A general deterministic time-inconsistent optimal control problem is formulated for ordinary differential equations. To find a time-consistent equilibrium value function and the corresponding time-consistent equilibrium control, a non-cooperative N-person differential game (but essentially cooperative in some sense) is introduced. Under certain conditions, it is proved that the open-loop Nash equilibrium value function of the N-person differential game converges to a time-consistent equilibrium value function of the original problem, which is the value function of a time-consistent optimal control problem. Moreover, it is proved that any optimal control of the time-consistent limit problem is a time-consistent equilibrium control of the original problem.
This paper continues to study the asymptotic behavior of Gerber-Shiu expected discounted penalty functions in the renewal risk model as the initial capital becomes large. Under the assumption that the claim-size distribution is exponential, we establish an explicit asymptotic formula. Some straightforward consequences of this formula match existing results in the field.
In this paper, we are interested in investigating the causal relationships among futures sugar prices in the Zhengzhou futures exchange market (ZF), the spot sugar prices in Zhengzhou (ZS) and the futures sugar prices in New York futures exchange market (NF). A useful tool called Bayesian network is introduced to analyze the problem. Since there are only three variables in our Bayesian network, the algorithm is straightforward: we display all the 25 possible network structures and adopt certain scoring metrics to evaluate them. We applied five different scoring metrics in total. Firstly, for each metric, we obtained 24 scores, each calculated from one of the 24 possible structures i.e. a Directed Acyclic Graph (DAG). Then we eliminated the network structure which represents the independence of the three variables according to our prior knowledge concerning the futures sugar market. After that, the optimal network structure which implies the causal relationships was selected according to the corresponding scoring metric. Finally, after comparing the results from different scoring metrics, we obtained the relatively affirmative conclusion that ZS causes ZF from both the Bayesian Dirichlet (BD) metric, Bayesian Dirichlet-Akaike Information Criterion (BD-AIC) metric, Bayesian Dirichlet-Bayesian Information Criterion (BD-BIC) metric and Bayesian Information Criterion (BIC) metric. The conclusions that NF causes ZF and ZF causes ZS from the Akaike Information Criterion (AIC) metric and ZF causes ZS from the BIC metric were useful and significant to our investigation.
The aim of this paper is to study the tests for variance heterogeneity and/or autocorrelation in nonlinear regression models with elliptical and AR(1) errors. The elliptical class includes several symmetric multivariate distributions such as normal, Student-t, power exponential, among others. Several diagnostic tests using score statistics and their adjustment are constructed. The asymptotic properties, including asymptotic chi-square and approximate powers under local alternatives of the score statistics, are studied. The properties of test statistics are investigated through Monte Carlo simulations. A data set previously analyzed under normal errors is reanalyzed under elliptical models to illustrate our test methods.
When the role of network topology is taken into consideration, one of the objectives is to understand the possible implications of topological structure on epidemic models. As most real networks can be viewed as complex networks, we propose a new delayed SE^{τ}IR^{ω}S epidemic disease model with vertical transmission in complex networks. By using a delayed ODE system, in a small-world (SW) network we prove that, under the condition R_{0} ≤ 1, the disease-free equilibrium (DFE) is globally stable. When R_{0} > 1, the endemic equilibrium is unique and the disease is uniformly persistent. We further obtain the condition of local stability of endemic equilibrium for R_{0} > 1. In a scale-free (SF) network we obtain the condition R_{1} > 1 under which the system will be of non-zero stationary prevalence.
The present paper proposes a semiparametric reproductive dispersion nonlinear model (SRDNM) which is an extension of the nonlinear reproductive dispersion models and the semiparameter regression models. Maximum penalized likelihood estimates (MPLEs) of unknown parameters and nonparametric functions in SRDNM are presented. Assessment of local influence for various perturbation schemes are investigated. Some local influence diagnostics are given. A simulation study and a real example are used to illustrate the proposed methodologies.
Based on the modified homotopy perturbation method (MHPM), exact solutions of certain partial differential equations are constructed by separation of variables and choosing the finite terms of a series in p as exact solutions. Under suitable initial conditions, the PDE is transformed into an ODE. Some illustrative examples reveal the efficiency of the proposed method.
In this paper, we investigate the model checking problem for a general linear model with nonignorable missing covariates. We show that, without any parametric model assumption for the response probability, the least squares method yields consistent estimators for the linear model even if only the complete data are applied. This makes it feasible to propose two testing procedures for the corresponding model checking problem: a score type lack-of-fit test and a test based on the empirical process. The asymptotic properties of the test statistics are investigated. Both tests are shown to have asymptotic power 1 for local alternatives converging to the null at the rate n^{-r}, 0 ≤ r < (1/2). Simulation results show that both tests perform satisfactorily.
This paper investigates the semi-online machine covering problem on three special uniform machines with the known largest size. Denote by s_{j} the speed of each machine, j = 1, 2, 3. Assume 0 < s_{1} = s_{2} = r ≤ t = s_{3}, and let s = t/r be the speed ratio. An algorithm with competitive ratio max{2, (3s+6)/(s+6)} is presented. We also show the lower bound is at least max{2, (3s)/(s+6)}. For s ≤ 6, the algorithm is an optimal algorithm with the competitive ratio 2. Besides, its overall competitive ratio is 3 which matches the overall lower bound. The algorithm and the lower bound in this paper improve the results of Luo and Sun.
Numerical quadrature schemes of a non-conforming finite element method for general second order elliptic problems in two dimensional (2-D) and three dimensional (3-D) space are discussed in this paper. We present and analyze some optimal numerical quadrature schemes. One of the schemes contains only three sampling points, which greatly improves the efficiency of numerical computations. The optimal error estimates are derived by using some traditional approaches and techniques. Lastly, some numerical results are provided to verify our theoretical analysis.
For k = (k_{1}, … , k_{n}) ∈ N^{n}, 1 ≤ k_{1} ≤ … ≤ k_{n}, let L_{k}^{r} be the family of labeled r-sets on k given by L_{k}^{r} := {{(a_{1}, l_{a1}), … , (a_{r}, l_{ar})} : {a_{1}, … , a_{r}} ⊆ [n], l_{ai} ∈ [k_{ai}], i = 1, … , r}. A family A of labeled r-sets is intersecting if any two sets in A intersect. In this paper we give the sizes and structures of intersecting families of labeled r-sets.
In this article, we study the stabilization problem of a nonuniform Euler-Bernoulli beam with locally distributed feedbacks. Firstly, using the semi-group theory, we establish the well-posedness of the associated closed loop system. Then by proving the uniqueness of the solution of a related ordinary differential equations, we derive the asymptotic stability of the closed loop system. Finally, by means of the piecewise frequency domain multiplier method, we prove that the corresponding closed loop system can be exponentially stabilized by only one of the two distributed feedback controls proposed in this paper.
This paper is concerned with the aging and dependence properties in the additive hazard mixing models including some stochastic comparisons. Further, some useful bounds of reliability functions in additive hazard mixing models are obtained.
This paper studies the symmetry, with respect to the real axis, of the point spectrum of the upper triangular infinite dimensional Hamiltonian operator H. Note that the point spectrum of H can be described as σ_{p}(H) = σ_{p}(A)∪σ_{p}^{1}(-A*). Using the characteristic of the set σ_{p}^{1}(-A*), we divide the point spectrum σ_{p}(A) of A into three disjoint parts. Then, a necessary and sufficient condition is obtained under which σ_{p}^{1}(-A*) and one part of σ_{p}(A) are symmetric with respect to the real axis each other. Based on this result, the symmetry of σ_{p}(H) is completely given. Moreover, the above result is applied to thin plates on elastic foundation, plane elasticity problems and harmonic equations.
The Dirichlet problem for a quasilinear sub-critical inhomogeneous elliptic equation with critical potential and singular coefficients, which has indefinite weights in R^{N}, is studied in this paper. We discuss the corresponding eigenvalue problems by the variational techniques and Picone’s identity, and obtain the existence of non-trivial solutions for the inhomogeneous Dirichlet problem by using Hardy inequality, Mountain Pass Lemma in conjunction with the property of eigenvalues.
The present paper investigates the asymptotic behavior of solutions for stochastic non-Newtonian fluids in a two-dimensional domain. Firstly, we prove the existence of random attractors Λ_{H}(ω) in H; Secondly, we prove the existence of random attractors Λ_{V} (ω) in V. Then we verify regularity of the random attractors by showing that Λ_{H}(ω) = Λ_{V} (ω), which implies the smoothing effect of the fluids in the sense that solution becomes eventually more regular than the initial data.
In this paper, we study Cohen-Grossberg neural networks (CGNN) with time-varying delay. Based on Halanay inequality and continuation theorem of the coincidence degree, we obtain some sufficient conditions ensuring the existence, uniqueness, and global exponential stability of periodic solution. Our results complement previously known results.
Let G_{n}^{k} denote a set of graphs with n vertices and k cut edges. In this paper, we obtain an order of the first four graphs in G_{n}^{k} in terms of their spectral radii for 6 ≤ k ≤ (n-2)/3 .
The problem of complementary cycles in tournaments and bipartite tournaments was completely solved. However, the problem of complementary cycles in semicomplete n-partite digraphs with n ≥ 3 is still open. Based on the definition of componentwise complementary cycles, we get the following result. Let D be a 2-strong n-partite (n ≥ 6) tournament that is not a tournament. Let C be a 3-cycle of D and D \ V (C) be nonstrong. For the unique acyclic sequence D_{1},D_{2}, … ,D_{α} of D\V (C), where α ≥ 2, let D_{c} = {D_{i}|D_{i} contains cycles, i = 1, 2, … , α}, D_{c} = {D_{1},D_{2}, … ,D_{α}} \ D_{c}. If D_{c} ≠ Ø, then D contains a pair of componentwise complementary cycles.