An entirely analytical cosmological model

The purpose of the present study is to show that in a particular cosmological model, with an affine equation of state, one can obtain, besides the background given by the scale factor, Hubble and deceleration parameters, a representation in terms of scalar fields and, more important, explicit mathematical expressions for the density contrast and the power spectrum. Although the model so obtained is not realistic, it reproduces features observed in some previous numerical studies and, therefore, it may be useful in the testing of numerical codes and as a pedagogical tool.Comment: 4 pages (revtex4), 4 figure

Analytical solutions for the Rabi model

The Rabi model that describes the fundamental interaction between a two-level system with a quantized harmonic oscillator is one of the simplest and most ubiquitous models in modern physics. However, this model has not been solved exactly because it is hard to find a second conserved quantity besides the energy. Here we present a unitary transformation to map this unsolvable Rabi model into a solvable Jaynes-Cummings-like model by choosing a proper variation parameter. As a result, the analytical energy spectrums and wavefunctions including both the ground and the excited states can be obtained easily. Moreover, these explicit results agree well with the direct numerical simulations in a wide range of the experimental parameters. In addition, based on our obtained energy spectrums, the recent experimental observation of Bloch-Siegert in the circuit quantum electrodynamics with the ultrastrong coupling can be explained perfectly. Our results have the potential application in the solid-state quantum information processing.Comment: 5 pages, 4 figure

Analytical solution of a generalized Penna model

In 1995 T.J.Penna introduced a simple model of biological aging. A modified Penna model has been demonstrated to exhibit behaviour of real-life systems including catastrophic senescence in salmon and a mortality plateau at advanced ages. We present a general steady-state, analytic solution to the Penna model, able to deal with arbitrary birth and survivability functions. This solution is employed to solve standard variant Penna models studied by simulation. Different Verhulst factors regulating both the birth rate and external death rate are considered.Comment: 6 figure

Geosynthetic-encased stone columns: analytical calculation model

This paper presents a newly developed design method for non-encased and encased stone columns. The developed analytical closed-form solution is based on previous solutions, initially developed for non-encased columns and for non-dilating rigid-plastic column material. In the present method, the initial stresses in the soil/column are taken into account, with the column considered as an elasto-plastic material with constant dilatancy, the soil as an elastic material and the geosynthetic encasement as a linear-elastic material. To check the validity of the assumptions and the ability of the method to give reasonable predictions of settlements, stresses and encasement forces, comparative elasto-plastic finite element analyses have been performed. The agreement between the two methods is very good, which was the reason that the new method was used to generate a parametric study in order to investigate various parameters, such as soil/column parameters, replacement ratio, load level and geosynthetic encasement stiffness on the behaviour of the improved ground. The results of this study show the influence of key parameters and provide a basis for the rational predictions of settlement response for various encasement stiffnesses, column arrangements and load levels. The practical use of the method is illustrated through the design chart, which enables preliminary selection of column spacing and encasement stiffness to achieve the desired settlement reduction for the selected set of the soil/column parameters. (C) 2010 Elsevier Ltd. All rights reserved

Analytical approximation for single-impurity Anderson model

We have applied the recently developed dual fermion technique to the spectral properties of single-band Anderson impurity problem (SIAM). In our approach a series expansion is constructed in vertices of the corresponding atomic Hamiltonian problem. This expansion contains a small parameter in two limiting cases: in the weak coupling case ($U/t \to 0$), due to the smallness of the irreducible vertices, and near the atomic limit ($U/t \to \infty$), when bare propagators are small. Reasonable results are obtained also for the most interesting case of strong correlations ($U \approx t$). The atomic problem of the Anderson impurity model has a degenerate ground state, so the application of the perturbation theory is not straightforward. We construct a special approach dealing with symmetry-broken ground state of the renormalized atomic problem. Formulae for the first-order dual diagram correction are obtained analytically in the real-time domain. Most of the Kondo-physics is reproduced: logarithmic contributions to the self energy arise, Kondo-like peak at the Fermi level appears, and the Friedel sum rule is fulfilled. Our approach describes also renormalization of atomic resonances due to hybridization with a conduction band. A generalization of the proposed scheme to a multi-orbital case can be important for the realistic description of correlated solids.Comment: 6 pages, 5 figure

Analytical expressions for the deprojected Sersic model

The Sersic model has become the standard to parametrize the surface brightness distribution of early-type galaxies and bulges of spiral galaxies. A major problem is that the deprojection of the Sersic surface brightness profile to a luminosity density cannot be executed analytically for general values of the Sersic index. Mazure & Capelato (2002) used the Mathematica computer package to derive an expression of the Sersic luminosity density in terms of the Meijer G function for integer values of the Sersic index. We generalize this work using analytical means and use Mellin integral transforms to derive an exact, analytical expression for the luminosity density in terms of the Fox H function for all values of the Sersic index. We derive simplified expressions for the luminosity density, cumulative luminosity and gravitational potential in terms of the Meijer G function for all rational values of the Sersic index and we investigate their asymptotic behaviour at small and large radii. As implementations of the Meijer G function are nowadays available both in symbolic computer algebra packages and as high-performance computing code, our results open up the possibility to calculate the density of the Sersic models to arbitrary precision.Comment: 9 pages, accepted for publication in Astronomy and Astrophysic

The Immediate Exchange model: an analytical investigation

We study the Immediate Exchange model, recently introduced by Heinsalu and Patriarca [Eur. Phys. J. B 87: 170 (2014)], who showed by simulations that the wealth distribution in this model converges to a Gamma distribution with shape parameter $2$. Here we justify this conclusion analytically, in the infinite-population limit. An infinite-population version of the model is derived, describing the evolution of the wealth distribution in terms of iterations of a nonlinear operator on the space of probability densities. It is proved that the Gamma distributions with shape parameter $2$ are fixed points of this operator, and that, starting with an arbitrary wealth distribution, the process converges to one of these fixed points. We also discuss the mixed model introduced in the same paper, in which exchanges are either bidirectional or unidirectional with fixed probability. We prove that, although, as found by Heinsalu and Patriarca, the equilibrium distribution can be closely fit by Gamma distributions, the equilibrium distribution for this model is {\it{not}} a Gamma distribution

Analytical model for flux saturation in sediment transport

The transport of sediment by a fluid along the surface is responsible for dune formation, dust entrainment and for a rich diversity of patterns on the bottom of oceans, rivers, and planetary surfaces. Most previous models of sediment transport have focused on the equilibrium (or saturated) particle flux. However, the morphodynamics of sediment landscapes emerging due to surface transport of sediment is controlled by situations out-of-equilibrium. In particular, it is controlled by the saturation length characterizing the distance it takes for the particle flux to reach a new equilibrium after a change in flow conditions. The saturation of mass density of particles entrained into transport and the relaxation of particle and fluid velocities constitute the main relevant relaxation mechanisms leading to saturation of the sediment flux. Here we present a theoretical model for sediment transport which, for the first time, accounts for both these relaxation mechanisms and for the different types of sediment entrainment prevailing under different environmental conditions. Our analytical treatment allows us to derive a closed expression for the saturation length of sediment flux, which is general and can thus be applied under different physical conditions