Branch cuts of Stokes wave on deep water. Part I: Numerical solution and Pad\'e approximation

Complex analytical structure of Stokes wave for two-dimensional potential flow of the ideal incompressible fluid with free surface and infinite depth is analyzed. Stokes wave is the fully nonlinear periodic gravity wave propagating with the constant velocity. Simulations with the quadruple and variable precisions are performed to find Stokes wave with high accuracy and study the Stokes wave approaching its limiting form with $2\pi/3$ radians angle on the crest. A conformal map is used which maps a free fluid surface of Stokes wave into the real line with fluid domain mapped into the lower complex half-plane. The Stokes wave is fully characterized by the complex singularities in the upper complex half-plane. These singularities are addressed by rational (Pad\'e) interpolation of Stokes wave in the complex plane. Convergence of Pad\'e approximation to the density of complex poles with the increase of the numerical precision and subsequent increase of the number of approximating poles reveals that the only singularities of Stokes wave are branch points connected by branch cuts. The converging densities are the jumps across the branch cuts. There is one branch cut per horizontal spatial period $\lambda$ of Stokes wave. Each branch cut extends strictly vertically above the corresponding crest of Stokes wave up to complex infinity. The lower end of branch cut is the square-root branch point located at the distance $v_c$ from the real line corresponding to the fluid surface in conformal variables. The limiting Stokes wave emerges as the singularity reaches the fluid surface. Tables of Pad\'e approximation for Stokes waves of different heights are provided. These tables allow to recover the Stokes wave with the relative accuracy of at least $10^{-26}$. The tables use from several poles to about hundred poles for highly nonlinear Stokes wave with $v_c/\lambda\sim 10^{-6}.$Comment: 38 pages, 9 figures, 4 tables, supplementary material

On dispersive energy transport and relaxation in the hopping regime

A new method for investigating relaxation phenomena for charge carriers hopping between localized tail states has been developed. It allows us to consider both charge and energy {\it dispersive} transport. The method is based on the idea of quasi-elasticity: the typical energy loss during a hop is much less than all other characteristic energies. We have investigated two models with different density of states energy dependencies with our method. In general, we have found that the motion of a packet in energy space is affected by two competing tendencies. First, there is a packet broadening, i.e. the dispersive energy transport. Second, there is a narrowing of the packet, if the density of states is depleting with decreasing energy. It is the interplay of these two tendencies that determines the overall evolution. If the density of states is constant, only broadening exists. In this case a packet in energy space evolves into Gaussian one, moving with constant drift velocity and mean square deviation increasing linearly in time. If the density of states depletes exponentially with decreasing energy, the motion of the packet tremendously slows down with time. For large times the mean square deviation of the packet becomes constant, so that the motion of the packet is ``soliton-like''.Comment: 26 pages, RevTeX, 10 EPS figures, submitted to Phys. Rev.

Majorana and the path-integral approach to Quantum Mechanics

We give, for the first time, the English translation of a manuscript by Ettore Majorana, which probably corresponds to the text for a seminar delivered at the University of Naples in 1938, where he lectured on Theoretical Physics. Some passages reveal a physical interpretation of the Quantum Mechanics which anticipates of several years the Feynman approach in terms of path integrals, independently of the underlying mathematical formulation.Comment: revtex, 9 pages, 2 figures; a contribution in the centenary of the birth of Ettore Majoran

Force autocorrelation function in linear response theory and the origin of friction

Vanishing of the equilibrium Green-Kubo fluctuation expression for the friction coefficient of a massive particle moving in a finite-volume liquid is usually interpreted as an unphysical consequence of the finite volume. Here I show that it is a physical consequence of the finite mass of the rest of the system, which allows it to be dragged by the moving particle. As a consequence, it is sufficient to have two infinite masses in the liquid for the friction coefficient to be finite. In addition, I give the physical interpretation of different friction coefficients for two infinite-mass particles moving in the liquid.Comment: 24 pages text and figure

Reduction and Reconstruction Aspects of Second-Order Dynamical Systems with Symmetry

We examine the reduction process of a system of second-order ordinary differential equations which is invariant under a Lie group action. With the aid of connection theory, we explain why the associated vector field decomposes in three parts and we show how the integral curves of the original system can be reconstructed from the reduced dynamics. An illustrative example confirms the results

Efficient MaxCount and threshold operators of moving objects

Calculating operators of continuously moving objects presents some unique challenges, especially when the operators involve aggregation or the concept of congestion, which happens when the number of moving objects in a changing or dynamic query space exceeds some threshold value. This paper presents the following six d-dimensional moving object operators: (1) MaxCount (or MinCount), which finds the Maximum (or Minimum) number of moving objects simultaneously present in the dynamic query space at any time during the query time interval. (2) CountRange, which finds a count of point objects whose trajectories intersect the dynamic query space during the query time interval. (3) ThresholdRange, which finds the set of time intervals during which the dynamic query space is congested. (4) ThresholdSum, which finds the total length of all the time intervals during which the dynamic query space is congested. (5) ThresholdCount, which finds the number of disjoint time intervals during which the dynamic query space is congested. And (6) ThresholdAverage, which finds the average length of time of all the time intervals when the dynamic query space is congested. For these operators separate algorithms are given to find only estimate or only precise values. Experimental results from more than 7,500 queries indicate that the estimation algorithms produce fast, efficient results with error under 5%