Flat

Abstract: We explore the geometry of nonpositively curved spaces with isolated flats, and its consequences for groups that act properly discontinuously, cocompactly, and isometrically on such spaces. We prove that the geometric boundary of the space is an invariant of the group up to equivariant homeomorphism. We also prove that any such group is relatively hyperbolic, biautomatic, and satisfies the Tits Alternative. The main step in establishing these results is a characterization of spaces with isolated flats as relatively hyperbolic with respect to flats. Finally we show that a CAT(0) space has isolated flats if and only if its Tits boundary is a disjoint union of isolated points and standard Euclidean spheres. In an appendix written jointly with Hindawi, we extend many of the results of this article to a more general setting in which the isolated subspaces are not required to be flats. AMS Classification numbers Primary: 20F67 Secondary: 20F69

Abstract: Existence theorems in analysis deal with functional transformations. This suggests that such existence theorems may be obtained from known theorems on point transformations in space of two or of three dimensions by generalization, first to space of n dimensions, and then to function space by a limiting process. This direction of attack has been followed out and has resulted in the theorems given below. For instance it is found that theorems on invariant points for the sphere or for its surface yield respectively by generalization existence theorems in analysis of non-homogeneous and of homogeneous type. The treatment is here confined to the case of real functions of a real variable, although extensions to real functions of several real variables are indicated. Only the case of a single unknown function is considered. In many cases, of course, apparently more general problems can be reduced to this case by a process which is familiar in the theory of integral equations, namely the juxtaposition of intervals. The applications include the classical existence theorems for differential and integral equations, linear and non-linear. Incidentally, it is proved that an algebraic manifold/i = Ci, ft = ct, .. ., fm = cm> where /i, ft, . .., fm are real polynomials in the real variables Xi, Xt, . . .,xn, has no singularity for general values of the real constants C\, c2, ..., cm. The authors have not been able to find any earlier proof of this simple and important theorem. The literature on the subject of invariant points does not appear to be extensive. For a geometric treatment of one-valued transformations with one-valued inverses, we may refer to LE. J. Brouwer.f Some existence theorems of un-

Abstract: This is the first part of a three part work, the common setting being E d , Euclidean space of d dimensions. Many random physical phenomena, often in the form of structures, admit models which are assemblages of random s -flats in E d . Indeed, taking s = 0, any n -sample from a d -dimensional distribution may of course be so regarded! For static phenomena generally 0 ≦ s d ≦ 3, while 4 is to be substituted for 3 if time variation is allowed. By postulating a high degree of stochastic independence and uniformity, a variety of “simple” models is defined. However, their investigation poses problems in geometrical probability of a wide range of difficulty; the solutions of many of which are still far from complete.

Abstract: Part I [21] treated the case of a finite number of independent random uniform s-flats in an 'admissible' subset of Ed (s = 0, ,d 1). In this second part, the natural and fruitful 'Poisson extension' to a 'countable number of independent random uniform s-flats in Ed itself' is considered. It is worth mentioning at the outset that to have read Part I is not a prerequisite for reading the present paper. Although results of that part are often applied here, they serve only in an auxiliary capacity, thereby allowing the main thread of the theory to be developed without interruption.