This paper is concerned with the global existence and uniform boundedness of solutions for two classes of chemotaxis models in two or three dimensional spaces. Firstly, by using detailed energy estimates, special interpolation relation and uniform Gronwall inequality, we prove the global existence of uniformly bounded solutions for a class of chemotactic systems with linear chemotactic-sensitivity terms and logistic reaction terms. Secondly, by applying detailed analytic semigroup estimates and special iteration techniques, we obtain the global existence of uniformly bounded solutions for a class of chemotactic systems with nonlinear chemotactic-sensitivity terms, which extends the global existence results of [6] to other general cases.
In the framework of games with coalition structure, we introduce probabilistic Owen value which is an extension of the Owen value and probabilistic Shapley value by considering the situation that not all priori unions are able to cooperate with others. Then we use five axioms of probabilistic efficiency, symmetric within coalitions, symmetric across coalitions applying to unanimity games, strong monotone property and linearity to axiomatize the value.
For a graph G and an integer r ≥ 1, G is r-EKR if no intersecting family of independent r-sets of G is larger than the largest star (a family of independent r-sets containing some fixed vertex in G), and G is strictlyr-EKR if every extremal intersecting family of independent r-sets is a star. Recently, Hurlbert and Kamat gave a preliminary result about EKR property of ladder graphs. They showed that a ladder graph with n rungs is 3-EKR for all n ≥ 3. The present paper proves that this graph is r-EKR for all 1 ≤r ≤ n, and strictly r-EKR except for r = n - 1.
In this paper, we study the existence and uniqueness of solutions to stochastic differential equations driven by G-Brownian motion (GSDEs) with integral-Lipschitz coefficients.
A class of singularly perturbed boundary value problem with singularities is considered. Introducing the stretched variables, the boundary layer corrective terms near x = 0 and x = 1 are constructed. Under suitable conditions, by using the theory of differential inequalities the existence and asymptotic behavior of solution for boundary value problem are proved, uniformly valid asymptotic expansion of solution with boundary layers are obtained.
In this paper, we consider the following chemotaxis model with ratio-dependent logistic reaction term
It is shown that the solution to the problem exists globally if b + β ≥ 0 and will blow up or quench if b + β < 0 by means of function transformation and comparison method. Various asymptotic behavior related to different coefficients and initial data is also discussed.
In this paper, we study the local bifurcation of critical periods near the nondegenerate center (the origin) of a class of Liénard equations with degree 2n, and prove that at most 2n-2 critical periods (taken into account multiplicity) can be produced from a weak center of finite order. We also prove that it can have exactly 2n-2 critical periods near the origin.
A hybrid triple system of order v and index λ, denoted by HTS(v, λ), is a pair (X, B) where X is a v-set and B is a collection of cyclic triples and transitive triples on X, such that every ordered pair of X belongs to λ triples of B. An overlarge set of disjoint HTS(v,λ), denoted by OLHTS(v,λ), is a collection {(Y \{y},A_{i})}_{i}, such that Y is a (v + 1)-set, each (Y \{y},A_{i}) is an HTS(v,λ) and all A_{i}s form a partition of all cyclic triples and transitive triples on Y. In this paper, we shall discuss the existence problem of OLHTS(v,λ) and give the following conclusion: there exists an OLHTS(v,λ) if and only if λ= 1, 2, 4, v≡0, 1 (mod 3) and v≥ 4.
In this paper, we investigate a smoothing-type algorithm for solving the symmetric cone linear program ((SCLP) for short) by making use of an augmented system of its optimality conditions. The algorithm only needs to solve one system of linear equations and to perform one line search at each iteration. It is proved that the algorithm is globally convergent without assuming any prior knowledge of feasibility/infeasibility of the problem. In particular, the algorithm may correctly detect solvability of (SCLP). Furthermore, if (SCLP) has a solution, then the algorithm will generate a solution of (SCLP), and if the problem is strongly infeasible, the algorithm will correctly detect infeasibility of (SCLP).
Epidemiologic studies use outcome-dependent sampling (ODS) schemes where, in addition to a simple random sample, there are also a number of supplement samples that are collected based on outcome variable. ODS scheme is a cost-effective way to improve study efficiency. We develop a maximum semiparametric empirical likelihood estimation (MSELE) for data from a two-stage ODS scheme under the assumption that given covariate, the outcome follows a general linear model. The information of both validation samples and nonvalidation samples are used. What is more, we prove the asymptotic properties of the proposed MSELE.
Let G = G_{n,p} be a binomial random graph with n vertices and edge probability p = p(n), and f be a nonnegative integer-valued function defined on V (G) such that 0 < a ≤ f(x) ≤ b < np-2√np logn for every x ∈ V (G). An fractional f-indicator function is an function h that assigns to each edge of a graph G a number h(e) in [0, 1] so that for each vertex x, we have d^{Gh}(x) = f(x), where d^{Gh}(x) = h(e) is the fractional degree of x in G. Set E_{h} = {e : e ∈ E(G) and h(e) ≠ 0}. If G_{h} is a spanning subgraph of G such that E(G_{h}) = E_{h}, then G_{h} is called an fractional f-factor of G. In this paper, we prove that for any binomial random graph G_{n,p} with p ≥ n^{-2/3}, almost surely G_{n,p} contains an fractional f-factor.
In [13], Schaubel et al. proposed a semiparametric partially linear rate model for the statistical analysis of recurrent event data. But they only considered the model with time-independent covariate effects. In this paper, rate function of the recurrent event is modeled by a semiparametric partially linear function which can include the time-varying effects. We propose the method of generalized estimating equations to make inferences about both the time-varying effects and time-independent effects. The large sample properties are established, while extensive simulation studies are carried out to examine the proposed procedures. At last, we apply the procedures to the well-known bladder cancer study.
The minimax optimization model introduced in this paper is an important model which has received some attention over the past years. In this paper, the application of minimax model on how to select the distribution center location is first introduced. Then a new algorithm with nonmonotone line search to solve the non-decomposable minimax optimization is proposed. We prove that the new algorithm is global convergent. Numerical results show the proposed algorithm is effective.
In this paper, two PVD-type algorithms are proposed for solving inseparable linear constraint optimization. Instead of computing the residual gradient function, the new algorithm uses the reduced gradients to construct the PVD directions in parallel computation, which can greatly reduce the computation amount each iteration and is closer to practical applications for solve large-scale nonlinear programming. Moreover, based on an active set computed by the coordinate rotation at each iteration, a feasible descent direction can be easily obtained by the extended reduced gradient method. The direction is then used as the PVD direction and a new PVD algorithm is proposed for the general linearly constrained optimization. And the global convergence is also proved.
In this paper we consider the problem of maximizing the total discounted utility of dividend payments for a Cramér-Lundberg risk model subject to both proportional and fixed transaction costs. We assume that dividend payments are prohibited unless the surplus of insurance company has reached a level b. Given fixed level b, we derive a integro-differential equation satisfied by the value function. By solving this equation we obtain the analytical solutions of the value function and the optimal dividend strategy when claims are exponentially distributed. Finally we show how the threshold b can be determined so that the expected ruin time is not less than some T. Also, numerical examples are presented to illustrate our results.
In this paper, we present a new computational approach for solving an internal optimal control problem, which is governed by a linear parabolic partial differential equation. Our approach is to approximate the PDE problem by a nonhomogeneous ordinary differential equation system in higher dimension. Then, the homogeneous part of ODES is solved using semigroup theory. In the next step, the convergence of this approach is verified by means of Toeplitz matrix. In the rest of the paper, the optimal control problem is solved by utilizing the solution of homogeneous part. Finally, a numerical example is given.
The modified simple equation method is employed to find the exact solutions of the nonlinear Kolmogorov-Petrovskii-Piskunov (KPP) equation. When certain parameters of the equations are chosen to be special values, the solitary wave solutions are derived from the exact solutions. It is shown that the modified simple equation method provides an effective and powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.
Under the assumption that μ is a non-doubling measure on R^{d} which only satisfies the polynomial growth condition, the authors obtain the boundedness of the multilinear fractional integrals on Morrey spaces, weak-Morrey spaces and Lipschitz spaces associated with μ, which, in the case when μ is the d-dimensional Lebesgue measure, also improve the known results.
In 2003, Borodin and Raspaud proved that if G is a plane graph without 5-circuits and without triangles of distance less than four, then G is 3-colorable. In this paper, we prove that if G is a plane graph without 5- and 6-circuits and without triangles of distance less than 2, then G is 3-colorable.
In constructing two-level fractional factorial designs, the so-called doubling method has been employed. In this paper, we study the problem of uniformity in double designs. The centered L_{2}-discrepancy is employed as a measure of uniformity. We derive results connecting the centered L_{2}-discrepancy value of D(X) and generalized wordlength pattern of X, which show the uniformity relationship between D(X) and X. In addition, we also obtain lower bounds of centered L_{2}-discrepancy value of D(X), which can be used to assess uniformity of D(X).
For the planar Z_{2}-equivariant cubic systems having two elementary focuses, the characterization of a bi-center problem and shortened expressions of the first six Lyapunov constants are completely solved. The necessary and sufficient conditions for the existence of the bi-center are obtained. On the basis of this work, in this paper, we show that under small Z_{2}-equivariant cubic perturbations, this cubic system has at least 13 limit cycles with the scheme 1⊂6∪6.
In this paper we determine all the bipartite graphs with the maximum sum of squares of degrees among the ones with a given number of vertices and edges.
This paper considers the dividend optimization problem for an insurance company under the consideration of internal competition between different units inside the company. The objective is to find a reinsurance policy and a dividend payment scheme so as to maximize the expected discounted value of the dividend payment, and the expected present value of an amount which the insurer earns until the time of ruin. By solving the corresponding constrained Hamilton-Jacobi-Bellman (HJB) equation, we obtain the value function and the optimal reinsurance policy and dividend payment.
In this paper we extend the Christoffel functions to the case of power orthogonal polynomials. The existence and uniqueness as well as some properties are given.
In this paper, we investigate the variable selection problem of the generalized regression models. To estimate the regression parameter, a procedure combining the rank correlation method and the adaptive lasso technique is developed, which is proved to have oracle properties. A modified IMO (iterative marginal optimization) algorithm which directly aims to maximize the penalized rank correlation function is proposed. The effects of the estimating procedure are illustrated by simulation studies.