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Displaying 1021 –
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In [12] Petrunin proves that a compact metric space X admits an intrinsic isometry into En if and only if X is a pro-Euclidean space of rank at most n, meaning that X can be written as a “nice” inverse limit of polyhedra. He also shows that either case implies that X has covering dimension at most n. The purpose of this paper is to extend these results to include both embeddings and spaces which are proper instead of compact. The main result of this paper is that any pro-Euclidean space of rank...
The paper studies the diagrams of woven fabrics consisting of white and black squares as geometrical objects and described their symmetries. The concepts of isonemality and mononemality due to B. Grünbaum and G. C. Shephard are used. A conjecture of these authors is proved in a particular case.
We consider an isoperimetric problem for product measures with respect to the uniform enlargement of sets. As an example, we find (asymptotically) extremal sets for the infinite product of the exponential measure.
We show that the unique isoperimetric regions in Rn with density rp for n ≥ 3 and p > 0 are balls with boundary through the origin.
Models of random sets and of point processes are introduced to simulate some specific clustering of points, namely on random lines in and and on random planes in . The corresponding point processes are special cases of Cox processes. The generating distribution function of the probability distribution of the number of points in a convex set and the Choquet capacity are given. A possible application is to model point defects in materials with some degree of alignment. Theoretical results...
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