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Point-ellipse and some other exotic configurationsMar 14 2019In this paper we introduce point-ellipse configurations and point-conic configurations. We study some of their basic properties and describe two interesting families of balanced point-ellipse, respectively point-conic $6$-configurations. The construction ... More

On rich and poor directions determined by a subset of a finite planeMar 09 2019We generalize to sets with cardinality more than $p$ a theorem of R\'edei and Sz\H{o}nyi on the number of directions determined by a subset $U$ of the finite plane $\mathbb F_p^2$. A $U$-rich line is a line that meets $U$ in at least $\#U/p+1$ points, ... More

Splittable and unsplittable graphs and configurationsMar 17 2018We prove that there exist infinitely many splittable and also infinitely many unsplittable cyclic $(n_3)$ configurations. We also present a complete study of trivalent cyclic Haar graphs on at most 60 vertices with respect to splittability. Finally, we ... More

A variation principle for ground spacesApr 25 2017Mar 21 2018The ground spaces of a vector space of hermitian matrices, partially ordered by inclusion, form a lattice constructible from top to bottom in terms of intersections of maximal ground spaces. In this paper we characterize the lattice elements and the maximal ... More

Paley and the Paley graphsJan 31 2017This paper discusses some aspects of the history of the Paley graphs and their automorphism groups.

Spectral data for simply-periodic solutions of the sinh-Gordon equationJan 11 2017This note summarizes results that were obtained by the author in his habilitation thesis (arXiv:1607.08792) concerning the development of a spectral theory for simply periodic, 2-dimensional, complex-valued solutions of the sinh-Gordon equation. Spectral ... More

A spectral theory for simply periodic solutions of the sinh-Gordon equationJul 29 2016In this work a spectral theory for 2-dimensional, simply periodic, complex-valued solutions u of the sinh-Gordon equation is developed. Spectral data for such solutions are defined (following Hitchin and Bobenko) and the space of spectral data is described ... More

The order of the automorphism group of a binary $q$-analog of the Fano plane is at most twoMay 12 2016It is shown that the automorphism group of a binary $q$-analog of the Fano plane is either trivial or of order $2$.

Least Capacity Point of TrianglesJul 15 2014Let D be a compact convex domain in the plane. P\'olya & Szeg\"o and, independently, Levi & Pan defined the point p in D that is "best insulated from the boundary C of D". We compute p in the case when C is an isosceles right triangle, revisiting exact ... More

An integral representation, complete monotonicity, and inequalities of Cauchy numbers of the second kindFeb 11 2014May 21 2014In the paper, the author establishes an integral representation for Cauchy numbers of the second kind, finds the complete monotonicity, minimality, and logarithmic convexity of Cauchy numbers of the second kind, and presents some inequalities for determinants ... More

Danzer's configuration revisitedJan 06 2013Jan 06 2015We revisit the configuration of Danzer DCD(4), a great inspiration for our work. This configuration of type (35_4) falls into an infinite series of geometric point-line configurations DCD(n). Each DCD(n) is characterized combinatorially by having the ... More

Kronecker covers, V-construction, unit-distance graphs and isometric point-circle configurationsJun 24 2012Jul 29 2012We call a polytope P of dimension 3 admissible if it has the following two properties: (1) for each vertex of P the set of its first-neighbours is coplanar; (2) all planes determined by the first-neighbours are distinct. It is shown that the Levi graph ... More

Weighted Projective Spaces and a Generalization of Eves' TheoremApr 07 2012For a certain class of configurations of points in space, Eves' Theorem gives a ratio of products of distances that is invariant under projective transformations, generalizing the cross-ratio for four points on a line. We give a generalization of Eves' ... More

Core-Free, Rank Two Coset Geometries from Edge-Transitive Bipartite GraphsJun 28 2011Jun 29 2011It is known that the Levi graph of any rank two coset geometry is an edge-transitive graph, and thus coset geometries can be used to construct many edge transitive graphs. In this paper, we consider the reverse direction. Starting from edge- transitive ... More

Concepts of relative velocityMar 30 2011The central concept of the theory of relativity is the relativity of velocity. The velocity of a material body is not an intrinsic property of the body; it depends on a free choice of reference system. Relative velocity is thus reference-dependent, it ... More

Ternary relative velocity; astonishing conflict of the Lorentz group with relativityMar 29 2011We are proving that the Lorentz boost entails the relative velocity to be ternary: ternary relative velocity is a velocity of a body with respect to an interior observer as seen by a preferred exterior-observer. The Lorentz boosts imply non-associative ... More

Some theoretical results for a class of neural mass equationsMay 04 2010We study the neural field equations introduced by Chossat and Faugeras in their article to model the representation and the processing of image edges and textures in the hypercolumns of the cortical area V1. The key entity, the structure tensor, intrinsically ... More

The inversion formula and holomorphic extension of the minimal representation of the conformal groupJun 30 2006The minimal representation $\pi$ of the indefinite orthogonal group $O(m+1,2)$ is realized on the Hilbert space of square integrable functions on $\mathbb R^m$ with respect to the measure $|x|^{-1} dx_1... dx_m$. This article gives an explicit integral ... More

A Gessel-Viennot-type method for cycle systems in a directed graphDec 02 2005We introduce a new determinantal method to count cycle systems in a directed graph that generalizes Gessel and Viennot's determinantal method on path systems. The method gives new insight into the enumeration of domino tilings of Aztec diamonds, Aztec ... More