Role of fractal dimension in random walks on scale-free networks

Fractal dimension is central to understanding dynamical processes occurring on networks; however, the relation between fractal dimension and random walks on fractal scale-free networks has been rarely addressed, despite the fact that such networks are ubiquitous in real-life world. In this paper, we study the trapping problem on two families of networks. The first is deterministic, often called $(x,y)$-flowers; the other is random, which is a combination of $(1,3)$-flower and $(2,4)$-flower and thus called hybrid networks. The two network families display rich behavior as observed in various real systems, as well as some unique topological properties not shared by other networks. We derive analytically the average trapping time for random walks on both the $(x,y)$-flowers and the hybrid networks with an immobile trap positioned at an initial node, i.e., a hub node with the highest degree in the networks. Based on these analytical formulae, we show how the average trapping time scales with the network size. Comparing the obtained results, we further uncover that fractal dimension plays a decisive role in the behavior of average trapping time on fractal scale-free networks, i.e., the average trapping time decreases with an increasing fractal dimension.Comment: Definitive version published in European Physical Journal

PCA Based Robust Motion Data Recovery.

Human motion tracking is a prevalent technique in many fields. A common difficulty encountered in motion tracking is the corrupted data is caused by detachment of markers in 3D motion data or occlusion in 2D tracking data. Most methods for missing markers problem may quickly become ineffective when gaps exist in the trajectories of multiple markers for an extended duration. In this paper, we propose the principal component eigenspace based gap filling methods that leverage a training sample set for estimation. The proposed method is especially beneficial in the scenario of motion data with less predictable or repeated movement patterns, and that of even missing entire frames within an interval of a sequence. To highlight algorithm robustness, we perform algorithms on twenty test samples for comparison. The experimental results show that our methods are numerical stable and fast to work

Transition from fractal to non-fractal scalings in growing scale-free networks

Real networks can be classified into two categories: fractal networks and non-fractal networks. Here we introduce a unifying model for the two types of networks. Our model network is governed by a parameter $q$. We obtain the topological properties of the network including the degree distribution, average path length, diameter, fractal dimensions, and betweenness centrality distribution, which are controlled by parameter $q$. Interestingly, we show that by adjusting $q$, the networks undergo a transition from fractal to non-fractal scalings, and exhibit a crossover from `large' to small worlds at the same time. Our research may shed some light on understanding the evolution and relationships of fractal and non-fractal networks.Comment: 7 pages, 3 figures, definitive version accepted for publication in EPJ

The effects of pre-ignition turbulence by gas jets on the explosion behavior of methane-oxygen mixtures

Most of the previous studies investigating explosion characteristics of combustible mixtures were performed at quiescent state. However, in realistic accidental explosion scenarios, the ignition of the combustible mixture usually occurs under a turbulent environment. In this study, we examine the maximum explosion pressure pmax and explosion time τe of CH4-2O2 mixtures under the pre-ignition turbulence condition in a spherical closed chamber at a room temperature of 298 K. Turbulence is generated using fluidic jet of three different gases (O2, CO2 and N2) and its intensity is controlled by changing the initial pressure of the gas jet pJ0 (i.e., 200 and 500 kPa) and the explosion chamber pressure p0 (i.e., 40 and 60 kPa). The dual effects of turbulence and gas dilution on the explosion behavior of CH4-2O2 mixtures are investigated in detail. The results indicate that by adding O2 into CH4-2O2 mixture at quiescent condition, pmax increases but the rate of overpressure rise is reduced. By introducing turbulence through gas jets into the combustible mixture, the explosion behavior is affected by both the turbulence and gas dilution. With O2 injection, turbulence overall enhances the explosion, but the amount of O2 dilution increases at higher pJ0/p0 and longer jet duration time (tJ0), rendering the mixture to tend toward fuel-lean side and slow down the explosion rate. The present results also demonstrate that the turbulence effect of CO2 is more profound than that of N2 jet. Both pmax and τe are enhanced by CO2 jet turbulence when tJ0 is relative short (tJ0 < 400 ms). However, for longer tJ0, the dominance of CO2 dilution becomes more noticeably than N2 dilution with a longer explosion time τe

Rotational symmetry of self-similar solutions to the Ricci flow

Let (M,g) be a three-dimensional steady gradient Ricci soliton which is non-flat and \kappa-noncollapsed. We prove that (M,g) is isometric to the Bryant soliton up to scaling. This solves a problem mentioned in Perelman's first paper.Comment: Final version, to appear in Invent. Mat

From semiclassical transport to quantum Hall effect under low-field Landau quantization

The crossover from the semiclassical transport to quantum Hall effect is studied by examining a two-dimensional electron system in an AlGaAs/GaAs heterostructure. By probing the magneto-oscillations, it is shown that the semiclassical Shubnikov-de Haas (SdH) formulation can be valid even when the minima of the longitudinal resistivity approach zero. The extension of the applicable range of the SdH theory could be due to the damping effects resulting from disorder and temperature. Moreover, we observed plateau-plateau transition like behavior with such an extension. From our study, it is important to include the positive magnetoresistance to refine the SdH theory.Comment: 11 pages, 5 figure

Fourier Acceleration of Langevin Molecular Dynamics

Fourier acceleration has been successfully applied to the simulation of lattice field theories for more than a decade. In this paper, we extend the method to the dynamics of discrete particles moving in continuum. Although our method is based on a mapping of the particles' dynamics to a regular grid so that discrete Fourier transforms may be taken, it should be emphasized that the introduction of the grid is a purely algorithmic device and that no smoothing, coarse-graining or mean-field approximations are made. The method thus can be applied to the equations of motion of molecular dynamics (MD), or its Langevin or Brownian variants. For example, in Langevin MD simulations our acceleration technique permits a straightforward spectral decomposition of forces so that the long-wavelength modes are integrated with a longer time step, thereby reducing the time required to reach equilibrium or to decorrelate the system in equilibrium. Speedup factors of up to 30 are observed relative to pure (unaccelerated) Langevin MD. As with acceleration of critical lattice models, even further gains relative to the unaccelerated method are expected for larger systems. Preliminary results for Fourier-accelerated molecular dynamics are presented in order to illustrate the basic concepts. Possible extensions of the method and further lines of research are discussed.Comment: 11 pages, two illustrations included using graphic

Laser-induced collective excitations in a two-component Fermi gas

We consider the linear density response of a two-component (superfluid) Fermi gas of atoms when the perturbation is caused by laser light. We show that various types of laser excitation schemes can be transformed into linear density perturbations, however, a Bragg spectroscopy scheme is needed for transferring energy and momentum into a collective mode. This makes other types of laser probing schemes insensitive for collective excitations and therefore well suited for the detection of the superfluid order parameter. We show that for the special case when laser light is coupled between the two components of the Fermi gas, density response is always absent in a homogeneous system.Comment: 6 pages, no figure

Numerical analysis of the radio-frequency single-electron transistor operation

We have analyzed numerically the response and noise-limited charge sensitivity of a radio-frequency single-electron transistor (RF-SET) in a non-superconducting state using the orthodox theory. In particular, we have studied the performance dependence on the quality factor Q of the tank circuit for Q both below and above the value corresponding to the impedance matching between the coaxial cable and SET.Comment: 14 page