应用数学的拓展——用一篇关于蛋白质分子的结构和功能的动理论发展的论文来说明

近来,人们喜欢从历史的角度来探讨数学在生物学中的应用问题.这种趋势似乎可以看作是应用数学范围逐渐拓展的一种自然现象.但它所用到的数学概念和方法与以往应用数学所用的概念和方法并无本质区别.例如,我们在本文中简单描述了如何将耗散系统中的动理学原理用于研究蛋白质分子的结构和功能.在这些研究中,传统统计物理学的概念和方法可以用来成功地建立与经验数据相对应的假设及理论.      

A new pansystems research about binary relation’s transitivity

Because of the great needs both for the research of pansystem mathematics and for the analysis of general things’ mechanism, this paper discusses the binary relation’s transitivity confined to multirelation. The so-called “g-transitivity” a completely new concept about transitivity—is introduced and its basic properties are investigated. The study shows that it is not only g-transitivity a generalization of traditional transitivity, quasi-transitivity, semi-order, quasi-order and other panorder, but also includes many important concepts such as closenss, convexity, topology, duality as its special cases.     

Nonlocal modeling of a randomly distributed and aligned long-fiber composite material

The work under study is about the variational and stochastic modeling of randomly distributed and aligned long-fiber composites. Its objective is to derive a homogenized behavior that exhibits the nonlocal phenomenon of this type of material at the macroscopic scale. Several methods of applied mathematics are used in order to keep the maximum information about the nonlocal behavior after homogenization.     

Principal difference between stability and structural stability (robustness) as used in systems biology

The concepts stability and structural stability (robustness) are used often in systems biology. According to Kitano (2004) robustness is a fundamental property of evolvable complex biological systems. For that reason, the purpose of this review is to clarify: (a) how are strictly formulated concepts, such as stability and robustness of a dynamical system, used in computational systems biology; (b) what is meant by structural stability (robustness) in contemporary biology and how are stability and robustness distinguished from each other; and (c) why is it necessary to investigate whether a cell signal pathway is stable. We formulate the two concepts stability and structural stability (robustness) of a dynamical system with an arbitrary dimensionality, in the way they are known in mathematics and mechanics, and clarify the principal difference between them. We also consider how these two concepts are used in the analysis of a concrete biological system in systems biology. In the last section we formulate when, according to us, in biology (and in systems biology in particular), it should be said that a system (process) is stable, and when it is structurally stable.     

同伦分析方法进展综述

本文简述同伦分析方法基本思想、最新理论进展及其在流体力学、固体力学、一般力学、量子力学、应用数学、金融等科学和工程领域的应用.同伦分析方法不依赖物理小参数, 适用范围更广,而且提供了一种简单的途径确保级数解收敛, 适用于强非线性问题.同伦分析方法已被成功应用于求解一些具有挑战性的力学问题,并获得一些全新的、 从未见报道的解. 这些成功的应用,证明了同伦分析方法的普遍有效性和原创性.      

凸轮机构系统磨损及可靠性寿命的数值仿真

基于离散数学理论和计算机技术提出了平底从动件凸轮机构系统磨损及可靠性寿命的数值仿真模型，引入了位置矢量和磨损步长的概念，使连续的磨损过程离散化，对凸轮的磨损寿命及其可靠度进行了预测，并对从动件的运动规律进行了动态跟踪．结果表明，复杂的磨损过程可用数值法进行模拟仿真，从而摆脱了传统的仅依赖实验的相关研究方法，通过一系列离散性准静态模型解决了经典微积分数学方法无法解决的动态和非线性磨损问题．所建立的方法具有良好的工程应用前景．     

植物生长、发育、演化动力学过程的数学-物理模型研究

自20世纪中叶开始, 在生物学, 物理学, 力学, 数学的边缘领域逐渐形成了一门新兴的交 叉学科------数学生物学, 它将生物系统视为一个复杂的动力学系统, 从应用数学角度进行 深入研究, 其研究课题目前已远远超出人们熟知的生物力学领域. 这门学科的迅速发展正深 刻地改变着古老的, 以观察、分类、实验为特征的传统生物学的面貌. 本文旨在探讨数学生 物学中的一个具有重大理论与实际意义的课题方向? 植物系统生长、发育、演化动力学过程 的数学、物理模型研究; 介绍近年来国内、外在这一方向的研究动态; 并说明在对此课题的 研究方法上, 动力学数学模型与统计学模型以及计算机模拟的根本区别. 进而提出以植 物根系生长过程的研究为突破口, 说明若将生长系统处理成一个热力学开放系统, 从不可逆 过程热力学、植物细胞学与植物生理学原理出发, 结合植物内部组织的微观结构的确立, 人 们有可能建立起一套合适的描述系统内部物质、能量与信号输运的控制微分方程; 再通过运 用强有力的非线性科学、应用数学的方法求解所得微分方程式, 并将理论预测与实验观察结 果进行比较, 人们可深入探索与更好地理解根系对土壤中的水分与营养物质吸收机制和根系 生长过程中所出现的种种生物界面不稳定性现象.     

Simultaneous characterization of multiple properties of solid and liquid phases in crystallization processes using NIR

Near infrared spectroscopy (NIR) is now probably the most popular process analytical technology (PAT) for pharmaceutical and some other industries. However, unlike mid-IR, NIR is known to have difficulties in monitoring crystallization or precipitation processes because the existence of solids could cause distortion of the spectra. This phenomenon, seen as unfavorable previously, is however an indication that NIR spectra contain rich information about both solids and liquids, giving the possibility of using the same instrument for multiple property characterization. In this study, transflectance NIR calibration data was obtained using solutions and slurries of varied solution concentration, particle size, solid concentration and temperature. The data was used to build calibration models for prediction of the multiple properties of both phases. Predictive models were developed for this challenging application using an approach that combines genetic algorithm (GA) and support vector machine (SVM). GA is used for wavelength selection and SVM for mode building. The new GA–SVM approach is shown to outperform other methods including GA–PLS (partial least squares) and traditional SVM. NIR is thus successfully applied to monitoring seeded and unseeded cooling crystallization processes of l-glutamic acid.     

A brief review of the homotopy analysis method

本文简述同伦分析方法基本思想、最新理论进展及其在流体力学、固体力学、一般力学、量子力学、应用数学、金融等科学和工程领域的应用.同伦分析方法不依赖物理小参数, 适用范围更广,而且提供了一种简单的途径确保级数解收敛, 适用于强非线性问题.同伦分析方法已被成功应用于求解一些具有挑战性的力学问题,并获得一些全新的、 从未见报道的解. 这些成功的应用,证明了同伦分析方法的普遍有效性和原创性.     

加速度计加速退化数据统计分析方法

加速退化试验广泛应用于加速度计的可靠性评估或寿命预测,然而加速退化数据统计分析方法的研究不足制约了可靠性评估或寿命预测的准确性。提出了一种基于Wiener退化模型的加速退化数据统计分析方法,利用加速因子不变原则推导退化模型各参数的变化规律,在此基础上建立加速退化模型,从而克服依赖假定进行加速退化建模容易造成较大误差的不足。针对传统的极大似然估计方法适用性不强的问题,提出了利用fminsearch函数结合最小二乘法求解极大似然估计值的方法。针对寿命预测时无法得到可靠寿命解析式的问题,提出了利用fzero函数求取可靠寿命的方法。仿真试验证明了加速因子不变原则推导结论的正确性,案例应用验证了所提方法的有效性。研究工作在加速退化建模、参数估计、寿命预测三个方面提出了有效的方法,具有一定的工程应用价值。     

梁波传播的固有特性

以分布参数的理论为基础，分析了梁波传播的一些固有特性，说明了它们与周期性结构的关系，讨论了支持条件对波传播特性的影响，指出了如果把近似方法应用于波传播分析中，可能引起的一些问题．本文的结果可直接应用于周期性结构的分析中     

Symmetry-based optimal portfolio for a DC pension plan under a CEV model with power utility

In this article, explicit representation of solution for the Hamilton–Jacobi–Bellman (HJB) equation associated with the portfolio optimization problem for an investor who seeks to maximize the expected power (CRRA) utility of the terminal wealth in a defined-contribution pension plan under a constant elasticity of variance model is derived based on the application of the Lie symmetry method to the partial differential equation and its associated terminal condition. Compared with the ingenious ansatz techniques used before, here we present a group theoretical analysis of the terminal value problem for the solution following the algorithmic procedure of the Lie symmetry analysis. It shows that the interesting properties of the group structures of the original HJB equation and its successive similarity reduced equations lead to an elegant resolution of the problem. Moreover, we identify the meaningful range of risk aversion coefficient which is ignored in the previous work. At last, the properties and sensitivity analysis of the derived optimal strategy are demonstrated by numerical simulations and several figures. The method used here is quite general and can be applied to other equations obtained in financial mathematics.

     

Magnetorheological damping and semi-active control of an autoparametric vibration absorber

A numerical study of an application of magnetorheological (MR) damper for semi-active control is presented in this paper. The damper is mounted in the suspension of a Duffing oscillator with an attached pendulum. The MR damper with properties modelled by a hysteretic loop, is applied in order to control of the system response. Two methods for the dynamics control in the closed-loop algorithm based on the amplitude and velocity of the pendulum and the impulse on–off activation of MR damper are proposed. These concepts allow the system maintaining on a desirable attractor or, if necessary, to change a position from one attractor to another. Additionally, the detailed bifurcation analysis of the influence of MR damping on the number of periodic solutions and their stability is shown by continuation method. The influence of MR damping on the chaotic behavior is studied, as well.     

Variation transforming analysis (I)

This paper is concerned with operator variation from the transforming point of view, and presents some new concepts and new relations. Related problems and concepts include: convex operator, reciprocity set, reciprocity principles, unilateral variation principles, (H, H ₁,H ₂) — generalized solution and operator-differential equation, etc..Variation principle and variation method is a sort of basic concept and method in the analysis of many problems concerning mathematics, mechanics, physics and control theory. The change and transformation of variation or stationary value relations is the most fundamental form and process of these princeples and methods. This paper analyzes some properties of operator variation and stationary value from the transforming point of view and presents some new concepts, unifiedly treats and extends the reciprocity theorem of classical variation, variation theorem of quadratic functional and unilateral variation principles. Furthermore, we investigate the calculus of normed ring and the solution of a sort of operator differential equation.     

PARALLEL COMPUTING FOR STATIC RESPONSE ANALYSIS OF STRUCTURES WITH UNCERTAIN-BUT-BOUNDED PARAMETERS

The vertex solution for estimation on the static displacement bounds of structures with uncertain-but-bounded parameters is studied in this paper. For the linear static problem, when there are uncertain interval parameters in the stiffness matrix and the vector of applied forces, the static response may be an interval. Based on the interval operations, the interval solution obtained by the vertex solution is more accurate and more credible than other methods （such as the perturbation method）. However, the vertex solution method by traditional serial computing usually needs large computational efforts, especially for large structures. In order to avoid its disadvantages of large calculation and much runtime, its parallel computing which can be used in large-scale computing is presented in this paper. Two kinds of parallel computing algorithms are proposed based on the vertex solution. The parallel computing will solve many interval problems which cannot be resolved by traditional interval analysis methods.     

Experimental study of vortex–structure interaction noise radiated from rod–airfoil configurations

Vortex–structure interaction noise radiated from an airfoil embedded in the wake of a rod is investigated experimentally in an anechoic wind tunnel by means of a phased microphone array for acoustic tests and particle image velocimetry (PIV) for the flow field measurements. The rod–airfoil configuration is varied by changing the rod diameter (D), adjusting the cross-stream position (Y) of the rod and the streamwise gap (L) between the rod and the airfoil leading edge. Two noise control concepts, including “air blowing” on the upstream rod and a soft-vane leading edge on the airfoil, are applied to control the vortex–structure interaction noise. The motivation behind this study is to investigate the effects of the three parameters on the characteristics of the radiated noise and then explore the influences of the noise control concepts. Both the vortex–structure interaction noise and the rod vortex shedding tonal noise are analysed. The acoustic test results show that both the position and magnitude of the dominant noise source of the rod–airfoil model are highly dependent on the parameters considered. In the case where the vortex–structure interaction noise is dominant, the application of the air blowing and the soft vane can effectively attenuate the interaction noise. Flow field measurements suggest that the intensity of the vortex–structure interaction and the flow impingement on the airfoil leading edge are suppressed by the control methods, giving a reduction in noise.     

The Potential Scope of the Ultrasonic Surface Reflection Method Towards Mechanical Characterisation of Isotropic Materials. Part 1. A Theoretical Analysis

Background

This paper deals with the possible field of application of ultrasonic Surface Reflection Method (SRM) to achieve the mechanical characteristics of isotropic materials. This method is based on the measurement of the amplitude of the reflected wave at the interface between reference material and the material to be characterised. Objective: The purpose of Part 1 of this paper is to establish the theoretical conditions for the applicability of SRM.

Methods

First, the theoretical formulas necessary to obtain the mechanical properties of the material to be tested will be established. Then, on the basis of these analytical formulas, the validity of the results for the material to be studied will be discussed according to the choice of the mechanical properties of the reference material through uncertainty calculations. The measurand error of SRM is then compared to that of traditional methods (transmission, transmission in water bath, pulse-echo).

Results

The analytical solution to the inverse problem (the mechanical characteristics of the tested medium based on those of the reference medium and the waves’ amplitude) will be given. From this analytical solution, an analysis of the measurand error will be performed and a method for choosing the reference material will be proposed.

Conclusions

It appears that SRM is better suited than traditional methods in two specific cases: measurement of small deviations of mechanical properties from a reference material or characterisation of high damping materials. In Part 2 of this paper, the practical conditions of applicability of the method are described and then applied to different kinds of materials.

     

Stochastic methods in dispersion theory

In this study we show how methods from the theory of stochastic processes can be applied to problems in dispersion theory.First, we show that Taylor dispersion with adsorbing boundaries is easily transformed into a new Taylor dispersion problem without adsorbing boundaries. The transformed problem can then be solved using any of the traditional methods used for Taylor dispersion.Secondly, we consider the dispersion of particles in a channel (between parallel plates) with one partially adsorbing surface and one perfectly reflecting boundary. We determine the exact law of the position of adsorption for an arbitrary channel flow in terms of an infinite series of iterated integrals of the flow field, which is assumed to be a function of the cross-channel coordinate only. We also consider the case of shear flow over an adsorbing plane, by taking the limit where one of the boundaries is taken to infinity     

An upper bound for validity limits of asymptotic analytical approaches based on normal form theory

Perturbation methods are routinely used in all fields of applied mathematics where analytical solutions for nonlinear dynamical systems are searched. Among them, normal form theory provides a reliable method for systematically simplifying dynamical systems via nonlinear change of coordinates, and is also used in a mechanical context to define Nonlinear Normal Modes (NNMs). The main recognized drawback of perturbation methods is the absence of a criterion establishing their range of validity in terms of amplitude. In this paper, we propose a method to obtain upper bounds for amplitudes of changes of variables in normal form transformations. The criterion is tested on simple mechanical systems with one and two degrees-of-freedom, and for complex as well as real normal form. Its behavior with increasing order in the normal transform is established, and comparisons are drawn between exact solutions and normal form computations for increasing levels of amplitudes. The results clearly establish that the criterion gives an upper bound for validity limit of normal transforms.     

基于静动态混合重构的DG/FV混合格式

通过比较紧致格式和间断Galerkin(DG)格式, 提出了``静态重构'和``动态重构'的概念,对有限体积方法和DG有限元方法进行统一的表述. 借鉴有限体积的思想, 发展了基于``混合重构'技术的一类新的DG格式, 称之为间断Galerkin有限元/有限体积混合格式(DG/FV格式). 该类混合格式通过适当地扩展模板(拓展至紧邻单元)重构单元内的高阶多项式分布, 在提高精度的同时, 减少了传统DG格式的计算量和存储量. 通过典型一维和二维标量方程的计算发现新的混合格式在有些情况下具有超收敛(superconvergence)性质.