Length-biased data arise in many important fields, including epidemiological cohort studies, cancer screening trials and labor economics. Analysis of such data has attracted much attention in the literature. In this paper we propose a quantile regression approach for analyzing right-censored and length-biased data. We derive an inverse probability weighted estimating equation corresponding to the quantile regression to correct the bias due to length-bias sampling and informative censoring. This method can easily handle informative censoring induced by length-biased sampling. This is an appealing feature of our proposed method since it is generally difficult to obtain unbiased estimates of risk factors in the presence of length-bias and informative censoring. We establish the consistency and asymptotic distribution of the proposed estimator using empirical process techniques. A resampling method is adopted to estimate the variance of the estimator. We conduct simulation studies to evaluate its finite sample performance and use a real data set to illustrate the application of the proposed method.

Sampling from a truncated multivariate normal distribution (TMVND) constitutes the core computational module in fitting many statistical and econometric models. We propose two efficient methods, an iterative data augmentation (DA) algorithm and a non-iterative inverse Bayes formulae (IBF) sampler, to simulate TMVND and generalize them to multivariate normal distributions with linear inequality constraints. By creating a Bayesian incomplete-data structure, the posterior step of the DA algorithm directly generates random vector draws as opposed to single element draws, resulting obvious computational advantage and easy coding with common statistical software packages such as S-PLUS, MATLAB and GAUSS. Furthermore, the DA provides a ready structure for implementing a fast EM algorithm to identify the mode of TMVND, which has many potential applications in statistical inference of constrained parameter problems. In addition, utilizing this mode as an intermediate result, the IBF sampling provides a novel alternative to Gibbs sampling and eliminates problems with convergence and possible slow convergence due to the high correlation between components of a TMVND. The DA algorithm is applied to a linear regression model with constrained parameters and is illustrated with a published data set. Numerical comparisons show that the proposed DA algorithm and IBF sampler are more efficient than the Gibbs sampler and the accept-reject algorithm.  

A new expression is established for the common solution to six classical linear quaternion matrix equations A₁X = C₁, XB₁ = C₃, A₂X = C₂, XB₂ = C₄, A₃XB₃ = C₅, A₄XB₄ = C₆ which was investigated recently by Wang, Chang and Ning (Q. Wang, H. Chang, Q. Ning, The common solution to six quaternion matrix equations with applications, Appl. Math. Comput. 195: 721-732 (2008)). Formulas are derived for the maximal and minimal ranks of the common solution to this system. Moreover, corresponding results on some special cases are presented. As an application, a necessary and sufficient condition is presented for the invariance of the rank of the general solution to this system. Some known results can be regarded as the special cases of the results in this paper.  

A combined mixed finite element and discontinuous Galerkin method for a compressible miscible displacement problem which includes molecular diffusion and dispersion in porous media is investigated. That is to say, the mixed finite element method with Raviart-Thomas space is applied to the flow equation, and the transport one is solved by the symmetric interior penalty discontinuous Galerkin (SIPG) approximation. Based on projection interpolations and induction hypotheses, a superconvergence estimate is obtained. During the analysis, an extension of the Darcy velocity along the Gauss line is also used in the evaluation of the coefficients in the Galerkin procedure for the concentration.  

In this paper we first investigate zero-sum two-player stochastic differential games with reflection, with the help of theory of Reflected Backward Stochastic Differential Equations (RBSDEs). We will establish the dynamic programming principle for the upper and the lower value functions of this kind of stochastic differential games with reflection in a straightforward way. Then the upper and the lower value functions are proved to be the unique viscosity solutions to the associated upper and the lower Hamilton-Jacobi-Bellman-Isaacs equations with obstacles, respectively. The method differs significantly from those used for control problems with reflection, with new techniques developed of interest on its own. Further, we also prove a new estimate for RBSDEs being sharper than that in the paper of El Karoui, Kapoudjian, Pardoux, Peng and Quenez (1997), which turns out to be very useful because it allows us to estimate the L^p-distance of the solutions of two different RBSDEs by the p-th power of the distance of the initial values of the driving forward equations. We also show that the unique viscosity solution to the approximating Isaacs equation constructed by the penalization method converges to the viscosity solution of the Isaacs equation with obstacle.  

When the role of network topology is taken into consideration, one of the objectives is to understand the possible implications of topological structure on epidemic models. As most real networks can be viewed as complex networks, we propose a new delayed SE^τIR^ωS epidemic disease model with vertical transmission in complex networks. By using a delayed ODE system, in a small-world (SW) network we prove that, under the condition R₀ ≤ 1, the disease-free equilibrium (DFE) is globally stable. When R₀ > 1, the endemic equilibrium is unique and the disease is uniformly persistent. We further obtain the condition of local stability of endemic equilibrium for R₀ > 1. In a scale-free (SF) network we obtain the condition R₁ > 1 under which the system will be of non-zero stationary prevalence.  

Based on the homogeneous balance method, the Jacobi elliptic expansion method and the auxiliary equation method, the first elliptic function equation is used to get a new kind of solutions of nonlinear evolution equations. New exact solutions to the Jacobi elliptic function of MKdV equations and Benjamin-Bona-Mahoney (BBM) equations are obtained with the aid of computer algebraic system Maple. The method is also valid for other (1+1)-dimensional and higher dimensional systems.

Expanded mixed finite element approximation of nonlinear reaction-diffusion equations is discussed. The equations considered here are used to model the hydrologic and bio-geochemical phenomena. To linearize the mixed-method equations, we use a two-grid method involving a small nonlinear system on a coarse gird of size H and a linear system on a fine grid of size h. Error estimates are derived which demonstrate that the error is O(Δt + h^k+1 + H ^2k+2-d/2) (k ≥ 1), where k is the degree of the approximating space for the primary variable and d is the spatial dimension. The above estimates are useful for determining an appropriate H for the coarse grid problems.  

We propose a new algorithm for the total variation based on image denoising problem. The split Bregman method is used to convert an unconstrained minimization denoising problem to a linear system in the outer iteration. An algebraic multi-grid method is applied to solve the linear system in the inner iteration. Furthermore, Krylov subspace acceleration is adopted to improve convergence in the outer iteration. Numerical experiments demonstrate that this algorithm is efficient even for images with large signal-to-noise ratio.

Based on the modified homotopy perturbation method (MHPM), exact solutions of certain partial differential equations are constructed by separation of variables and choosing the finite terms of a series in p as exact solutions. Under suitable initial conditions, the PDE is transformed into an ODE. Some illustrative examples reveal the efficiency of the proposed method.  

We consider an optimization problem of an insurance company in the diffusion setting, which controls the dividends payout as well as the capital injections. To maximize the cumulative expected discounted dividends minus the penalized discounted capital injections until the ruin time, there is a possibility of (cheap or non-cheap) proportional reinsurance. We solve the control problems by constructing two categories of suboptimal models, one without capital injections and one with no bankruptcy by capital injection. Then we derive the explicit solutions for the value function and totally characterize the optimal strategies. Particularly, for cheap reinsurance, they are the same as those in the model of no bankruptcy.

In this paper, we examine the best time to sell a stock at a price being as close as possible to its highest price over a finite time horizon [0, T], where the stock price is modelled by a geometric Brownian motion and the ‘closeness’ is measured by the relative error of the stock price to its highest price over [0, T]. More precisely, we want to optimize the expression:
where (V_t)_t≥0 is a geometric Brownian motion with constant drift α and constant volatility  is the running maximum of the stock price, and the supremum is taken over all possible stopping times 0 ≤ τ ≤ T adapted to the natural filtration (F_t)_t≥0 of the stock price. The above problem has been considered by Shiryaev, Xu and Zhou (2008) and Du Toit and Peskir (2009). In this paper we provide an independent proof that when α = 1/2σ², a selling strategy is optimal if and only if it sells the stock either at the terminal time T or at the moment when the stock price hits its maximum price so far. Besides, when α > 1/2σ², selling the stock at the terminal time T is the unique optimal selling strategy. Our approach to the problem is purely probabilistic and has been inspired by relating the notion of dominant stopping ρ_τ of a stopping time τ to the optimal stopping strategy arisen in the classical “Secretary Problem”.

In this paper, we are concerned with the asymptotic behaviour of a weak solution to the Navier-Stokes equations for compressible barotropic flow in two space dimensions with the pressure function satisfying p(ρ) = aρ log^d(ρ) for large ρ. Here d > 2, a>0. We introduce useful tools from the theory of Orlicz spaces and construct a suitable function which approximates the density for time going to infinity. Using properties of this function, we can prove the strong convergence of the density to its limit state. The behaviour of the velocity field and kinetic energy is also briefly discussed.  

The present paper proposes a semiparametric reproductive dispersion nonlinear model (SRDNM) which is an extension of the nonlinear reproductive dispersion models and the semiparameter regression models. Maximum penalized likelihood estimates (MPLEs) of unknown parameters and nonparametric functions in SRDNM are presented. Assessment of local influence for various perturbation schemes are investigated. Some local influence diagnostics are given. A simulation study and a real example are used to illustrate the proposed methodologies.  

In this article, we establish the existence of at least two positive solutions for the semi-positone m-point boundary value problem with a parameter
 
where λ > 0 is a parameter, 0 < ξ1 < ξ2 < · · · < ξm-2 < 1 with 0 < and f(t, u) ≥ -M with M is a positive constant. The method employed is the Leggett-Williams fixed-point theorem. As an application, an example is given to demonstrate the main result.  

Histogram and kernel estimators are usually regarded as the two main classical data-based nonparametric tools to estimate the underlying density functions for some given data sets. In this paper we will integrate them and define a histogram-kernel error based on the integrated square error between histogram and binned kernel density estimator, and then exploit its asymptotic properties. Just as indicated in this paper, the histogram-kernel error only depends on the choice of bin width and the data for the given prior kernel densities. The asymptotic optimal bin width is derived by minimizing the mean histogram-kernel error. By comparing with Scott’s optimal bin width formula for a histogram, a new method is proposed to construct the data-based histogram without knowledge of the underlying density function. Monte Carlo study is used to verify the usefulness of our method for different kinds of density functions and sample sizes.

This paper gives a dynamic concept and a new non-parametric method for evaluating returns to scale (RTS) of economic units with multiple inputs and outputs. It is frequently noticed that when we increase the input of a decision making unit (DMU) with a certain status of RTS, different status of RTS is observed. For example, when we increase the input of a DMU with constant RTS under the traditional method, a decreasing RTS is often observed instead of the expected constant RTS. We thus define the RTS of each DMU in both input expansion and contraction regions respectively. The research starts from transferring the production possibility set into the intersection form, by giving the explicit linear inequality representation of production frontiers. The RTS structural characteristics of DMUs' on the production frontier are described. Status of RTS of those DMUs on the production frontier include increasing RTS, constant RTS, decreasing RTS, saturated RTS and evidence of congestion. Necessary and sufficient conditions for RTS evaluation are provided. The definition and evaluation method given here provide more detailed economic characteristics of DMU for policy makers.  

In this paper we propose a new method of local linear adaptive smoothing for nonparametric conditional quantile regression. Some theoretical properties of the procedure are investigated. Then we demonstrate the performance of the method on a simulated example and compare it with other methods. The simulation results demonstrate a reasonable performance of our method proposed especially in situations when the underlying image is piecewise linear or can be approximated by such images. Generally speaking, our method outperforms most other existing methods in the sense of the mean square estimation (MSE) and mean absolute estimation (MAE) criteria. The procedure is very stable with respect to increasing noise level and the algorithm can be easily applied to higher dimensional situations.  

In this paper, we propose a dimensional splitting method for the three dimensional (3D) rotating Navier-Stokes equations. Assume that the domain is a channel bounded by two surfaces Im and is decomposed by a series of surfaces Im_i into several sub-domains, which are called the layers of the flow. Every interface Im_i between two sub-domains shares the same geometry. After establishing a semi-geodesic coordinate (S-coordinate) system based on Im_i, Navier-Stoke equations in this coordinate can be expressed as the sum of two operators, of which one is called the membrane operator defined on the tangent space on Im_i, another one is called the bending operator taking value in the normal space on Im_i. Then the derivatives of velocity with respect to the normal direction of the surface are approximated by the Euler central difference, and an approximate form of Navier-Stokes equations on the surface Im_i is obtained, which is called the two-dimensional three-component (2D-3C) Navier-Stokes equations on a two dimensional manifold. Solving these equations by alternate iteration, an approximate solution to the original 3D Navier-Stokes equations is obtained. In addition, the proof of the existence of solutions to 2D-3C Navier-Stokes equations is provided, and some approximate methods for solving 2D-3C Navier-Stokes equations are presented.

In this paper, we define a class of domains in Rⁿ. Using the synchronous coupling of reflecting Brownian motion, we obtain the monotonicity property of the solution of the heat equation with the Neumann boundary conditions. We then show that the hot spots conjecture holds for this class of domains.  

Numerical quadrature schemes of a non-conforming finite element method for general second order elliptic problems in two dimensional (2-D) and three dimensional (3-D) space are discussed in this paper. We present and analyze some optimal numerical quadrature schemes. One of the schemes contains only three sampling points, which greatly improves the efficiency of numerical computations. The optimal error estimates are derived by using some traditional approaches and techniques. Lastly, some numerical results are provided to verify our theoretical analysis.  

A variation of parameters formula and Gronwall type integral inequality are proved for a differential equation involving general piecewise alternately advanced and retarded argument.  

In this paper, we consider hybrid algorithms for finding common elements of the set of common fixed points of two families quasi-φ-non-expansive mappings and the set of solutions of an equilibrium problem. We establish strong convergence theorems of common elements in uniformly smooth and strictly convex Banach spaces with the property (K).

In this paper, we study the blow-up criterion of smooth solutions to the 3D magneto-hydrodynamic system in B_∞,∞⁰. We show that a smooth solution of the 3D MHD equations with zero kinematic viscosity in the whole space R³ breaks down if and only if certain norm of the vorticity blows up at the same time.

In this paper, we consider a BMAP/G/1 G-queue with setup times and multiple vacations. Arrivals of positive customers and negative customers follow a batch Markovian arrival process (BMAP) and Markovian arrival process (MAP) respectively. The arrival of a negative customer removes all the customers in the system when the server is working. The server leaves for a vacation as soon as the system empties and is allowed to take repeated (multiple) vacations. By using the supplementary variables method and the censoring technique, we obtain the queue length distributions. We also obtain the mean of the busy period based on the renewal theory.  

The problem of complementary cycles in tournaments and bipartite tournaments was completely solved. However, the problem of complementary cycles in semicomplete n-partite digraphs with n ≥ 3 is still open. Based on the definition of componentwise complementary cycles, we get the following result. Let D be a 2-strong n-partite (n ≥ 6) tournament that is not a tournament. Let C be a 3-cycle of D and D \ V (C) be nonstrong. For the unique acyclic sequence D₁,D₂, … ,D_α of D\V (C), where α ≥ 2, let D_c = {D_i|D_i contains cycles, i = 1, 2, … , α}, D_c = {D₁,D₂, … ,D_α} \ D_c. If D_c ≠ Ø, then D contains a pair of componentwise complementary cycles.  

In this article, we study the stabilization problem of a nonuniform Euler-Bernoulli beam with locally distributed feedbacks. Firstly, using the semi-group theory, we establish the well-posedness of the associated closed loop system. Then by proving the uniqueness of the solution of a related ordinary differential equations, we derive the asymptotic stability of the closed loop system. Finally, by means of the piecewise frequency domain multiplier method, we prove that the corresponding closed loop system can be exponentially stabilized by only one of the two distributed feedback controls proposed in this paper.  

In this paper, we consider the dividend problem in a two-state Markov-modulated dual risk model, in which the gain arrivals, gain sizes and expenses are influenced by a Markov process. A system of integrodifferential equations for the expected value of the discounted dividends until ruin is derived. In the case of exponential gain sizes, the equations are solved and the best barrier is obtained via numerical example. Finally, using numerical example, we compare the best barrier and the expected discounted dividends in the two-state Markov-modulated dual risk model with those in an associated averaged compound Poisson risk model. Numerical results suggest that one could use the results of the associated averaged compound Poisson risk model to approximate those for the two-state Markov-modulated dual risk model.  

By using some results of pseudo-monotone operator, we discuss the existence and uniqueness of the solution of one kind nonlinear Neumann boundary value problems involving the p-Laplacian operator. We also construct an iterative scheme converging strongly to this solution.  

Let A be a function with derivatives of order m and D ^γA ∈ _β (0 < β < 1, |γ| = m). The authors in the paper prove that if Ω(x, z) ∈ L^∞(Rⁿ) × L^s(S^n-1) (s ≥ n/(n-β)) is homogenous of degree zero and satisfies the mean value zero condition about the variable z, then both the generalized commutator for Marcinkiewicz type integral μ_Ω^A and its variation  are bounded from L^p(Rⁿ) to L^q(Rⁿ), where 1 < p < n/β and 1/q = 1/p-β/n. The authors also consider the boundedness of μ_Ω^A and its variation  on Hardy spaces.  

In this paper, we obtain the Painlevé-Kuratowski Convergence of the efficient solution sets, the weak efficient solution sets and various proper efficient solution sets for the perturbed generalized system with a sequence of mappings converging in a real locally convex Hausdorff topological vector spaces.

In this paper, we consider regularity criteria for solutions to the 3D MHD equations with incompressible conditions. By using some classical inequalities, we obtain the regularity of strong solutions to the three-dimensional MHD equations under certain sufficient conditions in terms of one component of the velocity field and the magnetic field respectively.  

We give more efficient criteria to characterise prime ideal or primary ideal. Further, we obtain the necessary and sufficient conditions that an ideal is prime or primary in real field from the Gröbner bases directly.  

In the present paper we introduce the q analogue of the Baskakov Beta operators. We establish some direct results in the polynomial weighted space of continuous functions defined on the interval [0, ∞). Then we obtain point-wise estimate, using the Lipschitz type maximal function.  

Let G be a graph of maximum degree at most four. By using the overlap matrix method which is introduced by B. Mohar, we show that the average genus of G is not less than (1/3) of its maximum genus, and the bound is best possible. Also, a new lower bound of average genus in terms of girth is derived.  

Let B_R be the ball centered at the origin with radius R in R^N (N≥2). In this paper we study the existence of solution for the following elliptic system
 
where λ>0, μ>0 p≥2, q≥2, υ is the unit outward normal at the boundary ∂BR. Under certain assumptions on κ(|x|), using variational methods, we prove the existence of a positive and radially increasing solution for this problem without growth conditions on the nonlinearity.

In this paper, we combine Leimer’s algorithm with MCS-M algorithm to decompose graphical models into marginal models on prime blocks. It is shown by experiments that our method has an easier and faster implementation than Leimer’s algorithm.

Let Ω∋0 be an open bounded domain in R^N (N≥3) and 2^*(s)=2(N-s)/N-2 , 0<s<2. We consider the following elliptic system of two equations in H₀¹(Ω)×H₀¹(Ω):
,
where λ, μ>0 and α,β> 1 satisfy α+β=2^*(s). Using the Moser iteration, we prove the asymptotic behavior of solutions at the origin. In addition, by exploiting the Mountain-Pass theorem, we establish the existence of solutions.

This paper studies the symmetry, with respect to the real axis, of the point spectrum of the upper triangular infinite dimensional Hamiltonian operator H. Note that the point spectrum of H can be described as σ_p(H) = σ_p(A)∪σ_p¹(-A*). Using the characteristic of the set σ_p¹(-A*), we divide the point spectrum σ_p(A) of A into three disjoint parts. Then, a necessary and sufficient condition is obtained under which σ_p¹(-A*) and one part of σ_p(A) are symmetric with respect to the real axis each other. Based on this result, the symmetry of σ_p(H) is completely given. Moreover, the above result is applied to thin plates on elastic foundation, plane elasticity problems and harmonic equations.