ON THE SPECTRUM OF OCTAGON QUADRANGLE SYSTEMS OF ANY INDEX

An \emph{octagon quadrangle} is the graph consisting of a length $8$ cycle $(x_{1},x_{2},\dots,x_{8})$ and two chords, $\{x_{1},x_{4}\}$ and $\{x_{5},x_{8}\}$. An \emph{octagon quadrangle system} of order $v$ and index $\lambda$ is a pair $(X,\mathcal B)$, where $X$ is a finite set of $v$ vertices and $\mathcal B$ is a collection of octagon quadrangles (called blocks) which partition the edge set of $\lambda K_{v}$, with $X$ as vertex set. In this paper we determine completely the spectrum of octagon quadrangle systems for any index $\lambda$, with the only possible exception of $v=20$ for $\lambda=1$

Zoology of Atlas-groups: dessins d'enfants, finite geometries and quantum commutation

Every finite simple group P can be generated by two of its elements. Pairs of generators for P are available in the Atlas of finite group representations as (not neccessarily minimal) permutation representations P. It is unusual but significant to recognize that a P is a Grothendieck's dessin d'enfant D and that most standard graphs and finite geometries G-such as near polygons and their generalizations-are stabilized by a D. In our paper, tripods P -- D -- G of rank larger than two, corresponding to simple groups, are organized into classes, e.g. symplectic, unitary, sporadic, etc (as in the Atlas). An exhaustive search and characterization of non-trivial point-line configurations defined from small index representations of simple groups is performed, with the goal to recognize their quantum physical significance. All the defined geometries G' s have a contextuality parameter close to its maximal value 1.Comment: 19 page

Perfect Octagon Quadrangle Systems with an upper C4-system and a large spectrum

An octagon quadrangle is the graph consisting of an 8-cycle (x1, x2,..., x8) with two additional chords: the edges {x1, x4} and {x5, x8}. An octagon quadrangle system of order ν and index λ [OQS] is a pair (X,H), where X is a finite set of ν vertices and H is a collection of edge disjoint octagon quadrangles (called blocks) which partition the edge set of λKν defined on X. An octagon quadrangle system Σ=(X,H) of order ν and index λ is said to be upper C4-perfect if the collection of all of the upper 4-cycles contained in the octagon quadrangles form a μ-fold 4-cycle system of order ν; it is said to be upper strongly perfect, if the collection of all of the upper 4-cycles contained in the octagon quadrangles form a μ-fold 4-cycle system of order ν and also the collection of all of the outside 8-cycles contained in the octagon quadrangles form a ρ-fold 8-cycle system of order ν. In this paper, the authors determine the spectrum for these systems, in the case that it is the largest possible

strongly balanced 4 kite designs nested into oq systems

In this paper we determine the spectrum for octagon quadrangle systems [OQS] which can be partitioned into two strongly balanced 4-kitedesigns

Bifurcation of hyperbolic planforms

Motivated by a model for the perception of textures by the visual cortex in primates, we analyse the bifurcation of periodic patterns for nonlinear equations describing the state of a system defined on the space of structure tensors, when these equations are further invariant with respect to the isometries of this space. We show that the problem reduces to a bifurcation problem in the hyperbolic plane D (Poincar\'e disc). We make use of the concept of periodic lattice in D to further reduce the problem to one on a compact Riemann surface D/T, where T is a cocompact, torsion-free Fuchsian group. The knowledge of the symmetry group of this surface allows to carry out the machinery of equivariant bifurcation theory. Solutions which generically bifurcate are called "H-planforms", by analogy with the "planforms" introduced for pattern formation in Euclidean space. This concept is applied to the case of an octagonal periodic pattern, where we are able to classify all possible H-planforms satisfying the hypotheses of the Equivariant Branching Lemma. These patterns are however not straightforward to compute, even numerically, and in the last section we describe a method for computation illustrated with a selection of images of octagonal H-planforms.Comment: 26 pages, 11 figure

Pattern formation for the Swift-Hohenberg equation on the hyperbolic plane

We present an overview of pattern formation analysis for an analogue of the Swift-Hohenberg equation posed on the real hyperbolic space of dimension two, which we identify with the Poincar\'e disc D. Different types of patterns are considered: spatially periodic stationary solutions, radial solutions and traveling waves, however there are significant differences in the results with the Euclidean case. We apply equivariant bifurcation theory to the study of spatially periodic solutions on a given lattice of D also called H-planforms in reference with the "planforms" introduced for pattern formation in Euclidean space. We consider in details the case of the regular octagonal lattice and give a complete descriptions of all H-planforms bifurcating in this case. For radial solutions (in geodesic polar coordinates), we present a result of existence for stationary localized radial solutions, which we have adapted from techniques on the Euclidean plane. Finally, we show that unlike the Euclidean case, the Swift-Hohenberg equation in the hyperbolic plane undergoes a Hopf bifurcation to traveling waves which are invariant along horocycles of D and periodic in the "transverse" direction. We highlight our theoretical results with a selection of numerical simulations.Comment: Dedicated to Klaus Kirchg\"assne

Incidence geometry from an algebraic graph theory point of view

The goal of this thesis is to apply techniques from algebraic graph theory to finite incidence geometry. The incidence geometries under consideration include projective spaces, polar spaces and near polygons. These geometries give rise to one or more graphs. By use of eigenvalue techniques, we obtain results on these graphs and on their substructures that are regular or extremal in some sense. The first chapter introduces the basic notions of geometries, such as projective and polar spaces. In the second chapter, we introduce the necessary concepts from algebraic graph theory, such as association schemes and distance-regular graphs, and the main techniques, including the fundamental contributions by Delsarte. Chapter 3 deals with the Grassmann association schemes, or more geometrically: with the projective geometries. Several examples of interesting subsets are given, and we can easily derive completely combinatorial properties of them. Chapter 4 discusses the association schemes from classical finite polar spaces. One of the main applications is obtaining bounds for the size of substructures known as partial m- systems. In one specific case, where the partial m-systems are partial spreads in the polar space H(2d − 1, q^2) with d odd, the bound is new and even tight. A variant of the famous Erdős-Ko-Rado problem is considered in Chapter 5, where we study sets of pairwise non-trivially intersecting maximal totally isotropic subspaces in polar spaces. A combination of geometric and algebraic techniques is used to obtain a classification of such sets of maximum size, except for one specific polar space, namely H(2d − 1, q^2) for odd rank d ≥ 5. Near polygons, including generalized polygons and dual polar spaces, are studied in the last chapter. Several results on substructures in these geometries are given. An inequality of Higman on the parameters of generalized quadrangles is generalized. Finally, it is proved that in a specific dual polar space, a highly regular substructure would yield a distance- regular graph, generalizing a result on hemisystems. The appendix consists of an alternative proof for one of the main results in the thesis, a list of open problems and a summary in Dutch

Assessing Positional Accuracy and Correcting Point Data for Digital Soil Mapping at Varying Scales

Accuracy, timeliness, and the effect of scale of soil maps are rarely assessed. The recent increase in the use of GIS technologies and modelling software in natural resources and land management, has increased the demand for soil information at a finer resolution worldwide. Most of the world\u27s developing countries rely on soils information at a scale that is too coarse for practical planning, and have obstacles impeding collection of new data, such as civil war and a lack of collection resources. The United States has an exhaustive collection of soils data at a fine scale. However, its location information is replete with errors and inconsistencies which, if unaccounted for, can affect predictive model estimates. An integrated digital soil mapping methodology is necessary to extract the wealth of knowledge stored in soil survey data for building detailed soil maps and for assessing the positional accuracy of soil pedon data. Two studies were conducted using public data contained in the U.S. Soil Survey databases. The first study tested the development of an accurate regional-scale digital soil class map by combining new elevation data and satellite imagery. As a result, a model design was created that may be applied in countries with limited soil data. In the second study, several models were developed to assess the locational accuracy of the U.S. Soil Survey pedon points for Indiana. The study resulted in the creation of a more detailed Public Land Survey System grid, as well as several ArcGIS tools to assign a margin of error to existing soil pedon point locations, which separately or together can be adopted on a national scale

Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page