Related papers: Synthetic Approach to the Singularity Problem
Symmetry is conventionally described in a contrariety manner that the system is either completely symmetric or completely asymmetric. Using group theoretical approach to overcome this dichotomous problem, we introduce the degree of symmetry…
We develop the theory of derived differential geometry in terms of bundles of curved $L_\infty[1]$-algebras, i.e. dg manifolds of positive amplitudes. We prove the category of derived manifolds is a category of fibrant objects. Therefore,…
A so-called special generic map is by definition a map of smooth manifolds all of whose singularities are definite fold points. It is in general an open problem posed by Saeki in 1993 to determine the set of integers $p$ for which a given…
DifferentialGeometry is a Maple software package which symbolically performs fundamental operations of calculus on manifolds, differential geometry, tensor calculus, Lie algebras, Lie groups, transformation groups, jet spaces, and the…
Ordered phases resulting from spontaneously broken continuous symmetries are effectively described by sigma models of maps to the coset space of Goldstone modes. A classic problem is to classify the topological sectors of the sigma model.…
In this article, we introduce the notion of stochastic symmetry of a differential equation. It consists in a stochastic flow that acts over a solution of a differential equation and produces another solution of the same equation. In the…
We study the fundamental properties of curvature in groupoids within the framework of synthetic differential geometry. As is usual in synthetic differential geometry, its combinatorial nature is emphasized. In particular, the classical…
Groupoids provide a more appropriate framework for differential geometry than principal bundles. Synthetic differential geometry is the avant-garde branch of differential geometry, in which nilpotent infinitesimals are available in…
The class of special generic maps contains Morse functions with exactly two singular points, characterizing spheres topologically which are not $4$-dimensional and the $4$-dimensional unit sphere. This class is for higher dimensional…
A stochastic nonlinear partial differential equation is built for two different models exhibiting self-organized criticality, the Bak, Tang, and Wiesenfeld (BTW) sandpile model and the Zhang's model. The dynamic renormalization group (DRG)…
According to the basic idea of category theory, any Einstein algebra, essentially an algebraic formulation of general relativity, can be considered from the point of view of any object of the category of smooth algebras; such an object is…
When analyzing parametric statistical models, a useful approach consists in modeling geometrically the parameter space. However, even for very simple and commonly used hierarchical models like statistical mixtures or stochastic deep neural…
Differential forms is a highly geometric formalism for physics used from field theories to General Relativity (GR) which has been a great upgrade over vector calculus with the advantages of being coordinate-free and carrying a high degree…
In this paper we develop a geometric approach to convex subdifferential calculus in finite dimensions with employing some ideas of modern variational analysis. This approach allows us to obtain natural and rather easy proofs of basic…
Differential geometric approaches to the analysis and processing of data in the form of symmetric positive definite (SPD) matrices have had notable successful applications to numerous fields including computer vision, medical imaging, and…
An algorithmic method to exploit a general class of infinitesimal symmetries for reducing stochastic differential equations is presented and a natural definition of reconstruction, inspired by the classical reconstruction by quadratures, is…
We present an abstract unifying framework for interpreting Stone-type dualities; several known dualities are seen to be instances of just one topos-theoretic phenomenon, and new dualities are introduced. In fact, infinitely many new…
We apply methods of nonstandard mathematics in order to regard analytic geometry in a very different way. For example, complex spaces are seen to be the "standard part" of certain algebraic nonstandard schemes. We construct a category of…
These notes are designed for those who either plan to work in differential geometry, or at least want to have a good reason not to do it. We discuss smooth curves and surfaces -- the main gate to differential geometry. We focus on the…
Since the time when the first optical instruments have been invented, an idea that the visible image of an object under observation depends on tools of observation became commonly assumed in physics. A way to formalize it in mathematics is…