Smooth diameter and eigenvalue rigidity in positive Ricci curvature

A recent injectivity radius estimate and previous sphere theorems yield the following smooth diameter sphere theorem for manifolds of positive Ricci curvature: For any given and there exists a positive constant 0$">such that any -dimensional complete Riemannian manifold with Ricci curvature , sectional curvature and diameter is Lipschitz close and diffeomorphic to the standard unit -sphere. A similar statement holds when the diameter is replaced by the first eigenvalue of the Laplacian.

     

Surface integral and Gauss-Ostrogradskij theorem from the viewpoint of applications

Making use of a surface integral defined without use of the partition of unity, trace theorems and the Gauss-Ostrogradskij theorem are proved in the case of three-dimensional domains with a Lipschitz-continuous boundary for functions belonging to the Sobolev spaces H ^1,p () (1 p < ). The paper is a generalization of the previous author's paper which is devoted to the line integral.     

Normality criteria for families of holomorphic mappings of several complex variables into

By applying the heuristic principle in several complex variables obtained by Aladro and Krantz, we shall prove some normality criteria for families of holomorphic mappings of several complex variables into , the complex N-dimensional projective space, related to Green's and Nochka's Picard type theorems. The equivalence of normality to being uniformly Montel at a point will be obtained. Some examples will be given to complement our theory in this paper.

     

Zur Reaktion eines Bis(amino)germylens mit Germaniumaziden: Abfangreaktionen von instabilen Germa-Iminen

Reaction of a Cyclic Bis(amino)germylene with Germaniumazides: Trapping-Reactions of Unstable Germa-Imines . The cyclic bis(amino)germylene 1 reacts with different germaniumazides of the type Me₂Si(NtBu)₂Ge(R)N₃ (R = Me ( 2 ), tBu ( 3 ), N(SiMe₃)₂ ( 4 ), R = N₃ ( 5 )). With the exception of 4 all azides lose dinitrogen when treated with 1 and the Ge^II center coordinates the α-nitrogen of the azide group. It seems to be reasonable to assume a transient germaimine (nitride) which is trapped by further reaction with the azide molecules 2 and 5 or by reaction with the solvent pyridine ( 3 ). In the case of 2 the germatetrazole [Me₂Si(NtBu)₂]GeN₄[Ge(NtBu)₂SiMe₂]₂ ( 6 ) is formed, the tetrazole nitrogens being exclusively substituted by germanium atoms (point symmetry of the molecule C_s(m)). When 1 is treated with 5 a tris(germa)amine [Me₂Si(NtBu)₂Ge(N₃)]₃N ( 8 ) is formed, which has an azide group attached to each Ge-atom. X-ray analysis reveals that the nine nitrogen atoms of the azide groups are coplanar with the trigonal planar Ge₃N moiety (crystallographic symmetry: 3/m). The reaction of 1 with 3 is very surprising: the pyridine in the product Me₂Si(NtBu)₂Ge(C₅H₄N)? N(H)Ge(tBu)(NtBu)₂SiMe₂ ( 7 ) is bonded via an α-carbon atom while the remaining hydrogen has added to the nitride-nitrogen. 6 crystallizes in the monoclinic system space group C2/m, a = 24.306(9), b = 10.933(6), c = 19.420(9) Å, β = 91.81(2)° and Z = 4. 7 crystallizes in the hexagonal system space group P6₃/m with a = b = 16.73(1), c = 11.006(8) Å, γ = 120° and Z = 2, and 8 crystallizes in the monoclinic system space group P2₁/n, a = 11.341(6), b = 26.086(9), c = 13.244(7) Å, β = 98. I2(2)° mit Z = 4.     

Limit theorems for records from discrete distributions

Weak and strong functional limit theorems are obtained for record values and record epochs in a sequence of independent random variables with common distribution F. The emphasis is on the case in which F is concentrated on the non-negative integers. For contrast, the well-known case of continuous F is also considered. Analogues of results obtained earlier by Resnick, de Haan and the author for continuous F are presented here for F concentrated on the non-negative integers. Also is investigated under which circumstances the latter case is so close to the continuous F case that the resulting limit theorems are the same.     

Nondifferentiable mathematical programming and convex-concave functions

For a convex-concave functionL(x, y), we define the functionf(x) which is obtained by maximizingL with respect toy over a specified set. The minimization problem with objective functionf is considered. We derive necessary conditions of optimality for this problem. Based upon these necessary conditions, we define its dual problem. Furthermore, a duality theorem and a converse duality theorem are obtained. It is made clear that these results are extensions of those derived in studies on a class of nondifferentiable mathematical programming problems.This work was supported by the Japan Society for the Promotion of Sciences.