J un 1 99 9 Uniform partitions of 3 - space , their relatives and embedding ∗

. M G ] 6 J un 1 99 9 Uniform partitions of 3 - space , their relatives and embedding ∗

We review 28 uniform partitions of 3-space in order to find out which of them have graphs (skeletons) embeddable isometrically (or with scale 2) into some cubic lattice Zn. We also consider some relatives of those 28 partitions, including Achimedean 4-polytopes of Conway-Guy, non-compact uniform partitions, Kelvin partitions and those with unique vertex figure (i.e. Delaunay star). Among last o...

Uniform Partitions of 3-space, their Relatives and Embedding

We review 28 uniform partitions of 3-space in order to find out which of them have graphs (skeletons) embeddable isometrically (or with scale 2) into some cubic lattice Zn . We also consider some relatives of those 28 partitions, including Archimedean 4-polytopes of Conway–Guy, non-compact uniform partitions, Kelvin partitions and those with a unique vertex figure (i.e., Delaunay star). Among t...

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On the Wadge reducibility of k-partitions

We establish some results on the Wadge degrees and on the Boolean hierarchy of k-partitions of some spaces, where k is a natural number. The main attention is paid to the Baire space, Baire domain and their close relatives. For the case of Δ2-measurable k-partitions the structures of Wadge degrees are characterized completely. For many degree structures, undecidability of the first-order theori...

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Computing distances in high-dimensional spaces is deemed with the empty space phenomenon, which may harm distance-based algorithms for data visualization. We focus on transforming high-dimensional numeric data for their visualization using the kernel PCA 2D projection. Gaussian and p-Gaussian kernels are often advocated when confronted to such data; we propose to give some insight of their prop...