Convergence of periodic wavetrains in the limit of large wavelength

The Korteweg-de Vries equation was originally derived as a model for unidirectional propagation of water waves. This equation possesses a special class of traveling-wave solutions corresponding to surface solitary waves. It also has permanent-wave solutions which are periodic in space, the so-called cnoidal waves. A classical observation of Korteweg and de Vries was that the solitary wave is obtained as a certain limit of cnoidal wavetrains.This result is extended here, in the context of the Korteweg-de Vries equation. It is demonstrated that a general class of solutions of the Korteweg-de Vries equation is obtained as limiting forms of periodic solutions, as the period becomes large.     

Probing the electronic structure and chemical bonding of gold oxides and sulfides in AuOn(-) and AuSn(-) (n = 1, 2)

The Au-O and Au-S interactions are essential in nanogold catalysis and nanotechnology, for which monogold oxide and sulfide clusters serve as the simplest molecular models. We report a combined photoelectron spectroscopy and ab initio study on AuO (-) and AuO 2 (-) and their valent isoelectronic AuS (-) and AuS 2 (-) species to probe their electronic structure and to elucidate the Au-O and Au-S chemical bonding. Vibrationally resolved spectra were obtained at different photon energies, providing a wealth of electronic structure information for each species. Similar spectra were observed for AuO (-) and AuS (-) and for the linear OAuO (-) and SAuS (-) species. A bent isomer was also observed as Au(S 2) (-) in the AuS 2 (-) spectra, whereas a similar Au(O 2) (-) complex was not observed in the case of AuO 2 (-). High-level ab initio calculations were conducted to aid spectral assignments and provide insight into the chemical bonding in the AuX (-) and AuX 2 (-) molecules. Excellent agreement is achieved between the calculated electronic excitations and the observed spectra. Configuration interactions and spin-orbit couplings were shown to be important and were necessary to achieve good agreement between theory and experiment. Strong covalent bonding was found in both the AuX (-) and the XAuX (-) species with multiple bonding characters. While Au(S 2) (-) was found to be a low-lying isomer with a significant binding energy, Au(O 2) (-) was shown to be unbound consistent with the experimental observation. The latter is understood in the context of the size-dependent reactivity of Au n (-) clusters with O 2.     

Theoretical exploration of ultrafast spectroscopy of small clusters

We present the ultrafast multistate nuclear dynamics involving adiabatic and nonadiabatic excited states of non-stoichiometric halide deficient clusters (Na_nF_n-1) characterized by strong ionic bonding and one-excess electron for which the “frozen ionic bonds” approximation has been justified allowing to consider the optical response of the single excess electron in the effective field of the other electrons. We combined the Wigner-Moyal representation of the vibronic density matrix with the ab initio multi state molecular dynamics in the ground and excited electronic states including the nonadiabatic couplings calculated “on the fly” at low computational demand. This method allows the simulation of femtosecond pump-probe and pump-dump signals based on an analytical formulation, which utilizes temperature dependent ground state initial conditions, an ensemble of trajectories carried out on the electronic excited state as well as on the ground state after the passage through the conical intersection in the case of nonadiabatic dynamics and for probing either in the cationic state or in the ground state. The choice of the systems we presented has been made in order to determine the timescales of the fast geometric relaxation leaving the bonding frame intact as during the dynamics in the first excited state of Na₄F₃, and of the bond breaking processes leading to conical intersection between the first excited state and the ground state as in Na₃F₂. The former is the smallest finite system prototype for an surface F-center of bulk color centers. The latter allows to study the photo isomerization in full complexity taking into account all degrees of freedom. In the case of Na₄F₃ after the fast geometric relaxation in the excited state leading to deformed cuboidal structure without breaking of bonds, different types of internal vibrational redistribution (IVR) processes have been identified in pump-dump signals by tuning the dump laser. In contrast, from the analysis of the pump-probe signals of Na₃F₂ cluster, the timescales for the metallic and the ionic bond breaking, as well as for the passage through conical intersection have been determined. Finally the conditions under which these processes can be experimentally observed have been identified. Received 22 December 2000     

Fundamental groups and finite sheeted coverings

It is well known that for a connected locally path-connected semi-locally 1-connected space X, there exists a bi-unique correspondence between the pointed d-fold connected coverings and the transitive representations of the fundamental group of X in the symmetric group Σ_d of degree d.The classification problem becomes more difficult if X is a more general space, particularly if X is not locally connected. In attempt to solve the problem for general spaces, several notions of coverings have been introduced, for example, those given by Lubkin or by Fox. On the other hand, different notions of ‘fundamental group’ have appeared in the mathematical literature, for instance, the Brown-Grossman-Quigley fundamental group, the ?ech-Borsuk fundamental group, the Steenrod-Quigley fundamental group, the fundamental profinite group or the fundamental localic group.The main result of this paper determines different ‘fundamental groups’ that can be used to classify pointed finite sheeted connected coverings of a given space X depending on topological properties of X.     

Linear high-resolution schemes for hyperbolic conservation laws: TVB numerical evidence

The Osher–Chakrabarthy family of linear flux-modification schemes is considered. Improved lower bounds on the compression factors are provided, which suggest the viability of using the unlimited version. The LLF flux formula is combined with these schemes in order to obtain efficient finite-difference algorithms. The resulting schemes are applied to a battery of numerical tests, going from advection and Burgers equations to Euler and MHD equations, including the double Mach reflection and the Orszag–Tang 2D vortex problem. Total-variation-bounded (TVB) behavior is evident in all cases, even with time-independent upper bounds. The proposed schemes, however, do not deal properly with compound shocks, arising from non-convex fluxes, as shown by Buckley–Leverett test simulations.     

Magnetic anisotropy engineering in square magnetic elements

Square magnetic elements with side in the 100–500 nm range have been fabricated using the focused ion beam (FIB) milling technique from a 10 nm thick, single-crystal Fe film, epitaxially grown on MgO(0 0 1). Thanks to the good crystal quality of the film, magnetic elements with well-defined magnetocrystalline anisotropy have been prepared, while the fine control of the size and shape of the magnets allows for the effective engineering of the anisotropic behavior of the magnetostatic energy that determines the so-called configurational anisotropy. Micromagnetic calculations and experiments show that the angular dependence of the transverse susceptibility has a strong dependence on the material parameters as well as on the static applied field. This allows the effective engineering of the total anisotropy of the magnets.