Generalized iterated wreath products of cyclic groups and rooted trees correspondence

Consider the generalized iterated wreath product $\mathbb{Z}_{r_1}\wr \mathbb{Z}_{r_2}\wr \ldots \wr \mathbb{Z}_{r_k}$ where $r_i \in \mathbb{N}$. We prove that the irreducible representations for this class of groups are indexed by a certain type of rooted trees. This provides a Bratteli diagram for the generalized iterated wreath product, a simple recursion formula for the number of irreducible representations, and a strategy to calculate the dimension of each irreducible representation. We calculate explicitly fast Fourier transforms (FFT) for this class of groups, giving literature's fastest FFT upper bound estimate.Comment: 15 pages, to appear in Advances in the Mathematical Science

Generalized iterated wreath products of symmetric groups and generalized rooted trees correspondence

Consider the generalized iterated wreath product $S_{r_1}\wr \ldots \wr S_{r_k}$ of symmetric groups. We give a complete description of the traversal for the generalized iterated wreath product. We also prove an existence of a bijection between the equivalence classes of ordinary irreducible representations of the generalized iterated wreath product and orbits of labels on certain rooted trees. We find a recursion for the number of these labels and the degrees of irreducible representations of the generalized iterated wreath product. Finally, we give rough upper bound estimates for fast Fourier transforms.Comment: 18 pages, to appear in Advances in the Mathematical Sciences. arXiv admin note: text overlap with arXiv:1409.060

Maximum gradient embeddings and monotone clustering

Let (X,d_X) be an n-point metric space. We show that there exists a distribution D over non-contractive embeddings into trees f:X-->T such that for every x in X, the expectation with respect to D of the maximum over y in X of the ratio d_T(f(x),f(y)) / d_X(x,y) is at most C (log n)^2, where C is a universal constant. Conversely we show that the above quadratic dependence on log n cannot be improved in general. Such embeddings, which we call maximum gradient embeddings, yield a framework for the design of approximation algorithms for a wide range of clustering problems with monotone costs, including fault-tolerant versions of k-median and facility location.Comment: 25 pages, 2 figures. Final version, minor revision of the previous one. To appear in "Combinatorica

Loss of heterozygosity (LOH), malignancy grade and clonality in microdissected prostate cancer

The aim of the present study was to find out whether increasing malignancy of prostate carcinoma correlates with an overall increase of loss of heterozygosity (LOH), and whether LOH typing of microdissected tumour areas can help to distinguish between multifocal or clonal tumour development. In 47 carcinomas analysed at 25 chromosomal loci, the overall LOH rate was found to be significantly lower in grade 1 areas (2.2%) compared with grade 2 (9.4%) and grade 3 areas (8.3%, P = 0.007). A similar tendency was found for the mean fractional allele loss (FAL, 0.043 for grade 1, 0.2 for grade 2 and 0.23 for grade 3, P = 0.0004). Of 20 tumours (65%) with LOH in several microdissected areas, 13 had identical losses at 1–4 loci within two or three areas, suggesting clonal development of these areas. Markers near RB, DCC, BBC1, TP53 and at D13S325 (13q21–22) showed higher loss rates in grades 2 and 3 (between 25% and 44.4%) compared with grade 1 (0–6.6%). Tumour-suppressor genes (TSGs) near these loci might, thus, be important for tumour progression. TP53 mutations were detected in 27%, but BBC1 mutations in only 7%, of samples with LOH. Evaluation of all 25 loci in every tumour made evident that each prostate cancer has its own pattern of allelic losses. © 1999 Cancer Research Campaig