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Velocity hodographs are of importance in weather analysis. A historically important case is the hodograph in the Kepler problem. Here, we explore the analytical solutions for the velocity hodograph for the projectile trajectories under a polynomial, in velocity, hydrodynamic dissipation. General solutions in terms of hypergeometric functions are found for odd powers in monomial terms. The even powers have analytical hodographs but we do not find a general expression for them. For completeness we include the constant or Coulomb case, as well as its sum with the linear and quadratic cases. Both shown explicit solutions as functions of the angle. Usually the linear case is discussed in textbooks and articles, here we find that more general cases admit analytical solutions, even when the position and time cannot be obtained explicitly and they need of numerical integration. Furthermore, motion with constant and, additional linear or quadratic drag could be used to design new university lab experiments.

A study of quantum-classical correspondence for the onset of thermalization in an isolated system of four interacting particles is presented. Provided that classical dynamics of the system is chaotic, thermalization is present and fits the expected statistical mechanics results. Quantum classical correspondence is established for the classical analogue of the single particle distributions of occupation numbers. The latter quantity is useful to test thermalization and it allows to obtain the microcanonical Boltzmann factor in the quantum and semi-classical cases. The correspondence between them and the statistical mechanics results are surprisingly good considering that we have only four interacting particles. The resulting Boltzmann factor for the occupation number distribution depends on the eigenfunction width, which is a semi-classical parameter explicitly calculated and consistent with the quantum one.

Quantum classical correspondence, QCC, is a wide subject of interest from the historical approaches to current studies in simple and complex systems, including time dependent Hamiltonians. Quantum eigenfunctions, or wave-functions, are part of the quantities analysed in recent years, including the implementation due to M V Berry and A Voros in 1977 of a suitable quantity to be considered as the classical eigenfunction counterpart. Here we present the calculation of the statistical moments of eigenfunctions in energy representation for anharmonic oscillators. The classical counter part of the system is fully chaotic. The comparison of the quantum version in order to test for quantum classical correspondence is analysed using a slightly modified formula of the proposed by Berry and Voros. The agreement is good with the corresponding classical counterpart being at the bottom of the quantum version for all the energy range considered.

Dissipation is of major interest in physics and engineering in all energy scales and, hence, it devoted a large amount of investigation. For quantum systems, an usual model for study is the Bateman dual system. Here we use the same model to explore dissipation in angular momentum instead of energy. For that matter, we start with an effective 2-D Bateman model in polar (cylindrical) coordinates and we find the expression for the Hamiltonian. Two constants of motion are found related to modified angular momenta and we test if they are in involution with the Hamiltonian, i.e., if they are integrals of motion. This is in order to verify that the corresponding quantum versions of these constants are good candidates to be a complete set of commuting observables.

We present an analytical solution for the projectile coplanar motion under constant drag parametrised by the velocity angle. We found the locus formed by the apices of the projectile trajectories. The range and time of flight are obtained numerically and we find that the optimal launching angle is smaller than in the free drag case. This is a good example of problems with constant dissipation of energy that includes curvature, and it is proper for intermediate courses of mechanics., Comment: 8 pages, 4 figures, two columns version

We show that the nearest neighbour distribution of distances between basis pairs of some intron-less and intron-containing coding regions are the same when a procedure, called {\em unfolding}, is applied. Such a procedure consists in separating the secular variations from the oscillatory terms. The form of the distribution obtained is quite similar to that of a random, i.e. Poissonian, sequence. This is done for the HUMBMYH7CD, DROMYONMA, HUMBMYH7 and DROMHC sequences. The first two correspond to highly coding regions while the last two correspond to non-coding regions. We also show that the distributions before the unfolding procedure depend on the secular part but, after the unfolding procedure we obtain an striking result: all distributions are similar to each other. The result becomes independent of the content of introns or the species we have chosen. This is in contradiction with the results obtained with the detrended fluctuation analysis in which the correlations yield different results for intron-less and intron-containing regions., Comment: 8 pages. elesart style.Alberto Robledo's 60th Anniversary Symposium - Nonlinearity, nonequilibrium and complexity: Questions and perspectives in statistical physics. Accepted in Physica A

The Sturm-Coulomb problem is an integrable one since its symmetry group is O(4). When we apply to it a magnetic field, this symmetry is broken and reduced to the O(2) group. The problem is then nonintegrable, but we can derive its matrix representation in a basis in which the Sturm-Coulomb problem alone is diagonal. We use this matrix representation to obtain the corresponding eigenvalues and their nearest neighbor spacing distribution. From the histogram of the latter, we discuss the presence or absence of quantum chaos as a function of the intensity H of the magnetic field and the angular momentum m in the direction of this field.

A Hamiltonian approach is presented to study the two dimensional motion of damped electric charges in time dependent electromagnetic fields. The classical and the corresponding quantum mechanical problems are solved for particular cases using canonical transformations applied to Hamiltonians for a particle with variable mass. The Green's function is constructed and, from it, the motion of a Gaussian wave packet is studied in detail., Comment: 14 pages, 4 figures, misprints and figures corrected

The Presidential Election in Mexico of July 2012 has been the third time that PREP, Previous Electoral Results Program works. PREP gives voting outcomes based in electoral certificates of each polling station that arrive to capture centers. In previous ones, some statistical regularities had been observed, three of them were selected to make predictions and were published in \texttt{arXiv:1207.0078 [physics.soc-ph]}. Using the database made public in July 2012, two of the predictions were completely fulfilled, while, the third one was measured and confirmed using the database obtained upon request to the electoral authorities. The first two predictions confirmed by actual measures are: (ii) The Partido Revolucionario Institucional, PRI, is a sprinter and has a better performance in polling stations arriving late to capture centers during the process. (iii) Distribution of vote of this party is well described by a smooth function named a Daisy model. A Gamma distribution, but compatible with a Daisy model, fits the distribution as well. The third prediction confirms that {\it errare humanum est}, since the error distributions of all the self-consistency variables appeared as a central power law with lateral lobes as in 2000 and 2006 electoral processes. The three measured regularities appeared no matter the political environment., Comment: All the predictions were verified. English corrected (I hope)

We present an analysis on the geometrical place formed by the set of maxima of the trajectories of a projectile launched in a media with linear drag. Such a place, the locus of apexes, is written in term of the Lambert $W$ function in polar coordinates, confirming the special role played by this function in the problem. In order to characterize the locus, a study of its curvature is presented in two parameterizations, in terms of the launch angle and in the polar one. The angles of maximum curvature are compared with other important angles in the projectile problem. As an addendum, we find that the synchronous curve in this problem is a circle as in the drag-free case., 7 pages, 6 color eps figures. Synchronous curve added. Typos and style corrected.

The distribution of votes of one of the corporate parties in Mexico during elections of 2000, 2003 and 2006 is analyzed. After proper normalization and unfolding the agreement of the votes distributions with those of daisy models of several ranks is good. These models are generated by retaining each $r+1$ level in a sequence which follows a Poisson distribution.Beyond the fact that rank 2 daisy model resembles the distribution of the quasi-optimal distances for the Traveling Salesman Problem no clear explanation exists for this behavior, but the agreement is not fortuitous and the possibility of a universal phenomena for corporate vote is discussed., To be send to Phys. Rev. E. 4 pages. 3 color figures. Unfolding procedure explained

We analyze the fluctuating dynamics at the golden-mean transition to chaos in the critical circle map and find that trajectories within the critical attractor consist of infinite sets of power laws mixed together. We elucidate this structure assisted by known renormalization group (RG) results. Next we proceed to weigh the new findings against Tsallis' entropic and Mori's thermodynamic theoretical schemes and observe behavior to a large extent richer than previously reported. We find that the sensitivity to initial conditions has the form of families of intertwined q-exponentials, of which we determine the q-indexes and the generalized Lyapunov coefficient spectra. Further, the dynamics within the critical attractor is found to consist of not one but a collection of Mori's q-phase transitions with a hierarchical structure. The value of Mori's `thermodynamic field' variable q at each transition corresponds to the same special value for the entropic index q. We discuss the relationship between the two formalisms and indicate the usefulness of the methods involved to determine the universal trajectory scaling function and/or the ocurrence and characterization of dynamical phase transitions., Resubmitted to Physical Review E. The title has been changed slightly and the abstract has been extended. There is a new subsection following the conclusions that explains the role and usefulness of the q-statistics in the problem studied. Other minor changes througout the text

The author replies to a comment made on a previous paper (Hernandez-Saldana 2010 Eur. J. Phys. 31 1319).

Semi-Poisson statistics are shown to be obtained by removing every other number from a random sequence. Retaining every $(r+1)\mathrm{th}$ level we obtain a family of sequences, which we call daisy models. Their statistical properties coincide with those of Bogomolny's nearest-neighbor interaction Coulomb gas if the inverse temperature coincides with the integer r. In particular, the case $r=2$ reproduces closely the statistics of quasioptimal solutions of the traveling salesman problem.

We present an experimental setup based on the normal modes of vibrating soap films which shows quantum features of integrable and chaotic billiards. In particular, we obtain the so-called scars -narrow linear regions with high probability along classical periodic orbits- for the classically chaotic billiards. We show that these scars are also visible at low frequencies. Finally, we suggest some applications of our experimental setup in other related two-dimensional wave phenomena., Comment: 5 pages, 7 figures. Better Postscript figures available on request

The effect on spectral statistics and on the revival probability of intruder states in a random background is analysed numerically and with perturbative methods. For random coupling the intruder does not affect the GOE spectral statistics of the background significantly, while a constant coupling causes very strong correlations at short range with a fourth power dependence of the spectral two-point function at the origin.The revival probability is significantly depressed for constant coupling as compared to random coupling., Comment: 18 pages, 10 Postscript figures