Related papers: Simplifying the axiomatization for the ordered aff…
Classification of cubics (that is, third order planar curves in the $R^2$ up to certain transformations is interested since Newton, and treated by several authors. We classify cubics up to affine transformations, in seven class, and give a…
We study generically split octonion algebras over schemes using techniques of ${\mathbb A}^1$-homotopy theory. By combining affine representability results with techniques of obstruction theory, we establish classification results over…
Many proofs of the fundamental theorem of algebra rely on the fact that the minimum of the modulus of a complex polynomial over the complex plane is attained at some complex number. The proof then follows by arguing the minimum value is…
Linear topological spaces with partial ordering (linear kinematics) are studied. They are defined by a set of 8 axioms implying that topology, linear structure and ordering are compatible with each other. Most of the results are valid for…
The pursue of what are properties that can be identified to permit an automated reasoning program to generate and find new and interesting theorems is an interesting research goal (pun intended). The automatic discovery of new theorems is a…
An ordinary plane of a finite set of points in real 3-space with no three collinear is a plane intersecting the set in exactly three points. We prove a structure theorem for sets of points spanning few ordinary planes. Our proof relies on…
The fixed point index of topological fixed point theory is a well studied integer-valued algebraic invariant of a mapping which can be characterized by a small set of axioms. The coincidence index is an extension of the concept to…
We deal with generalizations of the Fundamental Theorem of Projective Geometry to other related geometries (of dimension $\geq 3$) and non bijective maps. We consider locally projective geometries and locally affino-projective geometries…
We prove that, if the closed unit ball of a normed space $X$ has sufficiently many extreme points, then every mapping $\Phi$ from $X$ into itself with the following property is affine: For any pair of points in $X$, there exists a (not…
First order formulas in a relational signature can be considered as operations on the relations of an underlying set, giving rise to multisorted algebras we call first order algebras. We present universal axioms so that an algebra satisfies…
Jacques Tits gave a general recipe for producing an abstract geometry from a semisimple algebraic group. This expository paper describes a uniform method for giving a concrete realization of Tits's geometry and works through several…
Geometrization says `` any closed oriented three-manifold which is prime (not a connected sum) carries one of the eight Thurston geometries OR it has incompressible torus walls whose complementary components each carry one of four…
We prove that the theory of the Farey graph is pseudofinite by constructing a sequence of finite structures that satisfy increasingly large subsets of its first-order axiomatization. This graph is an important object in the study of curve…
The paper gives a succinct appraisal of the properties of the projective line defined over the direct product ring $R\_{\triangle} \equiv$ GF(2)$\otimesGF(2)\otimes$GF(2). The ring is remarkable in that except for unity, all the remaining…
Theory of Riemann Extensions of the spaces with constant affine connection for the studying of the properties of nonlinear the first order systems of differential equations is proposed. Quadratic planar system of equations and the Lorenz…
In this article we prove a theorem about the number of directions determined by less then $q$ affine points, similar to the result of Blokhuis et al. (in J. Comb. Theory Ser. A 86(1), 187-196, 1999) on the number of directions determined by…
A classical theorem of De Bruijn and Erd\H{o}s asserts that any noncollinear set of n points in the plane determines at least n distinct lines. We prove that an analogue of this theorem holds for graphs. Restricting our attention to…
Any stretching of Ringel's non-Pappus pseudoline arrangement when projected into the Euclidean plane, implicitly contains a particular arrangement of nine triangles. This arrangement has a complex constraint involving the sines of its…
We combine several folklore observations to provide a working framework for iterating constructions which contradict the axiom of choice. We use this to define a model in which any kind of structural failure must fail with a proper class of…
The book is designed for a semester-long course in Foundations of Geometry and meant to be rigorous, conservative, elementary and minimalist. List of topics: Euclidean geometry: The Axioms / Half-planes / Congruent triangles / Perpendicular…