What is the suitable Laplace operator on vector fields for the Navier-Stokes equation on a Riemannian manifold? In this note, by considering Nash embedding, we will try to elucidate different aspects of different Laplace operators such as de Rham-Hodge Laplacian as well as Ebin-Marsden's Laplacian. A probabilistic representation formula for Navier-Stokes equations on a general compact Riemannian manifold is obtained when de Rham-Hodge Laplacian is involved.
By using a split argument due to[1], the transportation cost inequality is established on the free path space of Markov processes. The general result is applied to stochastic reaction diffusion equations with random initial values.
A new approach to modeling populations incorporating stochasticity, a random environment, and individual behavior is illustrated with a specific example of two interacting populations:rabbits and grass. The derivation of the system of stochastic partial differential equations (SPDEs) to show how the individual mechanisms of both populations are included. This model also has an unusual feature of a nonlocal term. The harvesting of the rabbit population is introduced as a control variable.
Let X_{1}, X_{2}, …, X_{n}, … be a sequence of i.i.d. random variables uniformly distributed on[0; 1], and denote by L_{n} the length of the longest increasing subsequences of X_{1}, X_{2}, …, X_{n}. Consider the poissonized version H_{n} based on Hammersley's representation in the 2-dimensional space. A law of the iterated logarithm for H_{n} is established using the well-known subsequence method and Borel-Cantelli lemma. The key technical ingredients in the argument include superadditivity, increment independence and precise tail estimates for the H_{n}'s. The work was motivated by recent works due to Ledoux (J. Theoret. Probab. 31, (2018)). It remains open to establish an analog for the L_{n} itself.
Let {x_{n}, n ≥ 0} be a Markov chain with a countable state space S and let f(·) be a measurable function from S to R and consider the functionals of the Markov chain y_{n}:=f(x_{n}). We construct a new type of self-normalized sums based on the random-block scheme and establish a Cramér-type moderate deviations for self-normalized sums of functionals of the Markov chain.
This paper discusses robust nonparametric estimators of location regression function for errorsin-variables model with de-convolution kernel. The local constant smoother is used for the estimation of the nonparametric function, and the local linear smoother is proposed to deal with the boundary problem, as well as to improve the local constant smoother. We establish the asymptotic properties of the estimator, the influence function of the statistical functional and the breakdown point. A simulation study is carried out to demonstrate robust performance of the proposed estimator. The motorcycle data is presented to illustrate the application of the robust estimator further.
We combine the robust criterion with the lasso penalty together for the high-dimensional threshold model. It estimates regression coefficients as well as the threshold parameter robustly that can be resistant to outliers or heavy-tailed noises and perform variable selection simultaneously. We illustrate our approach with the absolute loss, the Huber's loss, and the Tukey's loss, it can also be extended to any other robust losses. Simulation studies are conducted to demonstrate the usefulness of our robust approach. Finally, we use our estimators to investigate the presence of a shift in the effect of debt on future GDP growth.
In this paper, we study the stochastic partial differential equation with two reflecting smooth walls h^{1} and h^{2}, driven by a fractional noise, which is fractional in time and white in space. The large deviation principle for the law of the solution to this equation, will be established through developing a classical method. Furthermore, we obtain the Hölder continuity of the solution.
For a positive continuous function f satisfying some standard conditions, we study the f-moments of continuous-state branching processes with or without immigration. The main results give criteria for the existence of the f-moments. The characterization of the processes in terms of stochastic equations plays an essential role in the proofs.
In functional data analysis, the collected data are often assumed to be fully observed on the domain. However, in dealing with real data (for example, environmental pollution data), we are often faced with the scenario that some functional data are fully observed on dense lattice while others are incompletely observed. In this paper, we propose a method for testing equivalence of mean functions of two samples under this scenario. Some asymptotic results of the proposed methods are established. The proposed test is employed to analyze an environmental pollution study in Liuzhou City of China. Simulations show that the proposed test has a good control of the type-I error, and is more powerful than the complete case test in most cases.
In this paper, we study the periodic solutions to a type of differential delay equations with 2k-1 lags. The 4k-periodic solutions are obtained by using the variational method and the method of Kaplan-Yorke coupling system. This is a new type of differential delay equations compared with all the previous researches. And this paper provides a theoretical basis for the study of differential delay equations. An example is given to demonstrate our main results.
Riccati equation approach is used to look for exact travelling wave solutions of some nonlinear physical models. Solitary wave solutions are established for the modified KdV equation, the Boussinesq equation and the Zakharov-Kuznetsov equation. New generalized solitary wave solutions with some free parameters are derived. The obtained solutions, which includes some previously known solitary wave solutions and some new ones, are expressed by a composition of Riccati differential equation solutions followed by a polynomial. The employed approach, which is straightforward and concise, is expected to be further employed in obtaining new solitary wave solutions for nonlinear physical problems.
In this paper, we first consider the classical p-median location problem on a network in which the vertex weights and the distances between vertices are uncertain variables. The uncertainty distribution of the optimal objective value of the p-median problem is given and the concepts of the α-p-median, the most p-median and the expected p-median are introduced. Then, it is shown that the uncertain p-median problem is NP-hard on general networks. However, if the underlying network is a tree, an efficient algorithm for the uncertain 1-median problem with linear time complexity is proposed. Finally, we investigate the inverse 1-median problem on a tree with uncertain vertex weights and present a programming model for the problem. Then, it is shown that the proposed model can be reformulated into a deterministic programming model.
The vertex-arboricity a(G) of a graph G is the minimum number of colors required for a vertex coloring of G such that no cycle is monochromatic. The list vertex-arboricity a_{l}(G) is the list-coloring version of this concept. In this paper, we prove that every planar graph G without intersecting 5-cycles has a_{l}(G) ≤ 2. This extends a result by Raspaud and Wang[On the vertex-arboricity of planar graphs, European J. Combin. 29 (2008), 1064-1075] that every planar graph G without 5-cycles has a(G) ≤ 2.
In this article, we put forward a new approach to estimate multiple conditional regression quantiles simultaneously. Unlike the double summation method in most of the literatures, our proposed model allows continuous variety for the quantile level over (0,1). As a result, all the quantile curves can be obtained via a 2-dimensional surface simultaneously. Most importantly, the proposed minimizing criterion can be readily transformed to a linear programming problem. We use tensor product bi-linear quantile smoothing B-splines to fit it. The asymptotic property of the estimator is derived and a real data set is analyzed to demonstrate the proposed method.
A robust and efficient shrinkage-type variable selection procedure for varying coefficient models is proposed, selection consistency and oracle properties are established. Furthermore, a BIC-type criterion is suggested for shrinkage parameter selection and theoretical property is discussed. Numerical studies and real data analysis also are included to illustrate the finite sample performance of our method.
In this paper, nonconforming finite element methods (FEMs) are proposed for the constrained optimal control problems (OCPs) governed by the nonsmooth elliptic equations, in which the popular EQ_{1}^{rot} element is employed to approximate the state and adjoint state, and the piecewise constant element is used to approximate the control. Firstly, the convergence and superconvergence properties for the nonsmooth elliptic equation are obtained by introducing an auxiliary problem. Secondly, the goal-oriented error estimates are obtained for the objective function through establishing the negative norm error estimate. Lastly, the methods are extended to some other well-known nonconforming elements.
This paper presents a novel genetic algorithm for globally solving un-constraint optimization problem. In this algorithm, a new real coded crossover operator is proposed firstly. Furthermore, for improving the convergence speed and the searching ability of our algorithm, the good point set theory rather than random selection is used to generate the initial population, and the chaotic search operator is adopted in the best solution of the current iteration. The experimental results tested on numerical benchmark functions show that this algorithm has excellent solution quality and convergence characteristics, and performs better than some algorithms.
Let p be a prime, q be a power of p, and let F_{q} be the field of q elements. For any positive integer n, the Wenger graph W_{n}(q) is defined as follows:it is a bipartite graph with the vertex partitions being two copies of the (n+1)-dimensional vector space F_{q}^{n+1}, and two vertices p=(p(1), …, p(n+1)) and l=[l(1), …, l(n+1)] being adjacent if p(i) + l(i)=p(1)l(1)^{i-1}, for all i=2, 3, …, n + 1. In 2008, Shao, He and Shan showed that for n ≥ 2, W_{n}(q) contains a cycle of length 2k where 4 ≤ k ≤ 2p and k≠ 5. In this paper we extend their results by showing that (i) for n ≥ 2 and p ≥ 3, W_{n}(q) contains cycles of length 2k, where 4 ≤ k ≤ 4p + 1 and k≠ 5; (ii) for q ≥ 5, 0 < c < 1, and every integer k, 3 ≤ k ≤ q^{c}, if 1 ≤ n < (1-c-7/3 log_{q} 2)k-1, then W_{n}(q) contains a 2k-cycle. In particular, W_{n}(q) contains cycles of length 2k, where n + 2 ≤ k ≤ q^{c}, provided q is sufficiently large.
In this paper, we study the dispersive properties of multi-symplectic discretizations for the nonlinear Schrödinger equations. The numerical dispersion relation and group velocity are investigated. It is found that the numerical dispersion relation is relevant when resolving the nonlinear Schrödinger equations.
In this paper, by using an extension of Mawhin's continuation theorem and some analysis methods, we study the existence of periodic solutions for the following prescribed mean curvature system d/dtφ(x') + ▽W(x)=p(t), where x ∈ R^{n}, W ∈ C^{1}(R^{n}, R), p ∈ C(R, R^{n}) is T-periodic and φ(x)=(x/(√1+|x|^{2})).
In this paper, the modified projective synchronization between two fractional-order chaotic systems with different dimensions is investigated. The added-order scheme and the reduced-order scheme are proposed, respectively. Based on the Laplace transformation and feedback control theory, controllers are designed such that two chaotic systems with different dimensions could be synchronized asymptotically under the presented schemes. Corresponding numerical simulations are given to show the effectiveness of the proposed schemes.