Existence of Positive Solutions of Second-order Periodic Boundary Value Problems with Sign-Changing Green's Function

Cheng-hua GAO, Fei ZHANG, Ru-yun MA

In this paper, we consider the existence of positive solutions of second-order periodic boundary value problem u"+((1)/(2)+ε)^{2}u=λg(t)f(u), t∈[0, 2π], u(0)=u(2π), u'(0)=u'(2π), where 0 < ε < (1)/(2), g:[0, 2π]→R is continuous, f:[0, ∞)→R is continuous and λ > 0 is a parameter.

Solving the 2-D Elliptic Monge-Ampère Equation by a Kansa's Method

Qin LI, Zhi-yong LIU

In this paper, a Kansa's method is designed to solve numerically the Monge-Ampère equation. The primitive Kansa's method is a meshfree method which applying the combination of some radial basis functions (such as Hardy's MQ) to approximate the solution of the linear parabolic, hyperbolic and elliptic problems. But this method is deteriorated when is used to solve nonlinear partial differential equations. We approximate the solution in some local triangular subdomains by using the combination of some cubic polynomials. Then the given problems can be computed in each subdomains independently. We prove the stability and convergence of the new method for the elliptic Monge-Ampère equation. Finally, some numerical experiments are presented to demonstrate the theoretical results.

The Klein-bottle fullerene is a finite trivalent graph embedded on the Klein-bottle such that each face is a hexagon. The paper deals with the problem of labeling the vertices, edges and faces of the Klein-bottle fullerene in such a way that the label of a face and the labels of the vertices and edges surrounding that face add up to a weight of that face and the weights of all 6-sided faces constitute an arithmetic progression of difference d. In this paper we study the existence of such labelings for several differences d.

Existence of Three Solutions for Quasilinear Elliptic Equations: an Orlicz-Sobolev Space Setting

Fei FANG, Zhong TAN

In this paper, we establish the existence of three weak solutions for quasilinear elliptic equations in an Orlicz-Sobolev space via an abstract result recently obtained by Ricceri in[13].

Numerical Algorithm for Solving Multi-Pantograph Delay Equations on the Half-line Using Jacobi Rational Functions with Convergence Analysis

Eid H. DOHA, Ali H. BHRAWY, Ramy M. HAFEZ

A new spectral Jacobi rational-Gauss collocation (JRC) method is proposed for solving the multipantograph delay differential equations on the half-line. The method is based on Jacobi rational functions and Gauss quadrature integration formula. The main idea for obtaining a semi-analytical solution for these equations is essentially developed by reducing the pantograph equations with their initial conditions to systems of algebraic equations in the unknown expansion coefficients. The convergence analysis of the method is analyzed. The method possesses the spectral accuracy. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented. Indeed, the present method is compared favorably with other methods.

Green Function of Fourth-order Differential Operator with Eigenparameter-dependent Boundary and Transmission Conditions

Xin-yan ZHANG, Jiong SUN

We investigate a class of fourth-order regular differential operator with transmission conditions at an interior discontinuous point and the eigenparameter appears not only in the differential equation but also in the boundary conditions. We prove that the operator is symmetric, construct basic solutions of differential equation, and give the corresponding Green function of the operator is given.

Partial Penalized Empirical Likelihood Ratio Test Under Sparse Case

Shan-shan WANG, Heng-jian CUI

A consistent test via the partial penalized empirical likelihood approach for the parametric hypothesis testing under the sparse case, called the partial penalized empirical likelihood ratio (PPELR) test, is proposed in this paper. Our results are demonstrated for the mean vector in multivariate analysis and regression coefficients in linear models, respectively. And we establish its asymptotic distributions under the null hypothesis and the local alternatives of order n^{-1/2} under regularity conditions. Meanwhile, the oracle property of the partial penalized empirical likelihood estimator also holds. The proposed PPELR test statistic performs as well as the ordinary empirical likelihood ratio test statistic and outperforms the full penalized empirical likelihood ratio test statistic in term of size and power when the null parameter is zero. Moreover, the proposed method obtains the variable selection as well as the p-values of testing. Numerical simulations and an analysis of Prostate Cancer data confirm our theoretical findings and demonstrate the promising performance of the proposed method in hypothesis testing and variable selection.

Group Classification of Differential-difference Equations: Low-dimensional Lie Algebras

Shou-feng SHEN, Yong-yang JIN

Differential-difference equations of the form =F_{n}(t, u_{n-1}, u_{n}, u_{n+1}, 2???-1, , +1) are classified according to their intrinsic Lie point symmetries, equivalence group and some low-dimensional Lie algebras including the Abelian symmetry algebras, nilpotent nonAbelian symmetry algebras, solvable symmetry algebras with nonAbelian nilradicals, solvable symmetry algebras with Abelian nilradicals and nonsolvable symmetry algebras. Here F_{n} is a nonlinear function of its arguments and the dot over u denotes differentiation with respect to t.

In this paper we establish asymptotic results and a generalized uniform law of the iterated logarithm (LIL) for the increments of a strictly stationary random process, whose results are proved by separating linearly positive quadrant dependent (LPQD) random process and linearly negative quadrant dependent (LNQD) one, respectively.

Variable Selection for the Partial Linear Single-Index Model

Wu WANG, Zhong-yi ZHU

In this paper, we consider the issue of variable selection in partial linear single-index models under the assumption that the vector of regression coefficients is sparse. We apply penalized spline to estimate the nonparametric function and SCAD penalty to achieve sparse estimates of regression parameters in both the linear and single-index parts of the model. Under some mild conditions, it is shown that the penalized estimators have oracle property, in the sense that it is asymptotically normal with the same mean and covariance that they would have if zero coefficients are known in advance. Our model owns a least square representation, therefore standard least square programming algorithms can be implemented without extra programming efforts. In the meantime, parametric estimation, variable selection and nonparametric estimation can be realized in one step, which incredibly increases computational stability. The finite sample performance of the penalized estimators is evaluated through Monte Carlo studies and illustrated with a real data set.

On Global Existence for Mass-supercritical Nonlinear Fractional Hartree Equations

Dan WU

In this paper, we consider the nonlinear fractional Schrödinger equations with Hartree type nonlinearity in mass-supercritical and energy-subcritical case. By sharp Hardy-Littlewood-Sobolev inequality and the Pohozaev identity, we established a threshold condition, which leads to a global existence of solutions in energy space.

General Energy Decay of Solutions for a Weakly Dissipative Kirchhoff Equation with Nonlinear Boundary Damping

Amir Peyravi

In this article, we study the weak dissipative Kirchhoff equation u_{tt}-M(||∂u||_{2}^{2})△u+b(x)u_{t}+f(u)=0,
under nonlinear damping on the boundary
(∂u)/(∂ν)+α(t)g(u_{t})=0. We prove a general energy decay property for solutions in terms of coefficient of the frictional boundary damping. Our result extends and improves some results in the literature such as the work by Zhang and Miao (2010) in which only exponential energy decay is considered and the work by Zhang and Huang (2014) where the energy decay has been not considered.

A Stabilized Crank-Nicolson Mixed Finite Element Method for Non-stationary Parabolized Navier-Stokes Equations

Yan-jie ZHOU, Fei TENG, Zhen-dong LUO

In this study, a time semi-discrete Crank-Nicolson (CN) formulation with second-order time accuracy for the non-stationary parabolized Navier-Stokes equations is firstly established. And then, a fully discrete stabilized CN mixed finite element (SCNMFE) formulation based on two local Gauss integrals and parameterfree with the second-order time accuracy is established directly from the time semi-discrete CN formulation. Thus, it could avoid the discussion for semi-discrete SCNMFE formulation with respect to spatial variables and its theoretical analysis becomes very simple. Finaly, the error estimates of SCNMFE solutions are provided.

The Existence of Semiclassical States for Some P-Laplacian Equation with Critical Exponent

Ji-xiu WANG

In this paper, we study the existence of semiclassical states for some p-Laplacian equation. Under given conditions and minimax methods, we show that this problem has at least one positive solution provided that ε≤Ε; for any m∈N, it has m pairs solutions if ε≤Ε_{m}, where Ε, Ε_{m} are sufficiently small positive numbers. Moreover, these solutions are closed to zero in W_{1,p}(R_{N}) as ε→0.

Numerical Approximation of Solution for the Coupled Nonlinear Schrödinger Equations

Juan CHEN, Lu-ming ZHANG

In this article, a compact finite difference scheme for the coupled nonlinear Schrödinger equations is studied. The scheme is proved to conserve the original conservative properties. Unconditional stability and convergence in maximum norm with order O(τ^{2}+h^{4}) are also proved by the discrete energy method. Finally, numerical results are provided to verify the theoretical analysis.

Some Inequalities and Limit Theorems Under Sublinear Expectations

Ze-Chun HU, Yan-Zhi YANG

In this note, we study inequality and limit theory under sublinear expectations. We mainly prove Doob's inequality for submartingale and Kolmogrov's inequality. By Kolmogrov's inequality, we obtain a special version of Kolmogrov's law of large numbers. Finally, we present a strong law of large numbers for independent and identically distributed random variables under one-order type moment condition.

The mortality of ovarian cancer is higher than any other female genital malignant tumors, while there exists a strong correlation between early-stage detection and cure for it. CA125 and HE4 are two most common and effective serum markers in recent screening research of ovarian cancer. This paper derives a sequential screening strategy for ovarian cancer by jointly modeling the longitudinal profiles of CA125 and HE4. We construct a Bayesian hierarchical mixture model with changepoint, and propose two approaches for diagnosis:the risk of cancer index and the hypothesis test on the true incidence time. We simulated a 7-year sequential screening research and compared with the standard approach based on a fixed cutoff level. Our approach achieves a 15% higher sensitivity for a fixed specificity, indicating that the sequential strategy combining multiple markers is more effective in the early-stage detection of ovarian cancer.

Generalized Competition Index of Primitive Digraphs

Li-hua YOU, Fang CHEN, Jian SHEN, Bo ZHOU

For any positive integers k and m, the k-step m-competition graph C_{m}^{k}(D) of a digraph D has the same set of vertices as D and there is an edge between vertices x and y if and only if there are distinct m vertices v_{1}, v_{2}, …, v_{m} in D such that there are directed walks of length k from x to v_{i} and from y to vi for all 1≤i≤m. The m-competition index of a primitive digraph D is the smallest positive integer k such that C_{m}^{k}(D) is a complete graph. In this paper, we obtained some sharp upper bounds for the m-competition indices of various classes of primitive digraphs.

Bayesian Lasso with Neighborhood Regression Method for Gaussian Graphical Model

Fan-qun LI, Xin-sheng ZHANG

In this paper, we consider the problem of estimating a high dimensional precision matrix of Gaussian graphical model. Taking advantage of the connection between multivariate linear regression and entries of the precision matrix, we propose Bayesian Lasso together with neighborhood regression estimate for Gaussian graphical model. This method can obtain parameter estimation and model selection simultaneously. Moreover, the proposed method can provide symmetric confidence intervals of all entries of the precision matrix.

Stationary Patterns of a Ratio-dependent Prey-predator Model with Cross-diffusion

Jing-fu ZHAO, Hong-tao ZHANG, Jing YANG

This paper is concerned with a ratio-dependent predator-prey system with diffusion and crossdiffusion in a bounded domain with no flux boundary condition. We show that under certain hypotheses, the cross-diffusion can create non-constant positive steady states even though the corresponding model without cross-diffusion fails.

Some New Inequalities Involving the Hadamard Product of an M-matrix and Its Inverse

Feng WANG, Jian-xing ZHAO, Chao-qian LI

For the Hadamard product of an M-matrix and its inverse, some new lower bounds on the minimum eigenvalue are given. These bounds can improve considerably some previous results.

Self-converse Large Sets of Pure Hybrid Triple Systems

Bing-li FAN

A hybrid triple system of order v, briefly by HTS (v), is a pair (X, B) where X is a v-set and B is a collection of cyclic and transitive triples (called blocks) on X such that every ordered pair of X belongs to exactly one block of B. An HTS (v) is called pure and denoted by PHTS (v) if one element of the block set {(x, y, z), (z, y, x), (z, x, y), (y, x, z), (y, z, x), (x, z, y), <x, y, z>, <z, y, x>} is contained in B then the others will not be contained in B. A self-converse large set of disjoint PHTS (v)s, denoted by LPHTS^{*}(v), is a collection of 4(v-2) disjoint PHTS (v)s which contains exactly (v-2)/2 converse octads of PHTS (v)s. In this paper, some results about the existence for LPHTS^{*}(v) are obtained.

Efficient Estimation of Longitudinal Data Additive Varying Coefficient Regression Models

Shu LIU

We consider a longitudinal data additive varying coefficient regression model, in which the coefficients of some factors (covariates) are additive functions of other factors, so that the interactions between different factors can be taken into account effectively. By considering within-subject correlation among repeated measurements over time and additive structure, we propose a feasible weighted two-stage local quasi-likelihood estimation. In the first stage, we construct initial estimators of the additive component functions by B-spline series approximation. With the initial estimators, we transform the additive varying coefficients regression model into a varying coefficients regression model and further apply the local weighted quasi-likelihood method to estimate the varying coefficient functions in the second stage. The resulting second stage estimators are computationally expedient and intuitively appealing. They also have the advantages of higher asymptotic efficiency than those neglecting the correlation structure, and an oracle property in the sense that the asymptotic property of each additive component is the same as if the other components were known with certainty. Simulation studies are conducted to demonstrate finite sample behaviors of the proposed estimators, and a real data example is given to illustrate the usefulness of the proposed methodology.

Regularity and Radial Symmetry of Positive Solutions for a Higher Order Elliptic System

Huai-yu ZHOU, Su-fang TANG

We discuss the properties of solutions for the following elliptic partial differential equations system in R^{n},
(-△)^{(α/(2))}u=u^{p1}v^{p2},
(-△)^{(α/(2))}v=u^{q1}v^{q2},
where 0 < α < n, p_{i} and q_{i} (i=1, 2) satisfy some suitable assumptions. Due to equivalence between the PDEs system and a given integral system, we prove the radial symmetry and regularity of positive solutions to the PDEs system via the method of moving plane in integral forms and Regularity Lifting Lemma. For the special case, when p1+p2=q1+q2=(n+α)/(n-α), we classify the solutions of the PDEs system.