Super-Brownian motion with extra birth at one point

A super-Brownian motion in two and three dimensions is constructed where "particles" give birth at a higher rate, if they approach the origin. Via a log-Laplace approach, the construction is based on Albeverio et al. (1995) who calculated the fundamental solutions of the heat equation with one-point potential in dimensions less than four

On the large scale behavior of super-Brownian motion in three dimensions with a single point source

In a recent work, Fleischmann and Mueller (2004) showed the existence of a super-Brownian motion in R^d, d=2,3, with extra birth at the origin. Their construction made use of an analytical approach based on the fundamental solution of the heat equation with a one point potential worked out by Albeverio et al. (1995). The present note addresses two properties of this measure-valued process in the three-dimensional case, namely the scaling of the process and the large scale behavior of its mean

Force Statistics and Correlations in Dense Granular Packings

In dense, static, polydisperse granular media under isotropic pressure, the probability density and the correlations of particle-wall contact forces are studied. Furthermore, the probability density functions of the populations of pressures measured with different sized circular pressure cells is examined. The questions answered are: (i) What is the number of contacts that has to be considered so that the measured pressure lies within a certain error margin from its expectation value? (ii) What is the statistics of the pressure probability density as function of the size of the pressure cell? Astonishing non-random correlations between contact forces are evidenced, which range at least 10 to 15 particle diameter. Finally, an experiment is proposed to tackle and better understand this issue.Comment: 10 pages, 12 figure

Two interacting particles in a random potential: mapping onto one parameter localization theories without interaction

We consider two models for a pair of interacting particles in a random potential: (i) two particles with a Hubbard interaction in arbitrary dimensions and (ii) a strongly bound pair in one dimension. Establishing suitable correpondences we demonstrate that both cases can be described in terms familiar from theories of noninteracting particles. In particular, these two cases are shown to be controlled by a single scaling variable, namely the pair conductance $g_2$. For an attractive or repulsive Hubbard interaction and starting from a certain effective Hamiltonian we derive a supersymmetric nonlinear $\sigma$ model. Its action turns out to be closely related to the one found by Efetov for noninteracting electrons in disordered metals. This enables us to describe the diffusive motion of the particle pair on scales exceeding the one-particle localization length $L_1$ and to discuss the corresponding level statistics. For tightly bound pairs in one dimension, on the other hand, we follow early work by Dorokhov and exploit the analogy with the transfer matrix approach to quasi 1d conductors. Extending our study to M particles we obtain a M-particle localization length scaling like the Mth power of the one-particle localization length.Comment: 29 pages, Revtex, no figure

Beyond saliency: understanding convolutional neural networks from saliency prediction on layer-wise relevance propagation

Despite the tremendous achievements of deep convolutional neural networks (CNNs) in many computer vision tasks, understanding how they actually work remains a significant challenge. In this paper, we propose a novel two-step understanding method, namely Salient Relevance (SR) map, which aims to shed light on how deep CNNs recognize images and learn features from areas, referred to as attention areas, therein. Our proposed method starts out with a layer-wise relevance propagation (LRP) step which estimates a pixel-wise relevance map over the input image. Following, we construct a context-aware saliency map, SR map, from the LRP-generated map which predicts areas close to the foci of attention instead of isolated pixels that LRP reveals. In human visual system, information of regions is more important than of pixels in recognition. Consequently, our proposed approach closely simulates human recognition. Experimental results using the ILSVRC2012 validation dataset in conjunction with two well-established deep CNN models, AlexNet and VGG-16, clearly demonstrate that our proposed approach concisely identifies not only key pixels but also attention areas that contribute to the underlying neural network's comprehension of the given images. As such, our proposed SR map constitutes a convenient visual interface which unveils the visual attention of the network and reveals which type of objects the model has learned to recognize after training. The source code is available at https://github.com/Hey1Li/Salient-Relevance-Propagation.Comment: 35 pages, 15 figure