Two observations about free graded hopf algebras

This paper records two results about graded Hopf algebras that do not appear to be stated explicitly in the literature. Let B be a graded set, graded by the positive integers. Let V be the graded vector space with basis B over a field K of characteristic zero and V^'=K⊕V, where K is in grading zero. Let L ne the free graded Lie algebra on B over K and let T be the free graded tensor algebra on B. The first result is the "graded Witt formula" giving the dimension of the subspace of L in each grading. The second result is the observation that any graded coassociative, co-commutative comultiplication Δ:V^'→V^'⊗V^', with co-unit the projection V¹→K. extends to a graded Hopf algebra structure on T that is in fact isomorphic to the natural graded Hopf algebra structure on T. In the ungraded case the statement analogous to the second result is false.     

Stability of solutions to the Cauchy problem for a plane hyperbolic system with time-periodic coefficients

Considering a plane hyperbolic system with time-periodic coefficients, we construct a version of the direct Lyapunov method with the condition on the derivative of the Lyapunov functional along the trajectories of the system which is weakened by use of periodicity of the coefficients. We exhibit an illustrative example.     

Partition of Euclidean space into polyhedra induced by a small deformation of the densest lattice sphere packing

The number of combinatorially nonequivalent Dirichlet-Voronoi diagrams constructed for the centers of balls in the packing obtained from the densest lattice packing of equal spheres by a small displacement of the spheres is estimated. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 5, pp. 79–84, 2005.     

Convergence of optimal controls in some optimization problems

We consider the behavior of bundles of optimal controls when the initial state of the system goes to some given vector. We investigate three types of optimization problems: problems with fixed process length and a free right endpoint; problems with fixed process length and a fixed right endpoint; time-optimal problems. The article is a review of recent results obtained by the author. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 4, pp. 301–314, 2004.     

Colorings of spaces,and random graphs

This work deals with some problems on the embeddings of finite geometric graphs into the random ones. In particular, we study here applications of the random graph theory to the Nelson-Erdös-Hadwiger problem on coloring spaces.     

Necessary first-order conditions for optimal crossing of a given region

We consider a nonlinear optimal control problem with an integral functional in which the integrand is the characteristic function of a closed set in the phase space. An approximation method is applied to prove the necessary conditions of optimality in the form of a Pontryagin maximum principle without any prior assumptions on the behavior of the optimal trajectory. Similarly to phase-constrained problems, we derive conditions of nondegeneracy and pointwise nontriviality of the maximum principle. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 4, pp. 179–204, 2004.     

Magnetic structure and transport properties of atomically disordered crystals of two-dimensional manganites La2−2x Sr1+2x Mn2O7

The magnetic structure and transport properties of partially disordered crystals of two-dimensional manganites La_2?2x Sr_1+2x Mn₂O₇ (x = 0.3, 0.4) are studied over a wide range of temperatures. The crystals are transformed into an atomically disordered state under irradiation with fast neutrons at a dose of 2 × 10¹⁹ cm^?2. The average concentration of substitutional defects in the crystal is ≈4%. It is found that substitutional defects are responsible for the transition of these manganites from the ferromagnetic metal state to the insulator state with a spin glass structure. The results obtained are discussed in terms of the ratio between the kinetic energy of charge carriers and the exchange energy of localized spins.     

Anomalous electric-field effect and glassy behaviour in granular aluminium thin films: electron glass?

We present a study of non-equilibrium phenomena observed in the electrical conductance of insulating granular aluminium thin films. An anomalous field effect and its slow relaxation are studied in some detail. The phenomenology is very similar to the one already observed in indium oxide. The origin of the phenomena is discussed. In granular systems, the present experiments can naturally be interpreted along two different lines. One relies on a slow polarisation in the dielectric surrounding the metallic islands. The other one relies on a purely electronic mechanism: the formation of an electron Coulomb glass in the granular metal. More selective experiments and/or quantitative predictions about the Coulomb glass properties are still needed to definitely distinguish between the two scenarios.     

A model for the interaction of near-infrared laser pulses with metal powders in selective laser sintering

A thermal model of the interaction of pulsed near-infrared laser radiation from a Nd:YAG laser was made, taking the measured powder properties such as reflectance, optical penetration depth and thermal conductivity into account. It allows an estimation of the evolution of two different temperatures: the average temperature of the powder (taken over the grains in a volume given by the laser beam diameter and the optical penetration depth) and the temperature distinction within a single grain. It showed that in pulsed mode consolidation can be achieved at much lower average power as the surface of the powder particles are molten but their cores remain at nearly room temperature. This leads to a much lower average temperature and therefore a dramatic decrease in residual thermal stresses in the finished piece. The results of the model were experimentally tested and confirmed. Received: 26 July 2001 / Accepted: 23 November 2001 / Published online: 23 January 2002     

Geometrically qualified ESPI vibration analysis of an engine

This paper presents an application in the automotive industry where a combination of electronic speckle-pattern interferometry and laser doppler velocimetry were used at a critical stage in the design process of an internal combustion engine. Combined deformation and surface relief measurements were used to study the phase and amplitude of deformation of a vibrating engine. The relief data was combined with the interferometer geometry and used to geometrically correct the deformation data, in an effort to improve accuracy. The measurements allowed rapid identification and quantification of design weaknesses, particularly those causing undesirable resonances. This led to a significant reduction in the design time and lowering of costs, when compared with existing design optimisation methods.