Related papers: Old and new about equidistant sets and generalized…
Formulas about the side lengths, diagonal lengths or radius of the circumcircle of a cyclic polygon in Euclidean geometry, hyperbolic geometry or spherical geometry can be unified.
To understand how the human visual system analyzes images, it is essential to know the structure of the visual environment. In particular, natural images display consistent statistical properties that distinguish them from random luminance…
In this article we review some recent developments in heterotic compactifications. In particular we review an ``inherently toric'' description of certain sheaves, called equivariant sheaves, that has recently been discussed in the physics…
This is a paper about triangle cubics and conics in classical geometry with elements of projective geometry. In recent years, N.J. Wildberger has actively dealt with this topic using an algebraic perspective. Triangle conics were also…
We study the concept of universal sets from the additive--combinatorial point of view. Among other results we obtain some applications of this type of uniformity to sets avoiding solutions to linear equations, and get an optimal upper bound…
This article describes an entirely algebraic construction for developing conformal geometries, which provide models for, among others, the Euclidean, spherical and hyperbolic geometries. On one hand, their relationship is usually shown…
In this work we reproduce the characterization of $\Gg^s$-sets from the euclidean setting [J. London Math. Soc. 49:267-280,1994] to more general metric spaces. These sets have Hausdorff dimension at least $s$ and are closed by countable…
The compactness phenomenon is one of the featured aspects of structuralism in mathematics. In simple and broad words, a compactness property holds in a structure if a related property is satisfied by sufficiently many substructures of that…
The hypothesis concerning the off-site continuum existence is investigated from the point of view of the mathematical theory of sets. The principles and methods of the mathematical description of the physical objects from different off-site…
We investigate helicoidal (screw) surfaces generated not only by regular curves but also by curves with singular points. For curves with singular points, it is useful to use frontals in the Euclidean plane. The helicoidal surface of a…
The notion of a coherent space is a nonlinear version of the notion of a complex Euclidean space: The vector space axioms are dropped while the notion of inner product is kept. Coherent spaces provide a setting for the study of geometry in…
The process of cognition is analysed to adjust the set theory to physical description. Postulates and basic definitions are revised. The specific sets of predicates, called presets, corresponding to the physical objects identified by an…
This introductory survey deals with mathematical and physical properties of discrete structures such as point sets and tilings. The emphasis is on proper generalizations of concepts and ideas from classical crystallography. In particular,…
Projective geometry provides the preferred framework for most implementations of Euclidean space in graphics applications. Translations and rotations are both linear transformations in projective geometry, which helps when it comes to…
There are many papers studying properties of point sets in the Euclidean space $\mathbb{E}^m$ or on integer grids $\mathbb{Z}^m$, with pairwise integral or rational distances. In this article we consider the distances or coordinates of the…
The aim of this paper is twofold. On one hand the generalized Minkowski sets are defined and characterized. On the other hand, the Motzkin decomposable sets, along with their epigraphic versions are considered and characterized in new ways.…
In this work, we review the concept of center of a geometric object as an equivariant map, unifying and generalizing different approaches followed by authors such as C. Kimberling or A. Edmonds. We provide examples to illustrate that this…
We develop a unified framework for the study of properties involving diagonalizations of dense families in topological spaces. We provide complete classification of these properties. Our classification draws upon a large number of methods…
We consider the collection of uniformly discrete point sets in Euclidean space equipped with the vague topology. For a point set in this collection, we characterise minimality of an associated dynamical system by almost repetitivity of the…
This report consists of two parts. The first part is a brief exposition of classical descriptive set theory. This part introduces some fundamental concepts, motivations and results from the classical theory and ends with a section on the…