In these papers, classical Morse lemma about canonical form of the function in its nondegenerate critical point is extended to integral functionals of the one-dimensional (papers 1 and 2) and multi-dimensional (paper 3) calculus of variations.
Harmonic balance method uses trigonometric-polynomial approximation for numerical finding of periodic solutions in systems of differential equations. The papers give estimates for the degree of a trigonometric polynomial sufficient to approximate the solution with the accuracy prescribed. A stability criterion for the periodic solution using only information about its approximations, is established.
For a boundary-value problem for an n-th order ordinary differential equation, an algorithm is studied allowing to construct a piecewise constant control with a minimal number of switchings.
A general projection method for approximation of a fixed point of an infinity-dimensional operator A replaces the initial operator PA where P is a projector to a finite-dimensional space. The paper contains an estimate for the dimension of this space sufficient for approximation with the accuracy prescribed.
The book contains a thorough description of numerical procedures used for approximation of periodic solutions of systems of differential equations.
A notion of Poisson bracket compatible with a structure of an associative algebra is studied. Such brackets are always quadratic; their relation to the classical Yang--Baxter equation is studied. A description of Poisson Lie structures on groups whose Lie algebras come from associative ones is obtained. Paper 10 contains general theory, and paper 9, description of a particular case (algebra of quaternions).
Connected components of the space of functions without critical points on a 2-manifold, taking only values 0 and 1 on the boundary. Paper 11 solves the problem for 2-manifolds of special type.
Smooth immersed curves on the plane are classified by a single invariant, the winding number. This number is given by an explicit integral formula; for generic curves it is equal to the sum of signs of its self-intersection points. The paper contains an analogous integral formula (proved using asymptotic expansion of certain Laplace integrals) for immersions of the n-sphere into R^{2n}. This allows to write an explicit formula for generator of the n-th cohomology group of Stiefel variety V(n,2n).
Generalization of the Whitney's formula for a plane curve. The Whitney's equality is split into a series of identities possessing independent geometrical meanings. The formula is generalized to the curves on a 2-torus. (Article)
Describes a way to code parking functions by pairs of permutations satisfying certain admissibility condition. A hypothesis relating this construction to a basis of M.Haiman's module of diagonal harmonics is formulated. (Article)
Generalization of the Cayley's matrix-tree theorem: the number (more precisely, the statistical sum) of subgraphs of a given graph having fixed 2-core is expressed via Cayley matrix of the graph. ( Article)
The standard module of the rational Cherednik algebra is studied, especially for the Cherednik algebra of the dihedral group. We introduce the notion of a quasi-harmonic and compute the characteristic polynomial of the Frobenius (apolar) algebra generated by them.
To every factorization of a cycle into transpositions we assign a monomial in variables w_{ij} retaining information about the transpositions used, but not about their order. Summing over all possible factorizations of n-cycles we obtain a generating function generalizing «polynomial» Hurzitz numbers. The functions happens to admit a closed expression. From this expression we deduce a formula for the number of 1-faced embeddings of a given graph.
If a smooth mapping of a Hilbert space to itself does not decrease distances between points and has nondegenerate differential in some point then its image is the whole space.
The book contains description representations of various infinity-dimensional Lie algebras (Heisenberg algebra, Virasoro algebra, algebras of semiinfinite matrices, etc.) in the space of semiinfinite forms. Vertex operators and relations between them are introduced and studied. Applications of the theory to algebra and combinatorics are explained.