Rolling of a non-homogeneous ball over a sphere without slipping and twisting

Consider the problem of rolling a dynamically asymmetric balanced ball (the Chaplygin ball) over a sphere. Suppose that the contact point has zero velocity and the projection of the angular velocity to the normal vector of the sphere equals zero. This model of rolling differs from the classical one. It can be realized, in some approximation, if the ball is rubber coated and the sphere is absolutely rough. Recently, J. Koiller and K. Ehlers pointed out the measure and the Hamiltonian structure for this problem. Using this structure we construct an isomorphism between this problem and the problem of the motion of a point on a sphere in some potential field. The integrable cases are found.      

Titanium nanocarbides: Synthesis and modeling

The present state of research on the production and modeling of nanostructures based on titanium carbide-a typical representative of an extensive class of carbides of d-and f-metals-is reviewed. Methods for the synthesis of various Ti-C nanostructures (molecular clusters, nanocrystallites, nanospheres, nanofibers, nanowires) are examined, and their morphology, atomic structure, and known physicochemical characteristics are described. Theoretical models of the atomic structure and properties of new types of nanostructures in the titanium-carbon system (endo-and exohedral titanofullerenes, “hybrid” structures based on carbon nanotubes, the so-called peapods, nanocables, and a number of others) and the prospects for their application as components of nanoceramics, hydrogen accumulators, materials for spintronics, etc. are discussed. __________ Translated from Teoreticheskaya i éksperimental’naya Khimiya, Vol. 43, No. 1, pp. 1–23, January–February, 2007.     

Infinite-dimensional linear groups with restrictions on subgroups that are not soluble <Emphasis Type="Italic">A</Emphasis><Subscript>3</Subscript>-groups

We are concerned with infinite-dimensional locally soluble linear groups of infinite central dimension that are not soluble A₃-groups and all of whose proper subgroups, which are not soluble A₃-groups, have finite central dimension. The structure of groups in this class is described. The case of infinite-dimensional locally nilpotent linear groups satisfying the specified conditions is treated separately. A similar problem is solved for infinite-dimensional locally soluble linear groups of infinite fundamental dimension that are not soluble A₃-groups and all of whose proper subgroups, which are not soluble A₃-groups, have finite fundamental dimension. __________ Translated from Algebra i Logika, Vol. 46, No. 5, pp. 548–559, September–October, 2007.     

On the existence and the stability of solutions for higher-order semilinear Dirichlet problems

We investigate the existence and stability of solutions for higher-order two-point boundary value problems in case the differential operator is not necessarily positive definite, i.e. with superlinear nonlinearities. We write an abstract realization of the Dirichlet problem and provide abstract existence and stability results which are further applied to concrete problems.     

On potentially <Emphasis Type="Italic">H</Emphasis>-graphic sequences

For given a graph H, a graphic sequence π = (d ₁, d ₂,..., d _n) is said to be potentially H-graphic if there is a realization of π containing H as a subgraph. In this paper, we characterize the potentially (K ₅ − e)-positive graphic sequences and give two simple necessary and sufficient conditions for a positive graphic sequence π to be potentially K ₅-graphic, where K _r is a complete graph on r vertices and K _r-e is a graph obtained from K _r by deleting one edge. Moreover, we also give a simple necessary and sufficient condition for a positive graphic sequence π to be potentially K ₆-graphic. Project supported by National Natural Science Foundation of China (No. 10401010).     

Stability of shock waves in the modified quintic complex Ginzburg-Landau equation

We consider a shock-type wave solution of the modified quintic complex Ginzburg-Landau equation and make a numerical study of its spatiotemporal stability. Discussions related to the behavior of this front wave are introduced and it is shown how the velocities of the wave can be used to collect information concerning the pattern formation in the system. Published in Neliniini Kolyvannya, Vol. 10, No. 2, pp. 270–276, April–June, 2007.     

On the uniform decay for the Korteweg–de Vries equation with weak damping

The aim of this work is to consider the Korteweg–de Vries equation in a finite interval with a very weak localized dissipation namely the H^?1‐norm. Our main result says that the total energy decays locally uniform at an exponential rate. Our analysis improves earlier works on the subject (Q. Appl. Math. 2002; LX (1):111–129; ESAIM Control Optim. Calculus Variations 2005; 11 (3):473–486) and gives a satisfactory answer to a problem suggested in (Q. Appl. Math. 2002; LX (1):111–129). Copyright © 2007 John Wiley & Sons, Ltd.     

Norms on unitizations of banach algebras revisited

Let A be an algebra without unit. If ∥ ∥ is a complete regular norm on A it is known that among the regular extensions of ∥ ∥ to the unitization of A there exists a minimal (operator extension) and maximal (ℓ₁-extension) which are known to be equivalent. We shall show that the best upper bound for the ratio of these two extensions is exactly 3. This improves the results represented by A. K. Gaur and Z. V. Kovářík and later by T. W. Palmer. The second author was partially supported by the grant No. 201/03/0041 of GAČR.     

Continuous dependence on the geometry and on the initial time for a class of parabolic problems I

In this paper, we investigate the continuous dependence on the geometry and the initial time for solutions u( x , t) of a class of nonlinear parabolic initial‐boundary value problems. Copyright © 2007 John Wiley & Sons, Ltd.     

The effect of a parameter in a boundary condition on the stability of difference schemes

A nonlocal difference scheme for the heat conduction equation with a real parameter γ > 1 or γ < 0 in the boundary condition is considered. Thus, the analysis of this problem is generalized to all real values of the parameter in the boundary condition. The spectrum of the spatial operator of the scheme is studied. The results obtained are used to formulate a stability criterion for the difference scheme. The case γ = ?1 proves to be special, because the system of eigenfunctions of the operator does not form a basis in the space of grid functions.