Roughness correction to the Casimir force at short separations: Contact distance and extreme value statistics

So far there has been no reliable method to calculate the Casimir force at separations comparable to the root-mean-square of the height fluctuations of the surfaces. Statistical analysis of rough gold samples has revealed the presence of peaks considerably higher than the root-mean-square roughness. These peaks redefine the minimum separation distance between the bodies and can be described by extreme value statistics. Here we show that the contribution of the high peaks to the Casimir force can be calculated with a pairwise additive summation, while the contribution of asperities with normal height can be evaluated perturbatively. This method provides a reliable estimate of the Casimir force at short distances, and it solves the significant, so far unexplained discrepancy between measurements of the Casimir force between rough surfaces and the results of perturbation theory. Furthermore, we illustrate the importance of our results in a technologically relevant situation.Comment: 29 pages, 11 figures, to appear in Phys. Rev.

Resonances in a spring-pendulum: algorithms for equivariant singularity theory

A spring-pendulum in resonance is a time-independent Hamiltonian model system for formal reduction to one degree of freedom, where some symmetry (reversibility) is maintained. The reduction is handled by equivariant singularity theory with a distinguished parameter, yielding an integrable approximation of the Poincaré map. This makes a concise description of certain bifurcations possible. The computation of reparametrizations from normal form to the actual system is performed by Gröbner basis techniques.

Bifurcation curves of subharmonic solutions

We revisit a problem considered by Chow and Hale on the existence of subharmonic solutions for perturbed systems. In the analytic setting, under more general (weaker) conditions, we prove their results on the existence of bifurcation curves from the nonexistence to the existence of subharmonic solutions. In particular our results apply also when one has degeneracy to first order -- i.e. when the subharmonic Melnikov function vanishes identically. Moreover we can deal as well with the case in which degeneracy persists to arbitrarily high orders, in the sense that suitable generalisations to higher orders of the subharmonic Melnikov function are also identically zero. In general the bifurcation curves are not analytic, and even when they are smooth they can form cusps at the origin: we say in this case that the curves are degenerate as the corresponding tangent lines coincide. The technique we use is completely different from that of Chow and Hale, and it is essentially based on rigorous perturbation theory.Comment: 29 pages, 2 figure

Gauge-invariant description of several (2+1)-dimensional integrable nonlinear evolution equations

We obtain new gauge-invariant forms of two-dimensional integrable systems of nonlinear equations: the Sawada-Kotera and Kaup-Kuperschmidt system, the generalized system of dispersive long waves, and the Nizhnik-Veselov-Novikov system. We show how these forms imply both new and well-known two-dimensional integrable nonlinear equations: the Sawada-Kotera equation, Kaup-Kuperschmidt equation, dispersive long-wave system, Nizhnik-Veselov-Novikov equation, and modified Nizhnik-Veselov-Novikov equation. We consider Miura-type transformations between nonlinear equations in different gauges.Comment: Talk given at the Workshop "Nonlinear Physics: Theory and Experiment. V", Gallipoli (Lecce, Italy), 12-21 June, 200

Dissipative Boussinesq System of Equations in the B\'enard-Marangoni Phenomenon

By using the long-wave approximation, a system of coupled evolution equations for the bulk velocity and the surface perturbations of a B\'enard-Marangoni system is obtained. It includes nonlinearity, dispersion and dissipation, and it can be interpreted as a dissipative generalization of the usual Boussinesq system of equations. As a particular case, a strictly dissipative version of the Boussinesq system is obtained. Finnaly, some speculations are made on the nature of the physical phenomena described by this system of equations.Comment: 15 Pages, REVTEX (Version 3.0), no figure

Die Zukunft der EU-Finanzen: Neue Wege der Finanzierung und der Verteilung?

Nicht zuletzt im Zusammenhang mit dem geplanten Austritt Großbritanniens aus der EU stellt sich die Frage nach neuen Wegen bei der Finanzierung des EU-Budgets und der Verwendung der Mittel. Für Thiess Büttner, Universität Erlangen-Nürnberg, erscheint es sinnvoll, die obsoleten Mehrwertsteuereigenmittel aufzugeben und sich auf die traditionellen Eigenmittel und die bewährten BNE-Eigenmittel zu beschränken. Da die EU weder ausschließlich noch überwiegend europaweite Aufgaben wahrnimmt, ist es nach Ansicht von Michael Broer, Ostfalia Hochschule für angewandte Wissenschaft, Wolfsburg, auch nicht erforderlich, dass sich diese Institution mittels eigener Einnahmen finanziert. Erst wenn sich die EU vermehrt Aufgaben mit einem europäischen Mehrwert zuwende, gewinne der Gedanke an eine EU-Steuer an Bedeutung. Clemens Fuest, ifo Institut, legt dar, dass unter den derzeitigen institutionellen Einrichtungen der EU die Abschaffung der aktuellen Mehrwertsteuereigenmittel und der nationalen Rabatte sowie eine Reform der Ausgaben erhebliche Verbesserungen bringen könnten. Christian Waldhoff, Humboldt-Universität zu Berlin, stellt die Frage, ob es in dem derzeitigen juristischen Rahmen möglich wäre, der EU eigene Besteuerungsbefugnisse zu übertragen. Margit Schratzenstaller, Österreichisches Institut für Wirtschaftsforschung, Wien, führt aus, dass das Potenzial von EU-Steuern als Instrumente zur Stärkung der Nachhaltigkeitsorientierung der Besteuerung in der EU bislang weitgehend vernachlässigt werde und plädiert für die Einführung von »nachhaltigkeitsorientierten steuerbasierten Eigenmittel«. Peter Becker, Stiftung Wissenschaft und Politik, Berlin, sieht das entscheidende Problem der derzeitigen EU-Haushaltspolitik in der einseitigen Fokussierung der Mitgliedstaaten auf eine rein fiskalische Kosten-Nutzen-Bilanz ihrer Mitgliedschaft anhand ihrer nationalen Nettosalden. Jörg Haas, Jacques Delors Institut, Berlin, möchte den Brexit als Chance für eine EU-Haushaltsreform nutzen, die

Modeling the dynamics of glacial cycles

This article is concerned with the dynamics of glacial cycles observed in the geological record of the Pleistocene Epoch. It focuses on a conceptual model proposed by Maasch and Saltzman [J. Geophys. Res.,95, D2 (1990), pp. 1955-1963], which is based on physical arguments and emphasizes the role of atmospheric CO2 in the generation and persistence of periodic orbits (limit cycles). The model consists of three ordinary differential equations with four parameters for the anomalies of the total global ice mass, the atmospheric CO2 concentration, and the volume of the North Atlantic Deep Water (NADW). In this article, it is shown that a simplified two-dimensional symmetric version displays many of the essential features of the full model, including equilibrium states, limit cycles, their basic bifurcations, and a Bogdanov-Takens point that serves as an organizing center for the local and global dynamics. Also, symmetry breaking splits the Bogdanov-Takens point into two, with different local dynamics in their neighborhoods

Aromaticity in a Surface Deposited Cluster: Pd4_4 on TiO2_2 (110)

We report the presence of \sigma-aromaticity in a surface deposited cluster, Pd$_4$ on TiO$_2$ (110). In the gas phase, Pd$_4$ adopts a tetrahedral structure. However, surface binding promotes a flat, \sigma-aromatic cluster. This is the first time aromaticity is found in surface deposited clusters. Systems of this type emerge as a promising class of catalyst, and so realization of aromaticity in them may help to rationalize their reactivity and catalytic properties, as a function of cluster size and composition.Comment: 4 pages, 3 figure