In this paper, a sequence of solutions to the one-dimensional motion of a radiating gas are constructed. Furthermore, when the absorption coefficient α tends to ∞, the above solutions converge to the rarefaction wave, which is an elementary wave pattern of gas dynamics, with a convergence rate α^{1/3}|lnα|^{2}.
In this paper, we propose a new separable fractional interpolation model which can be established by 2n interpolation points where n is the number of variables. Based on this model, a new direct search method is presented. In this method, a new iterate is determined by solving the fractional interpolation model in trust region. Under mild assumptions, the convergence results of this method are given and proved. Numerical experiments show that the new method is promising.
In lifetime data analysis, naturally recorded observations are length-biased data if the probability to select an item is proportional to its length. Based on i.i.d. observations of the true distribution, empirical likelihood (EL) procedure is proposed for the inference on mean residual life (MRL) of naturally recorded item. The limit distribution of the EL based log-likelihood ratio is proved to be the chi-square distribution. Under right censorship, since the EL based log-likelihood ratio leads to a scaled chi-square distribution and estimating the scale parameter leads to lower coverage of confidence interval, we propose an algorithm to calculate the likelihood ratio (LR) directly. The corresponding log-likelihood ratio converges to the standard chi-square distribution and the corresponding confidence interval has a better coverage. Simulation studies are used to support the theoretical results.
An important problem in a given dynamical system is to determine the existence of a homoclinic orbit. We improve the results of Qin and Xiao [Nonlinearity, 20 (2007), 2305-2317], who present some sufficient conditions for the existence of a homoclinic/heteroclinic orbit for the generalized Hénon map. Moreover, an algorithm is presented to locate these homoclinic orbits.
Two isomorphic groups R^{2} and M are firstly constructed. Then we extend them into the differential manifold R^{2n} and n products of the group M for which four kinds of Lie algebras are obtained. By using these Lie algebras and the Tu scheme, integrable hierarchies of evolution equations along with multi-component potential functions can be generated, whose Hamiltonian structures can be worked out by the variational identity. As application illustrations, two integrable Hamiltonian hierarchies with 4 component potential functions are obtained, respectively, some new reduced equations are followed to present. Specially remark that the integrable hierarchies obtained by taking use of the approach presented in the paper are not integrable couplings. Finally, we generalize an equation obtained in the paper to introduce a general nonlinear integrable equation with variable coefficients whose bilinear form, Bäcklund transformation, Lax pair and infinite conserved laws are worked out, respectively, by taking use of the Bell polynomials.
The authors study the empirical likelihood method for partially linear errors-in-variables model with covariate data missing at random. Empirical likelihood ratios for the regression coefficients and the baseline function are investigated, and the corresponding empirical log-likelihood ratios are proved to be asymptotically standard chi-squared, which can be used to construct confidence regions. The finite sample behavior of the proposed methods is evaluated by a simulation study which indicates that the proposed methods are comparable in terms of coverage probabilities and average length of confidence intervals. Finally, the Earthquake Magnitude dataset is used to illustrate our proposed method.
In this paper, we propose a susceptible-infected-susceptible (SIS) model on complex networks, small-world (WS) networks and scale-free (SF) networks, to study the epidemic spreading behavior with time delay which is added into the infected phase. Considering the uniform delay, the basic reproduction number R_{0} on WS networks and R_{0} on SF networks are obtained respectively. On WS networks, if R_{0}≤1, there is a disease-free equilibrium and it is locally asymptotically stable; if R_{0}>1, there is an epidemic equilibrium and it is locally asymptotically stable. On SF networks, if R_{0}≤1, there is a disease-free equilibrium; if R_{0}>1, there is an epidemic equilibrium. Finally, we carry out simulations to verify the conclusions and analyze the effect of the time delay τ, the effective rate λ, average connectivity <k> and the minimum connectivity m on the epidemic spreading.
In this paper, we study the nonexistence and longtime behavior of weak solution for the degenerate parabolic equation ∂_{t}u^{n}=u^{m}div(|∇u^{m}|^{p-2}∇u^{m})+γ|∇u^{m}|^{p}+βu^{n} with zero boundary condition. Blow-up time is derived when the blow-up does occur.
The impulsive solution for a semi-linear singularly perturbed differential-difference equation is studied. Using the methods of boundary function and fractional steps, we construct the formula asymptotic expansion of the problem. At the same time, Based on sewing techniques, the existence of the smooth impulsive solution and the uniform validity of the asymptotic expansion are proved.
Anticipated backward stochastic differential equation (ABSDE) studied the first time in 2007 is a new type of stochastic differential equations. In this paper, we establish a general comparison theorem for ABSDEs.
In this note, we give a necessary and sufficient condition for viability property of diffusion processes with jumps on closed submanifolds of R^{m}. Our result is the system is viable in a closed submanifold K iff the coefficients are tangent to K along K if the equation is in the sense of stratonovich integral and the solution jumps from K to K.
In this paper, a class of parameter-free filled functions is proposed for solving box-constrained system of nonlinear equations. Firstly, the original problem is converted into an equivalent global optimization problem. Subsequently, a class of parameter-free filled functions is proposed for solving the problem. Some properties of the new class of filled functions are studied and discussed. Finally, an algorithm which neither computes nor explicitly approximates gradients during minimizing the filled functions is presented. The global convergence of the algorithm is also established. The implementation of the algorithm on several test problems is reported with satisfactory numerical results.
For a graph G, we denote by p(G) and c(G) the number of vertices of a longest path and a longest cycle in G, respectively. For a vertex v in G, id(v) denotes the implicit degree of v. In this paper, we obtain that if G is a 2-connected graph on n vertices such that the implicit degree sum of any three independent vertices is at least n+1, then either G contains a hamiltonian path, or c(G)≥p(G)-1.
Count data with excess zeros are often encountered in many medical, biomedical and public health applications. In this paper, an extension of zero-inflated Poisson mixed regression models is presented for dealing with multilevel data set, referred as hierarchical mixture zero-inflated Poisson mixed regression models. A stochastic EM algorithm is developed for obtaining the ML estimates of interested parameters and a model comparison is also considered for comparing models with different latent classes through BIC criterion. An application to the analysis of count data from a Shanghai Adolescence Fitness Survey and a simulation study illustrate the usefulness and effectiveness of our methodologies.
Consider a ρ-mixing sequence of identically distributed random variables with the underlying distribution in the domain of attraction of the normal distribution. This paper proves that law of the iterated logarithm holds for ρ-mixing sequences of random variables. Our results generalize and improve Theorems 1.2-1.3 of Qi and Cheng (1996) from the i.i.d. case to ρ-mixing sequences.
This paper considers the reliable control design for T-S fuzzy systems with probabilistic actuators faults and random time-varying delays. The faults of each actuator occurs randomly and its failure rates are governed by a set of unrelated random variables satisfying certain probabilistic distribution. In terms of the probabilistic failures of each actuator and time-varying random delays, new fault model is proposed. Based on the new fuzzy model, reliable controller is designed and sufficient conditions for the exponentially mean square stability (EMSS) of T-S fuzzy systems are derived by using Lyapunov functional method and linear matrix inequality (LMI) technique. It should be noted that the obtained criteria depend on not only the size of the delay, but also the probability distribution of it. Finally, a numerical example is given to show the effectiveness of the proposed method.
In this paper we consider infinite horizon backward doubly stochastic differential equations (BDSDEs for short) coupled with forward stochastic differential equations, whose terminal functions are non-degenerate. For such kind of BDSDEs, we study the existence and uniqueness of their solutions taking values in weighted L^{p}(dx)⊗L^{2}(dx) space (p≥2), and obtain the stationary property for the solutions.
The unsteady flow of viscous incompressible shear-thinning non-Newtonian fluid with mixed boundary is investigated. The boundary condition on the outflow is the modified natural boundary condition, it contains the additional nonlinear term, which enables us to control the kinetic energy of the backward flow. The global existence of weak solution is proved. The fictitious domain method which consists in filling the moving rigid screws with the surrounding fluid and taking into account the boundary conditions on these bodies by introducing a well-chosen distribution of boundary forces is used.
One of the most interesting problems of nonlinear differential equations is the construction of partial solutions. A new method is presented in this paper to seek special solutions of nonlinear diffusion equations. This method is based on seeking suitable function to satisfy Bernolli equation. Many new special solutions are obtained.
As a special shift-invariant spaces, spline subspaces yield many advantages so that there are many practical applications for signal or image processing. In this paper, we pay attention to the sampling and reconstruction problem in spline subspaces. We improve lower bound of sampling set conditions in spline subspaces. Based on the improved explicit lower bound, a improved explicit convergence ratio of reconstruction algorithm is obtained. The improved convergence ratio occupies faster convergence rate than old one. At the end, some numerical examples are shown to validate our results.
In this work, it is aimed to find one-and two-soliton solutions to nonlinear Tzitzeica-Dodd-Bullough (TDB) equation. Since the double exp-function method has been widely used to solve several nonlinear evolution equations in mathematical physics, we have also used it with the help of symbolic computation for solving the present equation. The method seems to be easier and more accurate thanks to the recent developments in the field of symbolic computation.
For non-negative integers i, j and k, we denote the generalized net as N_{i, j, k}, which is a triangle with disjoint paths of length i, j and k, attached to distinct vertices of the triangle. In this paper, we prove that every 3-connected {K_{1, 3}, N_{8-i, i, 1}}-free graph is hamiltonian, where 1≤i≤4.
The class of population-size-dependent branching processes in independent identically distributed random environments is investigated. Under the critical case and appropriate moment assumption, we establish an asymptotic estimate of the survival probability at generation n.
In this paper, a pair of Mond-Weir type higher-order symmetric dual programs over arbitrary cones is formulated. The appropriate duality theorems, such as weak duality theorem, strong duality theorem and converse duality theorem, are established under higher-order (strongly) cone pseudoinvexity assumptions.
Recently, variable selection based on penalized regression methods has received a great deal of attention, mostly through frequentist's models. This paper investigates regularization regression from Bayesian perspective. Our new method extends the Bayesian Lasso regression (Park and Casella, 2008) through replacing the least square loss and Lasso penalty by composite quantile loss function and adaptive Lasso penalty, which allows different penalization parameters for different regression coefficients. Based on the Bayesian hierarchical model framework, an efficient Gibbs sampler is derived to simulate the parameters from posterior distributions. Furthermore, we study the Bayesian composite quantile regression with adaptive group Lasso penalty. The distinguishing characteristic of the newly proposed method is completely data adaptive without requiring prior knowledge of the error distribution. Extensive simulations and two real data examples are used to examine the good performance of the proposed method. All results confirm that our novel method has both robustness and high efficiency and often outperforms other approaches.
In this paper, for b∈(-∞, ∞) and b≠-1, -2, we investigate the explicit periodic wave solutions for the generalized b-equation u_{t}+2ku_{x}-u_{xxt}+(1+b)u^{2}u_{x}=bu_{x}u_{xx}+uu_{xxx}, which contains the generalized Camassa-Holm equation and the generalized Degasperis-Procesi equation. Firstly, via the methods of dynamical system and elliptic integral we obtain two types of explicit periodic wave solutions with a parametric variable α. One of them is made of two elliptic smooth periodic wave solutions. The other is composed of four elliptic periodic blow-up solutions. Secondly we show that there exist four special values for α. When α tends to these special values, these above solutions have limits. From the limit forms we get other three types of nonlinear wave solutions, hyperbolic smooth solitary wave solution, hyperbolic single blow-up solution, trigonometric periodic blow-up solution. Some previous results are extended. For b=-1 or b=-2, we guess that the equation does not have any one of above solutions.
This paper deals with the boundary integral method to study the Navier-Stokes equations around a rotating obstacle. The detail of this method is that the exterior domain is truncated into a bounded domain and a new exterior domain by introducing some open ball B_{R}, and the nonlinear problem in the bounded domain and the linearized problem in the new exterior domain are considered and the approximation coupled problem is obtained. We show that the error between the solution u of Navier-Stokes equations around a rotating obstacle and the solution u_{ε} of the approximation coupled problem is O(R^{-1/4}) in the H^{1}-seminorm when |ω| does not exceed some constant.
Let G=(V, E) be a graph and φ: V∪E→{1, 2, …, k} be a total-k-coloring of G. Let f(v)(S(v)) denote the sum(set) of the color of vertex v and the colors of the edges incident with v. The total coloring φ is called neighbor sum distinguishing if (f(u)≠f(v)) for each edge uv∈E(G). We say that φ is neighbor set distinguishing or adjacent vertex distinguishing if S(u)≠S(v) for each edge uv∈E(G). For both problems, we have conjectures that such colorings exist for any graph G if k≥Δ(G)+3. The maximum average degree of G is the maximum of the average degree of its non-empty subgraphs, which is denoted by mad (G). In this paper, by using the Combinatorial Nullstellensatz and the discharging method, we prove that these two conjectures hold for sparse graphs in their list versions. More precisely, we prove that every graph G with maximum degree Δ(G) and maximum average degree mad (G) has ch_{Σ}"(G)≤Δ(G)+3 (where ch_{Σ}"(G) is the neighbor sum distinguishing total choice number of G) if there exists a pair (k, m)∈{(6, 4), (5, 18/5), (4, 16/5)} such that Δ(G)≥k and mad (G)<m.
The DGMRES method for solving Drazin-inverse solution of singular linear systems is generally used with restarting. But the restarting often slows down the convergence and DGMRES often stagnates. We show that adding some eigenvectors to the subspace can improve the convergence just like the method proposed by R. Morgan in [R. Morgan, A restarted GMRES method augmented with eigenvectors, SIAM J. Matrix Anal. Appl., 16: 1154-1171, 1995. We derive the implementation of this method and present some numerical examples to show the advantages of this method.