Related papers: The bridge between Desargues' and Pappus' theorems
A differential geometrical and topological structure of Delsarte transmutation operators in multidimension is studied, the relationships with De Rham-Hodge-Skrypnik theory of generalized differential complexes is stated.
The review of modern study of algebraic, geometric and differential properties of quaternionic (Q) numbers with their applications. Traditional and "tensor" formulation of Q-units with their possible representations are discussed and groups…
In this paper, we clarify and build connections between various conjectures largely motivated by the works of Jean-Pierre Serre and John Tate. We closely study the Tate conjecture for algebraic cycles as well as their motivic…
In 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.…
We discuss a discretisation of the de Rham-Hodge theory in the two-dimensional case based on a discrete exterior calculus framework. We present discrete analogues of the Hodge-Dirac and Laplace operators in which key geometric aspects of…
In this paper we discuss several results about the structure of the configuration space of two-dimensional tensegrities with a small number of points. We briefly describe the technique of surgeries that is used to find geometric conditions…
We study tilings of the plane composed of two repeating tiles of different assigned areas relative to an arbitrary periodic lattice. We classify isoperimetric configurations (i.e., configurations with minimal length of the interfaces) both…
The relationship between mathematics and physics has long been an area of interest and speculation. Subscribing to the recent definition by Tegmark, we present a mathematical structure involving the only division rings - the real,…
We compute the fundamental group of various spaces of Desargues configurations in complex projective spaces: planar and non-planar configurations, with a fixed center and also with an arbitrary center.
Betweenness as a relation between three individual points has been widely studied in geometry and axiomatized by several authors in different contexts. The article proposes a more general notion of betweenness as a relation between three…
We introduce a new map between a (q,h)-deformed gauge theory and ordinary gauge theory in a full analogy with Seiberg-Witten map.
In this paper, we develop new techniques for understanding surfaces in $\mathbb{CP}^2$ via bridge trisections. Trisections are a novel approach to smooth 4-manifold topology, introduced by Gay and Kirby, that provide an avenue to apply…
We consider a thin and narrow rectangular plate where the two short edges are hinged whereas the two long edges are free. This plate aims to represent the deck of a bridge, either a footbridge or a suspension bridge. We study a nonlocal…
Suppose that a group $G$ acts transitively on the points of a non-Desarguesian plane, $\mathcal{P}$. We prove first that the Sylow 2-subgroups of $G$ are cyclic or generalized quaternion. We also prove that $\mathcal{P}$ must admit an odd…
In this paper we propose a well-justified synthetic approach of the projective space. We define the concepts of plane and space of incidence and also the Gallucci's axiom as an axiom to our classical projective space. To this purpose we…
We study the new ``gauge'' theories in 5+1 dimensions, and their non-commutative generalizations. We argue that the $\theta$-term and the non-commutative torus parameters appear on an equal footing in the non-critical string theories which…
An ansatz is presented for a possible non-associative deformation of the standard Yang-Mills type gauge theories. An explicit algebraic structure for the deformed gauge symmetry is put forward and the resulting gauge theory developed. The…
We present a description of two dimensional Yang-Mills gauge theory on the plane and on compact surfaces, examining the topological, geometric and probabilistic aspects.
In this paper, in the context of the ``Dessins d'enfants'' theory, we give a combinatorial criterion for a plane tree to cover a tree from the classes of "chains" or "stars''. Besides, we discuss some applications of this result which are…
We build and investigate a pure gauge theory on arbitrary discrete groups. A systematic approach to the construction of the differential calculus is presented. We discuss the metric properties of the models and introduce the action…