In this paper, we establish a novel unique continuation property for two-dimensional anisotropic elasticity systems with partial information. More precisely, given a homogeneous elasticity system in a connected open bounded domain, we investigate the unique continuation by assuming only the vanishing of one component of the solution in a subdomain. Using the corresponding Riemann function, we prove that the solution vanishes in the whole domain provided that the other component vanishes at one point up to its second derivatives. Further, we construct several examples showing the possibility of further reducing the additional information of the other component. This result possesses remarkable significance in both theoretical and practical aspects because the required data are almost halved for the unique determination of the whole solution.

The time-domain multiple cavity scattering problem, which arises in diverse scientific areas, has significant industrial and military applications. The multiple cavities, embedded in an infinite ground plane, is filled with inhomogeneous media characterized by variable dielectric permittivities and magnetic permeabilities. Corresponding to the transverse electric, the scattering problem can be studied by the Helmholtz equation in frequency domain and wave equation in time-domain respectively. A novel transparent boundary condition in time-domain is developed to reformulate the cavity scattering problem into an initial-boundary value problem in a bounded domain. The well-posedness and stability of the reduced problem are established. Moreover, a priori energy estimates for the electric field is obtained with minimum regularity requirement for the data and an explicit dependence on the time by studying the wave equation directly.

Based on sparse information recovery, we develop a new method for locating multiple multiscale acoustic scatterers. Firstly, with the prior information of the scatterers' shape, we reformulate the location identification problem into a sparse information recovery model which brought the power of sparse recovery method into this type of inverse scattering problems. Specifically, the new model can advance the judgment of the existence of alternative scatterers and, in the meantime, conclude the number and locating of each existing scatterers. Secondly, as well known, the core model (l₀-minimization) in sparse information recovery is an NP-hard problem. According to the characteristics of the proposed sparse model, we present a new substitute method and give a detailed theoretical analysis of the new substitute model. Relying on the properties of the new model, we construct a basic algorithm and an improved one. Finally, we verify the validity of the proposed method through two numerical experiments.

Consider the inverse problem of recovering a multi-layered fluid-solid medium from many acoustic measurements corresponding to time-harmonic acoustic plane waves. We prove that both the supports of the embedded solid obstacle and the surrounding layered fluid medium can be uniquely identified by means of acoustic far-field pattern for all incident wave fields at a fixed frequency. Our proof is based on the constructions of some well-posed partial differential equation systems in sufficiently small domains combined with the a priori estimates for the solutions of the forward scattering problem.

We consider the inverse fluid-solid interaction scattering of incident plane wave from the knowledge of the phased and phaseless far field patterns. For the phased data, one direct sampling method for location and shape reconstruction is proposed. Only inner product is involved in the computation, which makes it very simple and fast to be implemented. With the help of the factorization of the far field operator, we give a lower bound of the proposed indicator functional for the sampling points inside the elastic body. While for the sampling points outside, we show that the indicator functional decays like the Bessel function as the points go away from the boundaries of the elastic body. We also show that the proposed indicator functional continuously dependents on the far field patterns, which further implies that the novel sampling method is extremely stable with respect to data error. For the phaseless data, to overcome the translation invariance, we consider the scattering of point sources simultaneously. By adding a reference sound-soft obstacle into the scattering system, we show some uniqueness results with phaseless far field data. Numerically, we introduce a phase retrieval algorithm to retrieve the phased data without the additional obstacle. The novel phase retrieval algorithm can also be combined with the sampling method for phased data. We also design two novel direct sampling methods using the phaseless data directly. Finally, some numerical simulations in two dimensions are conducted with noisy data, and the results further verify the effectiveness and robustness of the proposed numerical methods.

This paper considers the scattering of a time-dependent electromagnetic plane wave by a bounded elastic body. An appropriate decomposition of the coupling interface conditions is proposed according to the Voigt's model between the electromagnetic and elastic medium. The original unbounded scattering problem is equivalently reduced into an initial-boundary value problem in a bounded domain by introducing an exact transparent boundary condition (TBC) on a sufficiently large sphere. Making use of the Lax-Milgram lemma, the abstract inversion theorem of Laplace transform and the energy method, we verify the well-posedness and stability for the reduced problem. Moreover, a priori estimates are established for the electromagnetic field and elastic displacement by taking special test functions directly in the time domain variational formulation.

In this paper, we consider the inverse acoustic scattering problem by an unbounded rough surface. A direct imaging method is proposed to reconstruct the rough surfaces from scattered-field data for incident plane waves and the performance analysis is also presented. The reconstruction method is very robust to noises of measured data and does't need to know the type of the boundary conditions of the surfaces in advance. Finally, numerical examples are carried out to illustrate that our method is fast, accurate and stable even for the case of multiple-scale profiles.

This paper is concerned with inverse acoustic source problems in an unbounded domain with dynamical boundary surface data of Dirichlet kind. The measurement data are taken at a surface far away from the source support. We prove uniqueness in recovering source terms of the form f(x)g(t) and f(x₁, x₂, t)h(x₃), where g(t) and h(x₃) are given and x=(x₁, x₂, x₃) is the spatial variable in three dimensions. Without these a priori information, we prove that the boundary data of a family of solutions can be used to recover general source terms depending on both time and spatial variables. For moving point sources radiating periodic signals, the data recorded at four receivers are prove sufficient to uniquely recover the orbit function. Simultaneous determination of embedded obstacles and source terms was verified in an inhomogeneous background medium using the observation data of infinite time period. Our approach depends heavily on the Laplace transform.

In this paper, we consider an inverse time-dependent source problem of heat conduction equation. Firstly, the ill-posedness and conditional stability of this inverse source problem is analyzed. Then, a finite difference inversion method is proposed for reconstructing the time-dependent source from a nonlocal measurement. The existence and uniqueness of the finite difference inverse solutions are rigorously analyzed, and the convergence is proved. Combined with the mollification method, the proposed finite difference inversion method can obtain more stable reconstructions from the nonlocal data with noise. Finally, numerical examples are given to illustrate the efficiency and convergence of the proposed finite difference inversion method.

Consider a linear inverse problem of determining the space-dependent source term in a diffusion equation with time fractional order derivative from the flux measurement specified in partial boundary. Based on the analysis on the forward problem and the adjoint problem with inhomogeneous boundary condition, a variational identity connecting the inversion input data with the unknown source function is established. The uniqueness and the conditional stability for the inverse problem are proven by weak unique continuation and the variational identity in some norm. The inversion scheme minimizing the regularizing cost functional is implemented by using conjugate gradient method, with numerical examples showing the validity of the proposed reconstruction scheme.

In this study, we consider an ensemble Kalman inversion (EKI) for the numerical solution of time fractional diffusion inverse problems (TFDIPs). Computational challenges in the EKI arise from the need for repeated evaluations of the forward model. We address this challenge by introducing a non-intrusive reduced basis (RB) method for constructing surrogate models to reduce computational cost. In this method, a reduced basis is extracted from a set of full-order snapshots by the proper orthogonal decomposition (POD), and a doubly stochastic radial basis function (DSRBF) is used to learn the projection coefficients. The DSRBF is carried out in the offline stage with a stochastic leave-one-out cross-validation algorithm to select the shape parameter, and the outputs for new parameter values can be obtained rapidly during the online stage. Due to the complete decoupling of the offline and online stages, the proposed non-intrusive RB method-referred to as POD-DSRBF-provides a powerful tool to accelerate the EKI approach for TFDIPs. We demonstrate the practical performance of the proposed strategies through two nonlinear time-fractional diffusion inverse problems. The numerical results indicate that the new algorithm can achieve significant computational gains without sacrificing accuracy.

Fluorescence imaging is a target-specific molecular imaging technology using absorption coefficient of fluorophore. For this imaging model governed by an inverse problem for the coupled diffusion system, which describes the interaction of the excitation field from several boundary sources and the corresponding emission field, we reformulate it as an optimization problem. For solving this non-quadratic optimizing problem, we propose a decomposition scheme, which extracts the horizontal information of the target from the boundary measurement data directly. The realizability of this hybrid imaging scheme is rigorously proved mathematically for cubic and ellipsoid targets, by constructing an indicator function for the horizontal location of the target explicitly. Then based on this horizonal location as a good initial guess for the iteration process, the cost functional is optimized efficiently using the trust domain scheme. Numerical implementations are provided to show the validity of the proposed scheme.

This paper deals with an inverse problem for recovering the piecewise constant viscoelasticity of a living body from MRE (Magnetic Resonance Elastography) data. Based on a scalar partial differential equation whose solution can approximately simulate MRE data, our inverse coefficient problem is considered as a statistical inverse problem of reconstructing the posterior distribution of unknown viscoelastic modulus. For sampling this distribution, one usually can use the Metropolis-Hastings Markov chain Monte Carlo (MHMCMC) algorithm. However, without an appropriate "proposal" distribution given artificially, the MH-MCMC algorithm is hard to draw samples efficiently. To avoid this, a so-called slice sampling algorithm is introduced in this paper and applied for solving our problem. The performance of these statistical inversion algorithms is numerically tested basing on simulated data.

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 X₁, X₂, …, 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 X₁, X₂, …, 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 h¹ and h², 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 EQ₁^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_qⁿ⁺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/dt φ(x') + ▽W(x)=p(t),
where x ∈ Rⁿ, W ∈ C¹(Rⁿ, R), p ∈ C(R, Rⁿ) 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.

Let a, b and k be nonnegative integers with a ≥ 2 and b ≥ a(k + 1) + 2. A graph G is called a k-Hamiltonian graph if after deleting any k vertices of G the remaining graph of G has a Hamiltonian cycle. A graph G is said to have a k-Hamiltonian[a, b]-factor if after deleting any k vertices of G the remaining graph of G admits a Hamiltonian[a, b]-factor. Let G is a k-Hamiltonian graph of order n with n ≥ a + k + 2. In this paper, it is proved that G contains a k-Hamiltonian[a, b]-factor if δ(G) ≥ a + k and δ(G) ≥ I(G) ≥ a-1 + ((a(k+1))/b-2).

In survival analysis, data are frequently collected by some complex sampling schemes, e.g., length biased sampling, case-cohort sampling and so on. In this paper, we consider the additive hazards model for the general biased survival data. A simple and unified estimating equation method is developed to estimate the regression parameters and baseline hazard function. The asymptotic properties of the resulting estimators are also derived. Furthermore, to check the adequacy of the fitted model with general biased survival data, we present a test statistic based on the cumulative sum of the martingale-type residuals. Simulation studies are conducted to evaluate the performance of proposed methods, and applications to the shrub and Welsh Nickel Refiners datasets are given to illustrate the methodology.

In this paper, we deduced an iteration formula for the computation of central composite discrepancy. By using the iteration formula, the computational complexity of uniform design construction in flexible region can be greatly reduced. And we also made a refinement to threshold accepting algorithm to accelerate the algorithm's convergence rate. Examples show that the refined algorithm can converge to the lower discrepancy design more stably.

This paper evaluates the performance of the F_W -test for testing part of p-regression coefficients in linear panel data model when p is divergent. The asymptotic power of the F_W -statistic is obtained under some regular conditions. The theoretical development are challenging since the number of covariates increases as the sample size increases. It is worth noting that the inference approach does not require any specification of the error distribution. Some simulation comparisons are conducted and show that the simulated power coincide with theoretical power well. The method is also illustrated using a renal cancer data example.