Alexey Remizov

• PhD
• Professor (Associate) at Moscow Institute of Physics and Technology

We consider a pseudo-Riemannian metric that changes signature along a smooth curve on a surface, called the discriminant curve. The discriminant curve separates the surface locally into a Riemannian and a Lorentzian domain. We study the local behaviour and properties of geodesics at a point on the discriminant where the isotropic direction is tange...

This book on linear algebra and geometry is based on a course given by renowned academician I.R. Shafarevich at Moscow State University. The book begins with the theory of linear algebraic equations and the basic elements of matrix theory and continues with vector spaces, linear transformations, inner product spaces, and the theory of affine and pr...

We study phase portraits at singular points of vector fields of the special type where all components are fractions with a common denominator vanishing on a smooth regular hypersurface in the phase space. Also we assume some additional conditions, which are fulfilled, for instance, if the vector field is divergence-free. This problem is motivated b...

This paper presents a semidiscrete alternative to the theory of neurogeometry of vision, due to Citti, Petitot, and Sarti. We propose a new ingredient, namely, working on the group of translations and discrete rotations $SE(2,N)$. The theoretical side of our study relates the stochastic nature of the problem with the Moore group structure of $SE(2,...

Smooth 2-surfaces with pseudo-Riemannian metric are considered, that is, ones with quadratic form in the tangent bundle that is not positive-definite. Degeneracy points of the form are said to be parabolic. Geodesic lines induced by this pseudo-Riemannian metric in a neighbourhood of typical parabolic points are considered, their phase portraits ar...

MATHEMATICAL EDUCATION. TEXT IN RUSSIAN. We consider the Newton equations and the Euler-Lagrange equations that admit reduction of the order via the conservation of energy law. Passing to the first-order equation of the constant energy, "extraneous solutions" that are not solutions of the initial second-order equation can appear. A careful study o...

The paper is devoted to the problem of local diagonalizability for pairs of quadratic differential forms depending on two real variables, which naturally generalizes the similar problem for algebraic quadratic forms. Here we have a new obstacle for diagonalization, which appears if a linear combination of the forms has an umbilic point, where all c...

We establish an interesting property of geodesics equations in signature changing (pseudo-Riemannian) 2D-metrics. On the way to it we briefly present main facts concerning singular points of such equations, as well as equations of a more general type, for which the notion of admissible directions appears.

Излагаются элементы теории динамических систем (качественной теории автономных дифференциальных уравнений) на плоскости. Основное внимание уделено предельным циклам, приводятся примеры естественно-научных задач с предельными циклами. Затрагиваются некоторые геометрические и топологические аспекты теории динамических систем, включая индекс Кронекера...

To the Centenary of I.R. Shafarevich [In Russian] The article is a selection of quotes from various works of the great mathematician and thinker Igor Rostislavovich Shafarevich, dedicated to the present and future of mathematics and science in general, with short comments.

Eight plots are presented related to various natural sciences, described by ordinary differential equations, a distinctive feature of which is the simplicity of the mathematical apparatus necessary for the study. The author hopes that they can be used when reading the course of differential equations and help make it more interesting and meaningful...

We study the behaviour of solutions of quasi-linear differential equations of the second order at their singular points, where the coefficient of the second-order derivative vanishes. We consider solutions entering a singular point either with definite tangential direction (proper solutions) or without definite tangential direction (oscillating sol...

The talk is devoted to a special class of vector (or direction) fields, whose equilibrium (singular) points are not isolated, but fill a submanifold of codimension two in the phase space. Similarly, just as frontal mappings are ungeneric in the space of all smooth mappings, so the considered class of vector fields is ungeneric in the whole space of...

Обложка (front cover)

Учебное пособие, изданное в МФТИ. Издательство: МФТИ (Москва) Объём: 181 страниц ISBN: 978-5-7417-0794-4

We study the behaviour of solutions of quasi-linear differential equations of the second order at their singular points, where the coefficient of the second-order derivative vanishes. We consider solutions entering a singular point either with definite tangential direction (proper solutions) or without definite tangential direction (oscillating sol...

A local normal form for Roussarie vector fields with degenerate quadratic part is presented.

In 1975, Roussarie studied a special class of vector fields, whose singular points fill a submanifold of codimension two and the ratio between two non-zero eigenvalues λ 1 : λ 2 = 1 : −1. He established a smooth orbital normal form for such fields at points where λ 1,2 are real and the quadratic part of the field satisfied a certain genericity cond...

We study the behaviour of solutions of ordinary differential equations of the second order with singular points, where the coefficients of the second-order derivative vanishes. In particular, we consider solutions entering a singular point without definite tangential direction. Great attention is paid to second-order equations, whose right-hand sid...

Это записки лекций по теории особенностей, читавшихся на Физтехе в 2020 и 2021 гг., исправленные и дополненные. В начале 2022 г. на основе этих записок выпущена небольшая книжка с тем же названием: https://www.researchgate.net/publication/358280768_Vvedenie_v_teoriu_osobennostej

Slightly corrected text. July 8, 2020.

Эта статья является заключительной в серии работ, посвященных типичным особенностям геодезических потоков в двумерных псевдоримановых метриках переменной сигнатуры и метриках на поверхностях с ребром возврата, индуцированных евклидовой метрикой объемлющего пространства. Исследованы локальные фазовые портреты и свойства геодезических в точках вырожд...

This is the final paper in a series of works dedicated to generic singularities of geodesic flows in 2-dimensional pseudo-Riemannian metrics with varying signature and metrics on surfaces with cuspidal edge induced by Euclidean metric of the ambient space. We study local phase portraits and properties of geodesic at degenerate points of a special t...

We present a new image inpainting algorithm, the Averaging and Hypoelliptic Evolution (AHE) algorithm, inspired by the one presented in [1] and based upon a (semi-discrete) variation of the Citti--Petitot--Sarti model of the primary visual cortex V1. In particular, we focus on reconstructing highly corrupted images (i.e. where more than the 80% of...

We present a survey on generic singularities of geodesic flows in smooth signature changing metrics (often called pseudo-Riemannian) in dimension 2. Generically, a pseudo-Riemannian metric on a 2-manifold $S$ changes its signature (degenerates) along a curve $S_0$, which locally separates $S$ into a Riemannian ($R$) and a Lorentzian ($L$) domain. T...

As is well known, for every orthogonal transformation of the Euclidean space there exists an orthogonal basis such that the matrix of the transformation is block-diagonal with first order blocks ±1 and second order blocks that are rotations of the Euclidean plane. There exists a natural generalization of this theorem for Lorentz transformations of...

In this paper we review several algorithms for image inpainting based on the hypoelliptic diffusion naturally associated with a mathematical model of the primary visual cortex. In particular, we present one algorithm that does not exploit the information of where the image is corrupted, and others that do it. While the first algorithm is able to re...

In this paper we review several algorithms for image inpainting based on the hypoelliptic diffusion naturally associated with a mathematical model of the primary visual cortex. In particular, we present one algorithm that does not exploit the information of where the image is corrupted, and others that do it. While the first algorithm is able to re...

This is the final part of the paper published in the same journal one year ago. We continue to acquaint the reader with basic facts of singularity theory, for instance, division theorem, Malgrange preparation theorem, and we give the proof of Whitney theorem on mappings of the plane into the plane.

In this paper, we present some elements of the theory of smooth mappings, which are appropriate for students in graduate. Most of the presented results and notions are connected with Hassler Whitney, one of the greatest specialists in this field. The full text (in Russian) is available here: http://matob.ru/files/nomer79.pdf

We study singularities of geodesics flows in two-dimensional generalized Finsler spaces (pseudo-Finsler spaces). Geodesics are defined as extremals of a certain auxiliary functional whose non-isotropic extremals coincide with extremals of the action functional. This allows us to consider isotropic lines as (unparametrized) geodesics.

We study singularities of geodesics flows in two-dimensional generalized Finsler spaces (pseudo-Finsler spaces). Geodesics are defined as extremals of a certain auxiliary functional whose non-isotropic extremals coincide with extremals of the action functional. This allows to consider isotropic lines as (unparametrized) geodesics.

The paper is a study of geodesics in two-dimensional pseudo-Riemannian metrics. Firstly, the local properties of geodesics in a neighborhood of generic parabolic points are investigated. The equation of the geodesic flow has singularities at such points that leads to a curious phenomenon: geodesics cannot pass through such a point in arbitrary tang...

Lecture Notes in USP (January 2015) _ Part 1

We compare the image inpainting results of two models of geometry of vision obtained through control the-oretic considerations (the semi-discrete versions of the Citti-Petitot-Sarti and Mumford Elastica models). The main feature described by these models is the lifting of 2D images to the 3D group of translations and discrete rotations on the plane...

Книга представляет собой курс линейной алгебры и геометрии, основанный на лекциях, которые на протяжении многих лет читались одним из авторов на механико-математическом факультете Московского государственного университета. Изложение предмета начинается с теории линейных уравнений и матриц и далее ведется на языке векторных пространств. В книге такж...

Keywords: neurogeometry, hypoelliptic diffusion, sub-Riemannian geometry, generalized Fourier transform

We study phase portraits of a first order implicit differential equation in a neighborhood of its pleated singular point that is a nondegenerate singular point of the lifted field. Although there is no a visible local classification of implicit differential equations at pleated singular points (even in the topological category), we show that there...

In the second chapter we deal with matrices and determinants. The chapter starts with determinants of second and third orders, which are defined through solutions of linear algebraic systems; determinants of arbitrary order are defined inductively. The basic properties of determinants are investigated. We then take a look at determinants from a mor...

The first chapter is devoted to linear functions and systems of linear algebraic equations. The basic properties of linear functions (defined on the set of rows of a fixed length) are considered, and the method of Gaussian elimination for linear systems is presented. The final part of the chapter contains several applications of theoretical results...

Quadratic and bilinear forms on vector spaces are considered. A connection between the notion of bilinear form and that of linear transformation is established, based on the isomorphism between the space of bilinear forms and the space of linear transformations of the vector space to the dual space. A theorem on reducing a quadratic form to canonic...

This chapter is devoted mainly to Euclidean vector spaces and their transformations. It starts with notions of inner product, length, angle, Gramian, orthogonality, orthonormal basis, etc. Orthogonal transformations are investigated, and orientation of Euclidean spaces is discussed. After that, symmetric linear transformations of real vector spaces...

Chapter 4 begins with the definition of eigenvalues, eigenvectors, eigenspaces, annihilator, and characteristic polynomials, etc. Then linear transformations of a complex or real vector space to itself are investigated in greater detail. In this chapter we consider the case in which a linear transformation is diagonalizable. Namely, for a complex v...

This chapter is concerned with hyperbolic (Lobachevskian) geometry, which is tightly linked to linear algebra through a model in which hyperbolic space is represented as the projectivization of the interior of the light cone in the corresponding pseudo-Euclidean space. The chapter begins with a detailed description of this model and a study of hype...

The goal of this chapter is a more complete study of linear transformations of a complex or real vector space to itself, including the investigation of nondiagonalizable transformations. The Jordan normal forms for complex and real vector spaces are established. The final part of the chapter contains applications of the Jordan normal form: raising...

This chapter presents an excursion in abstract algebra. It begins with the notions of group, subgroup, direct sum, homomorphism, isomorphism, etc., their basic properties, and numerous examples. This is justified by the main aim of the chapter: to establish the decomposition of a finite abelian group as a direct sum of cyclic subgroups, which is ve...

The final chapter of the book presents the basic elements of representation theory, especially finite-dimensional representations of finite groups. For instance, it is proved that every representation of a finite group is a direct sum of irreducible representations, and that a finite group has only a finite number of distinct (up to equivalence) ir...

The chapter begins with notion of the Plücker coordinates of a subspace in a vector space. Then the Plücker relations are derived, and the Grassmann varieties are described. Then an exterior product of vectors is defined, and the connection between the exterior product and Plücker coordinates is explained: the Plücker relations give necessary and s...

The theory of projective spaces and their transformations is presented. The notions of a projective space, projective subspace, homogeneous and inhomogeneous coordinates, projective algebraic variety, projective transformation, cross ratio, etc., are introduced and discussed. The principle of projective duality is presented. At the end of the chapt...

The theory of affine spaces and their transformations is presented. The case of affine Euclidean spaces is also considered, and their motions are investigated. For instance, it is proved that every motion (defined in the most general way, as an isometry of the affine Euclidean space as a metric space) is an affine transformation, and it can be repr...

The theory of quadrics in spaces of several different types is presented. The chapter begins with quadrics in projective spaces. The principle of projective duality is extended to nonsingular quadrics and is illustrated with Pascal’s and Brianchon’s theorems. The isotropic subspaces of maximum possible dimension on a nonsingular quadric in complex...

In the third chapter we move to a more abstract and general level. The notions of vector space, subspace, dimension, basis, linear transformations, isomorphism, etc. are introduced and discussed. At the end of this chapter, the notions of dual vector space and forms and polynomials in vectors are considered. Most of the abstract concepts are illust...

We study phase portraits and singular points of vector fields of a special type, that is, vector fields whose components are fractions with a common denominator vanishing on a smooth regular hypersurface in the phase space. We assume also some additional conditions, which are fulfilled, for instance, if the vector field is divergence-free. This pro...

This paper is a study of singularities of geodesic flows on surfaces with nonisolated singular points that form a smooth curve (like a cuspidal edge). The main results of the paper are normal forms of the corresponding direction field on the tangent bundle of the plane of local coordinates and the projection of its trajectories to the surface.

Книга представляет собой курс линейной алгебры и геометрии, основанный на лекциях, которые на протяжении многих лет читались одним из авторов на механико-математическом факультете Московского государственного университета. Изложение предмета начинается с теории линейных уравнений и матриц и далее ведется на языке векторных пространств. В книге такж...

Рассматриваются гладкие двумерные поверхности с севдоримановой метрикой, т.е. с квадратичной формой на касательном расслоении, не являющейся положительно определенной. Точки, в которых форма вырождается, называются параболическими. Рассматриваются геодезические, индуцированные данной псевдоримановой метрикой в окрестности типичных параболических то...

Generic properties of regular first integrals of systems of implicit differential equations are considered. In particular, for systems of two equations with two phase variables, a classification of generic bifurcations of integral level surfaces is described.

Codimension-two singularities of the field of totally singular extremal trajectories in 3D affine control systems with scalar control are investigated. These singularities can be of two types: the first is related to singularities of the field of the Hamiltonian system of the maximum principle itself, while the second is related to the degenerate p...

This paper is devoted to singular points of the so-called lifted vector fields, which arise in studying systems of implicit differential equations by using the method of lifting the equation to a surface, a generalization of the construction used by Poincaré for a single implicit equation. The author studies the phase portraits and renormal forms o...

Vector fields with singularities that are not isolated, but form a smooth submanifold of the phase space of codimension 2 are studied. Fields of this kind occur, for instance, in the analysis of implicit differential equations. Furthermore, under slight perturbations of the original problem the variety of singular points does not disappear or degen...

This paper is devoted to singular points of the so-called lifted vector fields, which arise in studying systems of implicit differential equations by using the method of lifting the equation to a surface, a generalization of the construction used by Poincaré for a single implicit equation. The author studies the phase portraits and renormal forms o...

Vector fields with singularities that are not isolated, but form a smooth submanifold of the phase space of codimension 2 are studied. Fields of this kind occur, for instance, in the analysis of implicit differential equations. Furthermore, under slight perturbations of the original problem the variety of singular points does not disappear or degen...

We consider a numerical method for a system of implicit differential equations in a domain without singular points. We get an estimation for the degree of convergence and for the local error. For finding the global error estimation we use the ellipsoid method. This way allows us to construct a neighborhood of the approximate solution in the phase s...

A numerical method is considered for solving the system of implicit ordinary differential equations in the region without singular points. The iteration convergence rate for searching the approximated values of the solution in the net points is estimated. The local error is estimated too. The global error is estimated by the ellipsoid method, permi...