Weakly-nonlocal Symplectic Structures, Whitham method, and weakly-nonlocal Symplectic Structures of Hydrodynamic Type

We consider the special type of the field-theoretical Symplectic structures called weakly nonlocal. The structures of this type are in particular very common for the integrable systems like KdV or NLS. We introduce here the special class of the weakly nonlocal Symplectic structures which we call the weakly nonlocal Symplectic structures of Hydrodynamic Type. We investigate then the connection of such structures with the Whitham averaging method and propose the procedure of "averaging" of the weakly nonlocal Symplectic structures. The averaging procedure gives the weakly nonlocal Symplectic Structure of Hydrodynamic Type for the corresponding Whitham system. The procedure gives also the "action variables" corresponding to the wave numbers of $m$-phase solutions of initial system which give the additional conservation laws for the Whitham system.Comment: 64 pages, Late

An update on the Hirsch conjecture

The Hirsch conjecture was posed in 1957 in a letter from Warren M. Hirsch to George Dantzig. It states that the graph of a d-dimensional polytope with n facets cannot have diameter greater than n - d. Despite being one of the most fundamental, basic and old problems in polytope theory, what we know is quite scarce. Most notably, no polynomial upper bound is known for the diameters that are conjectured to be linear. In contrast, very few polytopes are known where the bound $n-d$ is attained. This paper collects known results and remarks both on the positive and on the negative side of the conjecture. Some proofs are included, but only those that we hope are accessible to a general mathematical audience without introducing too many technicalities.Comment: 28 pages, 6 figures. Many proofs have been taken out from version 2 and put into the appendix arXiv:0912.423

New and simple algorithms for stable flow problems

Stable flows generalize the well-known concept of stable matchings to markets in which transactions may involve several agents, forwarding flow from one to another. An instance of the problem consists of a capacitated directed network, in which vertices express their preferences over their incident edges. A network flow is stable if there is no group of vertices that all could benefit from rerouting the flow along a walk. Fleiner established that a stable flow always exists by reducing it to the stable allocation problem. We present an augmenting-path algorithm for computing a stable flow, the first algorithm that achieves polynomial running time for this problem without using stable allocation as a black-box subroutine. We further consider the problem of finding a stable flow such that the flow value on every edge is within a given interval. For this problem, we present an elegant graph transformation and based on this, we devise a simple and fast algorithm, which also can be used to find a solution to the stable marriage problem with forced and forbidden edges. Finally, we study the stable multicommodity flow model introduced by Kir\'{a}ly and Pap. The original model is highly involved and allows for commodity-dependent preference lists at the vertices and commodity-specific edge capacities. We present several graph-based reductions that show equivalence to a significantly simpler model. We further show that it is NP-complete to decide whether an integral solution exists

From Fe3O4/NiO bilayers to NiFe2O4-like thin films through Ni interdiffusion

Ferrites with (inverse) spinel structure display a large variety of electronic and magnetic properties, making some of them interesting for potential applications in spintronics. We investigate the thermally induced interdiffusion of Ni$^{2+}$ ions out of NiO into Fe$_3$O$_4$ ultrathin films, resulting in off-stoichiometric nickel ferrite–like thin layers. We synthesized epitaxial Fe$_3$O$_4$ bilayers on Nb-doped SrTiO$_3$(001) substrates by means of reactive molecular beam epitaxy. Subsequently, we performed an annealing cycle comprising three steps at temperatures of 400$^\circ$C, 600$^\circ$C, and 800$^\circ$C under an oxygen background atmosphere. We studied the changes of the chemical and electronic properties as result of each annealing step with help of hard x-ray photoelectron spectroscopy and found a rather homogeneous distribution of Ni and Fe cations throughout the entire film after the overall annealing cycle. For one sample we observed a cationic distribution close to that of the spinel ferrite NiFe$_2$O$_4$. Further evidence comes from low-energy electron diffraction patterns indicating a spinel-type structure at the surface after annealing. Site- and element-specific hysteresis loops performed by x-ray magnetic circular dichroism uncovered the antiferrimagnetic alignment between the octahedral coordinated Ni$^{2+}$ and Fe$^{3+}$ ions and the Fe$^{3+}$ ion in tetrahedral coordination. We find a quite low coercive field of 0.02 T, indicating a rather low defect concentration within the thin ferrite films

A reconstruction of the initial conditions of the Universe by optimal mass transportation

Reconstructing the density fluctuations in the early Universe that evolved into the distribution of galaxies we see today is a challenge of modern cosmology [ref.]. An accurate reconstruction would allow us to test cosmological models by simulating the evolution starting from the reconstructed state and comparing it to the observations. Several reconstruction techniques have been proposed [8 refs.], but they all suffer from lack of uniqueness because the velocities of galaxies are usually not known. Here we show that reconstruction can be reduced to a well-determined problem of optimisation, and present a specific algorithm that provides excellent agreement when tested against data from N-body simulations. By applying our algorithm to the new redshift surveys now under way [ref.], we will be able to recover reliably the properties of the primeval fluctuation field of the local Universe and to determine accurately the peculiar velocities (deviations from the Hubble expansion) and the true positions of many more galaxies than is feasible by any other method. A version of the paper with higher-quality figures is available at http://www.obs-nice.fr/etc7/nature.pdfComment: Latex, 4 pages, 3 figure

Electrical resistivity, magnetism and electronic structure of the intermetallic 3d/4f Laves phase compounds ErNi2Mnx

The non-stoichiometric intermetallic compounds RENi2Mnx (RE = rare earth) with the cubic MgCu2-type structure display a large variety of magnetic properties which is due to a complex interplay between the degrees of freedom of the 3d and 4f electrons and their interactions. We performed a comprehensive study of the electrical resistivity, magnetic properties and the electronic structure of ErNi2Mnx (x =0, 0.25, 0.5, 0.75, 1, 1.25) compounds by employing a suitable set of complementary experimental approaches. We find an increase in electrical resistance compared to ErNi2 upon Mn doping, the residual resistivity ratio decreases with increasing manganese content. The Curie temperature exhibits a sharp increase to around 50 K for Mn concentrations x 0.5, whereas the saturation magnetization decreases with growing Mn content x 0.5. Valence band X-ray photoelectron spectroscopy reveals an increasing intensity of Mn 3d states near Fermi energy in dependence of Mn concentration and Curie temperature. Resonant photoelectron spectroscopy of ErNi2Mn0.75 reveals that the photoemission decay channels dominate the valence band spectra across the Er N5 and Mn L3 X-ray absorption maxima, whereas the L3VV Auger dictates the resonant valence band spectra close to and at the Ni L3 X-ray absorption edge