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.

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.

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.

A mathematical model of a predator-prey model with Ivlev's functional response concerning integrated pest management (IPM) is proposed and analyzed. We show that there exists a stable pest-eradication periodic solution when the impulsive period is less than some critical values. Further more, the conditions for the permanence of the system are given. By using bifurcation theory, we show the existence and stability of a positive periodic solution. These results are quite different from those of the corresponding system without impulses. Numerical simulation shows that the system we consider has more complex dynamical behaviors. Finally, it is proved that IPM stragey is more effective than the classical one.

This paper studies estimation and serial correlation
test of a semiparametric varying-coefficient partially linear EV
model of the form $Y=X^{\tau}\beta+Z^{\tau}\alpha(T)+\varepsilon,
\xi= X+\eta$ with the identifying condition
$E[(\varepsilon,\eta^{\tau})^{\tau}]=0,$
Cov$[(\varepsilon,\eta^{\tau})^{\tau}]=\sigma^{2}I_{p+1}$. The
estimators of interested regression parameters $\beta$ , and the
model error variance $\sigma^{2}$, as well as the nonparametric
components $\alpha(T)$, are constructed. Under some regular
conditions, we show that the estimators of the unknown vector
$\beta$ and the unknown parameter $\sigma^{2}$ are strongly
consistent and asymptotically normal and that the estimator of
$\alpha(T)$ achieves the optimal strong convergence rate of the
usual nonparametric regression. Based on these estimators and
asymptotic properties, we propose the $V_{N,p}$ test statistic and
empirical log-likelihood ratio statistic for testing serial
correlation in the model. The proposed statistics are shown to
have asymptotic normal or chi-square distributions under the null
hypothesis of no serial correlation. Some simulation studies are
conducted to illustrate the finite sample performance of the
proposed tests.

This article solved the asymptotic solution of a singularly perturbed boundary value problem with second order turning point, encountered in the dissipative equilibrium vector field of the coupled convection disturbance kinetic equations under the constrained filed and the gravity. Using the matching of asymptotic expansions, the formal asymptotic solution is constructed. By using the theory of differential inequality the uniform validity of the asymptotic expansion for the solution is proved.

Chaotic mixing of distinct fluids produces a convoluted
structure to the interface separating these fluids. For miscible
fluids (as considered here), this interface is defined as a $50\%$
mass concentration isosurface. For shock wave induced
(Richtmyer-Meshkov) instabilities, we find the interface to be
increasingly complex as the computational mesh is refined. This
interfacial chaos is cut off by viscosity, or by the computational
mesh if the Kolmogorov scale is small relative to the mesh. In a
regime of converged interface statistics, we then examine mixing,
i.e. concentration statistics, regularized by mass diffusion. For
Schmidt numbers significantly larger than unity, typical of a liquid
or dense plasma, additional mesh refinement is normally needed to
overcome numerical mass diffusion and to achieve a converged
solution of the mixing problem. However, with the benefit of front
tracking and with an algorithm that allows limited interface
diffusion, we can assure convergence uniformly in the Schmidt
number. We show that different solutions result from variation of
the Schmidt number. We propose subgrid viscosity and mass diffusion
parameterizations which might allow converged solutions at realistic
grid levels.

A general deterministic time-inconsistent optimal control problem is formulated for ordinary differential equations. To find a time-consistent equilibrium value function and the corresponding time-consistent equilibrium control, a non-cooperative N-person differential game (but essentially cooperative in some sense) is introduced. Under certain conditions, it is proved that the open-loop Nash equilibrium value function of the N-person differential game converges to a time-consistent equilibrium value function of the original problem, which is the value function of a time-consistent optimal control problem. Moreover, it is proved that any optimal control of the time-consistent limit problem is a time-consistent equilibrium control of the original problem.  

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.  

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.

This paper introduces the Tukey trimmed and Winsorized means for the transformed data based on a scaled deviation. The trimmed and Winsorized means and scale based on a scaled deviation are as special cases. Meanwhile, the trimmed and Winsorized skewness and kurtosis based on a scaled deviation are given. Furthermore, some of their robust properties (influence function, breakdown points) and asymptotic properties (asymptotic representation and limiting distribution) are also obtained.

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.  

In this paper, we propose a GL method for solving the
ordinary and the partial differential equation in mathematical
physics and chemics and engineering. These equations govern the
acustic, heat, electromagnetic, elastic, plastic, flow, and quantum
etc. macro and micro wave field in time domain and frequency
domain. The space domain of the differential equation is infinite
domain which includes a finite inhomogeneous domain. The
inhomogeneous domain is divided into finite sub domains. We present
the solution of the differential equation as an explicit recursive
sum of the integrals in the inhomogeneous sub domains. Actualy, we
propose an explicit representation of the inhomogeneous parameter
nonlinear inversion. The analytical solution of the equation in the
infinite homogeneous domain is called as an initial global field.
The global field is updated by local scattering field successively
subdomain by subdomain. Once all subdomains are scattered and the
updating process is finished in all the sub domains, the solution of
the equation is obtained. We call our method as Global and Local
field method, in short , GL method. It is different from FEM
method, the GL method directly assemble inverse matrix and gets
solution. There is no big matrix equation needs to solve in the GL
method. There is no needed artificial boundary and no absorption
boundary condition for infinite domain in the GL method. We proved
several theorems on relationships between the field solution and
Green's function that is the theoretical base of our GL method. The
numerical discretization of the GL method is presented. We proved
that the numerical solution of the GL method convergence to the
exact solution when the size of the sub domain is going to zero. The
error estimation of the GL method for solving wave equation is
presented. The simulations show that the GL method is accurate,
fast, and stable for solving elliptic, parabolic, and hyperbolic
equations. The GL method has advantages and wide applications in the
3D electromagnetic (EM) field, 3D elastic and plastic etc seismic
field, acoustic field, flow field, and quantum field. The GL method
software for the above 3D EM etc field are developed.

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.  

In this paper, we derive two general parameterized boundaries of finite difference scheme for Ve?e?'s PDE which is used to price both fixed and floating strike Asian options. Using these two boundaries, we can deal with all kinds of situations, especially, some extreme cases, like overhigh volatility, very small volatility, etc, under which the Asian option is usually mispriced in many existing numerical methods. Numerical results show that our boundaries are pretty efficient.

In this paper, we proposed a new efficient approach to construct confidence intervals for the location and scale parameters of the generalized Pareto distribution (GPD) when the shape parameter is known. The superiority of our method is that the distributions of pivots are exact, but not approximate distributions. The proposed interval estimation provides the shortest interval for the GPD parameter whether or not the confident distribution of the pivot is symmetric. We first estimate the location and scale parameters of the GPD using least squares and then, construct confidence intervals based on the equal probability density principle. The results of various simulation studies illustrate that our interval estimators show the better performance than competing method.

Many cellular and subcellular biological processes can be described in terms of diffusing and chemically reacting species (e.g. enzymes). In this paper, we will use reflected and absorbed Brownian motion and stochastic differential equations to construct a closed form solution to one dimensional Robin boundary problems. Meanwhile, we will give a reasonable explanation to the closed form solution from a stochastic point of view. Finally, we will extend the problem to Robin boundary problem with two boundary conditions and give a specific solution by resorting to a stopping time.

In this paper, we first introduce a special structure that
allows us to construct a large set of resolvable Mendelsohn triple
systems of orders $2q+2$, or LRMTS$(2q+2)$, where $q=6t+5$ is a
prime power. Using a computer, we find examples of such structure
for $t\in T=\{0,1,2,3,4,6,7,8,9,14,16,18,20,22,24\}$. Furthermore,
by a method we introduced in [13], large set of resolvable directed
triple systems with the same orders are obtained too. Finally, by
the tripling construction and product construction for LRMTS and
LRDTS introduced in [2, 20, 21], and by the new results for
$LR$-design in [8], we obtain the existence for LRMTS$(v)$ and
$LRDTS(v)$, where $v=12(t+1)\prod\limits_{m_i\geq0}(2\cdot7^{m_i}+1)
\prod\limits_{n_i\geq0}(2\cdot13^{n_i}+1)$ and $t\in T$, which
provides more infinite family for LRMTS and LRDTS of even orders.

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 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.  

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.  

In this paper, we prove a central limit theorem for m-dependent random variables under sublinear expectations. This theorem can be regarded as a generalization of Peng's central limit theorem.

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.  

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.  

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.  

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.

A recursive formula of the Gerber-Shiu discounted penalty function for a compound binomial risk model with by-claims is obtained. In the discount-free case, an explicit formula is given. Utilizing such an explicit expression, we derive some useful insurance quantities, including the ruin probability, the density of the deficit at ruin, the joint density of the surplus immediately before ruin and the deficit at ruin, and the density of the claim causing ruin.

It is well-known that the eigenvalues of stochastic matrices lie in the unit circle and at least one of them has the value one. Let {1, r₂, · · · , r_N} be the eigenvalues of stochastic matrix X of size N × N. We will present in this paper a simple necessary and sufficient condition for X such that |r_j| < 1, j = 2, · · ·,N. Moreover, such condition can be very quickly examined by using some search algorithms from graph theory.  

In this paper we investigate the existence of solutions of the nonhomogeneous three-point boundary value problem 

The coefficient functions a and b are continuous real-valued functions on [0, 1], η and ζ are some positive constants. Denote by E a Banach space and assume, that u belongs to an Orlicz space i.e., u(·)∈L_M([0, 1],R), where M is an N-function and c∈E.
We search for solutions of the above problem in the Banach space of continuous functions C([0, 1],E) with the Pettis integrability assumptions imposed on f. Some classes of Pettis-integrable functions are described in the paper and exploited in the proofs of main results. We stress on a class of pseudo-solutions of considered problem. Our results extend previous results of the same type for both Bochner and Pettis integrability settings. Similar results are also proved for differential inclusions i.e. when f is a multivalued function.

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.  

Duffing equation with damping and external excitations is investigated. By using Melnikov method and bifurcation theory, the criterions of existence of chaos under periodic perturbations are obtained. By using second-order averaging method, the criterions of existence of chaos in averaged system under quasi-periodic perturbations for Ω= nω +εσ, n = 2, 4, 6 (where σis not rational to ω) are investigated. However, the criterions of existence of chaos for n = 1, 3, 5, 7 - 20 can not be given. The numerical simulations verify the theoretical analysis, show the occurrence of chaos in the averaged system and original system under quasi- periodic perturbation for n = 1, 2, 3, 5, and expose some new complex dynamical behaviors which can not be given by theoretical analysis. In particular, the dynamical behaviors under quasi-periodic perturbations are different from that under periodic perturbations, and the period-doubling bifurcations to chaos has not been found under quasi-periodic perturbations.

We establish an improved GP iterative algorithm for the
extrapolation of band-limited function to fully 3-dimensional
image reconstruction by the convolution-backprojection algorithm.
Numerical experiments demonstrate that the image resolving power
of IGP algorithm is better than that of the original GP algorithm
for noisy data.

In this paper, we investigate the large-time behavior of solutions to the initial-boundary value problem for $n\times
n$ hyperbolic system of conservation laws with artificial viscosity in the half line $(0,\infty)$. We first show that a boundary
layer exists if the corresponding hyperbolic part contains at least one characteristic field with negative propagation speed. We
further show that such boundary layer is nonlinearly stable under small initial perturbation. The proofs are given by an
elementary energy method.

This paper investigates the semi-online machine covering problem on three special uniform machines with the known largest size. Denote by s_j the speed of each machine, j = 1, 2, 3. Assume 0 < s₁ = s₂ = r ≤ t = s₃, and let s = t/r be the speed ratio. An algorithm with competitive ratio max{2, (3s+6)/(s+6)} is presented. We also show the lower bound is at least max{2, (3s)/(s+6)}. For s ≤ 6, the algorithm is an optimal algorithm with the competitive ratio 2. Besides, its overall competitive ratio is 3 which matches the overall lower bound. The algorithm and the lower bound in this paper improve the results of Luo and Sun.  

For k = (k₁, … , k_n) ∈ Nⁿ, 1 ≤ k₁ ≤ … ≤ k_n, let L_k^r be the family of labeled r-sets on k given by L_k^r := {{(a₁, l_a1), … , (a_r, l_ar)} : {a₁, … , a_r} ⊆ [n], l_ai ∈ [k_ai], i = 1, … , r}. A family A of labeled r-sets is intersecting if any two sets in A intersect. In this paper we give the sizes and structures of intersecting families of labeled r-sets.  

The aim of this paper is to study the tests for variance heterogeneity and/or autocorrelation in nonlinear regression models with elliptical and AR(1) errors. The elliptical class includes several symmetric multivariate distributions such as normal, Student-t, power exponential, among others. Several diagnostic tests using score statistics and their adjustment are constructed. The asymptotic properties, including asymptotic chi-square and approximate powers under local alternatives of the score statistics, are studied. The properties of test statistics are investigated through Monte Carlo simulations. A data set previously analyzed under normal errors is reanalyzed under elliptical models to illustrate our test methods.  

Quantile regression is gradually emerging as a powerful tool for estimating models of conditional quantile functions, and therefore research in this area has vastly increased in the past two decades. This paper, with the quantile regression technique, is the first comprehensive longitudinal study on mathematics participation data collected in Alberta, Canada. The major advantage of longitudinal study is its capability to separate the so-called cohort and age effects in the context of population studies. One aim of this paper is to study whether the family background factors alter performance on the mathematical achievement of the strongest students in the same way as that of weaker students based on the large longitudinal sample of 2000, 2001 and 2002 mathematics participation longitudinal data set. The interesting findings suggest that there may be differential family background factor effects at different points in the mathematical achievement conditional distribution.

A genuine variational principle developed by Gyarmati, in the field of thermodynamics of irreversible processes
unifying the theoretical requirements of technical, environmental and biological sciences is employed to study the effects of
uniform suction and injection on MHD flow adjacent to an isothermal wedge with pressure gradient in the presence of a transverse
magnetic field. The velocity distribution inside the boundary layer has been considered as a simple polynomial function and the
variational principle is formulated. The Euler-Lagrange equation is reduced to a simple polynomial equation in terms of momentum
boundary layer thickness. The velocity profiles, displacement thickness and the coefficient of skin friction are calculated for
various values of wedge angle parameter $m$, magnetic parameter $\xi$ and suction/injection parameter $H$. The present results
are compared with known available results and the comparison is found to be satisfactory. The present study establishes high
accuracy of results obtained by this variational technique.

It is shown that self-similar $BV$ solutions of genuinely
nonlinear strictly hyperbolic systems of conservation laws are
special functions of bounded variation, with vanishing Cantor part.

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.