Exponential decay of correlations in the two-dimensional random field Ising model

We study random field Ising model on $\mathbb Z^2$ where the external field is given by i.i.d.\ Gaussian variables with mean zero and positive variance. We show that the effect of boundary conditions on the magnetization in a finite box decays exponentially in the distance to the boundary.Comment: 1, This document merges arXiv:1902.03302 and a previous version arXiv:1905.05651v2. The title and abstract are revised accordingly. 2, We also improved exposition by incorporating comments from two anonymous referee

Feynman-Kac formula for fractional heat equation driven by fractional white noise

In this paper we obtain a Feynman-Kac formula for the solution of a fractional stochastic heat equation driven by fractional noise. One of the main difficulties is to show the exponential integrability of some singular nonlinear functionals of symmetric stable L\'evy motion. This difficulty will be overcome by a technique developed in the framework of large deviation. This Feynman-Kac formula is applied to obtain the H\"older continuity and moment formula of the solution.Comment: 27 pages. arXiv admin note: substantial text overlap with arXiv:0906.307

Test of the cosmic evolution using Gaussian processes

Much focus was on the possible slowing down of cosmic acceleration under the dark energy parametrization. In the present paper, we investigate this subject using the Gaussian processes (GP), without resorting to a particular template of dark energy. The reconstruction is carried out by abundant data including luminosity distance from Union2, Union2.1 compilation and gamma-ray burst, and dynamical Hubble parameter. It suggests that slowing down of cosmic acceleration cannot be presented within 95\% C.L., in considering the influence of spatial curvature and Hubble constant. In order to reveal the reason of tension between our reconstruction and previous parametrization constraint for Union2 data, we compare them and find that slowing down of acceleration in some parametrization is only a "mirage". Although these parameterizations fits well with the observational data, their tension can be revealed by high order derivative of distance $D$. Instead, GP method is able to faithfully model the cosmic expansion history.Comment: 9 pages, 10 figures, 2 tables, accepted for publication in JCA

Temporal asymptotics for fractional parabolic Anderson model

In this paper, we consider fractional parabolic equation of the form $\frac{\partial u}{\partial t}=-(-\Delta)^{\frac{\alpha}{2}}u+u\dot W(t,x)$, where $-(-\Delta)^{\frac{\alpha}{2}}$ with $\alpha\in(0,2]$ is a fractional Laplacian and $\dot W$ is a Gaussian noise colored in space and time. The precise moment Lyapunov exponents for the Stratonovich solution and the Skorohod solution are obtained by using a variational inequality and a Feynman-Kac type large deviation result for space-time Hamiltonians driven by $\alpha$-stable process. As a byproduct, we obtain the critical values for $\theta$ and $\eta$ such that $\mathbb{E}\exp\left(\theta\left(\int_0^1 \int_0^1 |r-s|^{-\beta_0}\gamma(X_r-X_s)drds\right)^\eta\right)$ is finite, where $X$ is $d$-dimensional symmetric $\alpha$-stable process and $\gamma(x)$ is $|x|^{-\beta}$ or $\prod_{j=1}^d|x_j|^{-\beta_j}$

Is Extreme Learning Machine Feasible? A Theoretical Assessment (Part II)

An extreme learning machine (ELM) can be regarded as a two stage feed-forward neural network (FNN) learning system which randomly assigns the connections with and within hidden neurons in the first stage and tunes the connections with output neurons in the second stage. Therefore, ELM training is essentially a linear learning problem, which significantly reduces the computational burden. Numerous applications show that such a computation burden reduction does not degrade the generalization capability. It has, however, been open that whether this is true in theory. The aim of our work is to study the theoretical feasibility of ELM by analyzing the pros and cons of ELM. In the previous part on this topic, we pointed out that via appropriate selection of the activation function, ELM does not degrade the generalization capability in the expectation sense. In this paper, we launch the study in a different direction and show that the randomness of ELM also leads to certain negative consequences. On one hand, we find that the randomness causes an additional uncertainty problem of ELM, both in approximation and learning. On the other hand, we theoretically justify that there also exists an activation function such that the corresponding ELM degrades the generalization capability. In particular, we prove that the generalization capability of ELM with Gaussian kernel is essentially worse than that of FNN with Gaussian kernel. To facilitate the use of ELM, we also provide a remedy to such a degradation. We find that the well-developed coefficient regularization technique can essentially improve the generalization capability. The obtained results reveal the essential characteristic of ELM and give theoretical guidance concerning how to use ELM.Comment: 13 page

Cyclic Delay Transmission for Vector OFDM Systems

Single antenna vector OFDM (V-OFDM) system has been proposed and investigated in the past. It contains the conventional OFDM and the single carrier frequency domain equalizer (SC-FDE) as two special cases and is flexible to choose any number of symbols in intersymbol interference (ISI) by choosing a proper vector size. In this paper, we develop cyclic delay diversity (CDD) transmission for V-OFDM when there are multiple transmit antennas (CDD-V-OFDM). Similar to CDD-OFDM systems, CDD-V-OFDM can also collect both spatial and multipath diversities. Since V-OFDM first converts a single input single output (SISO) ISI channel to a multi-input and multi-output (MIMO) ISI channel of order/length $K$ times less, where $K$ is the vector size, for a given bandwidth, the CDD-V-OFDM can accommodate $K$ times more transmit antennas than the CDD-OFDM does to collect all the spatial and multipath diversities. This property will specially benefit a massive MIMO system. We show that with the linear MMSE equalizer at each subcarrier, the CDD-V-OFDM achieves diversity order $d_{\text{CDD-V-OFDM}}^{\text{MMSE}} = \min \{ \lfloor 2^{-R}K \rfloor, N_t L \} +1$, where $R$ is the transmission rate, $N_t$ is the number of transmit antennas, and $L$ is the ISI channel length between each transmit and receive antenna pair. Simulations are presented to illustrate our theory.Comment: 26 pages, 8 figures, Transaction on Wireless Communicatio

Distributed Solver for Discrete-Time Lyapunov Equations Over Dynamic Networks with Linear Convergence Rate

This paper investigates the problem of solving discrete-time Lyapunov equations (DTLE) over a multi-agent system, where every agent has access to its local information and communicates with its neighbors. To obtain a solution to DTLE, a distributed algorithm with uncoordinated constant step sizes is proposed over time-varying topologies. The convergence properties and the range of constant step sizes of the proposed algorithm are analyzed. Moreover, a linear convergence rate is proved and the convergence performances over dynamic networks are verified by numerical simulations

An optimal consensus tracking control algorithm for autonomous underwater vehicles with disturbances

The optimal disturbance rejection control problem is considered for consensus tracking systems affected by external persistent disturbances and noise. Optimal estimated values of system states are obtained by recursive filtering for the multiple autonomous underwater vehicles modeled to multi-agent systems with Kalman filter. Then the feedforward-feedback optimal control law is deduced by solving the Riccati equations and matrix equations. The existence and uniqueness condition of feedforward-feedback optimal control law is proposed and the optimal control law algorithm is carried out. Lastly, simulations show the result is effectiveness with respect to external persistent disturbances and noise

ECoPANN: A Framework for Estimating Cosmological Parameters using Artificial Neural Networks

In this work, we present a new method to estimate cosmological parameters accurately based on the artificial neural network (ANN), and a code called ECoPANN (Estimating Cosmological Parameters with ANN) is developed to achieve parameter inference. We test the ANN method by estimating the basic parameters of the concordance cosmological model using the simulated temperature power spectrum of the cosmic microwave background (CMB). The results show that the ANN performs excellently on best-fit values and errors of parameters, as well as correlations between parameters when compared with that of the Markov Chain Monte Carlo (MCMC) method. Besides, for a well-trained ANN model, it is capable of estimating parameters for multiple experiments that have different precisions, which can greatly reduce the consumption of time and computing resources for parameter inference. Furthermore, we extend the ANN to a multibranch network to achieve a joint constraint on parameters. We test the multibranch network using the simulated temperature and polarization power spectra of the CMB, Type Ia supernovae, and baryon acoustic oscillations, and almost obtain the same results as the MCMC method. Therefore, we propose that the ANN can provide an alternative way to accurately and quickly estimate cosmological parameters, and ECoPANN can be applied to the research of cosmology and even other broader scientific fields.Comment: 21 pages, 14 figures, and 7 tables, matches published version. The code repository is available at https://github.com/Guo-Jian-Wang/ecopan

Propagation Phenomena for Nonlocal Dispersal Equations in Exterior Domains

This paper is concerned with the spatial propagation of nonlocal dispersal equations with bistable or multistable nonlinearity in exterior domains. We obtain the existence and uniqueness of an entire solution which behaves like a planar wave front as time goes to negative infinity. In particular, some disturbances on the profile of the entire solution happen as the entire solution comes to the interior domain. But the disturbances disappear as the entire solution is far away from the interior domain. Furthermore, we prove that the solution can gradually recover its planar wave profile and continue to propagate in the same direction as time goes to positive infinity for compact convex interior domain. Our work generalizes the local (Laplace) diffusion results obtained by Berestycki et al. (2009) to the nonlocal dispersal setting by using new known Liouville results and Lipschitz continuity of entire solutions due to Li et al. (2010)