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We revisit the product convolution semigroup of probability densities e_c(t),c>0 on the positive half-line with moments (n!)^c and determine the asymptotic behaviour of e_c(t) for large and small t>0. This shows that (n!)^c is indeterminate as Stieltjes moment sequence if and only if c>2, Comment: 13 pages

In this communication, some economic models given by functional mappings are addressed. These are models for random markets where agents trade by pairs and exchange their money in a random and conservative way. They display the exponential wealth distribution as asymptotic equilibrium, independently of the effectiveness of the transactions and of the limitation of the total wealth. The entropy increases with time in these models and the existence of an H-theorem is computationally checked. Also, it is shown that any small perturbation of the models equations make them to lose the exponential distribution as an equilibrium solution., Comment: 11 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:1104.2187

The Gauss hypergeometric function ${}_2F_1(a,b,c;z)$ can be computed by using the power series in powers of $z, z/(z-1), 1-z, 1/z, 1/(1-z),(z-1)/z$. With these expansions ${}_2F_1(a,b,c;z)$ is not completely computable for all complex values of $z$. As pointed out in Gil, {\it et al.} [2007, \S2.3], the points $z=e^{\pm i\pi/3}$ are always excluded from the domains of convergence of these expansions. B\"uhring [1987] has given a power series expansion that allows computation at and near these points. But, when $b-a$ is an integer, the coefficients of that expansion become indeterminate and its computation requires a nontrivial limiting process. Moreover, the convergence becomes slower and slower in that case. In this paper we obtain new expansions of the Gauss hypergeometric function in terms of rational functions of $z$ for which the points $z=e^{\pm i\pi/3}$ are well inside their domains of convergence . In addition, these expansion are well defined when $b-a$ is an integer and no limits are needed in that case. Numerical computations show that these expansions converge faster than B\"uhring's expansion for $z$ in the neighborhood of the points $e^{\pm i\pi/3}$, especially when $b-a$ is close to an integer number., Comment: 18 pages, 6 figures, 4 tables. In Advances in Computational Mathematics, 2012 Second version with corrected typos in equations (18) and (19)

In simulations of some economic gas-like models, the asymptotic regime shows an exponential wealth distribution, independently of the initial wealth distribution given to the system. The appearance of this statistical equilibrium for this type of gas-like models is explained in a rigorous analytical way., Comment: 9 pages, 4 figures

Exponential distribution is ubiquitous in the framework of multi-agent systems. Usually, it appears as an equilibrium state in the asymptotic time evolution of statistical systems. It has been explained from very different perspectives. In statistical physics, it is obtained from the principle of maximum entropy. In the same context, it can also be derived without any consideration about information theory, only from geometrical arguments under the hypothesis of equiprobability in phase space. Also, several multi-agent economic models based on mappings, with random, deterministic or chaotic interactions, can give rise to the asymptotic appearance of the exponential wealth distribution. An alternative approach to this problem in the framework of iterations in the space of distributions has been recently presented. Concretely, the new iteration given by $ f_{n+1}(x) = \int\int_{u+v>x}{f_n(u)f_n(v)\over u+v} dudv.$. It is found that the exponential distribution is a stable fixed point of the former functional iteration equation. From this point of view, it is easily understood why the exponential wealth distribution (or by extension, other kind of distributions) is asymptotically obtained in different multi-agent economic models., Comment: 6 pages, 5 figures

Asymptotic expansions are given for large values of $n$ of the generalized Bessel polynomials $Y_n^\mu(z)$. The analysis is based on integrals that follow from the generating functions of the polynomials. A new simple expansion is given that is valid outside a compact neighborhood of the origin in the $z-$plane. New forms of expansions in terms of elementary functions valid in sectors not containing the turning points $z=\pm i/n$ are derived, and a new expansion in terms of modified Bessel functions is given. Earlier asymptotic expansions of the generalized Bessel polynomials by Wong and Zhang (1997) and Dunster (2001) are discussed., Comment: 22 pages, 1 figure

A step by step procedure to derive analytically the exact dynamical evolution equations of the probability density functions (PDF) of well known kinetic wealth exchange economic models is shown. This technique gives a dynamical insight into the evolution of the PDF, e.g., allowing the calculation of its relaxation times. Their equilibrium PDFs can also be calculated by finding its stationary solutions. This gives as a result an integro-differential equation, which can be solved analytically in some cases and numerically in others. This should provide some guidance into the type of probability density functions that can be derived from particular economic agent exchange rules, or for that matter, any other kinetic model of gases with particular collision physics., Comment: 8 pages, 4 figures

Asymptotic expansions are given for large values of $n$ of the generalized Bernoulli polynomials $B_n^\mu(z)$ and Euler polynomials $E_n^\mu(z)$. In a previous paper L\'opez and Temme (1999) these polynomials have been considered for large values of $\mu$, with $n$ fixed. In the literature no complete description of the large $n$ asymptotics of the considered polynomials is available. We give the general expansions, summarize known results of special cases and give more details about these results. We use two-point Taylor expansions for obtaining new type of expansions. The analysis is based on contour integrals that follow from the generating functions of the polynomials., Comment: 20 pages, 1 figure

The limit cycle of the van der Pol oscillator, $\ddot{x}+ \epsilon (x^2-1) \dot{x} + x =0$, is studied in the plane $(x,\dot{x})$ by applying the homotopy analysis method. A recursive set of formulas that approximate the amplitude and form of this limit cycle for the whole range of the parameter $\epsilon$ is obtained. These formulas generate the amplitude with an error less than 0.1%. To our knowledge, this is the first time where an analytical approximation of the amplitude of the van der Pol limit cycle, with validity from the weakly up to the strongly nonlinear regime, is given., Comment: 13 pages, 5 figures, 1 table

In this work, Lienard equations are considered. The limit cycles of these systems are studied by applying the homotopy analysis method. The amplitude and frequency obtained with this methodology are in good agreement with those calculated by computational methods. This puts in evidence that the homotopy analysis method is an useful tool to solve nonlinear differential equations., Comment: 16 pages, 9 figures, 1 table

We predict general trends for surface segregation in a binary metal cluster based on the difference between the atomic properties of the constituent elements. Considering the attractive and repulsive contributions of the cohesive energy of an atom in a cluster, energetically most favorable sites for a guest atom on a pure metal cluster is determined. It is predicted that for adjacent elements in a row of the periodic table, the bimetallic system would be more stable if the component with smallest valence electron density is placed on the surface. On the contrary, for well separated components in the periodic table, the bimetallic cluster would be more stable if the component with the smallest core electron density is placed inside. Such chemical {\bf ordering trends} in the lowest energy configurations of Pt-Au, Pt-Pd and Pt-Ni binary alloy clusters are verified for their 561 atom systems through simulated annealing process. It is predicted that the Ir, Rh, Ni, Pd atoms would tend to be located inside the Ir-Pt, Rh-Pd, Ni-Cu and Pd-Ag bimetallic nanoclusters, respectively., Comment: Submitted to Phys. Rev. Lett. on April 30th, 2008

Li\'enard equations of the form $\ddot{x}+\epsilon f(x)\dot{x}+x=0$, with $f(x)$ an even function, are considered in the weakly nonlinear regime ($\epsilon\to 0$). A perturbative algorithm for obtaining the number, amplitude and shape of the limit cycles of these systems is given. The validity of this algorithm is shown and several examples illustrating its application are given. In particular, an ${\mathcal O}(\epsilon^8)$ approximation for the amplitude of the van der Pol limit cycle is explicitly obtained., Comment: 17 text-pages, 1 table, 6 figures

Li\'{e}nard equations, $\ddot{x}+\epsilon f(x)\dot{x}+x=0$, with $f(x)$ an even continuous function are considered. In the weakly nonlinear regime ($\epsilon\to 0$), the number and an order zero in $\epsilon$ approximation of the amplitude of limit cycles present in this type of systems can be obtained by applying a methodology recently proposed by the authors [L\'opez-Ruiz R, L\'opez JL. Bifurcation curves of limit cycles in some Li\'enard systems. Int J Bifurcat Chaos 2000; 10:971-980]. In the present work, that method is carried forward to higher orders in $\epsilon$ and is embedded in a general recursive algorithm capable to approximate the form of the limit cycles and to correct their amplitudes as an expansion in powers of $\epsilon$. Several examples showing the application of this scheme are given., Comment: 24 pages, 6 figures, 1 table

We consider three examples of weekly perturbed centers which do not have {\it geometrical equivalence}: a linear center, a degenerate center and a non-hamiltonian center. In each case the number and amplitude of the limit cycles emerging from the period annulus are calculated following the same strategy: we reduce of all of them to locally equivalent perturbed integrable systems of the form: $dH(x,y)+\epsilon(f(x,y)dy-g(x,y)dx)=0$, with $H(x,y)={1/2}(x^2+y^2)$. This reduction allows us to find the Melnikov function, $M(h)=\int_{H=h}fdy-gdx$, associated to each particular problem. We obtain the information on the bifurcation curves of the limit cycles by solving explicitly the equation $M(h)=0$ in each case., Comment: 17 pages, 0 figures

Lienard systems of the form $\ddot{x}+\epsilon f(x)\dot{x}+x=0$, with f(x) an even continous function, are considered. The bifurcation curves of limit cycles are calculated exactly in the weak ($\epsilon\to 0$) and in the strongly ($\epsilon\to\infty$) nonlinear regime in some examples. The number of limit cycles does not increase when $\epsilon$ increases from zero to infinity in all the cases analyzed., Comment: 25 pages, 0 figures. Published in Int. Journal of Bifurcation and Chaos, vol. 10, 971-980 (2001)

Lienard systems of the form $\ddot{x}+\epsilon f(x)\dot{x}+x=0$, with f(x) an even function, are studied in the strongly nonlinear regime ($\epsilon\to\infty$). A method for obtaining the number, amplitude and loci of the limit cycles of these equations is derived. The accuracy of this method is checked in several examples. Lins-Melo-Pugh conjecture for the polynomial case is true in this regime., Comment: 22 pages, 0 figures. Published in Chaos, Solitons and Fractals, vol. 11, 747-756 (2001)