scholarly journals A Finite Word Poset

10.37236/1607 ◽  
2000 ◽  
Vol 8 (2) ◽  
Author(s):  
Péter L. Erdős ◽  
Péter Sziklai ◽  
David C. Torney
Keyword(s):  
Automorphism Groups ◽  
Partial Ordering ◽  
Finite Alphabet ◽  
Combinatorial Properties ◽  
Normalized Matching Property ◽  
Finite Word

Our word posets have finite words of bounded length as their elements, with the words composed from a finite alphabet. Their partial ordering follows from the inclusion of a word as a subsequence of another word. The elemental combinatorial properties of such posets are established. Their automorphism groups are determined (along with similar result for the word poset studied by Burosch, Frank and Röhl [4]) and a BLYM inequality is verified (via the normalized matching property).

Coinitial Grapfis and Whitehead Automorphisms

1979 ◽  
Vol 31 (1) ◽  
pp. 112-123 ◽  
Author(s):  
A. H. M. Hoare
Keyword(s):  
Automorphism Groups ◽  
Fuchsian Groups ◽  
Topological Methods ◽  
Combinatorial Properties ◽  
Algebraic Proofs

Coinitial graphs were used in [2; 3 ; 4] as a combinatorial tool in the Reidemeister- Schreier process in order to prove subgroup theorems for Fuchsian groups. Whitehead had previously introduced such graphs but used topological methods for his proofs [8; 9]. Subsequently Rapaport [7] and Iliggins and Lyndon [1] gave algebraic proofs of the results in [9], and AIcCool [5; 6] has further developed these methods so that presentations of automorphism groups could be found.In this paper it is shown that Whitehead automorphisms can be described by a “cutting and pasting” operation on coinitial graphs. Section 1 defines and gives some combinatorial properties of these operations, based on [1].

CROSS RATIO GRAPHS

2001 ◽  
Vol 64 (2) ◽  
pp. 257-272 ◽  
Author(s):  
A. GARDINER ◽  
CHERYL E. PRAEGER ◽  
SANMING ZHOU
Keyword(s):  
Automorphism Groups ◽  
Projective Line ◽  
Cross Ratio ◽  
Transitive Graph ◽  
Combinatorial Properties ◽  
The Family ◽  
The Cross ◽  
Transitive Graphs ◽  
Ordered Pairs

A family of arc-transitive graphs is studied. The vertices of these graphs are ordered pairs of distinct points from a finite projective line, and adjacency is defined in terms of the cross ratio. A uniform description of the graphs is given, their automorphism groups are determined, the problem of isomorphism between graphs in the family is solved, some combinatorial properties are explored, and the graphs are characterised as a certain class of arc-transitive graphs. Some of these graphs have arisen as examples in studies of arc-transitive graphs with complete quotients and arc-transitive graphs which ‘almost cover’ a 2-arc transitive graph.

On the automorphism group of minimal -adic subshifts of finite alphabet rank

2021 ◽  
pp. 1-23
Author(s):  
BASTIÁN ESPINOZA ◽  
ALEJANDRO MAASS
Keyword(s):  
Automorphism Group ◽  
Finite Number ◽  
Linear Growth ◽  
Automorphism Groups ◽  
Low Complexity ◽  
Combinatorial Analysis ◽  
Finite Alphabet

Abstract It has been recently proved that the automorphism group of a minimal subshift with non-superlinear word complexity is virtually $\mathbb {Z}$ [Cyr and Kra. The automorphism group of a shift of linear growth: beyond transitivity. Forum Math. Sigma3 (2015), e5; Donoso et al. On automorphism groups of low complexity subshifts. Ergod. Th. & Dynam. Sys.36(1) (2016), 64–95]. In this article we extend this result to a broader class proving that the automorphism group of a minimal $\mathcal {S}$ -adic subshift of finite alphabet rank is virtually $\mathbb {Z}$ . The proof is based on a fine combinatorial analysis of the asymptotic classes in this type of subshifts, which we prove are a finite number.

REPETITIONS, FULLNESS, AND UNIFORMITY IN TWO-DIMENSIONAL WORDS

2004 ◽  
Vol 15 (02) ◽  
pp. 355-383 ◽  
Author(s):  
ARTURO CARPI ◽  
ALDO de LUCA
Keyword(s):  
Essential Role ◽  
Finite Alphabet ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Fixed Size ◽  
Open Problems ◽  
One Dimensional ◽  
Combinatorial Properties ◽  
The Difference ◽  
The One

We consider some combinatorial properties of two-dimensional words (or pictures) over a given finite alphabet, which are related to the number of occurrences in them of words of a fixed size (m,n). In particular a two-dimensional word (briefly, 2D-word) is called (m,n)-full if it contains as factors (or subwords) all words of size (m,n). An (m,n)-full word such that any word of size (m,n) occurs in it exactly once is called a de Bruijn word of order (m,n). A 2D-word w is called (m,n)-uniform if the difference in the number of occurrences in w of any two words of size (m,n) is at most 1. A 2D-word is called uniform if it is (m,n)-uniform for all m,n>0. In this paper we extend to the two-dimensional case some results relating the notions above which were proved in the one-dimensional case in a preceding article. In this analysis the study of repeated factors in a 2D-word plays an essential role. Finally, some open problems and conjectures are discussed.

scholarly journals On Computing the Distinguishing Numbers of Trees and Forests

10.37236/1037 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Christine T. Cheng
Keyword(s):  
Automorphism Groups ◽  
Combinatorial Properties ◽  
Distinguishing Number ◽  
Minimum Number

Let $G$ be a graph. A vertex labeling of $G$ is distinguishing if the only label-preserving automorphism of $G$ is the identity map. The distinguishing number of $G$, $D(G)$, is the minimum number of labels needed so that $G$ has a distinguishing labeling. In this paper, we present $O(n \log n)$-time algorithms that compute the distinguishing numbers of trees and forests. Unlike most of the previous work in this area, our algorithm relies on the combinatorial properties of trees rather than their automorphism groups to compute for their distinguishing numbers.

Babbitt’s Beguiling Surfaces, Improvised Inside; Part I: Freedoms

2019 ◽  
Vol 5 (1) ◽  
Author(s):  
Joshua Banks Mailman
Keyword(s):  
Partial Ordering ◽  
Personal Traits ◽  
Milton Babbitt ◽  
Iconic Figure ◽  
Inside Part

Milton Babbitt has been a controversial and iconic figure, which has indirectly led to fallacious assumptions about how his music is made, and therefore to fundamental misconceptions about how it might be heard and appreciated. This video (the first of a three-part video essay) reconsiders his music in light of both his personal traits and a more precise examination of the constraints and freedoms entailed by his unusual and often misunderstood compositional practices, which are based inherently on partial ordering (as well as pitch repetition), which enables a surprising amount of freedom to compose the surface details we hear. The opening of Babbitt’s Composition for Four Instruments (1948) and three recompositions (based on re-ordering of pitches) demonstrate the freedoms intrinsic to partial ordering.

Minimal distributions in a stochastic partial ordering and bounds for Gaussian processes

1983 ◽  
Vol 20 (03) ◽  
pp. 529-536
Author(s):  
W. J. R. Eplett
Keyword(s):  
Gaussian Process ◽  
Gaussian Processes ◽  
Partial Ordering ◽  
Boundary Crossing ◽  
Conditional Probabilities ◽  
Life Distribution ◽  
Life Distributions ◽  
Bounded Below ◽  
Crossing Probabilities

A natural requirement to impose upon the life distribution of a component is that after inspection at some randomly chosen time to check whether it is still functioning, its life distribution from the time of checking should be bounded below by some specified distribution which may be defined by external considerations. Furthermore, the life distribution should ideally be minimal in the partial ordering obtained from the conditional probabilities. We prove that these specifications provide an apparently new characterization of the DFRA class of life distributions with a corresponding result for IFRA distributions. These results may be transferred, using Slepian's lemma, to obtain bounds for the boundary crossing probabilities of a stationary Gaussian process.

scholarly journals Further steps on the reconstruction of convex polyominoes from orthogonal projections

2021 ◽  
Author(s):  
Paolo Dulio ◽  
Andrea Frosini ◽  
Simone Rinaldi ◽  
Lama Tarsissi ◽  
Laurent Vuillon
Keyword(s):  
Discrete Geometry ◽  
Convex Subset ◽  
Convex Sets ◽  
A Priori ◽  
Orthogonal Projections ◽  
Reconstruction Process ◽  
Combinatorial Properties ◽  
The Family ◽  
Discrete Counterpart ◽  
Interior Part

AbstractA remarkable family of discrete sets which has recently attracted the attention of the discrete geometry community is the family of convex polyominoes, that are the discrete counterpart of Euclidean convex sets, and combine the constraints of convexity and connectedness. In this paper we study the problem of their reconstruction from orthogonal projections, relying on the approach defined by Barcucci et al. (Theor Comput Sci 155(2):321–347, 1996). In particular, during the reconstruction process it may be necessary to expand a convex subset of the interior part of the polyomino, say the polyomino kernel, by adding points at specific positions of its contour, without losing its convexity. To reach this goal we consider convexity in terms of certain combinatorial properties of the boundary word encoding the polyomino. So, we first show some conditions that allow us to extend the kernel maintaining the convexity. Then, we provide examples where the addition of one or two points causes a loss of convexity, which can be restored by adding other points, whose number and positions cannot be determined a priori.

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