一种新的变权向量及其应用

在对常权综合和变权综合进行分析的基础上,结合变权的初衷,构造了一个新的变权向量用于变权综合.构造的变权满足变权向量的公理条件和加型综合函数的定义,运用这个变权公式进行加权综合分析时,综合决策值等于加权均值减去加权方差的一半,具有很好的实际意义.     

变权综合与可加型标准综合函数

分析变权综合与可加型标准综合函数之间的关系，给出变权综合是可加型标准综合函数的一个必要条件，并讨论了几类重要状态变权向量构成的变权综合成为可加性标准综合函数的充分条件.     

基于几何均值的惩罚型状态变权向量的构造及应用

依据惩罚型状态变权向量的定义和对各因素间的均衡性要求,构造了含有几何平均值的惩罚型状态变权向量.给出了各因素的几何平均值相等和几何平均值不相等两种情况的构造方法,对多因素的变权综合提供了有效的理论支持.最后,用一个实例验证了该方法的可行性和有效性.     

状态变权向量的等效分析以及一类均衡函数的构造

进一步讨论变权综合理论的基本性质和均衡函数的构造方法。首先证明了一个状态变权向量等效的充要条件。然后，以两个反例分别说明变权向量不一定都能表示成因素常权向量与状态变权向量的归一化的Hadamard乘积；状态变权向量并非都是均衡函数的梯度向量。最后，引入因素状态向量离散度的定义，并由此得到了一类新的均衡函数。     

变权决策中确定状态变权向量的理想点法

利用因素状态向量构造正理想状态向量X+和负理想状态向量X-,然后由这两个理想状态向量分别构造两个极不均衡的状态向量RX+-和RX-+;根据状态变权向量对RX+-和RX-+的调权效果以及OWA算子中的主观偏好参数A-C(W)建立一个确定状态变权向量参数的数学模型,为解决变权综合过程中如何选用合适的状态变权向量提供了一个可操作性的方法.最后,通过一个实例对该方法的应用进行了分析、验证.     

属性值为区间数的变权综合决策方法

对属性值为区间数的多属性不确定性决策问题,先由区间数向量定义四组实数向量,然后利用实数状态变权向量和上述四组向量导出相应的区间数状态变权向量,由此得出区间数变权公式,进而建立区间数变权综合决策模型.最后给出一个应用区间数变权综合决策模型的实例.     

模糊推理的变权综合算法及其构造的模糊系统响应能力分析

将变权综合原理应用于模糊推理,提出一种新的模糊推理算法-变权综合推理算法,并给出利用模糊蕴涵算子确定规则变权的方法。证明了由一些正常蕴涵算子如Lukasiewicz蕴涵、Goguen蕴涵、Godel蕴涵、Dubois-Prade蕴涵等确定的变权综合算法是相容的。进一步分析了该变权综合算法构造的模糊系统的响应能力,结果表明:基于这些正常蕴涵算子的变权综合算法构造的模糊系统具有泛逼近性。     

因素空间理论与知识表示的数学框架（Ⅷ）─—变权综合原理

本文是文［１－７］的继续，研究变权综合问题，从确定变权的经验公式入手引出了变权原理，给出了变权的公理化定义，讨论了与之有关的均衡函数及其梯度向量。     

一种变权综合评价方法及其在膨胀土分类中的应用

针对膨胀土胀缩等级分类问题,构造了一种新的区间关联函数,提出了"分类特征体现度越大,权重越大"的权重计算原理,给出了一种新的变权计算方法,将单指标关联函数的最大值作为其评价类的靶心坐标,根据样本与靶心的贴近度,对样本进行分类.将该方法应用于膨胀土的胀缩等级分类,其合理性和有效性得到了验证.     

等效均衡函数的性质及均衡函数的构造

从均衡函数的等效性出发研究均衡函数的性质，证明了等效的均衡函数之和、积以及数乘仍为均衡函数，并且得到的新均衡函数与原均衡函数等效，这些性质表明等效的均衡函数关于加法和乘法运算均具有半群的代数结构。另外，进一步讨论了构造均衡函数的方法，给出了两个构造均衡函数的定理，该方法具有一般性，现有文献中的均衡函数几乎都能由其构造得到。     

Liberating the dimension for function approximation: Standard information

This is a follow-up paper of “Liberating the dimension for function approximation”, where we studied approximation of infinitely variate functions by algorithms that use linear information consisting of finitely many linear functionals. In this paper, we study similar approximation problems, however, now the algorithms can only use standard information consisting of finitely many function values. We assume that the cost of one function value depends on the number of active variables. We focus on polynomial tractability, and occasionally also study weak tractability. We present non-constructive and constructive results. Non-constructive results are based on known relations between linear and standard information for finitely variate functions, whereas constructive results are based on Smolyak’s construction generalized to the case of infinitely variate functions. Surprisingly, for many cases, the results for standard information are roughly the same as for linear information.     

A general result in stochastic optimal control of nonlinear discrete-time systems with quadratic performance criteria

It is known that the optimal controller for a linear dynamic system disturbed by additive, independently distributed in time, not necessarily Gaussian, noise is a linear function of the state variables if the performance criterion is the expected value of a quadratic form. This result is known to hold also when the noise is Gaussian and is multiplied by a linear function of the state and/or control variables.In this paper it is proved that the optimal controller for a discrete-time linear dynamic system with quadratic performance criterion is a linear function of the state variables when the additive random vector is a nonlinear function of the state and/or control variables and not necessarily Gaussian noise which is independently distributed in time, provided only that the mean value of the random vector is zero (there is no loss of generality in assuming this) and the covariance matrix of the random vector is a quadratic function of the state and/or control variables. The above-mentioned known results emerge as special cases and certain nonlinear other special cases are exhibited.     

On prediction rate in partial functional linear regression

We consider a prediction of a scalar variable based on both a function-valued variable and a finite number of real-valued variables. For the estimation of the regression parameters, which include the infinite dimensional function as well as the slope parameters for the real-valued variables, it is inevitable to impose some kind of regularization. We consider two different approaches, which are shown to achieve the same convergence rate of the mean squared prediction error under respective assumptions. One is based on functional principal components regression (FPCR) and the alternative is functional ridge regression (FRR) based on Tikhonov regularization. Also, numerical studies are carried out for a simulation data and a real data.     

A Discrete-Time American Put Option Model with Fuzziness of Stock Prices

To solve a mathematical model for American put option with uncertainty, we utilize two essentials, i.e., a λ-weighting function and a mean value of fuzzy random variables simultaneously. Estimation of randomness and fuzziness as uncertainty should be important when we deal with a reasonable and natural model extended from the original optimization/decision making. Three kinds of mean values by fuzzy measures, which are based on Possibility, Necessity and Credibility, are demonstrated particularly. We consider the optimal expected price of the American put option by dynamic programming under a reasonable assumption. A numerical example is given to illustrate our idea.     

多元函数中值定理中介值的渐近性

讨论了n元函数中值定理中介值的渐近性质.     

On the Synergistic Influence of Two Control Variables on the State of Nonlinear Optimal Control Models

Provided that the Hamiltonian maximizing conditions do not depend on the state variable, under certain assumptions, a synergistic influence of two control variables on the state of a nonlinear optimal control model can be identified. By such a synergism we mean that it is optimal to use both controls so as to influence the state variable ‘in the same direction’.     

A new approach to Bezout equations derived from multivariate polynomial matrices and real entire functions

This paper focuses on Bezout equations derived from multivariate polynomial matrices in which relationships between one primary variable and other variables are described by real entire functions. We propose a method for obtaining a solution belonging to a set of multivariate rational function matrices in which all entries are real entire functions with respect to the primary variable. The proposed method is based on a new approach that overcomes the constraints and difficulties due to many variables by expanding a class of solutions to multivariate rational function matrices.     

Calculating the mean growth rate of the vector of states of a stochastic system with synchronization of events

A stochastic dynamical system with synchronization is considered. The dynamics of the system is described by a linear vector equation with a second-order matrix in an idempotent semiring with the operations of taking maximum and addition. It is assumed that one diagonal entry of the matrix is an exponentially distributed random variable, whereas all other entries are equal to some nonnegative constant. To solve the problem of calculating the mean rate of growth of the state vector of the system, we make a change of variables: instead of the random coordinates of the state vector of the system we introduce new random variables which are more convenient to analyze. After that, the corresponding sequences of one-dimensional distribution functions are constructed and examined for convergence. The mean rate of growth is calculated as the mean value of the limit distribution. In addition, expressions for the limit probabilities of some events in the systems are derived.     

A partial Hamiltonian approach for current value Hamiltonian systems

We develop a partial Hamiltonian framework to obtain reductions and closed-form solutions via first integrals of current value Hamiltonian systems of ordinary differential equations (ODEs). The approach is algorithmic and applies to many state and costate variables of the current value Hamiltonian. However, we apply the method to models with one control, one state and one costate variable to illustrate its effectiveness. The current value Hamiltonian systems arise in economic growth theory and other economic models. We explain our approach with the help of a simple illustrative example and then apply it to two widely used economic growth models: the Ramsey model with a constant relative risk aversion (CRRA) utility function and Cobb Douglas technology and a one-sector AK model of endogenous growth are considered. We show that our newly developed systematic approach can be used to deduce results given in the literature and also to find new solutions.     

D'Alembert‐type solution of the Cauchy problem for the Boussinesq‐Klein‐Gordon equation

In this paper, we construct a weakly‐nonlinear d'Alembert‐type solution of the Cauchy problem for the Boussinesq‐Klein‐Gordon (BKG) equation. Similarly to our earlier work based on the use of spatial Fourier series, we consider the problem in the class of periodic functions on an interval of finite length (including the case of localized solutions on a large interval), and work with the nonlinear partial differential equation with variable coefficients describing the deviation from the oscillating mean value. Unlike our earlier paper, here we develop a novel multiple‐scales procedure involving fast characteristic variables and two slow time scales and averaging with respect to the spatial variable at a constant value of one or another characteristic variable, which allows us to construct an explicit and compact d'Alembert‐type solution of the nonlinear problem in terms of solutions of two Ostrovsky equations emerging at the leading order and describing the right‐ and left‐propagating waves. Validity of the constructed solution in the case when only the first initial condition for the BKG equation may have nonzero mean value follows from our earlier results, and is illustrated numerically for a number of instructive examples, both for periodic solutions on a finite interval, and localized solutions on a large interval. We also outline an extension of the procedure to the general case, when both initial conditions may have nonzero mean values. Importantly, in all cases, the initial conditions for the leading‐order Ostrovsky equations by construction have zero mean, while initial conditions for the BKG equation may have nonzero mean values.