In this paper, a domain in a cube is called a coverage hole if it is not covered by the largest component of the random geometric graph in this cube. We obtain asymptotic properties of the size of the largest coverage hole in the cube. In addition, we give an exponentially decaying tail bound for the probability that a line with length s do not intersect with the coverage of the infinite component of continuum percolation. These results have applications in communication networks and especially in wireless ad-hoc sensor networks.
In this paper, we propose a class of stable finite difference schemes for the initial-boundary value problem of the Cahn-Hilliard equation. These schemes are proved to inherit the total mass conservation and energy dissipation in the discrete level. The dissipation of the total energy implies boundness of the numerical solutions in the discrete H^{1} norm. This in turn implies boundedness of the numerical solutions in the maximum norm and hence the stability of the difference schemes. Unique existence of the numerical solutions is proved by the fixed-point theorem. Convergence rate of the class of finite difference schemes is proved to be O(h^{2} + Г^{2}) with time step Г and mesh size h. An efficient iterative algorithm for solving these nonlinear schemes is proposed and discussed in detail.
In this paper we provide a method to test the existence of the change points in the nonparametric regression function of partially linear models with conditional heteroscedastic variance. We propose the test statistic and establish its asymptotic properties under some regular conditions. Some simulation studies are given to investigate the performance of the proposed method in finite samples. Finally, the proposed method is applied to a real data for illustration.
This work focuses on stochastic Liénard equations with state-dependent switching. First, the existence and uniqueness of a strong solution are obtained by successive construction method. Next, strong Feller property is proved by introducing certain auxiliary processes and using the Radon-Nikodym derivatives and truncation arguments. Based on these results, positive Harris recurrence and exponential ergodicity are obtained under the Foster-Lyapunov drift conditions. Finally, examples using van der Pol equations are presented for illustrations, and the corresponding Foster-Lyapunov functions for the examples are constructed explicitly.
The two-component μ-Camassa-Holm equation, the μ-Camassa-Holm equation and the modified μ- Camassa-Holm equation are three nonlocal integrable models. In this paper, it is shown that the two-component μ-Camassa-Holm equation and the μ-Camassa-Holm equation with a linear dispersion admit a kind of nonlocal symmetries, and the modified μ-Camassa-Holm equation does not have such kind of nonlocal symmetry. An number of conservation laws to the modified μ-Camassa-Holm equation and μ-Camassa-Holm equation are obtained.
In this paper, we consider the uniform stability and uniformly asymptotical stability of nonlinear impulsive infinite delay differential equations. Instead of putting all components of the state variable x in one Lyapunov function, several Lyapunov-Razumikhin functions of partial components of the state variable x are used so that the conditions ensuring that stability are simpler and less restrictive; moreover, examples are discussed to illustrate the advantage of the results obtained.
In the system of m (m ≥ 2) seemingly unrelated regressions, we show that the Gauss-Markov estimator (GME) of any regression coefficients has unique simplified form, which exactly equals to the one- step covariance-adjusted estimator of the regression coefficients, and hence we conclude that for any finite k ≥ 2 the k-step covariance-adjusted estimator degenerates to the one-step covariance-adjusted estimator and the corresponding two-stage Aitken estimator has exactly one simplified form. Also, the unique simplified expression of the GME is just the estimator presented in the Theorem 1 of Wang' work [1988]. A new estimate of regression coefficients in seemingly unrelated regression system, Science in China, Series A 10, 1033-1040].
Let G = (V (G), E(G)) be a simple connected graph of order n. For any vertices u, v,w ∈ V (G) with uv ∈ E(G) and uw /∉ E(G), an edge-rotating of G means rotating the edge uv (around u) to the non-edge position uw. In this work, we consider how the least eigenvalue of a graph perturbs when the graph is performed by rotating an edge from the shorter hanging path to the longer one.
In this paper, we derive a law of large numbers under the nonlinear expectation generated by backward stochastic differential equations driven by G-Brownian motion.
Consider two linear models X_{i} = U_{i}^{'}β + ε_{i}, Y_{j} = V_{j}^{'}+η_{j} with response variables missing at random. In this paper, we assume that X, Y are missing at random (MAR) and use the inverse probability weighted imputation to produce ‘complete' data sets for X and Y. Based on these data sets, we construct an empirical likelihood (EL) statistic for the difference of X and Y (denoted as Δ), and show that the EL statistic has the limiting distribution of X_{1}^{2}, which is used to construct a confidence interval for Δ. Results of a simulation study on the finite sample performance of EL-based confidence intervals on Δ are reported.
The problem of decision making in an imprecise environment has found paramount importance in recent years. In this paper, we define vague soft relation and similarity measure of vague soft sets. Using these definitions, some novel methods of object recognition from an imprecise multiobserver data has been presented. Moreover, we introduce the notion of generalized vague soft sets and study some of its properties. The similarity measure of generalized vague soft sets is also presented and an application of this measure in decision making problems has been shown.
We consider a mathematical model which describes a contact between a deformable body and a foundation. The contact is bilateral and modelled with Tresca's friction law. The goal of this paper is to study an optimal control problem which consists of leading the stress tensor as close as possible to a given target, by acting with a control on the boundary of the body. We state an optimal control problem which admits at least one solution. We also introduce the regularized control problem for which we study the convergence when the regularization parameter tends to zero. Finally, an optimally condition is established for this problem.
Based on some recent results for interlacing eigenvalue intervals from 1-parameter families of se- quences of eigenvalue inequalities, a new method is given to solving the index problem for Sturm-Liouville eigenvalues for coupled self-adjoint boundary conditions in the regular case. The key is a new characteristic principle for indices for Sturm-Liouville eigenvalues. The algorithm corresponding on the characteristic princi- ple are discussed, and numerical examples are presented to illustrate the theoretical results and show that the algorithm is valid.
In this study, a new methodology based on the Hadamard matrix is proposed to construct quantum Boolean functions f such that f = I_{2}n-2P_{2}n, where I_{2}n is an identity matrix of order 2n and P_{2}n is a projective matrix with the same order as I_{2}n. The enumeration of this class of quantum Boolean functions is also presented.
In this paper, we consider existence and uniqueness of positive solutions to three coupled nonlinear Schrödinger equations which appear in nonlinear optics. We use the behaviors of minimizing sequences for a bound to obtain the existence of positive solutions for three coupled system. To prove the uniqueness of positive solutions, we use the radial symmetry of positive solutions to transform the system into an ordinary differential system, and then integrate the system. In particular, for N = 1, we prove the uniqueness of positive solutions when 0 ≤ β = μ1 = μ2 = μ3 or β > max{μ1, μ2, μ3}.
In this paper, we study the orbital stability of solitary waves of compound KdV-type equation in the form of u_{t} + au^{p}u_{x} + bu^{2p}u_{x} + u_{xxx} = 0 (b ≥ 0, p > 0). Our results imply that orbital stability of solitary waves is affected not only by the highest-order nonlinear term bu2pux, but also the nonlinear term au^{p}u_{x}. For the case of b > 0 and 0 < p ≤ 2, we obtain that the positive solitary wave u_{1}(x-ct) is stable when a > 0, while that unstable when a < 0. The stability for negative solitary wave u_{2}(x-ct) is on the contrary. In particular, we point that the nonlinear term with coefficient a makes contributes to the stability of the solitary waves when p = 2 and a > 0.
A physiological model with delay is considered. The time delay being regarded as a parameter, a group of conditions that guarantee the model have multiple periodic solutions is obtained by the global Hopf bifurcation theorem for FDE and Bendixson's criterion for high-dimensional ODE. The results are illustrated by some numerical simulations.
By using the refinement of the standard integral averaging technique, we obtain some oscillation criteria for second order mixed nonlinear elliptic equations. The results established in this paper extend and improve some existing oscillation criteria for half-linear PDE in the literature.
In this paper we consider the p-Laplace problem
-ε^{p}Δ_{p}^{u} + V (x)u^{p-1} = u^{p-1}, u > 0 in R^{N}
where 2 ≤ p < N, ε > 0 and p < q < p* = Np/N-p. V is a non-negative function satisfying certain conditions and ε is a small parameter. We obtain the existence of solutions concentrated near set consisting of disjoint components of zero set of V under certain assumptions on V when ε > 0 is small.
In this paper, the cycle's structure of embedded graphs in surfaces are studied. According to the method of fundamental cycles, the set C (C contains all shortest) is found. A undirected graph G with n vertices has at most O(n^{5}) many shortest cycles; If the shortest cycle of C is odd cycle, then G has at most O(n^{3}) many shortest cycles; If G has been embedded in a surface Sg (Ng, g is a constant), then it has at most O(n^{3}) shortest cycles, moreover, if the shortest cycle of G is odd cycle, then, G has at most O(n^{2}) many shortest cycles. We can find a cycle base of G, the number of odd cycles of G, the number of even cycles of G, the number of contractible cycles of G, the number of non-contractible cycles of G, are all decided. If the П-embedded graph G has П-twosided cycles, then, C contains a shortest П-twosided cycle of G, there is a polynomially bounded algorithm that finds a shortest П-twosided cycle of a П-embedded graph G, the new and simple solutions about the open problem of Bojan Mohar and Carsten Thomassen are obtained.
A graph G is called chromatic-choosable if its choice number is equal to its chromatic number, namely ch(G) =X (G). Ohba's conjecture states that every graph G with 2X (G)+1 or fewer vertices is chromaticchoosable. It is clear that Ohba's conjecture is true if and only if it is true for complete multipartite graphs. Recently, Kostochka, Stiebitz andWoodall showed that Ohba's conjecture holds for complete multipartite graphs with partite size at most five. But the complete multipartite graphs with no restriction on their partite size, for which Ohba's conjecture has been verified are nothing more than the graphs K_{t+2,3,2*(K-t-2),1*t} by Enotomo et al., and K_{t+2,3,2*(K-t-2),1*t} for t ≤ 4 by Shen et al.. In this paper, using the concept of f-choosable (or L0-size-choosable) of graphs, we show that Ohba's conjecture is also true for the graphs K_{t+2,3,2*(K-t-2),1*t} when t ≥ 5. Thus, Ohba's conjecture is true for graphs K_{t+2,3,2*(K-t-2),1*t} for all integers t ≥ 1.
We study the fleet size and mix vehicle routing problem with constraints on the capacity of each vehicle. The objective is to minimize the total cost including fixed utilization cost of vehicles and traveling cost by vehicles. We give differential approximation algorithms for the fleet size and mix vehicle routing problem (FSMVRP) with two kinds of vehicles, the capacities of which are respectively n_{1}k and n_{2}k, n_{2} > n_{1} ≥ 1, k ≥ 1. Using existing theories for vehicle routing problems and feature of the algorithms represented in the paper, we also prove that the algorithms give (1 - 6n+3/(n+1)2k+n+1) differential approximation ratio for (k, nk)VRP, n > 1 and (1 - 6n_{2}+3n_{1}/(n_{1}k+n_{2}k)2k) differential approximation ratio for (n_{1}k, n_{2}k)VRP, n_{2} > n_{1} > 1.
In this paper, we investigate some delay Gronwall type inequalities on time scales by using Gronwall's inequality. Our results unify and extend some delay integral inequalities and their corresponding discrete analogues. The inequalities given here can be used as handy tools in the qualitative theory of certain classes of delay dynamic equations on time scales.
Recurrent events data with a terminal event (e.g. death) often arise in clinical and observational studies. Most of existing models assume multiplicative covariate effects and model the conditional recurrent event rate given survival. In this article, we propose a general additive-multiplicative rates model for recurrent event data in the presence of a terminal event, where the terminal event stop the further occurrence of recurrent events. Based on the estimating equation approach and the inverse probability weighting technique, we propose two procedures for estimating the regression parameters and the baseline mean function. The asymptotic properties of the resulting estimators are established. In addition, some graphical and numerical procedures are presented for model checking. The finite-sample behavior of the proposed methods is examined through simulation studies, and an application to a bladder cancer study is also illustrated.
The relativistic problem of spin-1/2 fermions subject to vector hyperbolic (kink-like) potential (~ tanh kx) is investigated by using the parametric Nikiforov-Uvarov method. The energy eigenvalue equation and the corresponding normalized wave functions are obtained in terms of the Jacobi polynomials in two cases.
In a common authentication code with arbitration, the dishonest arbiter may make a threat to the security of authentication system. In this paper, an authentication code with double arbiters over symplectic geometry is constructed, and the relevant parameters and the probabilities of successful attacks are calculated. The model not only prevents deception from the opponent and members of the system, but also effectively limits the attacks of single arbiter. Moreover, the collusion attacks from arbiters and participators are difficult to succeed.
Hypothesis testing for the parametric component in the partial linear errors-in-variables (EV) regression models is discussed in this paper. Based on the corrected profile least square estimator, five test statistics are proposed and the asymptotic null distributions of them are deduced. Simulations have been done to show the performance of these test statistics under null and alternative hypothesis.
In this paper, by constructing the one-to-one correspondence between its IFS and the quaternary fractional expansion, the n-th iteration analytical expression and the limit representation of the family of the Koch-type curves with arbitrary angles are obtained. The distinction between our method and that of H. Sagan is that we provide the generation process analytically and represent it as a graph of a series function which looks like the Weierstrass function. With these arithmetic expressions, we further analyze and prove some of the fractal properties of the Koch-type curves such as the self-similarity, the Hölder exponent and with the property of continuous everywhere but differentiable nowhere. Then, we will show that the Kochtype curves can be approximated by different constructed generators. Based on the analytic transformation of the Koch-type curves, we also constructed more continuous but nowhere-differentiable curves represented by arithmetic expressions. This result implies that the analytical expression of a fractal has theoretical and practical significance.