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Enumeration and investigation of acute 0/1-simplices modulo the action of the hyperoctahedral group

100%
Brandts J., Cihangir A.
Special Matrices
|
2017
|
tom 5
|
nr 1
158-201
EN
The convex hull of n + 1 affinely independent vertices of the unit n-cube In is called a 0/1-simplex. It is nonobtuse if none its dihedral angles is obtuse, and acute if additionally none of them is right. In terms of linear algebra, acute 0/1-simplices in In can be described by nonsingular 0/1-matrices P of size n × n whose Gramians G = PTP have an inverse that is strictly diagonally dominant, with negative off-diagonal entries [6, 7]. The first part of this paper deals with giving a detailed description of how to efficiently compute, by means of a computer program, a representative from each orbit of an acute 0/1-simplex under the action of the hyperoctahedral group Bn [17] of symmetries of In. A side product of the investigations is a simple code that computes the cycle index of Bn, which can in explicit form only be found in the literature [11] for n ≤ 6. Using the computed cycle indices for B3, . . . ,B11 in combination with Pólya’s theory of enumeration shows that acute 0/1-simplices are extremely rare among all 0/1-simplices. In the second part of the paper, we study the 0/1-matrices that represent the acute 0/1-simplices that were generated by our code from a mathematical perspective. One of the patterns observed in the data involves unreduced upper Hessenberg 0/1-matrices of size n × n, block-partitioned according to certain integer compositions of n. These patterns will be fully explained using a so-called One Neighbor Theorem [4]. Additionally, we are able to prove that the volumes of the corresponding acute simplices are in one-to-one correspondence with the part of Kepler’s Tree of Fractions [1, 24] that enumerates ℚ ⋂ (0, 1). Another key ingredient in the proofs is the fact that the Gramians of the unreduced upper Hessenberg matrices involved are strictly ultrametric [14, 26] matrices.
2
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Permutations preserving sums of rearranged real series

81%
Wituła R.
Open Mathematics
|
2013
|
tom 11
|
nr 5
956-965
EN
The aim of this paper is to discuss one of the most interesting and unsolved problems of the real series theory: rearrangements that preserve sums of series. Certain hypothesis about combinatorial description of the corresponding permutations is presented and basic algebraic properties of the family $\mathfrak{S}_0 $, introduced by it, are investigated.
3
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Some new facts about group 𝒢 generated by the family of convergent permutations

81%
Wituła R., Hetmaniok E., Słota D.
Open Mathematics
|
2017
|
tom 15
|
nr 1
568-577
EN
The aim of this paper is to present some new and essential facts about group 𝒢 generated by the family of convergent permutations, i.e. the permutations on ℕ preserving the convergence of series of real terms. We prove that there exist permutations preserving the sum of series which do not belong to 𝒢. Additionally, we show that there exists a family G (possessing the cardinality equal to continuum) of groups of permutations on ℕ such that each one of these groups is different than 𝒢 and is composed only from the permutations preserving the sum of series. This result substantially strengthens some old Pleasants’ result.
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