Moorhouse's question on locally finite generalized quadrangles, part 1 -- the countable case

We settle a question posed by G. Eric Moorhouse on the model theory and existence of locally finite generalized quadrangles. In this paper, we completely handle the case in which the generalized quadrangles have a countable number of elements.Comment: 9 pages; submitted (June 2020

On semi-finite hexagons of order (2,t)(2, t) containing a subhexagon

The research in this paper was motivated by one of the most important open problems in the theory of generalized polygons, namely the existence problem for semi-finite thick generalized polygons. We show here that no semi-finite generalized hexagon of order $(2,t)$ can have a subhexagon $H$ of order $2$. Such a subhexagon is necessarily isomorphic to the split Cayley generalized hexagon $H(2)$ or its point-line dual $H^D(2)$. In fact, the employed techniques allow us to prove a stronger result. We show that every near hexagon $\mathcal{S}$ of order $(2,t)$ which contains a generalized hexagon $H$ of order $2$ as an isometrically embedded subgeometry must be finite. Moreover, if $H \cong H^D(2)$ then $\mathcal{S}$ must also be a generalized hexagon, and consequently isomorphic to either $H^D(2)$ or the dual twisted triality hexagon $T(2,8)$.Comment: 21 pages; new corrected proofs of Lemmas 4.6 and 4.7; earlier proofs worked for generalized hexagons but not near hexagon

The hyperplanes of the U (4)(3) near hexagon

Combining theoretical arguments with calculations in the computer algebra package GAP, we are able to show that there are 27 isomorphism classes of hyperplanes in the near hexagon for the group U (4)(3). We give an explicit construction of a representative of each class and we list several combinatorial properties of such a representative

Approximation and geometric modeling with simplex B-splines associated with irregular triangles

Bivariate quadratic simplical B-splines defined by their corresponding set of knots derived from a (suboptimal) constrained Delaunay triangulation of the domain are employed to obtain a C1-smooth surface. The generation of triangle vertices is adjusted to the areal distribution of the data in the domain. We emphasize here that the vertices of the triangles initially define the knots of the B-splines and do generally not coincide with the abscissae of the data. Thus, this approach is well suited to process scattered data.\ud \ud With each vertex of a given triangle we associate two additional points which give rise to six configurations of five knots defining six linearly independent bivariate quadratic B-splines supported on the convex hull of the corresponding five knots.\ud \ud If we consider the vertices of the triangulation as threefold knots, the bivariate quadratic B-splines turn into the well known bivariate quadratic Bernstein-Bézier-form polynomials on triangles. Thus we might be led to think of B-splines as of smoothed versions of Bernstein-Bézier polynomials with respect to the entire domain. From the degenerate Bernstein-Bézier situation we deduce rules how to locate the additional points associated with each vertex to establish knot configurations that allow the modeling of discontinuities of the function itself or any of its directional derivatives. We find that four collinear knots out of the set of five defining an individual quadratic B-spline generate a discontinuity in the surface along the line they constitute, and that analogously three collinear knots generate a discontinuity in a first derivative.\ud Finally, the coefficients of the linear combinations of normalized simplicial B-splines are visualized as geometric control points satisfying the convex hull property.\ud Thus, bivariate quadratic B-splines associated with irregular triangles provide a great flexibility to approximate and model fast changing or even functions with any given discontinuities from scattered data.\ud An example for least squares approximation with simplex splines is presented

On generalized hexagons of order (3, t) and (4, t) containing a subhexagon

We prove that there are no semi-finite generalized hexagons with $q + 1$ points on each line containing the known generalized hexagons of order $q$ as full subgeometries when $q$ is equal to $3$ or $4$, thus contributing to the existence problem of semi-finite generalized polygons posed by Tits. The case when $q$ is equal to $2$ was treated by us in an earlier work, for which we give an alternate proof. For the split Cayley hexagon of order $4$ we obtain the stronger result that it cannot be contained as a proper full subgeometry in any generalized hexagon.Comment: 13 pages, minor revisions based on referee reports, to appear in European Journal of Combinatoric

Characterizations of the Suzuki tower near polygons

In recent work, we constructed a new near octagon $\mathcal{G}$ from certain involutions of the finite simple group $G_2(4)$ and showed a correspondence between the Suzuki tower of finite simple groups, $L_3(2) < U_3(3) < J_2 < G_2(4) < Suz$, and the tower of near polygons, $\mathrm{H}(2,1) \subset \mathrm{H}(2)^D \subset \mathsf{HJ} \subset \mathcal{G}$. Here we characterize each of these near polygons (except for the first one) as the unique near polygon of the given order and diameter containing an isometrically embedded copy of the previous near polygon of the tower. In particular, our characterization of the Hall-Janko near octagon $\mathsf{HJ}$ is similar to an earlier characterization due to Cohen and Tits who proved that it is the unique regular near octagon with parameters $(2, 4; 0, 3)$, but instead of regularity we assume existence of an isometrically embedded dual split Cayley hexagon, $\mathrm{H}(2)^D$. We also give a complete classification of near hexagons of order $(2, 2)$ and use it to prove the uniqueness result for $\mathrm{H}(2)^D$.Comment: 20 pages; some revisions based on referee reports; added more references; added remarks 1.4 and 1.5; corrected typos; improved the overall expositio

Development of freeform optical systems

One of the essential properties of the freeform surface is that its asymmetric and locally variant surface profile breaks the symmetry of an optical system, thus provoking unique issues that have never been considered in rotationally symmetric optical systems. This thesis focuses on developing an optimization algorithm that automatically eliminates the obscuration in the non-rotationally symmetric reflective optical system, as well as defining and computing the generalized chromatic aberrations in the non-rotationally symmetric refractive optical system. Furthermore, a comprehensive model for the tolerancing of freeform surface is put forward. When optimizing a three-dimensional (3D) reflective optical system by tilting the mirrors, the mirrors can block the ray path and, in consequence, reduce the image brightness and contrast. To take the degree of obscuration into consideration, an error function that mathematically describes all the obscuration cases in 3D reflective systems is proposed. In order to analyze the generalized chromatic aberrations in 3D refractive systems, the reference axis and reference plane are clarified to figure out the precise definition of the generalized chromatic aberrations. Both ray-based and wavefront-based methods are proposed to calculate the generalized chromatic aberrations surface-by-surface. In addition, the influence of pupil aberration is discussed to improve calculation accuracy. The manufacturing error of the freeform surface can be transferred into the frequency domain by Fourier transform. The autocorrelation function (ACF) of the phase pattern is computed in different frequency ranges. By characterizing the width of ACF, the boundary frequency between the deterministic and statistic errors can be found. A comprehensive model representing different types of surface errors is proposed to construct a synthetic freeform surface. By performing the Monte-Carlo simulation, tolerancing of the freeform system can be realized

Categoric aspects of authentication

[no abstract available