scholarly journals Growth sequences of finite groups

1974 ◽  
Vol 17 (2) ◽  
pp. 133-141 ◽  
Author(s):  
James Wiegold
Keyword(s):  
Characteristic Property ◽  
Simple Consequence ◽  
Direct Power ◽  
Growth Sequence ◽  
Finitely Generated Groups ◽  
Number Of Generators ◽  
Special Cases ◽  
Different Types ◽  
Minimum Number ◽  
Positive Integers

During his investigation of the possible non-Hopf kernels for finitely generated groups in [1], Dey proves that the minimum number of generators d(Gn) of the n-th direct power Gn of a non-trival finite group G tends to infinity with n. This has prompted me to ask the question: what are the ways in which the sequence {d(Gn)} can tend to infinity? Let us call this the growth sequence for G; it is evidently monotone non-decreasing, and is at least logarithmic (Theorem 2.1). This paper is devoted to a proof that, broadly speaking, there are two different types of behaviour. If G has non-trivial abelian images (the imperfect case, § 3), then the growth sequence of G is eventually an arithmetic progression with common difference d(G/G'). In special cases (Theorem 5.2) the initial behaviour can be quite nasty. Our arguments in § 3 are totally elementary. If G has only trivial abelian images (the perfect case,§ 4), then the growth sequence of G is eventually bounded above by a sequence that grows logarithmically. It is a simple consequence of this fact that there are arbitrarily long blocks of positive integers on which the growth sequence takes constant values. This is a characteristic property of perfect groups, and indeed it was this feature in the growth sequences of large alternating groups (which I found by using ad hoc permutational arguments) that attracted me to the problem in the first place. The discussion of the perfect case rests on the lovely paper of Hall [2], which was brought to my notice by M. D. Atkinson.

scholarly journals Growth sequences of finite semigroups

1987 ◽  
Vol 43 (1) ◽  
pp. 16-20 ◽  
Author(s):  
James Wiegold ◽  
H. Lausch
Keyword(s):  
Real Number ◽  
Finite Semigroup ◽  
Piecewise Linear ◽  
The Other ◽  
Direct Power ◽  
Growth Sequence ◽  
Group Of Units ◽  
Number Of Generators ◽  
Finite Semigroups ◽  
Minimum Number

AbstractThe growth sequence of a finite semigroup S is the sequence {d(Sn)}, where Sn is the nth direct power of S and d stands for minimum generating number. When S has an identity, d(Sn) = d(Tn) + kn for all n, where T is the group of units and k is the minimum number of generators of S mod T. Thus d(Sn) is essentially known since d(Tn) is (see reference 4), and indeed d(Sn) is then eventually piecewise linear. On the other hand, if S has no identity, there exists a real number c > 1 such that d(Sn) ≥ cn for all n ≥ 2.

scholarly journals Growth sequences of finitely generated groups II

1989 ◽  
Vol 40 (2) ◽  
pp. 323-329 ◽  
Author(s):  
A.G.R. Stewart ◽  
James Wiegold
Keyword(s):  
Finitely Generated ◽  
Direct Power ◽  
Finitely Generated Groups ◽  
Number Of Generators ◽  
Minimum Number

A study is made of the minimum number of generators of the n-th direct power of certain finitely generated groups.

Growth sequences of 2-generator simple groups

1993 ◽  
Vol 123 (5) ◽  
pp. 839-855 ◽  
Author(s):  
V. N. Obraztsov
Keyword(s):  
Simple Groups ◽  
Direct Power ◽  
Number Of Generators ◽  
Minimum Number

SynopsisA study is made of the minimum number of generators of the n-th direct power of certain 2-generator groups.

scholarly journals Growth sequences of finite groups V

1981 ◽  
Vol 31 (3) ◽  
pp. 374-375 ◽  
Author(s):  
D. Meier ◽  
James Wiegold
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Finite Groups ◽  
The Other ◽  
Direct Power ◽  
Easy Proof ◽  
Number Of Generators ◽  
Other Hand ◽  
Minimum Number

AbstractA short and easy proof that the minimum number of generators of the nth direct power of a non-trival finite group of order s having automorphism group of order a is more than logsn + logsa, n > 1. On the other hand, for non-abelian simple G and large n, d(Gn) is within 1 + e of logsn + logsa.

Computation of Spatial Displacements From Redundant Geometric Features

1992 ◽  
Author(s):  
Q. J. Ge ◽  
B. Ravani
Keyword(s):  
Symmetric Matrix ◽  
Automatic Generation ◽  
Geometric Features ◽  
Least Eigenvalue ◽  
Special Cases ◽  
Minimum Number ◽  
Feature Information ◽  
Spatial Displacements ◽  
Simple Features ◽  
Cubic Function

Abstract This paper follows a previous one on the computation of spatial displacements (Ravani and Ge, 1992). The first paper dealt with the problem of computing spatial displacements from a minimum number of simple features of points, lines, planes, and their combinations. The present paper deals with the same problem using a redundant set of the simple geometric features. The problem for redundant information is formulated as a least squares problem which includes all simple features. A Clifford algebra is used to unify the handling of various feature information. An algorithm for determining the best orientation is developed which involves finding the eigenvector associated with the least eigenvalue of a 4 × 4 symmetric matrix. The best translation is found to be a rational cubic function of the best orientation. Special cases are discussed which yield the best orientation in closed form. In addition, simple algorithms are provided for automatic generation of body-fixed coordinate frames from various feature information. The results have applications in robot and world model calibration for off-line programming and computer vision.

scholarly journals TERRESTRIAL LASER SCANNERS SELF-CALIBRATION STUDY: DATUM CONSTRAINTS ANALYSES FOR NETWORK CONFIGURATIONS

2015 ◽  
Vol XL-2/W4 ◽  
pp. 1-9 ◽  
Author(s):  
M. A. Abbas ◽  
H. Setan ◽  
Z. Majid ◽  
A. K. Chong ◽  
L. Chong Luh ◽  
...  
Keyword(s):  
Laser Scanner ◽  
Network Configuration ◽  
Calibration Technique ◽  
Self Calibration ◽  
Laser Scanners ◽  
Different Types ◽  
Minimum Number ◽  
Calibration Study ◽  
Different Sources ◽  
Selection Of

Similar to other electronic instruments, terrestrial laser scanner (TLS) can also inherent with various systematic errors coming from different sources. Self-calibration technique is a method available to investigate these errors for TLS which were adopted from photogrammetry technique. According to the photogrammetry principle, the selection of datum constraints can cause different types of parameter correlations. However, the network configuration applied by TLS and photogrammetry calibrations are quite different, thus, this study has investigated the significant of photogrammetry datum constraints principle in TLS self-calibration. To ensure that the assessment is thorough, the datum constraints analyses were carried out using three variant network configurations: 1) minimum number of scan stations; 2) minimum number of surfaces for targets distribution; and 3) minimum number of point targets. Based on graphical and statistical, the analyses of datum constraints selection indicated that the parameter correlations obtained are significantly similar. In addition, the analysis has demonstrated that network configuration is a very crucial factor to reduce the correlation between the calculated parameters.

scholarly journals On Reay's Relaxed Tverberg Conjecture and Generalizations of Conway's Thrackle Conjecture

10.37236/6516 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Megumi Asada ◽  
Ryan Chen ◽  
Florian Frick ◽  
Frederick Huang ◽  
Maxwell Polevy ◽  
...  
Keyword(s):  
Geometric Property ◽  
Convex Sets ◽  
Convex Hulls ◽  
Open Problems ◽  
Higher Dimensional ◽  
Special Cases ◽  
Minimum Number ◽  
Point Set ◽  
Colored Version ◽  
Special Case

Reay's relaxed Tverberg conjecture and Conway's thrackle conjecture are open problems about the geometry of pairwise intersections. Reay asked for the minimum number of points in Euclidean $d$-space that guarantees any such point set admits a partition into $r$ parts, any $k$ of whose convex hulls intersect. Here we give new and improved lower bounds for this number, which Reay conjectured to be independent of $k$. We prove a colored version of Reay's conjecture for $k$ sufficiently large, but nevertheless $k$ independent of dimension $d$. Pairwise intersecting convex hulls have severely restricted combinatorics. This is a higher-dimensional analogue of Conway's thrackle conjecture or its linear special case. We thus study convex-geometric and higher-dimensional analogues of the thrackle conjecture alongside Reay's problem and conjecture (and prove in two special cases) that the number of convex sets in the plane is bounded by the total number of vertices they involve whenever there exists a transversal set for their pairwise intersections. We thus isolate a geometric property that leads to bounds as in the thrackle conjecture. We also establish tight bounds for the number of facets of higher-dimensional analogues of linear thrackles and conjecture their continuous generalizations.

scholarly journals On the 2-adic order of Stirling numbers of the second kind and their differences

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Tamás Lengyel
Keyword(s):  
Positive Integer ◽  
Stirling Numbers ◽  
Bell Polynomials ◽  
Binary Representation ◽  
Exact Order ◽  
Special Cases ◽  
International Audience ◽  
The Difference ◽  
Divisibility Properties ◽  
Positive Integers

International audience Let $n$ and $k$ be positive integers, $d(k)$ and $\nu_2(k)$ denote the number of ones in the binary representation of $k$ and the highest power of two dividing $k$, respectively. De Wannemacker recently proved for the Stirling numbers of the second kind that $\nu_2(S(2^n,k))=d(k)-1, 1\leq k \leq 2^n$. Here we prove that $\nu_2(S(c2^n,k))=d(k)-1, 1\leq k \leq 2^n$, for any positive integer $c$. We improve and extend this statement in some special cases. For the difference, we obtain lower bounds on $\nu_2(S(c2^{n+1}+u,k)-S(c2^n+u,k))$ for any nonnegative integer $u$, make a conjecture on the exact order and, for $u=0$, prove part of it when $k \leq 6$, or $k \geq 5$ and $d(k) \leq 2$. The proofs rely on congruential identities for power series and polynomials related to the Stirling numbers and Bell polynomials, and some divisibility properties.

scholarly journals The Smallest One-Realization of a Given Set

10.37236/1171 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Ping Zhao ◽  
Kefeng Diao ◽  
Kaishun Wang
Keyword(s):  
Open Problem ◽  
Feasible Set ◽  
Minimum Number ◽  
Chromatic Spectrum ◽  
Positive Integers ◽  
Mixed Hypergraph

For any set $S$ of positive integers, a mixed hypergraph ${\cal H}$ is a realization of $S$ if its feasible set is $S$, furthermore, ${\cal H}$ is a one-realization of $S$ if it is a realization of $S$ and each entry of its chromatic spectrum is either 0 or 1. Jiang et al. showed that the minimum number of vertices of a realization of $\{s,t\}$ with $2\leq s\leq t-2$ is $2t-s$. Král proved that there exists a one-realization of $S$ with at most $|S|+2\max{S}-\min{S}$ vertices. In this paper, we  determine the number  of vertices of the smallest one-realization of a given set. As a result, we partially solve an open problem proposed by Jiang et al. in 2002 and by Král  in 2004.

On the subsistence practices of the Shi’erqiao Culture: focused on zooarchaeology

2012 ◽  
Vol 12 (1) ◽  
Author(s):  
Kunyu He
Keyword(s):  
Early Stage ◽  
Subsistence Strategies ◽  
Chengdu Plain ◽  
Factors Influencing ◽  
Animal Remains ◽  
Different Types ◽  
Minimum Number ◽  
Minimum Number Of Individuals ◽  
Archaeological Cultures ◽  
Number Of Individuals

AbstractThrough the analyses of the number of identified specimens (NISP), minimum number of individuals (MNI) and estimation of meat available from the animal remains unearthed from representative sites of the Chengdu Plain Type and Eastern Chongqing Type of the Shi’erqiao Culture, this paper discusses the subsistence strategy of this culture. The results show that the meat resources of the Chengdu Plain Type were mainly domesticated animals, while those of the Eastern Chongqing Type were mainly hunted animals. This paper draws the conclusion that in the early stage of human civilization, the subsistence strategies of different types of the same archaeological culture might be different, while those of different archaeological cultures might be similar; the factors influencing the subsistence strategies are mainly the natural environment, regional economic traditions, and population pressures.

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