Acta Mathematicae Applicatae Sinica, English Series最新文献

This paper considers the random coefficient autoregressive model with time-functional variance noises, hereafter the RCA-TFV model. We first establish the consistency and asymptotic normality of the conditional least squares estimator for the constant coefficient. The semiparametric least squares estimator for the variance of the random coefficient and the nonparametric estimator for the variance function are constructed, and their asymptotic results are reported. A simulation study is presented along with an analysis of real data to assess the performance of our method in finite samples.

In this paper, we prove a local Hamilton type gradient estimate for positive solution of the nonlinear parabolic equation

on M × (−∞, ∞) with α ∈ R, where a and b are constants. As application, the Harnack inequalities are derived.

In this paper, we consider a system which has k statistically independent and identically distributed strength components and each component is constructed by a pair of statistically dependent elements with doubly type-II censored scheme. These elements (X₁, Y₁), (X₂, Y₂), ⋯, (X_k, Y_k) follow a bivariate Kumaraswamy distribution and each element is exposed to a common random stress T which follows a Kumaraswamy distribution. The system is regarded as operating only if at least s out of k (1 ≤ s ≤ k) strength variables exceed the random stress. The multicomponent reliability of the system is given by R_s,k=P(at least s of the (Z₁, ⋯, Z_k) exceed T) where Z_i = min(X_i, Y_i), i = 1, ⋯, k. The Bayes estimates of R_s,k have been developed by using the Markov Chain Monte Carlo methods due to the lack of explicit forms. The uniformly minimum variance unbiased and exact Bayes estimates of R_s,k are obtained analytically when the common second shape parameter is known. The asymptotic confidence interval and the highest probability density credible interval are constructed for R_s,k. The reliability estimators are compared by using the estimated risks through Monte Carlo simulations.

Motivated by a discrete-time process intended to measure the speed of the spread of contagion in a graph, the burning number b(G) of a graph G, is defined as the smallest integer k for which there are vertices x₁,…,x_k such that for every vertex u of G, there exists i ∈ {1,…,k} with d_G(u, x_i) ≤ k − i, and d_G(x_i, x_j) ≥ j − i for any 1 ≤ i < j ≤ k. The graph burning problem has been shown to be NP-complete even for some acyclic graphs with maximum degree three. In this paper, we determine the burning numbers of all short barbells and long barbells, respectively.

The existence and stability of stationary solutions for a reaction-diffusion-ODE system are investigated in this paper. We first show that there exist both continuous and discontinuous stationary solutions. Then a good understanding of the stability of discontinuous stationary solutions is gained under an appropriate condition. In addition, we demonstrate the influences of the diffusion coefficient on stationary solutions. The results we obtained are based on the super-/sub-solution method and the generalized mountain pass theorem. Finally, some numerical simulations are given to illustrate the theoretical results.

The classical Ambarzumyan’s theorem states that if the Neumann eigenvalues of the Sturm-Liouville operator ( - {{{d^2}} over {d{x^2}}} + q) with an integrable real-valued potential q on [0, π] are {n²: n ≥ 0}, then q = 0 for almost all x ∈ [0, π]. In this work, the classical Ambarzumyan’s theorem is extended to the Dirac operator on equilateral tree graphs. We prove that if the spectrum of the Dirac operator on graphs coincides with the unperturbed case, then the potential is identically zero.

The law of the iterated logarithm (LIL) for the performance measures of a two-station queueing network with arrivals modulated by independent queues is developed by a strong approximation method. For convenience, two arrival processes modulated by queues comprise the external system, all others are belong to the internal system. It is well known that the exogenous arrival has a great influence on the asymptotic variability of performance measures in queues. For the considered queueing network in heavy traffic, we get all the LILs for the queue length, workload, busy time, idle time and departure processes, and present them by some simple functions of the primitive data. The LILs tell us some interesting insights, such as, the LILs of busy and idle times are zero and they reflect a small variability around their fluid approximations, the LIL of departure has nothing to do with the arrival process, both of the two phenomena well explain the service station’s situation of being busy all the time. The external system shows us a distinguishing effect on the performance measures: an underloaded (overloaded, critically loaded) external system affects the internal system through its arrival (departure, arrival and departure together). In addition, we also get the strong approximation of the network as an auxiliary result.

Let G be a finite group generated by S and C(G, S) the Cayley digraphs of G with connection set S. In this paper, we give some sufficient conditions for the existence of hamiltonian circuit in C(G, S), where G = Z_m ⋊ H is a semiproduct of Z_m by a subgroup H of G. In particular, if m is a prime, then the Cayley digraph of G has a hamiltonian circuit unless G = Z_m × H. In addition, we introduce a new digraph operation, called φ-semiproduct of Γ₁ by Γ₂ and denoted by Γ₁ ⋊_φ Γ₂, in terms of mapping φ: V(Γ₂) → {1, −1}. Furthermore we prove that C(Z_m, {a}) ⋊_φC(H, S) is also a Cayley digraph if φ is a homomorphism from H to ({ 1, - 1} le Z_m^ * ), which produces some classes of Cayley digraphs that have hamiltonian circuits.

Aiming at the problem that the asset’s fluctuation influences the borrower’s repayment ability, a loan with a new and flexible repayment method is designed, which depends on the asset value of the borrower. The repayment method can reduce the loan default probability, but it causes the uncertainty of the pay off time. Because the repayment term is related to the regular repayment amount in this method, a boundary for the regular repayment amount is set up in order to avoid too long repayment term. This will balance the benefit of borrowers and lenders and improve the applicability of this method. By establishing a mathematical model of the residual value of the loan, this model can be transformed into an initial-boundary problem of a partial differential equation. The analytic solution and the expected time to pay off the loan are obtained. Finally, numerical analysis are shown.

The strong chromatic index of a graph is the minimum number of colors needed in a proper edge coloring so that no edge is adjacent to two edges of the same color. An outerplane graph with independent crossings is a graph embedded in the plane in such a way that all vertices are on the outer face and two pairs of crossing edges share no common end vertex. It is proved that every outerplane graph with independent crossings and maximum degree Δ has strong chromatic index at most 4Δ − 6 if Δ ≥ 4, and at most 8 if Δ ≤ 3. Both bounds are sharp.