Based on the fundamental commutator representation proposed by Cao  we established two explicit expressions for roots of a third order differential operator. By using those expressions we succeeded in clarifying the relationship between two major approaches in theory of integrable systems: the zero curvature and the Lax representations for the KdV and the Boussinesq hierarchies. The proposed procedure could be extended to the general case of higher order of differential operators that leads to the Gel'fand-Dickey hierarchy.
In this work, the Exp-function method is employed to find new wave solutions for the Sine-Gordon and Ostrovsky equation. The equations are simplified to the nonlinear partial differential equations and then different types of exact solutions are extracted by this method. It is shown that the Exp-function method is a powerful analytical method for solving other nonlinear equations occurring in nonlinear physical phenomena. Results are presented in contour plots that show the different values of effective parameters on the velocity profiles.
Two-dimensional Riemann problem for scalar conservation law are investigated and classification of global structure for its non-selfsimilar solution is given by analysis of structure and classification of envelope for non-selfsimilar 2D rarefaction wave. Initial data has two different constant states which are separated by initial discontinuity. We propose the concepts of plus envelope, minus envelope and mixed envelope, and some new structures and evolution phenomena are discovered by use of these concepts.
This paper analyzes a mathematical model of the photosynthetic carbon metabolism, which incorporates not only the Calvin-Benson cycle, but also another two important metabolic pathways: starch synthesis and photorespiratory pathway. Theoretically, the paper shows the existence of steady states, stability and instability of the steady states, the effects of CO2 concentration on steady states. Especially, a critical point is found, the system has only one steady state with CO2 concentration in the left neighborhood of the critical point, but has two with CO2 concentration in the right neighborhood. In addition, the paper also explores the influence of CO2 concentration on the efficiency of photosynthesis. These theoretical results not only provide insight to the kinetic behaviors of the photosynthetic carbon metabolism, but also can be used to help improving the efficiency of photosynthesis in plants.
Call a sequence of k Boolean variables or their negations a k-tuple. For a set V of n Boolean variables, let Tk(V) denote the set of all 2knk possible k-tuples on V. Randomly generate a set C of k-tuples by including every k-tuple in Tk(V) independently with probability p, and let Q be a given set of q “bad” tuple assignments. An instance I = (C,Q) is called satisfiable if there exists an assignment that does not set any of the k-tuples in C to a bad tuple assignment in Q. Suppose that θ, q > 0 are fixed and ε = ε(n) > 0 be such that εlnn/lnlnn → ∞. Let k ≥ (1 + θ) log2n and let p0 = ln2/qnk-1. We prove that [I is satisfiable] = 1, p ≤ (1 - ε) p0,
0, p ≥ (1 + ε) p0.
The Black-Scholes model does not account non-Markovian property and volatility smile or skew although asset price might depend on the past movement of the asset price and real market data can find a non-flat structure of the implied volatility surface. So, in this paper, we formulate an underlying asset model by adding a delayed structure to the constant elasticity of variance (CEV) model that is one of renowned alternative models resolving the geometric issue. However, it is still one factor volatility model which usually does not capture full dynamics of the volatility showing discrepancy between its predicted price and market price for certain range of options. Based on this observation we combine a stochastic volatility factor with the delayed CEV structure and develop a delayed hybrid model of stochastic and local volatilities. Using both a martingale approach and a singular perturbation method, we demonstrate the delayed CEV correction effects on the European vanilla option price under this hybrid volatility model as a direct extension of our previous work .
This paper obtains the solutions of the Kuramoto-Sivashinsky equation. The G'/G method is used to carry out the integration of this equation. Subsequently, its special case, will be integrated and topological 1- soliton solution will be obtained by the soliton ansatz method. The restrictions on the parameters and exponents are also identified.
In this paper, we study the ill posed Perona-Malik equation of image processing and the regu- larized P-M model i.e. C-model proposed by Catte et al.. The authors present the convex compound of these two models in the form of the system of partial differential equations. The weak solution for the equations is proved in detail. The additive operator splitting (AOS) algorithm for the proposed model is also given. Finally, we show some numeric experimental results on images.
In this paper, under the criterion of maximizing the expected exponential utility of terminal wealth, we study the optimal proportional reinsurance and investment policy for an insurer with the compound Poisson claim process. We model the price process of the risky asset to the constant elasticity of variance (for short, CEV) model, and consider net profit condition and variance reinsurance premium principle in our work. Using stochastic control theory, we derive explicit expressions for the optimal policy and value function. And some numerical examples are given.
The main purpose of this paper is using capture-recapture data to estimate the population size when some covariate values are missing, possibly non-ignorable. Conditional likelihood method is adopted, with a sub-model describing various missing mechanisms. The derived estimate is proved to be asymptotically normal, and simulation studies via a version of EM algorithm show that it is approximately unbiased. The proposed method is applied to a real example, and the result is compared with previous ones.
We investigate the instability of two-layer Phillips model in this paper, which is a prototypical geophysical fluid model. Using the results of Guo and Strauss et al, we obtained linear instability implies nonlinear instability provided the linearized system has an exponentially growing solution.
This paper is concerned with the application of generalized polynomial chaos (gPC) method to nonlinear random pantograph equations. An error estimation of gPC method is derived. The global error analysis is given for the error arising from finite-dimensional noise (FDN) assumption, projection error, aliasing error and discretization error. In the end, with several numerical experiments, the theoretical results are further illustrated.
This study is undertaken to apply a bootstrap method of controlling the false discovery rate (FDR) when performing pairwise comparisons of normal means. Due to the dependency of test statistics in pairwise comparisons, many conventional multiple testing procedures can't be employed directly. Some modified pro- cedures that control FDR with dependent test statistics are too conservative. In the paper, by bootstrap and goodness-of-fit methods, we produce independent p-values for pairwise comparisons. Based on these indepen- dent p-values, plenty of procedures can be used, and two typical FDR controlling procedures are applied here. An example is provided to illustrate the proposed approach. Extensive simulations show the satisfactory FDR control and power performance of our approach. In addition, the proposed approach can be easily extended to more than two normal, or non-normal, balance or unbalance cases.
The asymptotic variability analysis is studied for multi-server generalized Jackson network. It is characterized by law of the iterated logarithm (LIL), which quantifies the magnitude of asymptotic stochastic fluctuations of the stochastic processes compensated by their deterministic fluid limits. In the overloaded (OL) case, the asymptotic variability is studied for five performance measures: queue length, workload, busy time, idle time and number of departures. The proof is based on strong approximations, which approximate discrete performance processes with (reflected) Brownian motions. We conduct numerical examples to provide insights on these LIL results.
This paper deals with the global existence and energy decay of solutions to some coupled system of Kirchhoff type equations with nonlinear dissipative and source terms in a bounded domain. We obtain the global existence by defining the stable set in H01(Ω) × H01(Ω), and the energy decay of global solutions is given by applying a lemma of V. Komornik.
In this paper, we consider the estimation of the finite time survival probability in the classical risk model when the initial surplus is zero. We construct a nonparametric estimator by Fourier inversion and kernel density estimation method. Under some mild assumptions imposed on the kernel, bandwidth and claim size density, we derive the order of the bias and variance, and show that the estimator has asymptotic normality property. Some simulation studies show that the estimator performs quite well in the finite sample setting.
In this paper, we study the optimal investment and proportional reinsurance strategy for an insurer in a hidden Markov regime-switching environment. A risk-based approach is considered, where the insurer aims at selecting an optimal strategy with a view to minimizing the risk described by a convex risk measure of its terminal wealth. We solve the problem in two steps. First, we employ the filtering theory to turn the optimization problem with partial observations into one with complete observations. Second, by using BSDEs with jumps, we solve the problem with complete observations.
The aim of this paper is to study the exact controllability of the Petrovsky equation. Under some checkable geometric assumptions, we establish the observability inequality via the multiplier method for the Dirichlet control problem.
In this paper, we propose an efficient combination model of the second-order ROF model and a simple fourth-order partial differential equation (PDE) for image denoising. The split Bregman method is used to convert the nonlinear combination model into a linear system in the outer iteration, and an algebraic multigrid method is applied to solve the linear system in the inner iteration. Furthermore, Krylov subspace acceleration is adopted to improve convergence in the outer iteration. At the same time, we prove that the model is strictly convex and exists a unique global minimizer. We have also conducted a variety of numerical experiments to analyze the parameter selection criteria and discuss the performance of the fourth-order PDE in the combination model. The results show that our model can reduce blocky effects and our algorithm is efficient and robust to solve the proposed model.