Characterisations of finite egglike inversive planes

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From CT scans to 4-manifold topology

In this survey paper the ultrahyperbolic equation in dimension four is discussed from a geometric, analytic and topological point of view. The geometry centres on the canonical neutral metric on the space of oriented geodesics of 3-dimensional space-forms, the analysis discusses a mean value theorem for solutions of the equation and presents a new solution of the Cauchy problem over a certain family of null hypersurfaces, while the topology relates to generalizations of codimension two foliations of 4-manifolds.Comment: 26 pages, 3 figures. Results from the following papers on arXiv are discussed: 2304.05702, 2204.13163, 2210.08155, 2011.11330, 2110.03998, 1812.03562, 1812.00710, 2002.12787, 1812.00707, 1605.09576, 1404.5509, 0808.0851, math/0702276, math/0407490, math/0406185, 1207.5994, 0709.2441, 1806.05440, 0911.260

The Fundamental Theorem for locally projective geometries

We deal with generalizations of the Fundamental Theorem of Projective Geometry to other related geometries (of dimension $\geq 3$) and non bijective maps. We consider locally projective geometries and locally affino-projective geometries (which include the classical M\"obius, Laguerre and Minkowski geometries).Comment: arXiv admin note: text overlap with arXiv:2204.1059

Finite Laguerre Near-planes of Odd Order Admitting Desarguesian Derivations

AbstractWe introduce finite Laguerre near-planes and investigate such planes of odd order that admit a Desarguesian derivation

New Classes of Partial Geometries and Their Associated LDPC Codes

The use of partial geometries to construct parity-check matrices for LDPC codes has resulted in the design of successful codes with a probability of error close to the Shannon capacity at bit error rates down to $10^{-15}$. Such considerations have motivated this further investigation. A new and simple construction of a type of partial geometries with quasi-cyclic structure is given and their properties are investigated. The trapping sets of the partial geometry codes were considered previously using the geometric aspects of the underlying structure to derive information on the size of allowable trapping sets. This topic is further considered here. Finally, there is a natural relationship between partial geometries and strongly regular graphs. The eigenvalues of the adjacency matrices of such graphs are well known and it is of interest to determine if any of the Tanner graphs derived from the partial geometries are good expanders for certain parameter sets, since it can be argued that codes with good geometric and expansion properties might perform well under message-passing decoding.Comment: 34 pages with single column, 6 figure

Discrete Geometry in Normed Spaces

This work refers to ball-intersections bodies as well as covering, packing, and kissing problems related to balls and spheres in normed spaces. A quick introduction to these topics and an overview of our results is given in Section 1.1 of Chapter 1. The needed background knowledge is collected in Section 1.2, also in Chapter 1. In Chapter 2 we define ball-intersection bodies and investigate special classes of them: ball-hulls, ball-intersections, equilateral ball-polyhedra, complete bodies and bodies of constant width. Thus, relations between the ball-hull and the ball-intersection of a set are given. We extend a minimal property of a special class of equilateral ball-polyhedra, known as Theorem of Chakerian, to all normed planes. In order to investigate bodies of constant width, we develop a concept of affine orthogonality, which is new even for the Euclidean subcase. In Chapter 2 we solve kissing, covering, and packing problems. For a given family of circles and lines we find at least one, but for some families even all circles kissing all the members of this family. For that reason we prove that a strictly convex, smooth normed plane is a topological Möbius plane. We give an exact geometric description of the maximal radius of all homothets of the unit disc that can be covered by 3 or 4 translates of it. Also we investigate configurations related to such coverings, namely a regular 4-covering and a Miquelian configuration of circles. We find the concealment number for a packing of translates of the unit ball

Closed Affine Manifolds with an Invariant Line

Radiant manifolds are affine manifolds whose holonomy preserves a point. Here we discuss certain properties of closed affine manifolds whose holonomy preserves an affine line. Particular attention is given to the case wherein the holonomy acts on the invariant line by translations and reflections. We show that in this case, the developing image must avoid the translation invariant line providing a generalization to the well known fact that closed radiant affine manifolds cannot have their fixed points inside the developing image. We conclude by generalizing this result to translation and reflection invariant proper subspaces of the holonomy