-L2 and L1-L∞ theories that were not apparent earlier. Even in the most studied case, p = p′ = 2, the results below strengthen the theory.

Let f: A → B be a homomorphism of commutative noetherian rings. The main results of this paper are:

(a) Assume f is finite and induces a surjective map on the spectra. Then if B is quasi-excellent A is quasi-excellent and is excellent if it is universally catenarian (Th. 3.1); and

(b) If f is absolutely flat and A is excellent then B is excellent (Th. 5.3). In particular the strict henselization of an excellent local ring is excellent (Cor. 5.6.).

Let Ω be a bounded domain in n-dimensional Euclidian space En (n ≧ 2), and consider the space-time cylinder Q = Ω × (0, T] for some fixed T > 0. In this paper we deal with the Cauchy and Dirichlet problem for a second order quasi-linear equation

(1.1) ut — div A(x, t, u, ux) + B(x, t, u, ux) = 0 for (x, t) ∈ Q,

(1.2) u(x, 0) = (ϕ)(x) in Ω and u(x, t) = tψ(x, t) for (x, t) ∈ Γ = ∂Ω × (0, T] ,

where ∂Ω is a boundary of Ω which satisfies the following condition (A) : Condition (A). There exist constants ρ0 and »0 both in (0,1) such that, for any sphere K(ρ) with center on ∂Ω and radius ρ ≦ ρ0, the inequality meas [K(ρ) ∩ Ω] ≦ (1 — λ0) × meas E(ρ) holds, where meas E means the measure of a measurable set E.

0. Let p be a prime number or zero and let g be a non-negative integer. Then there is a coarse moduli space Mg for complete non-singular irreducible curves of genus g defined over fields of characteristic p, which is an irreducible variety over the algebraic closure F̅p of the prime field Fp. (Especially, F0 is also denoted by Q as usual.) ([8], [2]). The curve corresponding to a generic point of Mg over F̅p is called a generic curve of genus g.

Let Pm(C) be the complex projective space of dimension m. In a previous paper [2] we have proved

THEOREM A. Let f be a Kaehlerian immersion of a connected, complete Kaehler manifold Mn of dimension n into Pm(C). If the image f(τ) of each geodesic τ in Mn lies in a complex projective line P1(C) of Pm(C), then f(Mn) is a complex projective subspace of Pm(C), and f is totally geodesic.

Consider a diffusion process in Rd satisfying the stochastic differential equation

.