Given a connected graph G = (V,E) with a nonnegative cost on each edge in E, a nonnegative prize at each vertex in V, and a target set V'⊆ V, the Prize Collecting Steiner Tree (PCST) problem is to find a tree T in G interconnecting all vertices of V' such that the total cost on edges in T minus the total prize at vertices in T is minimized. The PCST problem appears frequently in practice of operations research. While the problem is NP-hard in general, it is polynomial-time solvable when graphs G are restricted to series-parallel graphs.
In this paper, we study the PCST problem with interval costs and prizes, where edge e could be included in T by paying cost x_e ∈ [c_e^-, c_e⁺] while taking risk (c_e⁺ -x_e)/(c_e⁺ -c_e^-) of malfunction at e, and vertex v could be asked for giving a prize y_v ∈ [p_v^-, p_v⁺] for its inclusion in T while taking risk (y_v - p_v^-)/(p_v⁺ - p_v^-) of refusal by v. We establish two risk models for the PCST problem with interval data. Under given budget upper bound on constructing tree T, one model aims at minimizing the maximum risk over edges and vertices in T and the other aims at minimizing the sum of risks over edges and vertices in T. We propose strongly polynomial-time algorithms solving these problems on series-parallel graphs to optimality. Our study shows that the risk models proposed have advantages over the existing robust optimization model, which often yields NP-hard problems even if the original optimization problems are polynomial-time solvable.

In this paper, we give a variational characterization for the growth rate of a multitype population modelled by a multitype Galton-Watson branching process. In particular, the so-called retrospective process plays an important role in the description of the equilibrium state used in the variational characterization. We define the retrospective process associated with a multitype Galton-Watson branching process and identify it with the mutation process describing the type evolution along typical lineages of the multitype Galton-Watson branching process.

In this paper, we propose a bias-corrected empirical likelihood (BCEL) ratio to construct a goodness-of-fit test for generalized linear mixed models. BCEL test maintains the advantage of empirical likelihood that is self scale invariant and then does not involve estimating limiting variance of the test statistic to avoid deteri- orating power of test. Furthermore, the bias correction makes the limit to be a process in which every variable is standard chi-squared. This simple structure of the process enables us to construct a Monte Carlo test procedure to approximate the null distribution. Thus, it overcomes a problem we encounter when classical empirical likelihood test is used, as it is asymptotically a functional of Gaussian process plus a normal shift function. The complicated covariance function makes it difficult to employ any approximation for the null distribution. The test is omnibus and power study shows that the test can detect local alternatives approaching the null at parametric rate. Simulations are carried out for illustration and for a comparison with existing method.

Multilevel (hierarchical) modeling is a generalization of linear and generalized linear modeling in which regression coefficients are modeled through a model, whose parameters are also estimated from data. Multilevel model fails to fit well typically by the use of the EM algorithm once one of level error variance (like Cauchy distribution) tends to infinity. This paper proposes a composite multilevel to combine the nested structure of multilevel data and the robustness of the composite quantile regression, which greatly improves the efficiency and precision of the estimation. The new approach, which is based on the Gauss-Seidel iteration and takes a full advantage of the composite quantile regression and multilevel models, still works well when the error variance tends to infinity. We show that even the error distribution is normal, the MSE of the estimation of composite multilevel quantile regression models nearly equals to mean regression. When the error distribution is not normal, our method still enjoys great advantages in terms of estimation efficiency.

In this paper, the order of approximation and Voronovskaja type results with quantitative estimate for complex q-Durrmeyer polynomials attached to analytic functions on compact disks are obtained.

In this paper, the infinite Prandtl number limit of Rayleigh-Bénard convection is studied. For well prepared initial data, the convergence of solutions in L^∞(0, t;H²(G)) is rigorously justified by analysis of asymptotic expansions.

Nagurney (1999) used variational inequalities to study economic equilibrium and financial networks and applied the modified projection method to solve the problem. In this paper, we formulate the problem as a nonlinear complementarity problem. The complementarity model is just the KKT condition for the model of Nagurney (1999). It is a simpler model than that of Nagurney (1999). We also establish sufficient conditions for existence and uniqueness of the equilibrium pattern, which are weaker than those in Nagurney (1999). Finally, we apply a smoothing Newton-type algorithm to solve the problem and report some numerical results.

Rarefaction wave solutions for a one-dimensional model system associated with non-Newtonian compressible fluid are investigated in terms of asymptotic stability. The rarefaction wave solution is proved to be asymptotically stable, provided the initial disturbance is suitably small. The proof is given by the elemental L² energy method.

The Landweber scheme is a method for algebraic image reconstructions. The convergence behavior of the Landweber scheme is of both theoretical and practical importance. Using the diagonalization of matrix, we derive a neat iterative representation formula for the Landweber schemes and consequently establish the convergence conditions of Landweber iteration. This work refines our previous convergence results on the Landweber scheme.

In this paper, we propose a class of varying coefficient seemingly unrelated regression models, in which the errors are correlated across the equations. By applying the series approximation and taking the contemporaneous correlations into account, we propose an efficient generalized least squares series estimation for the unknown coefficient functions. The consistency and asymptotic normality of the resulting estimators are established. In comparison with the ordinary least squares ones, the proposed estimators are more efficient with smaller asymptotical variances. Some simulation studies and a real application are presented to demonstrate the finite sample performance of the proposed methods. In addition, based on a B-spline approximation, we deduce the asymptotic bias and variance of the proposed estimators.

This paper is devoted to study the Crouzeix-Raviart (C-R) type nonconforming linear triangular finite element method (FEM) for the nonstationary Navier-Stokes equations on anisotropic meshes. By introducing auxiliary finite element spaces, the error estimates for the velocity in the L²-norm and energy norm, as well as for the pressure in the L²-norm are derived.

This article proposes a method for fitting models subject to a convex and log-convex constraint on the probability vector of a product multinomial (binomial) distribution. We present an iterative algorithm for finding the restricted maximum likelihood estimates (MLEs) of the probability vector and show that the algorithm converges to the true solution. Some examples are discussed to illustrate the method.

In this paper, we consider the following quasilinear differential equation (ø_p(u'))' + λf(t, u) = 0 subject to one of the two boundary conditions: u(0) = u'(1) = 0, u'(0) = u(1) = 0. After transforming them into a problem of symmetrical solutions, the existence of three solutions of the problem is obtained by using a recent critical point theorem of Recceri. An example is given to demonstrate our main result.

In this paper, we analyze the Bregman iterative model using the G-norm. Firstly, we show the convergence of the iterative model. Secondly, using the source condition and the symmetric Bregman distance, we consider the error estimations between the iterates and the exact image both in the case of clean and noisy data. The results show that the Bregman iterative model using the G-norm has the similar good properties as the Bregman iterative model using the L²-norm.

In this paper, we study the dynamic facility location problem with submodular penalties (DFLPSP). We present a combinatorial primal-dual 3-approximation algorithm for the DFLPSP.

This paper is concerned with the Cauchy problems of one-dimensional compressible Navier-Stokes equations with density-dependent viscosity coefficients. By assuming ρ₀ ∈ L¹(R), we will prove the existence of weak solutions to the Cauchy problems for θ > 0. This will improve results in Jiu and Xin's paper (Kinet. Relat. Models, 1(2): 313-330 (2008)) in which θ > 1/2 is required. In addition, We will study the large time asymptotic behavior of such weak solutions.

This paper deals with a discrete-time batch arrival retrial queue with the server subject to starting failures. Different from standard batch arrival retrial queues with starting failures, we assume that each customer after service either immediately returns to the orbit for another service with probability or leaves the system forever with probability 1 -θ (0 ≤θ < 1). On the other hand, if the server is started unsuccessfully by a customer (external or repeated), the server is sent to repair immediately and the customer either joins the orbit with probability q or leaves the system forever with probability 1 - q (0 ≤ q < 1). Firstly, we introduce an embedded Markov chain and obtain the necessary and sufficient condition for ergodicity of this embedded Markov chain. Secondly, we derive the steady-state joint distribution of the server state and the number of customers in the system/orbit at arbitrary time. We also derive a stochastic decomposition law. In the special case of individual arrivals, we develop recursive formulae for calculating the steady-state distribution of the orbit size. Besides, we investigate the relation between our discrete-time system and its continuous counterpart. Finally, some numerical examples show the influence of the parameters on the mean orbit size.

One of the main goals of machine learning is to study the generalization performance of learning algorithms. The previous main results describing the generalization ability of learning algorithms are usually based on independent and identically distributed (i.i.d.) samples. However, independence is a very restrictive concept for both theory and real-world applications. In this paper we go far beyond this classical framework by establishing the bounds on the rate of relative uniform convergence for the Empirical Risk Minimization (ERM) algorithm with uniformly ergodic Markov chain samples. We not only obtain generalization bounds of ERM algorithm, but also show that the ERM algorithm with uniformly ergodic Markov chain samples is consistent. The established theory underlies application of ERM type of learning algorithms.

This paper discuss the cusp bifurcation of codimension 2 (i.e. Bogdanov-Takens bifurcation) in a Leslie-Gower predator-prey model with prey harvesting, which was not revealed by Zhu and Lan [Phase portraits, Hopf bifurcation and limit cycles of Leslie-Gower predator-prey systems with harvesting rates, Discrete and Continuous Dynamical Systems Series B. 14(1) (2010), 289-306]. It is shown that there are different parameter values for which the model has a limit cycle or a homoclinic loop.

In this paper we mainly study the difference between the weak solutions generated by a wave front tracking algorithm to isentropic and non-isentropic isothermal Euler system of steady supersonic flow. Under the hypothesis that the initial data are of sufficiently small total variation, we prove that the difference between solutions to isentropic and non-isentropic isothermal Euler system of steady supersonic flow can be bounded by the cube of the total variation of the initial perturbation.

This paper is concerned with the existence and stability of steady states for a prey-predator system with cross diffusion of quasilinear fractional type. We obtain a sufficient condition for the existence of positive steady state solutions by applying bifurcation theory and a detailed priori estimate. In virtue of the principle of exchange of stability, we prove the stability of local bifurcating solutions near the bifurcation point.

We study a generalization of the vertex cover problem. For a given graph with weights on the vertices and an integer k, we aim to find a subset of the vertices with minimum total weight, so that at least k edges in the graph are covered. The problem is called the k-partial vertex cover problem. There are some 2-approximation algorithms for the problem. In the paper we do not improve on the approximation ratios of the previous algorithms, but we derive an iterative rounding algorithm. We present our technique in two algorithms. The first is an iterative rounding algorithm and gives a (2 + Q/OPT)-approximation for the k-partial vertex cover problem where Q is the largest finite weight in the problem definition and OPT is the optimal value for the instance. The second algorithm uses the first as a subroutine and achieves an approximation ratio of 2.

Prior empirical studies find positive and negative momentum effect across the global nations, but few focus on explaining the mixed results. In order to address this issue, we apply the quantile regression approach to analyze the momentum effect in the context of Chinese stock market in this paper. The evidence suggests that the momentum effect in Chinese stock is not stable across firms with different levels of performance. We find that negative momentum effect in the short and medium horizon (3 months and 9 months) increases with the quantile of stock returns. And the positive momentum effect is observed in the long horizon (12 months), which also intensifies for the high performing stocks. According to our study, momentum effect needs to be examined on the basis of stock returns. OLS estimation, which gives an exclusive and biased result, provides misguiding intuitions for momentum effect across the global nations. Based on the empirical results of quantile regression, effective risk control strategies can also be inspired by adjusting the proportion of assets with past performances.

The stochastic dissipative Zakharov equations with white noise are mainly investigated. The global random attractors endowed with usual topology for the stochastic dissipative Zakharov equations are obtained in the sense of usual norm. The method is to transform the stochastic equations into the corresponding partial differential equations with random coefficients by Ornstein-Uhlenbeck process. The crucial compactness of the global random attractors will be obtained by decomposition of solutions.

An acyclic edge coloring of a graph is a proper edge coloring such that every cycle contains edges of at least three distinct colors. The acyclic chromatic index of a graph G, denoted by a'(G), is the minimum number k such that there is an acyclic edge coloring using k colors. It is known that a'(G)≤16Δ for every graph G where Δ denotes the maximum degree of G. We prove that a'(G) < 13.8Δ for an arbitrary graph G. We also reduce the upper bounds of a'(G) to 9.8Δ and 9Δ with girth 5 and 7, respectively.

The backup 2-median problem is a location problem to locate two facilities at vertices with the minimum expected cost where each facility may fail with a given probability. Once a facility fails, the other one takes full responsibility for the services. Here we assume that the facilities do not fail simultaneously. In this paper, we consider the backup 2-median problem on block graphs where any two edges in one block have the same length and the lengths of edges on different blocks may be different. By constructing a tree-shaped skeleton of a block graph, we devise an O(n logn+m)-time algorithm to solve this problem where n and m are the number of vertices and edges, respectively, in the given block graph.

We investigate Chapman-Jouguet models in three-dimensional space by means of generalized characteristic analysis. The interaction of detonation, shock waves and contact discontinuity is discussed intensively in this paper. If contact discontinuity appears, the structure of global solutions becomes complex. We deal with this problem when strength of detonation is small.

We prove the existence and uniqueness of global strong solutions to the Cauchy problem of the three-dimensional magnetohydrodynamic equations in R³ when initial data are helically symmetric. Moreover, the large-time behavior of the strong solutions is obtained simultaneously.

The aim of this paper is to extend the semi-uniform ergodic theorem and semi-uniform sub-additive ergodic theorem to skew-product quasi-flows. Furthermore, more strict inequalities about these two theorems are established. By making use of these results, it is feasible to get uniform estimation of the Lyapunov exponent of some special systems even under non-uniform hypotheses.

Let [b, T] be the commutator of parabolic singular integral T. In this paper, the authors prove that the boundedness of [b, T] on the generalized Morrey spaces implies b ∈ BMO(Rⁿ, ρ). The results in this paper improve and extend the Komori and Mizuhara's results.

By modifying the procedure of binary nonlinearization for the AKNS spectral problem and its adjoint spectral problem under an implicit symmetry constraint, we obtain a finite dimensional system from the Lax pair of the nonlinear Schrödinger equation. We show that this system is a completely integrable Hamiltonian system.

Delay discrete integral inequalities with n nonlinear terms in two variables are discussed, which generalize some existing results and can be used as powerful tools in the analysis of certain partial difference equations. An application example is also given to show boundedness of solutions of a difference equation.

This paper suggests a modified serial correlation test for linear panel data models, which is based on the parameter estimates for an artificial autoregression modeled by differencing and centering residual vectors. Specifically, the differencing operator over the time index and the centering operator over the individual index are, respectively, used to eliminate the potential individual effects and time effects so that the resultant serial correlation test is robust to the two potential effects. Clearly, the test is also robust to the potential correlation between the covariates and the random effects. The test is asymptotically chi-squared distributed under the null hypothesis. Power study shows that the test can detect local alternatives distinct at the parametric rate from the null hypothesis. The finite sample properties of the test are investigated by means of Monte Carlo simulation experiments, and a real data example is analyzed for illustration.

In this paper, we derive a posteriori error estimators for the constrained optimal control problems governed by semi-linear parabolic equations under some assumptions. Then we use them to construct reliable and efficient multi-mesh adaptive finite element algorithms for the optimal control problems. Some numerical experiments are presented to illustrate the theoretical results.

This paper consider the (BMAP₁, BMAP₂)/(PH₁, PH₂)/N retrial queue with finite-position buffer. The behavior of the system is described in terms of continuous time multi-dimensional Markov chain. Arriving type I calls find all servers busy and join the buffer, if the positions of the buffer are insufficient, they can go to orbit. Arriving type Ⅱ calls find all servers busy and join the orbit directly. Each server can provide two types heterogeneous services with Phase-type (PH) time distribution to every arriving call (including types I and Ⅱ calls), arriving calls have an option to choose either type of services. The model is quite general enough to cover most of the systems in communication networks. We derive the ergodicity condition, the stationary distribution and the main performance characteristics of the system. The effects of various parameters on the system performance measures are illustrated numerically.

Suppose that we have a partially linear model Y_i = x'_iβ + g(t_i) + ε_i with independent zero mean errors ε_i, where {x_i, t_i, i = 1, …, n} are non-random and observed completely and {Y_i, i = 1, …, n} are missing at random(MAR). Two types of estimators of β and g(t) for fixed t are investigated: estimators based on semiparametric regression and inverse probability weighted imputations. Asymptotic normality of the estimators is established, which is used to construct normal approximation based confidence intervals on β and g(t). Results are reported of a simulation study on the finite sample performance of the estimators and confidence intervals proposed in this paper.

Distinguishability plays a crucial rule in studying observability of hybrid system such as switched system. Recently, for two linear systems, Lou and Si gave a condition not only necessary but also sufficient to the distinguishability of linear systems. However, the condition is not easy enough to verify. This paper will give a new equivalent condition which is relatively easy to verify.

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.

This paper deals with the boundedness of the solutions of the following dynamic equations (r(t)x^Δ (t))^Δ + a(t)f(x^σ(t)) + b(t)g(x^σ(t)) = 0 and (r(t)x^Δ(t))^Δ + a(t)x^σ(t) + b(t)f(x(t -τ(t))) = e(t) on a time scale T. By using the Bellman integral inequality, we establish some sufficient conditions for boundedness of solutions of the above equations. Our results not only unify the boundedness results for differential and difference equations but are also new for the q-difference equations.

In this paper, we investigate the a priori and a posteriori error estimates for the discontinuous Galerkin finite element approximation to a regularization version of the variational inequality of the second kind. We show the optimal error estimates in the DG-norm (stronger than the H¹ norm) and the L² norm, respectively. Furthermore, some residual-based a posteriori error estimators are established which provide global upper bounds and local lower bounds on the discretization error. These a posteriori analysis results can be applied to develop the adaptive DG methods.