Analysis of the transition from normal modes to local modes in a system of two harmonically coupled Morse oscillators

Summary The system consisting of two Morse oscillators coupled via either a potential or a kinetic quadratic term is considered. The corresponding classical equations of motion have been numerically integrated and the initial conditions have been systematically analyzed in the regime of low total excitation energy of the system. Particular attention was paid to the full characterization of an intermediate type of motion, herein called transition mode, which appears at total energy values in between those typical of normal modes and those where local and normal modes coexist. A previously proposed perturbative approach (Jaffé C, Brumer P (1980) J Chem Phys 73:5646) is reanalyzed and compared with the results of numerical experiments, with the purpose of lending further support to the existence of transition modes.     

Lower Bounds for the Nodal Length of Eigenfunctions of the Laplacian

We prove lower bounds for the length of the zero set of aneigenfunction of the Laplace operator on a Riemann surface; inparticular, in non-negative curvature, or when the associated eigenvalueis large, we give a lower bound which involves only the square root ofthe eigenvalue and the area of the manifold (modulo a numericalconstant, this lower bound is sharp).     

Phase coexistence in Ising, Potts and percolation models

We study phase coexistence (separation) phenomena in Ising, Potts and random cluster models in dimensions d3 below the critical temperature. The simultaneous occurrence of several phases is typical for systems with appropriately arranged (mixed) boundary conditions or for systems satisfying certain physically natural constraints (canonical ensembles). The various phases emerging in these models define a partition, called the empirical phase partition, of the space. Our main results are large deviations principles for (the shape of) the empirical phase partition. More specifically, we establish a general large deviation principle for the partition induced by large (macroscopic) clusters in the Fortuin–Kasteleyn model and transfer it to the Ising–Potts model where we obtain a large deviation principle for the empirical phase partition induced by the various phases. The rate function turns out to be the total surface free energy (associated with the surface tension of the model and with boundary conditions) which can be naturally assigned to each reasonable partition. These LDP-s imply a weak law of large numbers: asymptotically, the law of the phase partition is determined by an appropriate variational problem. More precisely, the empirical phase partition will be close to some partition which is compatible with the constraints imposed on the system and which minimizes the total surface free energy. A general compactness argument guarantees the existence of at least one such minimizing partition. Our results are valid for temperatures T below a limit of slab-thresholds conjectured to agree with the critical point T_c. Moreover, T should be such that there exists only one translation invariant infinite volume state in the corresponding Fortuin–Kasteleyn model; a property which can fail for at most countably many values and which is conjectured to be true for every T≠T_c.     

Some Remarks about the FM-partners of <Emphasis Type="Italic">K</Emphasis>3 Surfaces with Picard Numbers 1 and 2

In this paper we describe some results about K3 surfaces with Picard number 1 and 2. In particular, we give a new simple proof of a theorem due to Oguiso which shows that, given an integer N, there is a K3 surface with Picard number 2 and at least N non-isomorphic FM-partners. We describe also the Mukai vectors of the moduli spaces associated to the FM-partners of K3 surfaces with Picard number 1.     

Exotic smooth structures and symplectic forms on closed manifolds

In this paper we discuss relations between symplectic forms and smooth structures on closed manifolds. Our main motivation is the problem if there exist symplectic structures on exotic tori. This is a symplectic generalization of a problem posed by Benson and Gordon. We give a short proof of the (known) positive answer to the original question of Benson and Gordon that there are no Kähler structures on exotic tori. We survey also other related results which give an evidence for the conjecture that there are no symplectic structures on exotic tori.     

A method for solving problems on the limit equilibrium of reinforced shells of revolution

Rigid-plastic reinforced shells of revolution with a piecewise linear condition of plasticity are considered. It is shown that, in solving problems on their limit equilibrium, the application of linearized yield surfaces or the approximation of derivatives by finite differences restricts the set of possible solutions. In this paper, an asymptotic method for solving the problems by constructing a convergent sequence of solutions is offered. Each of these solutions is constructed numerically, and to approximate the derivatives, special finite differences coordinated with suppositions of the theory of thin shells are used. A feature of this method is that, with piecewise smooth yield surfaces, it is not necessary to determine a sequence of various plastic states, because the approximating yield surfaces are constructed during solution of the problem. Shells of revolution with positive and negative Gaussian curvatures and compound constructions of shells with various structures of reinforcement are examined. It is shown that the junction boundaries of rigid and plastic regions and the sequence of realization of plastic hinges greatly depend on the accuracy of approximation of the surfaces. With these approximations tending to the true yield surface, the sizes of the rigid regions decrease, and the range of structural and geometrical parameters of the shells grows when the yield state is reached through out their span. It is noted that, for closed constructions of shells reinforced only with spiral fibers at placement angles less that 55°, all possible mechanisms of plastic flow correspond to the direction of operating forces, whereas for other reinforcement structures, mechanisms of plastic flow with the opposite direction of velocities are possible. Translated from Mekhanika Kompozitnykh Materialov, Vol. 44, No. 5, pp. 613–632, September–October, 2008.     

Mean-field theory for polymer adsorption on curved surfaces

We discuss the influence of polymer adsorption on the curvature energy of an interface. Following an article by Clement and Joanny (J. Phys. II 7, 973 (1997)), a mean-field theory is used to calculate the surface tension, rigidity constants and spontaneous curvature associated with both reversible and irreversible polymer adsorption. In the case of irreversible polymer adsorption it is assumed that the amount of adsorbed polymer remains constant upon curving the interface. Unfortunately, constraining the amount of polymer by adding a Lagrange multiplier affects the thermodynamic state of the (free) polymer far away from the interface. Clement and Joanny solve this problem by removing the polymers in the bulk. We allow for the presence of free polymers, but to achieve this we have to apply a local external field to keep the adsorbed amount fixed. The results of the two approaches are compared and a physical interpretation is given. Received 25 July 2001 and Received in final form 5 December 2001     

Ultrafast-laser-assisted chemical restructuring of silicon and germanium surfaces

This article reports a comparative study on texturing in silicon and germanium surfaces after exposure to femtosecond laser irradiation in the gaseous environments of sulfur hexafluoride (SF₆) and hydrogen chloride (HCl). The surface texturing results from the combined effect of laser-assisted chemical etching and laser ablation. Optimized processing conditions have produced features on the order of nanometers in size. We demonstrate for the first time that regular conical pillars can be formed in Ge and that HCl can be used to form regular conical pillars in Si.     

Desorption of chlorine atoms on Si (1 1 1)-(7 × 7) surfaces induced by hole injection from scanning tunneling microscope tips

We investigated desorption of chlorine atoms on Si (1 1 1)-(7 × 7) surfaces induced by hole injection from scanning tunneling microscope tips. The hole-induced desorption of chlorine atoms had a threshold bias voltage corresponding to the energy position of the S3 surface band originated in Si backbonds. The chlorine atom desorption rate was almost proportional to the square of the tunneling current. We have discussed possible mechanisms that two holes injected into Si surface states get localized at the backbonds of chlorinated Si adatoms, which induces the rupture of Cl-Si bonds to result in chlorine atom desorption.