270 research outputs found
Sir George Cayley's Navigable Balloon
No full textSir George Cayley's article "Practical remarks on aerial navigation" cut from the March 4, 1837 issue of Mechanics' Magazine. Cayley's fourth published article, it contained several ideas which proved to be prophetic for the later development of the airplane. An accompanying illustration of his navigable balloon is pasted on the first page. Two additional illustrations are separated from these pages (see VBI_000408).For more information about this item, visit https://archivesspace.mit.edu/repositories/2/digital_objects/75
Planar Cayley graph
No full textTitle: Planar Cayley graph Author: Gloser David Department: Department of Algebra Supervisor: doc. Mgr. Pavel Růžička, Ph.D., Department of Algebra Abstract: In 1896, H. Maschke published an article on planar Cayley graphs, which connects group theory and graph theory. These graphs reveal the internal properties of groups. The aim of this bachelor's thesis is to expand Maschke's original article by providing a broader context and offering a more detailed perspective on the issue of planar Cayley graphs. This thesis provides proofs for theorems that were not proven in the original article but whose statements are used within it, as well as new theorems and their proofs that are necessary for the proper functioning of the original article. The thesis also includes specific examples of groups that generate planar Cayley graphs and the original illustrations of these drawings. Keywords: planar graph, Cayley graph, grou
Our birds /
No full text1st ed. Cover ill.; Also available online at: http://nla.gov.au/nla.aus-vn758015; Library's copy signed by the author, and inscribed: "With the publishers compliments"
On the geometry of Cayley automatic groups
No full text In contrast to being automatic, being Cayley automatic a priori has no geometric consequences. Specifically, Cayley graphs of automatic groups enjoy a fellow traveler property. Here, we study a distance function introduced by the first author and Trakuldit which aims to measure how far a Cayley automatic group is from being automatic, in terms of how badly the Cayley graph fails the fellow traveler property. The first author and Trakuldit showed that if it fails by at most a constant amount, then the group is in fact automatic. In this paper, we show that for a large class of non-automatic Cayley automatic groups this function is bounded below by a linear function in a precise sense defined herein. In fact, for all Cayley automatic groups which have super-quadratic Dehn function, or which are not finitely presented, we can construct a non-decreasing function which (1) depends only on the group and (2) bounds from below the distance function for any Cayley automatic structure on the group. </jats:p
Completing partial transversals of Cayley tables of Abelian groups
No full textIn 2003 Grüttmüller proved that if n ⩾ 3 is odd, then a partial transversal of the Cayley table of ℤₙ with length 2 is completable to a transversal. Additionally, he conjectured that a partial transversal of the Cayley table of ℤₙ with length k is completable to a transversal if and only if n is odd and either n ∈ {k, k + 1} or n ⩾ 3k - 1. Cavenagh, Hämäläinen, and Nelson (in 2009) showed the conjecture is true when k = 3 and n is prime. In this paper, we prove Grüttmüller's conjecture for k = 2 and k = 3 by establishing a more general result for Cayley tables of Abelian groups of odd order.Journal ArticleFinal article publishe
Colouring Cayley graphs
No full textI hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract We will discuss three ways to bound the chromatic number on a Cayley graph. (a) If the connection set contains information about a smaller graph, then these two graphs are related. Using this information, we will show that Cayley graphs cannot have chromatic number three. (b) We will prove a general statement that all vertex-transitive maximal triangle-free graphs on n vertices with valency greater than n/3 are 3-colourable. Since Cayley graphs are vertex-transitive, the bound of general graphs also applies to Cayley graphs. (c) Since Cayley graphs for abelian groups arise from vector spaces, we can view the connection set as a set of points in a projective geometry. We will give a characterization of all large complete caps, from which we derive that all maximal triangle-free cubelike graphs on 2n vertices and valency greater than 2n/4 are either bipartite or 4-colourable. ii
Cayley Transforms in Micromagnetics
No full textMethods of numerical integration of ordinary differential equations exploiting the Cayley transform arise in a variety of contexts, ranging from the classical mid-point rule to symplectic and (almost) Poisson integrators, to numerical methods on Lie Groups. In earlier work, the first author investigated the interplay between the Cayley transform and the Jacobi identity in establishing certain error formulas for the mid-point rule (with applications to coupled rigid bodies). In this paper, we use the Cayley transform to lift the Landau-Lifshitz-Gilbert equation of micromagnetics to the Lie algebra of the group of currents (on a compact magnetic body) with values in the 3-dimensional rotation group. This follows an idea of Arieh Iserles and, we use the lift to numerically integrate the Landau-Lifshitz-Gilbert equation conserving automatically the norm of the magnetization everywhere
On the Diameter of Cayley Digraphs
No full text证明了有关CAylEy图的直径的几个定理，并得到它们对有限群的问题的应用.The author proves several theorems on the diameter of Cayley Digraphs and gives its applications to the problem on finite groups.国家自然科学基金;福建省自然科学基
On 2-arc-transitivity of Cayley graphs
No full textAbstractThe classification of 2-arc-transitive Cayley graphs of cyclic groups, given in (J. Algebra. Combin. 5 (1996) 83–86) by Alspach, Conder, Xu and the author, motivates the main theme of this article: the study of 2-arc-transitive Cayley graphs of dihedral groups. First, a previously unknown infinite family of such graphs, arising as covers of certain complete graphs, is presented, leading to an interesting property of Singer cycles in the group PGL(2,q), q an odd prime power, among others. Second, a structural reduction theorem for 2-arc-transitive Cayley graphs of dihedral groups is proved, putting us—modulo a possible existence of such graphs among regular cyclic covers over a small family of certain bipartite graphs—a step away from a complete classification of such graphs. As a byproduct, a partial description of 2-arc-transitive Cayley graphs of abelian groups with at most three involutions is also obtained
Recommended from our members
An inductive proof of the Cayley-Hamilton theorem
No full textIn this article, the author investigates a computational proof of the Cayley-Hamilton theroem, based on induction
- …
