Continuous growth models in terms of generalized logarithm and exponential functions

Consider the one-parameter generalizations of the logarithmic and exponential functions which are obtained from the integration of non-symmetrical hyperboles. These generalizations coincide to the one obtained in the context of non-extensive thermostatistics. We show that these functions are suitable to describe and unify the great majority of continuous growth models, which we briefly review. Physical interpretation to the generalization function parameter is given for the Richards’ model, which has an underlying microscopic model to justify it.     

On the construction of complex networks with optimal Tsallis entropy

In this work, we first formulate the Tsallis entropy in the context of complex networks. We then propose a network construction whose topology maximizes the Tsallis entropy. The growing network model has two main ingredients: copy process and random attachment mechanism (C-R model). We show that the resulting degree distribution exactly agrees with the required degree distribution that maximizes the Tsallis entropy. We also provide another example of network model using a combination of preferential and random attachment mechanisms (P-R model) and compare it with the distribution of the Tsallis entropy. In this case, we show that by adequately identifying the exponent factor q, the degree distribution can also be written in the q-exponential form. Taken together, our findings suggest that both mechanisms, copy process and preferential attachment, play a key role for the realization of networks with maximum Tsallis entropy. Finally, we discuss the interpretation of q parameter of the Tsallis entropy in the context of complex networks.     

Epidemic outbreaks in growing scale-free networks with local structure

The class of generative models has already attracted considerable interest from researchers in recent years and much expanded the original ideas described in BA model. Most of these models assume that only one node per time step joins the network. In this paper, we grow the network by adding n interconnected nodes as a local structure into the network at each time step with each new node emanating m new edges linking the node to the preexisting network by preferential attachment. This successfully generates key features observed in social networks. These include power-law degree distribution p_k∼k^−(3+μ), where μ=(n−1)/m is a tuning parameter defined as the modularity strength of the network, nontrivial clustering, assortative mixing, and modular structure. Moreover, all these features are dependent in a similar way on the parameter μ. We then study the susceptible-infected epidemics on this network with identical infectivity, and find that the initial epidemic behavior is governed by both of the infection scheme and the network structure, especially the modularity strength. The modularity of the network makes the spreading velocity much lower than that of the BA model. On the other hand, increasing the modularity strength will accelerate the propagation velocity.     

Some exact results on the ultrametric overlap distribution in mean field spin glass models (I)

The mean field spin glass model is analyzed by a combination of exact methods and a powerful Ansatz. The method exploited is general, and can be applied to others disordered mean field models such as, e.g., neural networks.It is well known that the probability measure of overlaps among replicas carries the whole physical content of these models. A functional order parameter of Parisi type is introduced by rigorous methods, according to previous works by F. Guerra. By the Ansatz that the functional order parameter is the correct order parameter of the model, we explicitly find the full overlap distribution. The physical interpretation of the functional order parameter is obtained, and ultrametricity of overlaps is derived as a natural consequence of a branching diffusion process.It is shown by explicit construction that ultrametricity of the 3-replicas overlap distribution together with the Ghirlanda-Guerra relations determines the distribution of overlaps among s replicas, for any s, in terms of the one-overlap distribution.     

Physics of evolution: Selection without fitness

Traditionally evolution is seen as a process where from a pool of possible variations of a population (e.g. biological species or industrial goods) a few variations get selected which survive and proliferate, whereas the others vanish. Survival probabilities and proliferation rates are typically associated with the ‘fitness’ of particular variations. In this paper we argue that the notion of fitness is an a posteriori concept, in the sense that one can assign higher fitness to species that survive but one can generally not derive or predict fitness per se. Proliferation rates can be measured, whereas fitness landscapes, i.e. the inter-dependence of proliferation rates, cannot. For this reason we think that in a physical theory of evolution such notions should be avoided. In this spirit, here we propose a random matrix model of evolution where selection mechanisms are encoded in interaction matrices of species, thereby extending the previous work of ours by a control parameter describing suppressors in the system. We are able to recover some key facts of evolution dynamics endogenously, such as punctuated equilibrium, i.e. the existence of intrinsic large extinction events, and, at the same time, periods of dramatic diversification, as known e.g. from the fossil record. Further, we comment on two fundamental technical problems of a ‘physics of evolution’, the non-closedness of its phase space and the problem of co-evolving boundary conditions, apparent in all systems subject to evolution.     

Spin Hall effect in noncommutative coordinates

A semiclassical constrained Hamiltonian system which was established to study dynamical systems of matrix valued non-Abelian gauge fields is employed to formulate spin Hall effect in noncommuting coordinates at the first order in the constant noncommutativity parameter θ. The method is first illustrated by studying the Hall effect on the noncommutative plane in a gauge independent fashion. Then, the Drude model type and the Hall effect type formulations of spin Hall effect are considered in noncommuting coordinates and θ deformed spin Hall conductivities which they provide are acquired. It is shown that by adjusting θ different formulations of spin Hall conductivity are accomplished. Hence, the noncommutative theory can be envisaged as an effective theory which unifies different approaches to similar physical phenomena.     

Order-disorder transition in conflicting dynamics leading to rank-frequency generalized beta distributions

The behavior of rank-ordered distributions of phenomena present in a variety of fields such as biology, sociology, linguistics, finance and geophysics has been a matter of intense research. Often power laws have been encountered; however, their validity tends to hold mainly for an intermediate range of rank values. In a recent publication (Martínez-Mekler et al., 2009 [7]), a generalization of the functional form of the beta distribution has been shown to give excellent fits for many systems of very diverse nature, valid for the whole range of rank values, regardless of whether or not a power law behavior has been previously suggested. Here we give some insight on the significance of the two free parameters which appear as exponents in the functional form, by looking into discrete probabilistic branching processes with conflicting dynamics. We analyze a variety of realizations of these so-called expansion-modification models first introduced by Wentian Li (1989) [10]. We focus our attention on an order-disorder transition we encounter as we vary the modification probability p. We characterize this transition by means of the fitting parameters. Our numerical studies show that one of the fitting exponents is related to the presence of long-range correlations exhibited by power spectrum scale invariance, while the other registers the effect of disordering elements leading to a breakdown of these properties. In the absence of long-range correlations, this parameter is sensitive to the occurrence of unlikely events. We also introduce an approximate calculation scheme that relates this dynamics to multinomial multiplicative processes. A better understanding through these models of the meaning of the generalized beta-fitting exponents may contribute to their potential for identifying and characterizing universality classes.     

Some exact results on the ultrametric overlap distribution in mean field spin glass models (I)

The mean field spin glass model is analyzed by a combination of exact methods and a simple Ansatz. The method exploited is general, and can be applied to others disordered mean field models such as, e.g., neural networks. It is well known that the probability measure of overlaps among replicas carries the whole physical content of these models. A functional order parameter of Parisi type is introduced by rigorous methods, according to previous works by F. Guerra. By the Ansatz that the functional order parameter is the correct order parameter of the model, we explicitly find the full overlap distribution. The physical interpretation of the functional order parameter is obtained, and ultrametricity of overlaps is derived as a natural consequence of a branching diffusion process. It is shown by explicit construction that ultrametricity of the 3-replicas overlap distribution together with the Ghirlanda-Guerra relations determines the distribution of overlaps among s replicas, for any s, in terms of the one-overlap distribution. Received 14 February 2000     

Two dimensional fractional projectile motion in a resisting medium

In this paper we propose a fractional differential equation describing the behavior of a two dimensional projectile in a resisting medium. In order to maintain the dimensionality of the physical quantities in the system, an auxiliary parameter k was introduced in the derivative operator. This parameter has a dimension of inverse of seconds (sec)^?1 and characterizes the existence of fractional time components in the given system. It will be shown that the trajectories of the projectile at different values of γ and different fixed values of velocity v ₀ and angle θ, in the fractional approach, are always less than the classical one, unlike the results obtained in other studies. All the results obtained in the ordinary case may be obtained from the fractional case when γ = 1.     

Acoustic scattering of a high-order Bessel beam by an elastic sphere

The exact analytical solution for the scattering of a generalized (or “hollow”) acoustic Bessel beam in water by an elastic sphere centered on the beam is presented. The far-field acoustic scattering field is expressed as a partial wave series involving the scattering angle relative to the beam axis and the half-conical angle of the wave vector components of the generalized Bessel beam. The sphere is assumed to have isotropic elastic material properties so that the nth partial wave amplitude for plane wave scattering is proportional to a known partial-wave coefficient. The transverse acoustic scattering field is investigated versus the dimensionless parameter ka(k is the wave vector, a radius of the sphere) as well as the polar angle θ for a specific dimensionless frequency and half-cone angle β. For higher-order generalized beams, the acoustic scattering vanishes in the backward (θ = π) and forward (θ = 0) directions along the beam axis. Moreover it is possible to suppress the excitation of certain resonances of an elastic sphere by appropriate selection of the generalized Bessel beam parameters.     

Generalized analytic signal associated with linear canonical transform

Analytic signal is tightly associated with Hilbert transform and Fourier transform. The linear canonical transform is the generalization of many famous linear integral transforms, such as Fourier transform, fractional Fourier transform and Fresnel transform. Based on the parameter (a, b)-Hilbert transform and the linear canonical transform, in this paper, we develop some issues on generalized analytic signal. The generalized analytic signal can suppress the negative frequency components in the linear canonical transform domain. Furthermore, we prove that the kernel function of the inverse linear canonical transform satisfies the generalized analytic condition and get the generalized analytic pairs. We show the generalized Bedrosian theorem is valid in the linear canonical transform domain.     

A triangle model of criminality

This paper is concerned with a quantitative model describing the interaction of three sociological species, termed as owners, criminals and security guards, and denoted by X, Y and Z respectively. In our model, Y is a predator of the species X, and so is Z with respect to Y. Moreover, Z can also be thought of as a predator of X, since this last population is required to bear the costs of maintaining Z.We propose a system of three ordinary differential equations to account for the time evolution of X(t), Y(t) and Z(t) according to our previous assumptions. Out of the various parameters that appear in that system, we select two of them, denoted by H, and h, which are related with the efficiency of the security forces as a control parameter in our discussion. To begin with, we consider the case of large and constant owners population, which allows us to reduce (3), (4) and (5) to a bidimensional system for Y(t) and Z(t). As a preliminary step, this situation is first discussed under the additional assumption that Y(t)+Z(t) is constant. A bifurcation study is then performed in terms of H and h, which shows the key role played by the rate of casualties in Y and Z, that results particularly in a possible onset of bistability. When the previous restriction is dropped, we observe the appearance of oscillatory behaviours in the full two-dimensional system. We finally provide a exploratory study of the complete model (3), (4) and (5), where a number of bifurcations appear as parameter H changes, and the corresponding solutions behaviours are described.     

Surface flows of granular mixtures: II. Segregation with grains of different size

We present the generalization of a theoretical model for segregation of granular mixtures due to surface flows, published in J. Phys. I France 6, 1295 (1996). Our generalized model is valid for grains differing by their size and/or their surface properties; in the present paper, we describe the case of two species with the same surface properties but two different sizes. The rolling stream is assumed to be homogeneous. Exchanges between the grains at rest and the rolling stream are modelized via binary collisions. The model predicts that during the filling of a two-dimensional silo, continuous segregation appears inside the static phase: small (respectively large) grains tend to stop uphill (respectively downhill), although both species remain present everywhere. This fits the observations when the size difference between the species is small. When the size difference is large, a different regime is observed. We argue that in this case, segregation occurs directly inside the rolling stream. Received: 25 February 1998 / Received in final form and Accepted: 6 July 1998     

Higher gauge theory and a non-Abelian generalization of 2-form electrodynamics

In conventional gauge theory, a charged point particle is described by a representation of the gauge group. If we propagate the particle along some path, the parallel transport of the gauge connection acts on this representation. The Lagrangian density of the gauge field depends on the curvature of the connection which can be calculated from the holonomy around (infinitesimal) loops. For Abelian symmetry groups, say G=U(1), there exists a generalization, known as p-form electrodynamics, in which (p−1)-dimensional charged objects can be propagated along p-surfaces and in which the Lagrangian depends on a generalized curvature associated with (infinitesimal) closed p-surfaces. In this article, we use Lie 2-groups and ideas from higher category theory in order to formulate a discrete gauge theory which generalizes these models at the level p=2 to possibly non-Abelian symmetry groups. An important feature of our model is that it involves both parallel transports along paths and generalized transports along surfaces with a non-trivial interplay of these two types of variables. Our main result is the geometric picture, namely the assignment of non-Abelian quantities to geometrical objects in a coordinate free way. We construct the precise assignment of variables to the curves and surfaces, the generalized local symmetries and gauge invariant actions and we clarify which structures can be non-Abelian and which others are always Abelian. A discrete version of connections on non-Abelian gerbes is a special case of our construction. Even though the motivation sketched so far suggests applications mainly in string theory, the model presented here is also related to spin foam models of quantum gravity and may in addition provide some insight into the role of centre monopoles and vortices in lattice QCD.     

Phase transitions in self-dual generalizations of the Baxter–Wu model

We study two types of generalized Baxter–Wu models, by means of transfer-matrix and Monte Carlo techniques. The first generalization allows for different couplings in the up- and down-triangles, and the second generalization is to a q-state spin model with three-spin interactions. Both generalizations lead to self-dual models, so that the probable locations of the phase transitions follow. Our numerical analysis confirms that phase transitions occur at the self-dual points. For both generalizations of the Baxter–Wu model, the phase transitions appear to be discontinuous.     

Age-based model for weighted network with general assortative mixing

In this paper, we propose an evolutionary model for weighted networks by introducing an age-based mutual selection mechanism. Our model generates power-law distributions of degree, weight, and strength, which are confirmed by analytical predictions and are consistent with real observations. The investigation of the relationship between clustering and the connectivity of nodes suggests hierarchical organization in the weighted networks. Furthermore, both assortative and disassortative properties can be naturally obtained by tuning a parameter α, which controls the strength of age-based preferential attachments. Since the age information of nodes is easier to acquire than the degree and strength of nodes, and almost all empirically observed structural and weighted properties can be reproduced by the simple evolutionary regulation, our model may reveal some underlying mechanisms that are key for the evolution of weighted complex networks.     

Modules Over the Noncommutative Torus and Elliptic Curves

Using the Weil–Brezin–Zak transform of solid state physics, we describe line bundles over elliptic curves in terms of Weyl operators. We then discuss the connection with finitely generated projective modules over the algebra A _θ of the noncommutative torus. We show that such A _θ -modules have a natural interpretation as Moyal deformations of vector bundles over an elliptic curve E _τ , under the condition that the deformation parameter θ and the modular parameter τ satisfy a non-trivial relation.     

Cooperative behavior in evolutionary snowdrift game with bounded rationality

An evolutionary snowdrift game (SG) that incorporates bounded rationality and limited information in the evolutionary process is proposed and studied. Based on SG in a well-mixed population and defining the winning action at a turn to be the one that gets a higher payoff, the most recent m winning actions can be used as a public information based on which the competing agents decide their next actions. This defines a strategy pool from which each agent picks a number of strategies as their tool in adapting to the competing environment. The payoff parameter r in SG serves to set the maximum number of winners per turn. Due to the bounded rationality and limited information, the cooperative frequency shows steps and plateaux as a function of r and these features tend to be smoothed out as m increases. These features are results of an interplay between a restricted subset of m-bit histories that the system can visit at a value of r and the limited capacity that agents can adapt. The standard deviation in the number of agents taking the cooperative action is also studied. For general values of r, our model generates a realization of the binary-agent-resource model. The idea of introducing bounded rationality into a two-person game to realize the minority game or binary-agent-resource model could be a useful tool for future research.     

An entropic measure for the teaching-learning process

Few phenomena are as complex as the teaching-learning (TL) process. The instruction efficiency and information on the state of knowledge of the student group are some key variables in this process. Guided by Shannon’s definition of information we propose an entropy based performance index (S_p) for monitoring the teaching-learning process. Our index is based on item response curves (IRCs) which have been recently employed in physics education research. Our proposed index is an explicit function of the ability θ. A preliminary survey indicates that S_p is low (high) for high (low) ability student groups. We propose a simple model to explain this. We have also carried out a number of controlled studies to study the dependence of S_p on student ability, peer instruction and collaborative learning. Our studies indicate that S_p plays a role analogous to entropy in statistical mechanics, with student ability being akin to inverse temperature, peer instruction to an ordering (magnetic) field and collaborative learning to interacting components.