S. P. Novikov

Bio: S. P. Novikov is an academic researcher from Russian Academy of Sciences. The author has contributed to research in topics: Riemann surface & Dynamical systems theory. The author has an hindex of 41, co-authored 206 publications receiving 9898 citations. Previous affiliations of S. P. Novikov include Landau Institute for Theoretical Physics & University of Maryland, College Park.

Modern geometry--methods and applications

Abstract: The first € price and the £ and $ price are net prices, subject to local VAT. Prices indicated with * include VAT for books; the €(D) includes 7% for Germany, the €(A) includes 10% for Austria. Prices indicated with ** include VAT for electronic products; 19% for Germany, 20% for Austria. All prices exclusive of carriage charges. Prices and other details are subject to change without notice. All errors and omissions excepted. B.A. Dubrovin, A.T. Fomenko, S.P. Novikov Modern Geometry—Methods and Applications

Non-linear equations of korteweg-de vries type, finite-zone linear operators, and abelian varieties

Abstract: The basic content of this survey is an exposition of a recently developed method of constructing a broad class of periodic and almost-periodic solutions of non-linear equations of mathematical physics to which (in the rapidly decreasing case) the method of the inverse scattering problem is applicable. These solutions are such that the spectrum of their associated linear differential operators has a finite-zone structure. The set of linear operators with a given finite-zone spectrum is the Jacobian variety of a Riemann surface, which is determined by the structure of the spectrum. We give an explicit solution of the corresponding non-linear equations in the language of the theory of Abelian functions.

Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory

Abstract: CONTENTS Introduction Chapter I. Hamiltonian theory of systems of hydrodynamic type § 1. General properties of Poisson brackets § 2. Hamiltonian formalism of systems of hydrodynamic type and Riemannian geometry § 3. Generalizations: differential-geometric Poisson brackets of higher orders, differential-geometric Poisson brackets on a lattice, and the Yang-Baxter equation § 4. Riemann invariants and the Hamiltonian formalism of diagonal systems of hydrodynamic type. Novikov's conjecture. Tsarev's theorem. The generalized hodograph method Chapter II. Equations of hydrodynamics of soliton lattices § 5. The Bogolyubov-Whitham averaging method for field-theoretic systems and soliton lattices. The results of Whitham and Hayes for Lagrangian systems § 6. The Whitham equations of hydrodynamics of weakly deformed soliton lattices for Hamiltonian field-theoretic systems. The principle of conservation of the Hamiltonian structure under averaging § 7. Modulations of soliton lattices of completely integrable evolutionary systems. Krichever's method. The analytic solution of the Gurevich-Pitaevskii problem on the dispersive analogue of a shock wave § 8. Evolution of the oscillatory zone in the KdV theory. Multi-valued functions in the hydrodynamics of soliton lattices. Numerical studies § 9. Influence of small viscosity on the evolution of the oscillatory zone References

The Hamiltonian formalism and a many-valued analogue of Morse theory

Abstract: CONTENTS Introduction § 1. The Hamiltonian formalism. Simplest examples. Systems of Kirchhoff type. Factorization of the Hamiltonian formalism for the B-phase of 3He § 2. The Hamiltonian formalism of systems of hydrodynamic origin § 3. What is Morse (LSM) theory? § 4. Equations of Kirchhoff type and the Dirac monopole § 5. Many-valued functional and an analogue of Morse theory. The periodic problem for equations of Kirchhoff type. Chiral fields in an external field § 6. Many-valued functions on finite-dimensional manifolds. An analogue of Morse theory References

Geodesic active contours

Abstract: A novel scheme for the detection of object boundaries is presented. The technique is based on active contours deforming according to intrinsic geometric measures of the image. The evolving contours naturally split and merge, allowing the simultaneous detection of several objects and both interior and exterior boundaries. The proposed approach is based on the relation between active contours and the computation of geodesics or minimal distance curves. The minimal distance curve lays in a Riemannian space whose metric as defined by the image content. This geodesic approach for object segmentation allows to connect classical "snakes" based on energy minimization and geometric active contours based on the theory of curve evolution. Previous models of geometric active contours are improved as showed by a number of examples. Formal results concerning existence, uniqueness, stability, and correctness of the evolution are presented as well. >

Geodesic Active Contours

Abstract: A novel scheme for the detection of object boundaries is presented. The technique is based on active contours evolving in time according to intrinsic geometric measures of the image. The evolving contours naturally split and merge, allowing the simultaneous detection of several objects and both interior and exterior boundaries. The proposed approach is based on the relation between active contours and the computation of geodesics or minimal distance curves. The minimal distance curve lays in a Riemannian space whose metric is defined by the image content. This geodesic approach for object segmentation allows to connect classical “snakes” based on energy minimization and geometric active contours based on the theory of curve evolution. Previous models of geometric active contours are improved, allowing stable boundary detection when their gradients suffer from large variations, including gaps. Formal results concerning existence, uniqueness, stability, and correctness of the evolution are presented as well. The scheme was implemented using an efficient algorithm for curve evolution. Experimental results of applying the scheme to real images including objects with holes and medical data imagery demonstrate its power. The results may be extended to 3D object segmentation as well.

Flatness and defect of non-linear systems: introductory theory and examples

Abstract: We introduce flat systems, which are equivalent to linear ones via a special type of feedback called endogenous. Their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of Kalman's controllability. The distance to flatness is measured by a non-negative integer, the defect. We utilize differential algebra where flatness- and defect are best defined without distinguishing between input, state, output and other variables. Many realistic classes of examples are flat. We treat two popular ones: the crane and the car with n trailers, the motion planning of which is obtained via elementary properties of plane curves. The three non-flat examples, the simple, double and variable length pendulums, are borrowed from non-linear physics. A high frequency control strategy is proposed such that the averaged systems become flat.

Differentiable dynamical systems

Abstract: This is a survey article on the area of global analysis defined by differentiable dynamical systems or equivalently the action (differentiable) of a Lie group G on a manifold M. An action is a homomorphism G→Diff(M) such that the induced map G×M→M is differentiable. Here Diff(M) is the group of all diffeomorphisms of M and a diffeo- morphism is a differentiable map with a differentiable inverse. Everything will be discussed here from the C ∞ or C r point of view. All manifolds maps, etc. will be differentiable (C r , 1 ≦ r ≦ ∞) unless stated otherwise.