Department of Mathematics, National Taiwan University, Taipei, Taiwan Taida Institute for Mathematical Sciences, National Taiwan University, Taipei, Taiwan National Center for Theoretical Sciences, Taipei Office, Taiwan LaBRI UMR CNRS 5800, Univ. Bordeaux, Talence, France Department of Mathematics Education, Catholic University of Daegu, Kyongsan, Republic of Korea Department of Mathematics, Sh...

We investigate two problems concerning the complexity of evaluating a function f at a k-tuple of unrelated inputs by k parallel decision tree algorithms. In the product problem, for some xed depth bound d, we seek to maximize the fraction of input k-tuples for which all k decision trees are correct. Assume that for a single input to f, the best decision tree algorithm of depth d is correct on a...

Clustering is the procedure to group similar objects together. Several algorithms have been proposed for clustering. Among them, the K-means clustering method has less time complexity. But it is sensitive to extreme values and would cause less accurate clustering of the dataset. However, K-medoids method does not have such limitations. But this method uses user-defined value for K. Therefore, i...

Let k = (k1, k2, · · · , kt) be a t-tuple of integers, and m be a positive integer. For a subset A ⊂ Zm and any n ∈ Zm, let rk A(n) denote the number of solutions of the equation k1a1 + · · ·+ ktat = n with a1, · · · , at ∈ A. In this paper, we give a necessary and sufficient condition on (k,m) such that there exists a subset A ⊂ Zm satisifying rk A = r k Zm\A. This settles a problem of Yang an...

BACKGROUND The comparison of samples, or beta diversity, is one of the essential problems in ecological studies. Next generation sequencing (NGS) technologies make it possible to obtain large amounts of metagenomic and metatranscriptomic short read sequences across many microbial communities. De novo assembly of the short reads can be especially challenging because the number of genomes and the...

A k-extended Skolem-type 5-tuple difference set of order t is a set of t 5-tuples {(di,1, di,2, di,3, di,4, di,5) | i = 1, 2, . . . , t} such that di,1+di,2+di,3+di,4+di,5 = 0 for 1 ≤ i ≤ t and {|di,j| | 1 ≤ i ≤ t, 1 ≤ j ≤ 5} = {1, 2, . . . , 5t+1}\{k}. In this talk, we will give necessary and sufficient conditions on t and k for the existence of a k-extended Skolem-type 5-tuple difference set ...

Each of the elements of this list is itself a partition, and the numbers in each partition are referred to as parts . As one may clearly deduce, the larger the number, the more partitions it has. Also of note is the fact that is the same as , that is, order is not accounted for. Typically, the parts of a partition are written in nonincreasing order. It is also commonly accepted that the functio...

Grushko’s theorem [Mat. Sb. 8 (1940) 169–182] says that any generating tuple (g1, . . . , gm) of a free product H ∗K is Nielsen-equivalent to a tuple (h1, . . . , hl, kl+1, . . . , km) with hi ∈ H and ki ∈ K for all i. The hi and ki are clearly not unique. In this paper we address the extent of this non-uniqueness.

A covering array CA(N ; t, k, v) is anN×k array on v symbols such that everyN×t subarray contains as a row each t-tuple over the v symbols at least once. The minimum N for which a CA(N ; t, k, v) exists is called the covering array number of t, k, and v, and it is denoted by CAN(t, k, v). In this work we prove that if exists CA(N ; t+ 1, k + 1, v) it can be obtained from the juxtaposition of v ...