scholarly journals Transversals, Near Transversals, and Diagonals in Iterated Groups and Quasigroups

10.37236/9699 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Anna A. Taranenko
Keyword(s):  
Commutator Subgroup ◽  
Latin Hypercube ◽  
Cayley Table

Given a binary quasigroup $G$ of order $n$, a $d$-iterated quasigroup $G[d]$ is the $(d+1)$-ary quasigroup equal to the $d$-times composition of $G$ with itself. The Cayley table of every $d$-ary quasigroup is a $d$-dimensional latin hypercube. Transversals and diagonals in multiary quasigroups are defined so as to coincide with those in the corresponding latin hypercube. We prove that if a group $G$ of order $n$ satisfies the Hall–Paige condition, then the number of transversals in $G[d]$ is equal to $ \frac{n!}{ |G'| n^{n-1}} \cdot n!^{d}  (1 + o(1))$ for large $d$, where $G'$ is the commutator subgroup of $G$. For a general quasigroup $G$, we obtain similar estimations on the numbers of transversals and near transversals in $G[d]$  and develop a method for counting diagonals of other types in iterated quasigroups.

scholarly journals Transversals, Plexes, and Multiplexes in Iterated Quasigroups

10.37236/7304 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Anna Taranenko
Keyword(s):  
Latin Hypercube ◽  
Cayley Table ◽  
Recurrence Formulas

A $d$-ary quasigroup of order $n$ is a $d$-ary operation over a set of cardinality $n$ such that the Cayley table of the operation is a $d$-dimensional latin hypercube of the same order. Given a binary quasigroup $G$, the $d$-iterated quasigroup $G^{\left[d\right]}$ is a $d$-ary quasigroup that is a $d$-time composition of $G$ with itself. A $k$-multiplex (a $k$-plex) $K$ in a $d$-dimensional latin hypercube $Q$ of order $n$ or in the corresponding $d$-ary quasigroup is a multiset (a set) of $kn$ entries such that each hyperplane and each symbol of $Q$ is covered by exactly $k$ elements of $K$. It is common to call 1-plexes transversals. In this paper we prove that there exists a constant $c(G,k)$ such that if a $d$-iterated quasigroup $G$ of order $n$ has a $k$-multiplex then for large $d$ the number of its $k$-multiplexes is asymptotically equal to $c(G,k) \left(\frac{(kn)!}{k!^n}\right)^{d-1}$. As a corollary we obtain that if the number of transversals in the Cayley table of a $d$-iterated quasigroup $G$ of order $n$ is nonzero then asymptotically it is $c(G,1)  n!^{d-1}$.  In addition, we  provide limit constants and recurrence formulas for the numbers of transversals in two iterated quasigroups of order $5$, characterize a typical $k$-multiplex and estimate numbers of partial $k$-multiplexes and transversals in $d$-iterated quasigroups. 

scholarly journals Comparison of Factorial and Latin Hypercube Sampling Designs for Meta-Models of Building Heating and Cooling Loads

Energies ◽  
2021 ◽  
Vol 14 (2) ◽  
pp. 512
Author(s):  
Younhee Choi ◽  
Doosam Song ◽  
Sungmin Yoon ◽  
Junemo Koo
Keyword(s):  
Latin Hypercube Sampling ◽  
Building Design ◽  
Gain Coefficient ◽  
Design Parameters ◽  
Latin Hypercube ◽  
Heating And Cooling ◽  
Factor Design ◽  
Heating Loads ◽  
Cooling Loads ◽  
Interest In Research

Interest in research analyzing and predicting energy loads and consumption in the early stages of building design using meta-models has constantly increased in recent years. Generally, it requires many simulated or measured results to build meta-models, which significantly affects their accuracy. In this study, Latin Hypercube Sampling (LHS) is proposed as an alternative to Fractional Factor Design (FFD), since it can improve the accuracy while including the nonlinear effect of design parameters with a smaller size of data. Building energy loads of an office floor with ten design parameters were selected as the meta-models’ objectives, and were developed using the two sampling methods. The accuracy of predicting the heating/cooling loads of the meta-models for alternative floor designs was compared. For the considered ranges of design parameters, window insulation (WDI) and Solar Heat Gain Coefficient (SHGC) were found to have nonlinear characteristics on cooling and heating loads. LHS showed better prediction accuracy compared to FFD, since LHS considers the nonlinear impacts for a given number of treatments. It is always a good idea to use LHS over FFD for a given number of treatments, since the existence of nonlinearity in the relation is not pre-existing information.

scholarly journals The center and the commutator subgroup in hopfian groups

1971 ◽  
Vol 9 (1-2) ◽  
pp. 181-192 ◽  
Author(s):  
R. Hirshon
Keyword(s):  
Commutator Subgroup

scholarly journals The Monoid Consisting of Kuratowski Operations

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Szymon Plewik ◽  
Marta Walczyńska
Keyword(s):  
Systematic Study ◽  
Cayley Table ◽  
Paper And Pencil

The paper fills gaps in knowledge about Kuratowski operations which are already in the literature. The Cayley table for these operations has been drawn up. Techniques, using only paper and pencil, to point out all semigroups and its isomorphism types are applied. Some results apply only to topology, and one cannot bring them out, using only properties of the complement and a closure-like operation. The arguments are by systematic study of possibilities.

scholarly journals Transitive action on finite points of a full shift and a finitary Ryan’s theorem

2017 ◽  
Vol 39 (06) ◽  
pp. 1637-1667 ◽  
Author(s):  
VILLE SALO
Keyword(s):  
Automorphism Group ◽  
Commutator Subgroup ◽  
Turing Machines ◽  
Finite Support ◽  
Transitive Action ◽  
Group Invariant ◽  
Finite Set ◽  
Transitivity Property ◽  
Full Shift ◽  
Subshifts Of Finite Type

We show that on the four-symbol full shift, there is a finitely generated subgroup of the automorphism group whose action is (set-theoretically) transitive of all orders on the points of finite support, up to the necessary caveats due to shift-commutation. As a corollary, we obtain that there is a finite set of automorphisms whose centralizer is $\mathbb{Z}$ (the shift group), giving a finitary version of Ryan’s theorem (on the four-symbol full shift), suggesting an automorphism group invariant for mixing subshifts of finite type (SFTs). We show that any such set of automorphisms must generate an infinite group, and also show that there is also a group with this transitivity property that is a subgroup of the commutator subgroup and whose elements can be written as compositions of involutions. We ask many related questions and prove some easy transitivity results for the group of reversible Turing machines, topological full groups and Thompson’s  $V$ .

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