Related papers: Elementary schematic geometry
Consider briefly the equations of fluid dynamics-they describe the enormous wealth of detail in all the interacting physical elements of a fluid flow-whereas in applications we want to deal with a description of just that which is…
With a model of a geometric theory in an arbitrary topos, we associate a site obtained by endowing a category of generalized elements of the model with a Grothendieck topology, which we call the antecedent topology. Then we show that the…
We explore the canonical Grothendieck topology in some specific circumstances. First we use a description of the canonical topology to get a variant of Giraud's Theorem. Then we explore the canonical Grothendieck topology on the categories…
One considers geometry with the intransitive equaivalence relation. Such a geometry is a physical geometry, i.e. it is described completely by the world function, which is a half of the squared distance function. The physical geometry…
Geodesics become an essential element of the geometry of a semi-Riemannian manifold. In fact, their differences and similarities with the (positive definite) Riemannian case, constitute the first step to understand semi-Riemannian Geometry.…
We introduce orbifolds from the classical point of view, using charts, and present orbifold versions of elementary objects from Algebraic Topology, such as the fundamental group, coverings and Euler characteristic; Differential…
The Standard Model of the elementary particles is controlled by more than 20 parameters, of which it is not known today how they can be linked to deeper principles. Any attempt to clean up this theory, in general results in producing more…
A primary hypothesis that drives scientific and engineering studies is that data has structure. The dominant paradigms for describing such structure are statistics (e.g., moments, correlation functions) and signal processing (e.g.,…
In this article, we are concerned with various aspects of arcs on surfaces. In the first part, we deal with topological aspects of arcs and their complements. We use this understanding, in the second part, to construct interesting actions…
Universal algebraic geometry is generalised from solutions of equations in a single algebra to the study of $\varphi$- or $K$-spectra, akin to the prime spectrum of a ring. We explore their basic properties and constructions, give a…
An isometry is a geometric transformation that preserves distances between pairs of points. We present methods to classify isometries in the Euclidean plane, and extend these methods to spherical, single elliptical, and hyperbolic geometry.…
This is a brief review, in relatively non-technical terms, of recent advances in the theory of random field geometry. These advances have provided a collection of explicit new formulae describing mean values of a variety of geometric…
The presented work focuses on problems from determinant theory, set theory and topology. The term graph is the binding element that connects these problems. Graphs are distinguished by their geometrical simplicity, which helps in showing…
Geometric, topological and graph theory modeling and analysis of biomolecules are of essential importance in the conceptualization of molecular structure, function, dynamics, and transport. On the one hand, geometric modeling provides…
This paper is mainly a semi-tutorial introduction to elementary algebraic topology and its applications to Ising-type models of statistical physics, using graphical models of linear and group codes. It contains new material on systematic…
We study the homotopy theory of locally ordered spaces, that is manifolds with boundary whose charts are partially ordered in a compatible way. Their category is not particularly well-behaved with respect to colimits. However, this category…
We present and discuss applications of the category of probabilistic morphisms, initially developed in \cite{Le2023}, as well as some geometric methods to several classes of problems in statistical, machine and manifold learning which shall…
This paper develops the basic theory of formal schemes over fields in the supersymmetric setting. We introduce the notion of a formal superscheme and investigate some of its fundamental properties. Particular emphasis is placed on the study…
We study objects in triangulated categories which have a two-dimensional graded endomorphism algebra. Given such an object, we show that there is a unique maximal triangulated subcategory, in which the object is spherical. This general…
This paper is devoted to show the advantages of introducing a geometric viewpoint and a non extensive formulation in the description of apparently unrelated phenomena: combustion and earthquakes. Here, it is shown how the introduction of a…