Criticality in Two-Variable Earthquake Model on a Random Graph

A two-variable earthquake model on a quenched random graph is established here. It can be seen as a generalization of the OFC models. We numerically study the critical behavior of the model when the system is nonconservative： the result indicates that the model exhibits self-organized criticality deep within the nonconservative regime. The probability distribution for avalanche size obeys finite size scaling. We compare our mode/with the mode/ introduced by Stefano Lise and Maya Paczuski [Phys. Rev. Lett. 88 （2002） 228301], it is proved that they are not in the same universality class.     

Criticality in Two-Variable Earthquake Model on a Random Graph

A two-variable earthquake model on a quenched random graph is established here. It can be seen as a generalization of the OFC models. We numerically study the critical behavior of the model when the system is nonconservative: the result indicates that the model exhibits self-organized criticality deep within the nonconservative regime. The probability distribution for avalanche size obeys finite size scaling. We compare our model with the model introduced by Stefano Lise and Maya Paczuski [Phys. Rev. Lett. 88 (2002) 228301], it is proved that they are not in the same universality class.     

Random Graph Asymptotics on High-Dimensional Tori

We investigate the scaling of the largest critical percolation cluster on a large d-dimensional torus, for nearest-neighbor percolation in sufficiently high dimensions, or when d > 6 for sufficiently spread-out percolation. We use a relatively simple coupling argument to show that this largest critical cluster is, with high probability, bounded above by a large constant times V ^2/3 and below by a small constant times , where V is the volume of the torus. We also give a simple criterion in terms of the subcritical percolation two-point function on under which the lower bound can be improved to small constant times , i.e. we prove random graph asymptotics for the largest critical cluster on the high-dimensional torus. This establishes a conjecture by [1], apart from logarithmic corrections. We discuss implications of these results on the dependence on boundary conditions for high-dimensional percolation. Our method is crucially based on the results in [11, 12], where the scaling was proved subject to the assumption that a suitably defined critical window contains the percolation threshold on . We also strongly rely on mean-field results for percolation on proved in [17–20].     

Asymptotic Structure of Constrained Exponential Random Graph Models

In this paper, we study exponential random graph models subject to certain constraints. We obtain some general results about the asymptotic structure of the model. We show that there exists non-trivial regions in the phase plane where the asymptotic structure is uniform and there also exists non-trivial regions in the phase plane where the asymptotic structure is non-uniform. We will get more refined results for the star model and in particular the two-star model for which a sharp transition from uniform to non-uniform structure, a stationary point and phase transitions will be obtained.     

The Condensation Phase Transition in Random Graph Coloring

Based on a non-rigorous formalism called the “cavity method”, physicists have put forward intriguing predictions on phase transitions in diluted mean-field models, in which the geometry of interactions is induced by a sparse random graph or hypergraph. One example of such a model is the graph coloring problem on the Erd?s–Renyi random graph G(n, d/n), which can be viewed as the zero temperature case of the Potts antiferromagnet. The cavity method predicts that in addition to the k-colorability phase transition studied intensively in combinatorics, there exists a second phase transition called the condensation phase transition (Krzakala et al. in Proc Natl Acad Sci 104:10318–10323, 2007). In fact, there is a conjecture as to the precise location of this phase transition in terms of a certain distributional fixed point problem. In this paper we prove this conjecture for k exceeding a certain constant k₀.     

The Integrated Density of States of the Random Graph Laplacian

We analyse the density of states of the random graph Laplacian in the percolating regime. A symmetry argument and knowledge of the density of states in the nonpercolating regime allows us to isolate the density of states of the percolating cluster (DSPC) alone, thereby eliminating trivially localised states due to finite subgraphs. We derive a nonlinear integral equation for the integrated DSPC and solve it with a population dynamics algorithm. We discuss the possible existence of a mobility edge and give strong evidence for the existence of discrete eigenvalues in the whole range of the spectrum.     

On a Fractional Binomial Process

The classical binomial process has been studied by Jakeman (J. Phys. A 23:2815–2825, 1990) (and the references therein) and has been used to characterize a series of radiation states in quantum optics. In particular, he studied a classical birth-death process where the chance of birth is proportional to the difference between a larger fixed number and the number of individuals present. It is shown that at large times, an equilibrium is reached which follows a binomial process. In this paper, the classical binomial process is generalized using the techniques of fractional calculus and is called the fractional binomial process. The fractional binomial process is shown to preserve the binomial limit at large times while expanding the class of models that include non-binomial fluctuations (non-Markovian) at regular and small times. As a direct consequence, the generality of the fractional binomial model makes the proposed model more desirable than its classical counterpart in describing real physical processes. More statistical properties are also derived.     

Limiting Properties of Random Graph Models with Vertex and Edge Weights

This paper provides an overview of results, concerning longest or heaviest paths, in the area of random directed graphs on the integers along with some extensions. We study first-order asymptotics of heaviest paths allowing weights both on edges and vertices and assuming that weights on edges are signed. We aim at an exposition that summarizes, simplifies, and extends proof ideas. We also study sparse graph asymptotics, showing convergence of the weighted random graphs to a certain weighted graph that can be constructed in terms of Poisson processes. We are motivated by numerous applications, ranging from ecology to parallel computing models. It is the latter set of applications that necessitates the introduction of vertex weights. Finally, we discuss some open problems and research directions.     

Agglomeration Evolution of Nano-Particles Aluminium in Normal Incident Shock Wave

Agglomeration behaviour of nano-particle aluminium （nano-Al） in normal incident shock waves is investigated by our devised shock tube technology. The morphology, particle size, agglomeration process of nano-Al studied in normal incident shock waves are comprehensible evaluated by x-ray diffraction, transmission electron microscopy and scanning electron microscopy. The above-mentioned techniques show that the high strength and temperature of incident shock wave give a chance for activity of nano-Al in the reactions and decrease the agglomeration, and the morphology of agglomeration is affected by the temperature of nano-Al reaction region. The dynamic temperature of reaction region determined by the intensity ratio of two A10 bands is 2602K, which is closer to nano-Al actual reacted temperature than the determined temperature of ordinary methods （i.e. six channel instantaneous optical pyrometer; plank black body radiation law, etc.）     

Dynamical Evolution of Modified Chaplygin Gas

Based our previous work [Mod. Phys. Lett. A 22 （2007） 783, Gen. Relat. Gray. 39 （2007） 653], some properties of modified Chaplygin gas （MCG） as a dark energy model continue to be studied mainly in two aspects： one is the change rates of the energy density and energy transfer, and the other is the evolution of the growth index. It is pointed that the density of dark energy undergoes the change from decrease to increase no matter whether the interaction between dark energy and dark matter exists or not, but the corresponding transformation points are different from each other.Eurthermore, it is stressed that the MCG model even supports the existence of interaction between dark energy and dark matter, and the energy of transfer flows from dark energy to dark matter. The evolution of the interaction term with an ansatz 3Hc^2ρ is discussed with the MCG model. Moreover, the evolution of the growth index f in the MCG model without interaction is illustrated, from which we find that the evolutionary trajectory of f overlaps with that of the ACDM model when a 〉 0.7 and its theoretical value f ≈ 0.566 given by us at z = 0.15 is consistent with the observations.     

Influence of Inhomogeneity on Critical Behavior of Earthquake Model on Random Graph

We consider the earthquake model on a random graph. A detailed analysis of the probability distribution of the size of the avalanches will be given. The model with different inhomogeneities is studied in order to compare the critical behavior of different systems. The results indicate that with the increase of the inhomogeneities, the avalanche exponents reduce, i.e., the different numbers of defects cause different critical behaviors of the system. This is virtually ascribed to the dynamical perturbation.     

Influence of Inhomogeneity on Critical Behavior of Earthquake Model on Random Graph

We consider the earthquake model on a random graph. A detailed analysis of the probability distribution of the size of the avalanches will be given. The model with different inhomogeneities is studied in order to compare the critical behavior of different systems. The results indicate that with the increase of the inhomogeneities, the avalanche exponents reduce, i.e., the different numbers of defects cause different critical behaviors of the system. This is virtually ascribed to the dynamical perturbation.     

The Canopy Graph and Level Statistics for Random Operators on Trees

For operators with homogeneous disorder, it is generally expected that there is a relation between the spectral characteristics of a random operator in the infinite setup and the distribution of the energy gaps in its finite volume versions, in corresponding energy ranges. Whereas pure point spectrum of the infinite operator goes along with Poisson level statistics, it is expected that purely absolutely continuous spectrum would be associated with gap distributions resembling the corresponding random matrix ensemble. We prove that on regular rooted trees, which exhibit both spectral types, the eigenstate point process has always Poissonian limit. However, we also find that this does not contradict the picture described above if that is carefully interpreted, as the relevant limit of finite trees is not the infinite homogenous tree graph but rather a single-ended ‘canopy graph.’ For this tree graph, the random Schrödinger operator is proven here to have only pure-point spectrum at any strength of the disorder. For more general single-ended trees it is shown that the spectrum is always singular – pure point possibly with singular continuous component which is proven to occur in some cases.     

Weighted Exponential Random Graph Models: Scope and Large Network Limits

We study models of weighted exponential random graphs in the large network limit. These models have recently been proposed to model weighted network data arising from a host of applications including socio-econometric data such as migration flows and neuroscience. Analogous to fundamental results derived for standard (unweighted) exponential random graph models in the work of Chatterjee and Diaconis, we derive limiting results for the structure of these models as the number of nodes goes to infinity. Our results are applicable for a wide variety of base measures including measures with unbounded support. We also derive sufficient conditions for continuity of functionals in the specification of the model including conditions on nodal covariates. Finally we include a number of open problems to spur further understanding of this model especially in the context of applications.     

On Certain Perturbations of the Erdös–Renyi Random Graph

We study perturbations of the Erdös–Renyi model for which the statistical weight of a graph depends on the abundance of certain geometrical patterns. Using the formal correspondance with an exactly solvable effective model, we show the existence of a percolation transition in the thermodynamical limit and derive perturbatively the expression of the threshold. The free energy and the moments of the degree distribution are also computed perturbatively in that limit and the percolation criterion is compared with the Molloy–Reed criterion.     

Evolution of Number State to Density Operator of Binomial Distribution in the Amplitude Dissipative Channel

We show that passing through the amplitude dissipative channel the initial pure number state density operator is evolved into the density operator of binomial distribution （a mixed state）, and the binomial distribution parameter is just equal to e^-2kt, where k is the dissipative parameter of the channel. We solve the corresponding master equation to obtain the operator-sum representation of the density operator by virtue of the entangled state representation, which seems to be a convenient approach.     

Agglomeration of sub-micron particles by a non-thermal plasma electrostatic precipitator

Non-thermal Plasma agglomeration is presented as a promising process to reduce the number concentration of sub-micron particles in an acrylic duct, which included a saw-tooth electrode and a wire-plate electrostatic precipitator (ESP). The generated plasma by pulse-energized ESP, the particle agglomerations were controlled under operating conditions such as pulse voltages, pulse frequencies, dust loadings, and gas velocities. When gas velocity increased from 0.5 to 1 m s⁻¹ at 45 kV_p, 20 kHz, it was found that efficiency was increased. At gas velocity of 1 m s⁻¹, the sub-micron particle number reduction efficiency for all particle sizes was over 90% in ESP.     

Evolution Law of the Negative Binomial State in Laser Channel and its Photon-Number Decay Formula

For the first time we examine how a negative binomial state (NBS), whose density operator is \({\sum }_{n=0}^{\infty }\frac {\left (n+s\right ) !} {n!s!}\gamma ^{s+1}\left (1-\gamma \right )^{n}\left \vert n\right \rangle \left \langle n\right \vert ,\) evolves in a laser channel. By using a newly derived generating function formula about Laguerre polynomial we obtain the evolution law of NBS, which turns out to be an infinite operator-sum of photon-added negative binomial state with a new negative-binomial parameter, and the photon number of NBS decays with e ^?2(κ?g)t , where g and κ represent the cavity gain and loss respectively. The technique of integration (summation) within an ordered product of operators is used in our discussions.