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关键词： Cavitation modelling   thin film lubrication   Reynolds equation   Rayleigh-Plesset equation  

摘要： In lubrication problems, which concern thin film flow, cavitation has been considered as a fundamental element to correctly describe the characteristics of lubricated mechanisms. Here, the well-posedness of a cavitation model that can explain the interaction between viscous effects and micro-bubbles of gas is studied. This cavitation model consists of a coupled problem between the compressible Reynolds partial differential equation (PDE) (that describes the flow) and the Rayleigh-Plesset ordinary differential equation (that describes micro-bubbles evolution). A simplified form without bubbles convection is studied here. This coupled model seems never to be studied before from its mathematical aspects. Local times existence results are proved and stability theorems are obtained based on the continuity of the spectrum for bounded linear operators. Numerical results are presented to illustrate these theoretical results.

关键词： Segmentation   Boundary pixel   Mathematical model   Texture  

摘要： Optical Coherence Tomography (OCT) is a non-invasive technology which facilitates the process of capturing images from light-scattering organs like retina. Retina is a layered structure each layer of which has its own morphological properties. The segmentation of retinal layers helps to identify retinal diseases. In this paper, a novel mathematical model is proposed which can extract boundary pixels located on the borders between layers. The new model uses texture properties of pixels to extract distinguishing characteristics for boundary pixels. It is explored that boundary pixels provide certain values for texture indicators leading to the existence of special relation between neighbor pixels' intensities. Using the new model which is based on Laplace distribution, it is possible to compute the probability of being a boundary pixel for each pixel. The numerical results show that the proposed model is capable of identifying retinal layers' boundaries in normal cases with acceptable accuracy.

关键词： Credit default swap   counterparty risk   structure model   intensity model   CIR model  

摘要： A credit default swap (CDS) is an exchange of premium payments for a compensation for the occurrence of a credit event. Counterparty risks refer to defaults of parties holding CDS contracts. In this paper we develop a valuation/pricing model for a CDS subject to counterparty risks. Using the Cox [ngersoll Ross (CIR) model for interest rate and first arrival times of Poisson processes with variable intensities for the occurrences of credit default and counterparty defaults, we derive a mathematical formulation and make a full theoretical investigation. In addition, we develop a full theory for the corresponding infinite horizon problem and establish its connection with the asymptotic long expiry behaviour of finite horizon problem. Furthermore, we establish a connection between two major frameworks for default times: the structure model approach and the intensity model approach. We show that a solution of the structure model can be obtained as the limit of a sequence of solutions of intensity models. Regarded as an important theoretical development, we remove a constraint typically imposed on the parameters of the CIR model;that is, the well-posedness (existence, uniqueness and continuous dependence of parameters) of the mathematical model holds for any empirically calibrated parameters for the CIR model.

关键词： Stochastic noise   P-bifurcation   Bistability   Switching time   Disease index  

摘要： Alzheimer's disease is a worldwide disease of dementia and is characterized by beta-amyloid plaques. Increasing evidences show that there is a positive feedback loop between the level of beta-amyloid and the level of calcium. In this paper, stochastic noises are incorporated into a minimal model of Alzheimer's disease which focuses upon the evolution of beta-amyloid and calcium. Mathematical analysis indicates that solutions of the model without stochastic noises converge either to a unique equilibrium or to bistable equilibria. Analytical conditions for the stochastic P-bifurcation are derived by means of technique of slow-fast dynamical systems. A formula is presented to approximate the mean switching time from a normal state to a pathological state. A disease index is also proposed to predict the risk to transit from a normal state to a disease state. Further numerical simulations reveal how the parameters influence the evolutionary outcomes of beta-amyloid and calcium. These results give new insights on the strategies to slow the development of Alzheimer's disease. (C) 2020 Elsevier B.V. All rights reserved.

关键词： Leukemia   Treatment   Mathematical model   Optimal control   Numerical analysis  

摘要： Leukemia is a worldwide public health problem. It is the result of unconstrained growth of immature white blood cells in the blood. In this paper, we propose a compartmental model of leukemia in terms of a system of ordinary nonlinear differential equations and apply optimal control technique in obtaining optimal strategies to minimize the number of infected cells in the blood. Initially, a mathematical model of leukemia is formulated. We verify the stability of the equilibria (disease-free and endemic equilibrium point) of the model based on the basic reproduction number. After words, in order to minimize the number of infected cells as well as the costs of the control, we develop the proper optimal control model by using Pontryagin's maximum principle. Finally, numerical simulation of the proposed model and optimal control model are performed to show that the effectiveness of immune boosting drugs as well as immunotherapy to defend leukemia in the blood.

关键词： Biomass particles   Non-premixed combustion   Analytical modeling   Counter-flow structure   Flame   Devolatilization process  

摘要： In this study, non-premixed multi-zone combustion of porous lycopodium particles in a counter-flow configuration at steady state was modeled using a derived analytical model. To model the porosity, a reasonable series form is considered. In the derived analytical model, both convective and radiative heat losses were included. The overall combustion process was mathematically divided into pre-heating, drying, vaporization, flame, and post-flame zones and lycopodium particles and oxidizer are injected from opposite sides locating infinity away the stagnation plane. The particles are firstly dried and then vaporized in a limitedly finite region to produce a gaseous fuel of special chemical composition. Relevant conservation equations were formulated in both dimensional and dimensionless forms. Applying suitable jump and boundary conditions, the corresponding continuity and energy equations were solved using Mathematica and Matlab software. The derived analytical model was first validated through comparison against literature data and a satisfactory agreement was achieved. Then temperature distribution and mass fractions of fuel and oxidizer along the x direction were presented to describe the combustion characteristics. Finally, a series of parametric studies were conducted to study the effects of key operating conditions. (C) 2020 Elsevier Ltd. All rights reserved.

关键词： solid propellant combustion   nonlinear surface pyrolysis   travelling wave solution   dynamical system   shooting method  

摘要： We investigate a model of solid propellant combustion involving surface pyrolysis coupled to finite activation energy gas-phase combustion. Existence and uniqueness of a travelling wave solution are established by extending dynamical system tools classically used for premixed flames, dealing with the additional difficulty arising from the surface regression and pyrolysis. An efficient shooting method allows to solve the problem in phase space without resorting to space discretisation nor fixed-point Newton iterations. The results are compared to solutions from a CFD code developed at ONERA, assessing the efficiency and potential of the method, and the impact of the modelling assumptions is evaluated through parametric studies.

关键词： 合班   数学分析   教学方法  

摘要： 基于合班授课模式,以数学分析课程为例,通过对喀什大学数学与统计学院2018、2019级部分同学的问卷调查,剖析教学中存在的问题,进一步探讨如何在合班授课模式下提高教学质量,促进教学效果,以便对课程教育研究者有所帮助。

关键词： 高中数学   分析思想   应用策略  

摘要： 高中阶段的数学教学整体呈现出复杂化、多元化 的发展特点,要帮助学生在有限的时间内掌握数学知识,教师 应结合教育要求引入全新的教学方法。在现阶段的数学教育 活动中,数学分析思想正在被越来越多的教师与学生所重视。本文结合高中数学教学的相关特点展开论述,思考如何利用数 学分析思想帮助学生解决数学问题。

关键词： 数学分析习题课   教学效果   综合评价模型  

摘要： 基于主成分分析(PCA)和最小二乘线性回归分析(OLS)等统计模型的算法特点,构建PCA OLS综合评价模型,并将其应用在数学分析习题课的教学效果评价中.通过对成都师范学院学生数学分析习题课教学效果相关数据的实证分析,发现数学分析习题课对学生动手实践能力、创新能力和研究能力的培养至关重要.