Connected Hypergraphs with Small Spectral Radius

In 1970 Smith classified all connected graphs with the spectral radius at most $2$. Here the spectral radius of a graph is the largest eigenvalue of its adjacency matrix. Recently, the definition of spectral radius has been extended to $r$-uniform hypergraphs. In this paper, we generalize the Smith's theorem to $r$-uniform hypergraphs. We show that the smallest limit point of the spectral radii of connected $r$-uniform hypergraphs is $\rho_r=(r-1)!\sqrt[r]{4}$. We discovered a novel method for computing the spectral radius of hypergraphs, and classified all connected $r$-uniform hypergraphs with spectral radius at most $\rho_r$.Comment: 20 pages, fixed a missing class in theorem 2 and other small typo

Wave-equation based seismic multiple attenuation

Reflection seismology is widely used to map the subsurface geological structure of the Earth. Seismic multiples can contaminate seismic data and are therefore due to be removed. For seismic multiple attenuation, wave-equation based methods are proved to be effective in most cases, which involve two aspects: multiple prediction and multiple subtraction. Targets of both aspects are to develop and apply a fully datadriven algorithm for multiple prediction, and a robust technique for multiple subtraction. Based on many schemes developed by others regarding to the targets, this thesis addresses and tackles the problems of wave-equation based seismic multiple attenuation by several approaches. First, the issue of multiple attenuation in land seismic data is discussed. Multiple Prediction through Inversion (MPTI) method is expanded to be applied in the poststack domain and in the CMP domain to handle the land data with low S/N ratio, irregular geometry and missing traces. A running smooth filter and an adaptive threshold K-NN (nearest neighbours) filter are proposed to help to employ MPTI on land data in the shot domain. Secondly, the result of multiple attenuation depends much upon the effectiveness of the adaptive subtraction. The expanded multi-channel matching (EMCM) filter is proved to be effective. In this thesis, several strategies are discussed to improve the result of EMCM. Among them, to model and subtract the multiples according to their orders is proved to be practical in enhancing the effect of EMCM, and a masking filter is adopted to preserve the energy of primaries. Moreover, an iterative application of EMCM is proposed to give the optimized result. Thirdly, with the limitation of current 3D seismic acquisition geometries, the sampling in the crossline direction is sparse. This seriously affects the application of the 3D multiple attenuation. To tackle the problem, a new approach which applies a trajectory stacking Radon transform along with the energy spectrum is proposed in this thesis. It can replace the time-consuming time-domain sparse inversion with similar effectiveness and much higher efficiency. Parallel computing is discussed in the thesis so as to enhance the efficiency of the strategies. The Message-Passing Interface (MPI) environment is implemented in most of the algorithms mentioned above and greatly improves the efficiency

Existence results for some nonlinear elliptic systems on graphs

In this paper, several nonlinear elliptic systems are investigated on graphs. One type of the sobolev embedding theorem and a new version of the strong maximum principle are established. Then, by using the variational method, the existence of different types of solutions to some elliptic systems is confirmed. Such problems extend the existence results on closed Riemann surface to graphs and extend the existence results for one single equation on graphs [A. Grigor'yan, Y. Lin, Y. Yang, J. Differential Equations, 2016] to nonlinear elliptic systems on graphs. Such problems can also be viewed as one type of discrete version of the elliptic systems on Euclidean space and Riemannian manifold

Gradient estimates for the weighted porous medium equation on graphs

In this paper, we study the gradient estimates for the positive solutions of the weighted porous medium equation $$\Delta u^{m}=\delta(x)u_{t}+\psi u^{m}$$ on graphs for $m>1$, which is a nonlinear version of the heat equation. Moreover, as applications, we derive a Harnack inequality and the estimates of the porous medium kernel on graphs. The obtained results extend the results of Y. Lin, S. Liu and Y. Yang for the heat equation [8, 9]

Convergence and Consistency Analysis for A 3D Invariant-EKF SLAM

In this paper, we investigate the convergence and consistency properties of an Invariant-Extended Kalman Filter (RI-EKF) based Simultaneous Localization and Mapping (SLAM) algorithm. Basic convergence properties of this algorithm are proven. These proofs do not require the restrictive assumption that the Jacobians of the motion and observation models need to be evaluated at the ground truth. It is also shown that the output of RI-EKF is invariant under any stochastic rigid body transformation in contrast to $\mathbb{SO}(3)$ based EKF SLAM algorithm ($\mathbb{SO}(3)$-EKF) that is only invariant under deterministic rigid body transformation. Implications of these invariance properties on the consistency of the estimator are also discussed. Monte Carlo simulation results demonstrate that RI-EKF outperforms $\mathbb{SO}(3)$-EKF, Robocentric-EKF and the "First Estimates Jacobian" EKF, for 3D point feature based SLAM

Convergence analysis for extended Kalman filter based SLAM

The main contribution of this paper is a theoretical analysis of the Extended Kalman Filter (EKF) based solution to the simultaneous localisation and mapping (SLAM) problem. The convergence properties for the general nonlinear two-dimensional SLAM are provided. The proofs clearly show that the robot orientation error has a significant effect on the limit and/or the lower bound of the uncertainty of the landmark location estimates. Furthermore, some insights to the performance of EKF SLAM and a theoretical analysis on the inconsistencies in EKF SLAM that have been recently observed are given. © 2006 IEEE