In the last 30 days

• Open/close On the efficiency of the Orthogonal Matching Pursuit in compressed sensing

  Evgenii D Livshits 2012 Sb. Math. 203 183 Tag this article

  Article References Full text PDF (286 KB)

  The paper shows that if a matrix has the restricted isometry property (RIP) of order with isometry constant and if its coherence is less than , then the Orthogonal Matching Pursuit (the Orthogonal Greedy Algorithm) is capable to exactly recover an arbitrary -sparse signal from the compressed sensing in at most iterations. As a result, an arbitrary -sparse signal can be recovered by the Orthogonal Matching Pursuit from measurements.

  Bibliography: 23 titles.

• Open/close Asymptotic behaviour of an eigenvalue in the continuous spectrum of a narrowed waveguide

  Giuseppe Cardone et al 2012 Sb. Math. 203 153 Tag this article

  Article References Full text PDF (530 KB)

  The existence of an eigenvalue embedded in the continuous spectrum is proved for the Neumann problem for Helmholtz's equation in a two-dimensional waveguide with two outlets to infinity which are half-strips of width and , where is a small parameter. The width function of the part of the waveguide connecting these outlets is of order ; it is defined as a linear combination of three fairly arbitrary functions, whose coefficients are obtained from a certain nonlinear equation. The result is derived from an asymptotic analysis of an auxiliary object, the augmented scattering matrix.

  Bibliography: 29 titles.

• Open/close Parity in knot theory

  Vassily O Manturov 2010 Sb. Math. 201 693 Tag this article

  Article References Full text PDF (540 KB)

  In this work we study knot theories with a parity property for crossings: every crossing is declared to be even or odd according to a certain preassigned rule. If this rule satisfies a set of simple axioms related to the Reidemeister moves, then certain simple invariants solving the minimality problem can be defined, and invariant maps on the set of knots can be constructed.

  The most important example of a knot theory with parity is the theory of virtual knots. Using the parity property arising from Gauss diagrams we show that even a gross simplification of the theory of virtual knots, namely, the theory of free knots, admits simple and highly nontrivial invariants. This gives a solution to a problem of Turaev, who conjectured that all free knots are trivial.

  In this work we show that free knots are generally not invertible, and provide invariants which detect the invertibility of free knots.

  The passage to ordinary virtual knots allows us to strengthen known invariants (such as the Kauffman bracket) using parity considerations.

  We also discuss other examples of knot theories with parity.

  Bibliography: 27 items.

• Open/close Asymptotic behaviour of the first moment of the number of steps in the by-excess and by-deficiency Euclidean algorithms

  Dmitrii A Frolenkov 2012 Sb. Math. 203 288 Tag this article

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  The first moments for the number of steps in different Euclidean algorithms are considered. For these moments asymptotic formulae with new remainder terms are obtained using refined estimates for sums of fractional parts and some ideas in Selberg's elementary proof of the prime number theorem.

  Bibliography: 12 titles.

• Open/close The Dirichlet problem for a second-order elliptic equation with an L_p boundary function

  Anatolii K Gushchin 2012 Sb. Math. 203 1 Tag this article

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  We consider a Dirichlet problem in which the boundary value of a solution is understood as the -limit, , of traces of this solution on surfaces 'parallel' to the boundary. We suggest a setting of this problem which (in contrast to the notion of solution in ) enables us to study the solvability of the problem without making smoothness assumptions on the coefficients inside the domain. In particular, for an equation in selfadjoint form without lower-order terms, under the same conditions as those used for , we prove unique solvability and establish a bound for an analogue of the area integral.

  Bibliography: 37 titles.

• Open/close Dynamics of spatially chaotic solutions of parabolic equations

  A V Babin 1995 Sb. Math. 186 1389 Tag this article

  Article References Full text PDF (1.38 MB)

  We study parabolic systems with a potential non-linearity with one or many spatial variables. We describe a rather general and stable mechanism explaining the appearance and preservation of complicated stable spatial forms. The main idea consists in a description of the complexity of a solution in terms of its homotopy class. This class is a discrete-valued preserved quantity. The number of homotopy inequivalent solutions depends exponentially on the parameters of the equation. In our paper we discuss the connections between the dynamics of the solutions of parabolic systems with a complicated spatial structure and the properties of the Riemannian metric on the configuration space generated by the Jacobian variational functional. The relationships between the lengths of the geodesics are reflected in the complexity of the spatial forms and in such dynamical properties as the attraction and repulsion of solitons.
• Open/close Classification of singularities in the problem of motion of the Kovalevskaya top in a double force field

  Pavel E Ryabov and Mikhail P Kharlamov 2012 Sb. Math. 203 257 Tag this article

  Article References Full text PDF (714 KB)

  The problem of motion of the Kovalevskaya top in a double force field is investigated (the integrable case of A.G. Reyman and M.A. Semenov-Tian-Shansky without a gyrostatic momentum). It is a completely integrable Hamiltonian system with three degrees of freedom not reducible to a family of systems with two degrees of freedom. The critical set of the integral map is studied. The critical subsystems and bifurcation diagrams are described. The classification of all nondegenerate critical points is given. The set of these points consists of equilibria (nondegenerate singularities of rank 0), of singular periodic motions (nondegenerate singularities of rank 1), and also of critical two-frequency motions (nondegenerate singularities of rank 2).

  Bibliography: 32 titles.

• Open/close Parity and cobordism of free knots

  Vassily O Manturov 2012 Sb. Math. 203 196 Tag this article

  Article References Full text PDF (533 KB)

  A simple invariant is constructed which obstructs a free knot to be truncated. In particular, this invariant provides an obstruction to the truncatedness of curves immersed in two-dimensional surfaces. A curve on an oriented two-dimensional surface  is referred to as truncated (null-cobordant) if there exists a three-dimensional manifold  with boundary  and a smooth proper map of a two-disc to  such that the image of the boundary of the disc coincides with the curve.

  The problem of truncatedness for free knots is solved in this paper using the notion of parity recently introduced by the author.

  Bibliography: 12 titles.