Characterization of the finite weak singularities of quadratic systems via invariant theory

This article is about weak singularities of quadratic differential systems, that is, non-degenerate singular points with traces of the corresponding linearized systems at such points equal to zero. These could be foci, centers or saddles. Necessary and sufficient conditions for a real quadratic system to possess a fixed number of weak singularities of a specific order are given. The conditions are stated in terms of affine invariant polynomials in the 12-dimensional space of the coefficients.     

Decompositions of simple singularities of real functions

Leningrad Branch, Academy of Sciences of the USSR. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 22, No. 4, pp. 52–67, October–December, 1988.     

Geometric and analytic problems for a real submanifold in C~n with CR singularities

In this survey article, we present some known results and also propose some open questions related to the analytic and geometric aspects of Bishop submanifolds in a complex space. We mainly focus on those problems that the author and his coauthors have recently worked on. The article also contains an example of a Bishop submanifold in C~3 of real codimension two, which cannot be quadratically flattened at a CR singular point but is CR non-minimal at any CR point. This provides a counter-example to a question asked in a private communication by Zaistev(2013).     

Feasibility testing for systems of real quadratic equations

We consider the problem of deciding whether a given system of quadratic homogeneous equations over the reals has nontrivial solution. We design an algorithm which, for a fixed number of equations, uses a number of arithmetic operations bounded by a polynomial in the number of variables only. A preliminary version of this paper with a weaker result appeared in theProceedings of STOC '92. This research was supported by the Mittag-Leffler Institute and KTH, Stockholm.     

Rank-1 codimension one singularities of positive quadratic differential forms

We complete the study of first-order structural stability at singular points of positive quadratic differencial forms on two manifolds. For this, we consider the generic 1-parameter bifurcation of a D₂₃-singular point. This situation consists in having, before the bifurcation, two locally stable singular points (one of type D₂ and the other of type D₃) which collapse at the D₂₃-singular point when the bifurcation parameter is reached, and afterwards disappear. In local (x,y)-coordinates, such a point appears at the origin of a planar differential equation of the form with (b²-ac)(x,y)?0, such that

(1)

the first jet of the map (a,b,c) at the origin is T₁(a,b,c)(0,0)=(y,0,-y) and

(2)

     

Sums of distinct integral squares in real quadratic fields

We show that the elements of the ring of integers of real quadratic fields which are sums of integral squares are in fact sums of distinct squares, provided their norm is large enough.     

Geometric theory of singularities of solutions of nonlinear differential equations

The article is devoted to singularities of integral manifolds which realize solutions of nonlinear partial differential equations and to the algebraic geometric and topological questions related to them.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 20, pp. 207–247, 1988.     

On quadratic differential systems with equal reflecting functions

For a two-dimensional quadratic system, we obtain necessary conditions for the existence of a triangular quadratic system with the same Mironenko reflecting function as the original system. We suggest an algorithm that permits establishing the coincidence of the reflecting functions of a quadratic nonstationary system and some stationary system.     

Blow-up solutions of quadratic differential systems

The main aim of this paper is to investigate the existence problem for blow-up solutions of quadratic differential systems, Riccati equations, and Lotka-Volterra systems. For this purpose, we introduce the concept of negative and positive blow-up times of solutions of the above-mentioned systems and provide upper or lower bounds for these times. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 15, Differential and Functional Differential Equations. Part 1, 2006.     

Polynomial inverse integrating factors for quadratic differential systems

We characterize all the quadratic polynomial differential systems having a polynomial inverse integrating factor and provide explicit normal forms for such systems and for their associated first integrals. We also prove that these families of quadratic systems have no limit cycles.     

On some boundary value problems for linear generalized differential systems with singularities

For a linear Kurzweil generalized differential system with singularities, we consider two-point boundary value problems of two forms. The singularity is understood in the sense that the matrix and vector functions defining the system may have infinite total variation.     

A multipoint boundary value problem for systems of linear generalized differential equations with singularities

We study a multipoint boundary value problem for systems of Kurzweil generalized linear differential equations with singularities on a finite closed interval of the real line. We assume that the offdiagonal entries of the matrix function corresponding to the system, as well as the elements of the right-hand side of the system, have bounded variation on the entire interval; however, the diagonal entries of the matrix function are not assumed to have bounded variation on the entire interval. This is what we mean by saying that the system is singular. We study the unique solvability of the problem. We prove a general theorem and use it to we obtain efficient optimal (in particular, spectral) solvability conditions for the problem.     

On the resolution of singularities of ordinary differential systems

We show how some differential geometric ideas help to resolve some singularities of ordinary differential systems. Hence a singular problem is replaced by a regular one, which facilitates further analysis of the system. The methods employed are constructive and the regularized systems can also be used for numerical computations. This revised version was published online in June 2006 with corrections to the Cover Date.