In this paper, we focus on studying the asymptotic stability of the monotone decreasing kink profile solitary wave solutions for the generalized KdV-Burgers equation. We obtain the estimate of the firstorder and second-order derivatives for monotone decreasing kink profile solitary wave solutions, and overcome the difficulties caused by high-order nonlinear terms in the generalized KdV-Burgers equation in the estimate by using L^{2} energy estimating method and Young inequality. We prove that the monotone decreasing kink profile solitary wave solutions are asymptotically stable in H^{1}. Moreover, we obtain the decay rate of the perturbation ψ in the sense of L^{2} and L^{∞} norm, respectively, which are (1 + t)^{-1/2} and (1 + t)^{-1/4} by using Gargliado-Nirenberg inequality.
This paper considers a nonparametric M-estimator of a regression function for functional stationary ergodic data. More precisely, in the ergodic data setting, we consider the regression of a real random variable Y over an explanatory random variable X taking values in some semi-metric abstract space. Under some mild conditions, the weak consistency and the asymptotic normality of the M-estimator are established. Furthermore, a simulated example is provided to examine the finite sample performance of the M-estimator.
The objective of this paper is to study the issue of uniformity on three-level U-type designs in terms of the wrap-around L^{2}-discrepancy. Based on the known formula, we present a new lower bound of wrap-around L^{2}-discrepancy for three-level U-type designs and compare it with those existing ones through figures, numerical simulation and illustrative examples.
On the one hand, we investigate the Bahadur representation for sample quantiles under φ-mixing sequence with φ(n)=O(n^{-3}) and obtain a rate as O (n^{-3/4} log n), a.s. On the other hand, by relaxing the condition of mixing coefficients to Σ_{n=1}^{∞}φ^{1/2}(n) < ∞, a rate O(n^{-1/2}(log n)^{1/2}), a.s., is also obtained.
This paper considers the mathematical programs with equilibrium constraints (MPEC). It is wellknown that, due to the existence of equilibrium constraints, the Mangasarian-Fromovitz constraint qualification does not hold at any feasible point of MPEC and hence, in general, the developed numerical algorithms for standard nonlinear programming problems can not be applied to solve MPEC directly. During the past two decades, much research has been done to develop numerical algorithms and study optimality, stability, and sensitivity for MPEC. However, there are very few results on duality for MPEC in the literature. In this paper, we present a Wolfe-type duality for MPEC and, under some suitable conditions, we establish various duality theorems such as the weak duality, direct duality, converse duality, and strict converse duality theorems. We further show that a linear MPEC is equivalent to a linear programming problem in some sense.
Partly interval censored data frequently occur in many areas including clinical trials, epidemiology research, and medical follow-up studies. When data come from observational studies, we need to carefully adjust for the confounding bias in order to estimate the true treatment effect. Pair matching designs are popular for removing confounding bias without parametric assumptions. With time-to-event outcomes, there are some literature for hypothesis testing with paired right censored data, but not for interval censored data. O'Brien and Fleming extended the Prentice Wilcoxon test to right censored paired data by making use of the PrenticeWilcoxon scores. Akritas proposed the Akritas test and established its asymptotic properties. We extend Akritas test to partly interval censored data. We estimate the survival distribution function by nonparametric maximum likelihood estimation (NPMLE), and prove the asymptotic validity of the new test. To improve our test under small sample size or extreme distributions, we also propose a modified version using the rank of the score difference. Simulation results indicate that our proposed methods have very good performance.
In the article, we investigate a general class of semiparametric hazards regression models for recurrent gap times. The general class includes the proportional hazards model, the accelerated failure time model and the accelerated hazards models as special cases. The model is flexible in modelling recurrent gap times since a covariate effect is identified as having two separate components, namely a time-scale change on hazard progression and a relative hazards ratio. In order to infer the model parameters, the procedure is proposed based on estimating equations. The asymptotic properties of the proposed estimators are established and the finite sample properties are investigated via simulation studies. In addition, a lack of fit test is presented to assess the adequacy of the model and an application of data from a bladder cancer study is reported for illustration.
Using the Mönch fixed point theorem and progressive estimation method, we study the existence, uniqueness and regularity of mild solutions for damped second order impulsive functional differential equations with infinite delay in Banach spaces. The compactness assumption on associated family of operators and the impulsive term, some restrictive conditions on a priori estimation, noncompactness measure estimation and the impulsive term have not been used, our results are different from some known results. Finally, a noncompact semigroup example explains the obtained results.
The degree d(H) of a subgraph H of a graph G is|∪_{u∈V(H)}N(u)-V(H)|, where N(u) denotes the neighbor set of the vertex u of G. In this paper, we prove the following result on the condition of the degrees of subgraphs. Let G be a 2-connected claw-free graph of order n with minimum degree δ(G) ≥ 3. If for any three non-adjacent subgraphs H_{1}, H_{2}, H_{3} that are isomorphic to K_{1}, K_{1}, K_{2}, respectively, there is d(H_{1}) + d(H_{2}) + d(H_{3}) ≥ n + 3, then for each pair of vertices u, v ∈ G that is not a cut set, there exists a Hamilton path between u and v.
For the variational inequality with symmetric cone constraints problem, we consider using the inexact modified Newton method to efficiently solve it. It provides a unified framework for dealing with the variational inequality with nonlinear constraints, variational inequality with the second-order cone constraints, and the variational inequality with semi-definite cone constraints. We show that each stationary point of the unconstrained minimization reformulation based on the Fischer-Burmeister merit function is a solution to the problem. It is proved that the proposed algorithm is globally convergent under suitable conditions. The computation results show that the feasibility and efficiency of our algorithm.
Zero-utility principle is one of the main premium pricing principles, which has been widely used in insurance practice. In this paper, the nonparametric estimation of zero-utility premium is given. In addition, the consistency and asymptotic normality of the estimation are proved. Some special cases including linear, exponential and quadratic utility are discussed. Finally, the Monte Carlo method is used to show the convergence rate of premium estimation. Furthermore, the histogram and Normal-Probability-Plot are given to investigate the asymptotic normality of the estimators. The results show that our estimations are good enough to use in practice.
This paper deals with higher-order optimality conditions and duality theory for approximate solutions in vector optimization involving non-convex set-valued maps. Firstly, under the assumption of near cone-subconvexlikeness for set-valued maps, the higher necessary and sufficient optimality conditions in terms of Studniarski derivatives are derived for local weak approximate minimizers of a set-valued optimization problem. Then, applications to Mond-Weir type dual problem are presented.
A regular edge-transitive graph is said to be semisymmetric if it is not vertex-transitive. Let p be a prime. By Folkman[J. Combin. Theory 3 (1967), 215-232], there is no cubic semisymmetric graph of order 2p or 2p^{2}, and by Hua et al.[Science in China A 54 (2011), 1937-1949], there is no cubic semisymmetric graph of order 4p^{2}. Lu et al.[Science in China A 47 (2004), 11-17] classified connected cubic semisymmetric graphs of order 6p^{2}. In this paper, for p > q ≥ 5 two distinct odd primes, it is shown that the sufficient and necessary conditions which a connected cubic edge transitive bipartite graph of order 2qp^{2} is semisymmetric.
In present note, we apply the Leibniz formula with the nabla operator in discrete fractional calculus (DFC) due to obtain the discrete fractional solutions of a class of associated Bessel equations (ABEs) and a class of associated Legendre equations (ALEs), respectively. Thus, we exhibit a new solution method for such second order linear ordinary differential equations with singular points.
In this work, we study boundary value problems for second-order differential equations with singularities of second kind inside an interval. First, we investigate properties of the spectrum, then prove a theorem on completeness of eigenfunctions, and study the inverse problem of reconstructing the differential equation from spectral characteristics.
The paper studies an evolutionary p(x)-Laplacian equation with a convection term u_{t}=div(ρ^{α}|▽u|^{p(x)-2}▽u) +Σ_{i=1}^{N}∂b_{i}(u)/∂x_{i}, where ρ(x)=dist(x, ∂Ω), ess inf p(x)=p^{-} > 2. To assure the well-posedness of the solutions, the paper shows only a part of the boundary, Σ_{p} ⊂∂Ω), on which we can impose the boundary value. Σ_{p} is determined by the convection term, in particular, when 1 < α < p^{-}-2/2, Σ_{p}={x ∈∂Ω):b_{i}'(0)n_{i}(x) < 0}. So, there is an essential difference between the equation and the usual evolutionary p-Laplacian equation. At last, the existence and the stability of weak solutions are proved under the additional conditions α < p^{-}-2/2 and Σ_{p}=∂Ω).
The identification of within-subject dependence is important for constructing efficient estimation in longitudinal data models. In this paper, we proposed a flexible way to study this dependence by using nonparametric regression models. Specifically, we considered the estimation of varying coefficient longitudinal data model with non-stationary varying coefficient autoregressive error process over observational time quantum. Based on spline approximation and local polynomial techniques, we proposed a two-stage nonparametric estimation for unknown functional coefficients and didn't not drop any observations in a hybrid least square loss framework. Moreover, we showed that the estimated coefficient functions are asymptotically normal and derived the asymptotic biases and variances accordingly. Monte Carlo studies and two real applications were conducted for illustrating the performance of our proposed methods.
In this paper, a new Hilbert space is defined which is embedded into W_{0}^{1,q}(Ω) for 1 ≤ q < 2. By using sign-changing critical point theory, we prove the existence of infinitely many sign-changing solutions in this new space for nonlinear elliptic partial differential equations with Hardy potential and critical parameter in R^{2}.