Related papers: Basic differential geometry as a sequence of inter…
We study the systems of ordinary differential equations which are implicit with respect to the higher derivatives, appearing in the linear form, and their solutions near the singular points. The invertibility of the higher derivatives…
Differential geometry may be generalized to allow infinitesimals to any order. The purpose of the present contribution is to show that the theory so developed expands received geometrical ideas in an interesting way, rich in potential for…
This is an expository introduction to simplicial sets and simplicial homotopy theory with particular focus on relating the combinatorial aspects of the theory to their geometric/topological origins. It is intended to be accessible to…
A theory of principal bundles possessing quantum structure groups and classical base manifolds is presented. Structural analysis of such quantum principal bundles is performed. A differential calculus is constructed, combining differential…
The present article deals with general mechanics in an unconventional manner. At first, Newtonian mechanics for a point particle has been described in vectorial picture, considering Cartesian, polar and tangent-normal formulations in a…
A class of surfaces-graphs in a Riemannian 3-space with a prescribed projection of one field of principal directions onto a surface $\Pi$ is considered. A problem of determination of such surfaces when both principal curvatures are given…
These notes contain a survey of some aspects of the theory of differential modules and complexes as well as of their generalization, that is, the theory of $N$-differential modules and $N$-complexes. Several applications and examples coming…
This book encompasses both traditional and modern methods treating partial differential equation (PDE) of first order and second order. There is a balance in making a selfcontained mathematical text and introducing new subjects. The Lie…
We assume that the points in volumes smaller than an elementary volume (which may have a Planck size) are indistinguishable in any physical experiment. This naturally leads to a picture of a discrete space with a finite number of degrees of…
The elegance and usefulness of a complex formulation of the basic lensing equations is demonstrated with a number of applications. Using standard tools of complex function theory, we present, for instance, a new proof of the fact that the…
Physics-based differentiable rendering (PBDR) has become an efficient method in computer vision, graphics, and machine learning for addressing an array of inverse problems. PBDR allows patterns to be generated from perceptions which can be…
The geometric foundations of General Relativity are revisited, with particular attention to its gauge invariance, as a key to understanding the true nature of spacetime. Beyond the common image of spacetime as a deformable 'fabric' filling…
The paper is devoted to differential geometric invariants determining a Frenet curve in up to a direct similarity These invariants can be presented by the Euclidean curvatures in terms of an arc lengths of the spherical indicatrices. Then,…
This book introduces the new research area of Geometric Data Science, where data can represent any real objects through geometric measurements. The first part of the book focuses on finite point sets. The most important result is a complete…
The paper is dedicated to the close analogy between these two theories - some problems lying at the very root of Spectral Geometry are viewed in the context of Semiclassics, and vise versa. The treatment starts from a very basic level and…
Special Bohr - Sommerfeld geometry, first formulated for simply connected symplectic manifolds (or for simple connected algebraic varieties), gives rise to some natural problems for the simplest example in non simply connected case. Namely…
The proper handling of 3D orientations is a central element in many optimization problems in engineering. Unfortunately many researchers and engineers struggle with the formulation of such problems and often fall back to suboptimal…
A generic method for combinatorial constructions of intrinsic geometrical spaces is presented. It is based on the well known inverse sequences of finite graphs that determine (in the limit) topological spaces. If a pattern of the…
This book is addressed to students, professors and researchers of geometry, who will find herein many interesting and original results. The originality of the book The Geometry of Homological Triangles consists in using the homology of…
A new simple geometrical interpretation of complex numbers is presented. It differs from their usual interpretation as points in the complex plane. From the new point of view the complex numbers are rather operations on vectors than points.…