Let Γ be an undirected graph and G a subgroup of aut (Γ). We denote by ∂(x, y) the distance between two vertices x and y, by E(Γ) the edge set of Γ, by V(Γ) the vertex set of Γ, by Γ(x) the set of neighbors of the vertex x and by G(x)Γ(x) the permutation group induced by the stabilizer G(x) on Γ(x).

Let G be an algebraic group over a field k, and let ψ be an action of the multiplicative group km of k on G by automorphisms. We say ψ is an elementary action if it has only the weights 0, ±1; more precisely, if there exist subgroups H, U+, U- of G such that (i) H is fixed under ψ, (ii) U+ and U+ are vector groups and (iii) Ω = U+. H . U+ is open in G, and (iv) G is generated by H, U+, U+. This situation is characteristic for the complexifications of the automorphism groups of bounded symmetric domains (see, e.g., [9, 16]). A typical example is G = GLnwith (matrices being decomposed into 4 blocks) ψ given by

In 1972, Nomizu and Rodriguez [5] found the following characterization of the complete umbilic submanifolds of Euclidean space.

Theorem A. Let Mn, n ≥ 2, be a connected, complete Riemannian manifold isometrically immersed in a Euclidean space Em. Every Morse function of the form Lp has index 0 or n at all of its critical points if and only if Mnis embedded as a Euclidean n-subspace or a Euclidean n-sphere in Em.

In this paper we give a negative answer to the problem which is suggested in [3]: if a nonwandering set Ω is hyperbolic, are the periodic points dense in Ω?

Newhouse and Palis proved that on two dimensional closed manifolds the answer is positive ([1], [2]).

Consider a one-dimensional differential algebraic function field K over an algebraically closed ordinary differential field k of characteristic 0. We shall prove the following theorem:

Suppose that the group of all automorphisms of K over k is infinite. Then, K is either a differential elliptic function field over k or K = k(ν) with ν′ = ξ or ν′ = ην, where ξ, η ϵ k.

Let A(m), B(n) be positive definite integral matrices and suppose that B is represented by A over each p-adic integers ring Zp. Using the circle method or theory of modular forms in case of n = 1, B, if sufficiently large, is represented by A provided that m ≥ 5. The approach via the theory of modular forms has been extended by [7] to Siegel modular forms to obtain a partial result in the particular case when n = 2, m ≥ 7.

Im Zusammenhang mit dem von Grothendieck in [2] Chap. IV (7.4.8) gestellten Problem “1st die ideal-adische Komplettierung eines ausgezeichneten Ringes wieder ausgezeichnet?” scheint aufgrund der Arbeiten von Marot [3] und Valabrega [8] im lokalen Fall die folgende Frage intéressant zu sein: “R sei ein lokaler universell japanischer Ring. 1st der singuläre Ort jeder endlich erzeugten R- Algebra A abgeschlossen in Spec (A)?” Diese Frage wird hier negativ beantwortet, d.h. wir werden einen lokalen universell japanischen Ring A konstruieren, dessen singulärer Ort nicht abgeschlossen in der Zariski-Topologie von Spec (A) ist.

The celebrated Ito formula for the n-dimensional Brownian motion Xt and for u ∈ C2(Rn) runs as follows:

(0.1)

In § 6 of this paper we extend this to the case where u is any element of the Sobolev space H1R(n) and accordingly Δu is a tempered distribution which is not even a signed measure in general. As a consequence the second term of the right hand side of (0.1) may not be of bounded variation in t.