Related papers: Finite Euclidean and Non-Euclidean Geometries

The purpose of the present work is to show that an adequate basis for understanding the essentially nonlinear phenomena must also be essentially nonlinear however still simple enough to play the role of a basis. It is shown that such types…

We trace the development of arguments for the consistency of non-Euclidean geometries and for the independence of the parallel postulate, showing how the arguments become more rigorous as a formal conception of geometry is introduced. We…

We develop a rigorous theoretical framework for principal manifold estimation that recovers a latent low-dimensional manifold from a point cloud observed in a high-dimensional ambient space. Our framework accommodates manifolds with…

This is an outline of Erlangen Program at Large. Study of objects and properties, which are invariant under a group action, is very fruitful far beyond the traditional geometry. In this paper we demonstrate this on the example of the group…

These lectures review the classical Moebius-Lie geometry and recent work on its extension. The latter considers ensembles of cycles (quadrics), which are interconnected through conformal-invariant geometric relations (e.g. "to be…

A. Tarski uses in his system for the elementary geometry only the primitive concept of point, and the two primitive relations betweenness and equidistance. Another approach is the relations to be on lines instead of points. W.…

We study the Erdos distance problem over finite Euclidean and non-Euclidean spaces. Our main tools are graphs associated to finite Euclidean and non-Euclidean spaces that are considered in Bannai-Shimabukuro-Tanaka (2004, 2007). These…

Shape analysis and compuational anatomy both make use of sophisticated tools from infinite-dimensional differential manifolds and Riemannian geometry on spaces of functions. While comprehensive references for the mathematical foundations…

Euclidean geometry consists of straightedge-and-compass constructions and reasoning about the results of those constructions. We show that Euclidean geometry can be developed using only intuitionistic logic. We consider three versions of…

We refurbish our axiomatics of differential geometry introduced in [Mathematics for Applications,, 1 (2012), 171-182]. Then the notion of Euclideaness can naturally be formulated. The principal objective in this paper is to present an…

In this thesis I discuss combinatorial optimization problems, from the statistical physics perspective. The starting point are the motivations which brought physicists together with computer scientists and mathematicians to work on this…

At the beginning of the 20th Century there was a growing interest for the investigation of the action of linear groups on the geometry of surfaces. In that context of ideas, the quest for a connection between curvature and the behaviour of…

Conformal transformations of a Euclidean (complex) plane have some kind of completeness (sufficiency) for the solution of many mathematical and physical-mathematical problems formulated on this plane. There is no such completeness in the…

Three different representation of the proper Euclidean geometry are considered. They differ in the number of basic elements, from which the geometrical objects are constructed. In E-representation there are three basic elements (point,…

In this paper, we introduce an asymmetric metric on the space of marked Euclidean triangles, and we prove several properties of this metric, including two equivalent definitions of this metric, one of them comparing ratios of functions of…

Finite fields form an important chapter in abstract algebra, and mathematics in general. We aim to provide a geometric and intuitive model for finite fields, involving algebraic numbers, in order to make them accessible and interesting to a…

We discover suprising connections between three seemingly different problems: finding right triangles with rational sides in a non-Euclidean geometry, finding three integers such that the difference of the squares of any two is a square,…

These are lecture notes on the algebraic approach to regular languages. The classical algebraic approach is for finite words; it uses semigroups instead of automata. However, the algebraic approach can be extended to structures beyond…

The purpose of this paper is to prove that every finite set of points that can be constructed in the Euclidean plane by using a compass and a ruler can also be constructed by using unitary match-sticks in a non-simultaneous way and…

I summarize here the logic that leads us to a program for the Theory of the Total Field in Einstein's sense. The purpose is to show that this theory is a logical culmination of the developments of (fundamental) physical concepts and, hence,…