Über diej-ten Überdeckungsdichten konvexer Körper

Let for positive integersj,k,d and convex bodiesK of Euclideand-spaceE ^d of dimension at leastj V _{j, k} (K) denote the maximum of the intrinsic volumesV _j (C) of those convex bodies whosej-skeleton skel _j (C) can be covered withk translates ofK. Then thej-thk-covering density θ _j,k (K) is the ratiok V _j (K)/V _j,k (K). In particular, θ _d,k refers to the case of covering the entire convex bodiesC and the density is measured with respect to the volume while forj=d-1 the surface of the bodiesC is covered and accordingly the density is measured with respect to the surface area. The paper gives the estimate $$1 \leqslant \theta _{j,k} (K)< e (j + \sqrt {\pi /2} \sqrt {d - j} )< (d + 1) e$$ for thej-thk-covering density and some related results.     

Some estimates of theH ∞-uniform distribution

The sequences (y _n )=(a+blogn+ _n ) are uniformly distributed in the sense of the summation methodH . In the present paper the speed of convergence of this procedure is estimated for these, sequences and for some other sequences. For the sequences (a+blogn) theH -means converge considerably faster than logarithmic means.     

On the structure of a random sum-free set

Summary There is a natural probability measure on the set of all sum-free sets of natural numbers. If we represent such a set by its characteristic functions, then the zero-one random variabless(i) are far from independent, and we cannot expect a law of large numbers to hold for them. In this paper I conjecture a decomposition of into countably many more tractible pieces (up to a null set). I prove that each piece has positive measure, and show that, within each piece, a random set almost surely has a density which is a fixed rational number depending only on the piece. For example, the first such piece is made up of sets consisting entirely of odd numbers; it has probability 0.218 ..., and its members almost surely have density 1/4.     

On quadratic and sesquilinear functionals

LetX be a leftA-module, whereA is either a complex Banach *-algebra with an identity element or the field of quaternions. The main result of this note is that forQ, anA-quadratic functional defined onX, there exists a sesquilinear functionalB such thatB(x,x)=Q(x) holds for allxX.     

Vector two-point functions in maximally symmetric spaces

We obtain massive and massless vector two-point functions in maximally symmetric spaces (and vacua) of any number of dimensions. These include de Sitter space and anti-de Sitter space, and their Euclidean analogsS ⁿ andH ⁿ. Our method is based on a simple way of constructing every possible maximally symmetric bitensorT _a...bc...d(x, x) which carries tangent-space indicesa...b atx andc...d atx.     

Local limits and harmonic functions for nonisotropic random walks on free groups

Summary Nearest neighbour random walks on the homogeneous tree representing a free group withs generators (2s) are investigated. By use of generating functions and their analytic properties a local limit theorem is derived. A study of the harmonic functions corresponding to the random walk leads to properties that characterize ther-harmonic function connected with the local limits.