数学问题解答

数学问题解答１９９８年５月号问题解答（解答由问题提供人给出）１１３１试证：一元多项式Ｐ（ｘ）＝ｘｋ－ｘｋ－１－１的复根ｚ＝ｒｅｉθ一定满足ｒ２ｋ－２ｒ２ｋ－１ｃｏｓθ＋ｒ２ｋ－２－１＝０．证明我们有｜Ｐ（ｚ）｜｜ｚｋ－ｚｋ－１｜－１＝ｒｋ－１｜ｚ－...     

亚纯函数的幅角分布及其增长性

本文考虑了亚纯函数的幅角分布及其增长性的关系，得到了如下定理：设ｆ（ｚ）为亚纯函数，下级μ（μ〈＋∞，ａｒｇｚ＝θｋ（ｋ＝１，２，…，ｑ；０≤θ１〈θ２〈…〈θｑ〈２π，θｑ＋１＝θ１＋２π）为ｑ（１≤ｑ〈＋∞）条半直线使对Ａ↓ε〉０有：ｌｉｍｒ→∞↑－ｌｏｇ＋ｎ↑－（∪ｋ＝１↑ｑΩ↑－（θｋ＋ε，θｋ＋１－ε；ｒ），ｆ＝ｘ）／ｌｏｇｒ≤ρ〈＋∞ ｘ＝０，∞则当存在一非负整数ｌ使ｆ＾（ｌ）（ｚ）（     

1999年全国高中联赛第三题的简解

题目　已知当ｘ∈［０,１］时,不等式ｘ２ｃｏｓθ－ｘ（１－ｘ）＋（１－ｘ）２ｓｉｎθ＞０恒成立,试求θ的取值范围．这是１９９９年全国高中数学联合竞赛试题第三题,下面给出一种有别于“标准答案”的简单解法．解　若对一切ｘ∈［０,１］,恒有ｆ（ｘ）＝ｘ２ｃｏｓθ－ｘ（１－ｘ）＋（１－ｘ）２ｓｉｎθ＞０,则　ｓｉｎθ＝ｆ（０）＞０,ｃｏｓθ＝ｆ（１）＞０,∴　２ｋπ＜θ＜２ｋπ＋π２,ｋ∈Ｚ．（１）又　ｆ（ｘ）＝（１＋ｓｉｎθ＋ｃｏｓθ）ｘ２－（１＋２ｓｉｎθ）ｘ＋ｓｉｎθ＝（１＋ｓｉｎθ＋ｃｏｓθ）［…     

一个对称函数下界的加强

记ｆｋ（ｘ１…，ｘｎ）＝Ｅｋ（１－ｘ１，…，１－ｘｎ）－Ｅｋ（ｘ１，…，ｘｎ），ｋ＝１，…ｎ其中Ｅｋ（ｘ１，…，ｘｎ）为初等对称函数，并规定当ｋ＝０时，Ｅｋ（ｘ１，…，ｘｎ）＝１，当ｋ＜０或ｋ＞ｎ时，Ｅｋ（ｘ１，…，ｘｎ）＝０．笔者在文［１］证明了：...     

分组数据参数的单边估计与检验

设ｘ_０＜ｘ_１＜…＜ｘ_ｋ－１＜ｘ_ｋ是分布函数Ｆ（ｘ－θ）支撑集上的分点．令ｎ_ｉ表示落入区间（ｘ_ｉ－１,ｘ_ｉ）的观测值个数,并称其为分组数据．本文讨论了θ在θ≥θ０下的最大似然估计存在且唯一的条件及其渐近性质;给出了利用分组数据求最大似然估计的方法．最后;讨论了θ的单边检验问题．     

Opial－Olech型不等式的改进及其应用

本文给出：设ｆ（ｘ）在［０，ｈ］上绝对连续。ｆ（０）＝ｆ（ｈ）＝０，ｐ＞０，ｑ＞１和ｓ＝Ｐ／（ｐ＋ｑ－１），则有 其中θ（ｐ）＝1/2,p+q>0,θ（ｐ）＝P/2.当１＜ｐ＋ｑ＜２.若代（Ａ）右边为零。即为Ｏｐｉａｌ－Ｏｌｃｃｈ不等式。实际上本文所得结果还要广泛。     

两个不等式的简捷证法

下面给出的两类不等式问题,一般是通过代换的方法证明．本文给出直接简捷的证明．命题１　设ｘｉ∈Ｒ＋（ｉ＝１,２,…,ｎ）且ｘ２１１＋ｘ２１＋ｘ２２１＋ｘ２２＋…＋ｘ２ｎ１＋ｘ２ｎ＝ａ（０＜ａ＜ｎ）,求证：ｘ１１＋ｘ２＋ｘ２２１＋ｘ２２＋…＋ｘ２ｎ１＋ｘ２ｎ≤ａ（ｎ－ａ）①证　由题设易知：１１＋ｘ２１＋１１＋ｘ２２＋…＋１１＋ｘ２ｎ＝ｎ－ａ．由于　１１＋ｘ２ｋ＋ｎ－ａａ·ｘ２ｋ１＋ｘ２ｋ　　≥２１１＋ｘ２ｋ·ｎ－ａａ·ｘ２ｋ１＋ｋ２ｋ　　＝２ｎ－ａａ·ｘｋ１＋ｘ２ｋ）（ｋ＝１,２,…,ｎ）,此ｎ式相…     

圆锥曲线弦的中点与斜率的关系及其应用

关于圆锥曲线弦的中点问题,许多文章已有论述,本文综其为一体,给出圆锥曲线弦的一个重要性质．定理　圆锥曲线Ａｘ２＋Ｃｙ２＋Ｄｘ＋Ｅｙ＋Ｆ＝０的弦的斜率为ｋ,弦的中点为（ｘ０,ｙ０）,同有Ａｘ０＋Ｃｋｙ０＋１２Ｄ＋１２ｋＥ＝０．证　设弦的两端点为（ｘ１,ｙ１）,（ｘ２,ｙ２）斜率为ｋ,则有Ａｘ２１＋Ｃｙ２１＋Ｄｘ１＋Ｅｙ１＋Ｆ＝０,Ａｘ２２＋Ｃｙ２２＋Ｄｘ２＋Ｅｙ２＋Ｆ＝０．两式相减,得Ａ（ｘ２１－ｘ２２）＋Ｃ（ｙ２１－ｙ２２）＋Ｄ（ｘ１－ｘ２）　＋Ｅ（ｙ１－ｙ２）＝０．两边同除以ｘ１－ｘ２,注意到…     

一类线性errors－in－variables模型的参数强相合估计及其a．s．收敛速度

本文讨论了如下一类线性ｅｒｒｏｒｓ－ｉｎ－ｖａｒｉａｂｌｅｓ模型——多元线性结构关系模型β′ｘｋ＋α＝０，ξｋ＝ｘｋ＋εｋ．｛ｋ＝１，２，…，ｎ．其中，｛ｘｋ：ｋ＝１，２，…，ｎ｝为一组ｉ．ｉ．ｄ．的ｍ维随机向量，｛εｋ：ｋ＝１，２，…，ｎ｝是ｉ．ｉ．ｄ．的随机误差，Ｅ（ε１）＝０，Ｖａｒ（ε１）＝σ２Ｉｍ．且｛ｘｋ：ｋ＝１，２，…，ｎ｝与｛εｋ：ｋ＝１，２，…，ｎ｝相互独立．在一些条件下，我们证明了估计量β，α，σ２的强相合性、唯一性，并给出了估计量的收敛速度为ｏ（ｎ－１－１ｑ），这里ｑ∈［１，２）．对于Ｅ（ｘ１）ｕ１和Ｖａｒ（ｘ１）Ｖｘ的估计也得出了同样的结果     

区间上平顶单峰扩张自映射的周期轨道

设ｔ（０＜ｔ＜１）是一个常数，ｎ≥３是奇数，ｍ≥０及ｋ≥１是整数，Ｐ０（ｘ）＝ｘ－１，Ｐｉ（ｘ）＝（ｘ２ｉ－１－１）Ｐｉ－１（ｘ）（ｉ≥１），ｒｍｎ（ｔ）及ｒｋ（ｔ）分别是方程Ｐｍ（ｘ）（ｘ２ｍｎ－２ｘ２ｍ（ｎ－２）－１）－ｔ（ｘ２ｍｎ－１）（ｘ２ｍ＋１）＝０及Ｐｋ－１（ｘ）－ｔ（ｘ２ｋ－１＋１）＝０在（１，＋∞）上的唯一实根，ｆ是闭区间Ｉ＝［０，１］上的峰顶区间长度为ｔ的平顶单峰扩张自映射．本文证明了，若ｆ的扩张常数λ≥ｒｍｎ（ｔ）（或＞ｒｋ（ｔ）），，则ｆ有２ｍｎ（或２ｋ）周期点．此外，本文还指出，当１＜λ＜ｒｍｎ（ｔ）（或≤ｒｋ（ｔ）时，在Ｉ上存在着具有扩张常数λ及峰顶区间长度ｔ却无２ｍｎ（或２ｋ）周期点的平顶单峰扩张自映射     

线性指数危险率模型的贝叶斯判别分析

设有两个总体Π0和Π1，其危险率为具有不同参数的线性函数。对于待观测的寿命样本X，给出了相应的判别分析问题的Bayes停止判决法则，其中损失函数包括试验费用和误判损失两部分。      

随机截尾试验变量抽样接收方案的贝叶斯决策

本运用Bayes决策理论研究指数分布和随机截尾试验的抽样接收方案的一般模型,我们证明了最优Bayes法则具有单调性,并对二个特殊的决策损失函数给出了最优Bayes法则和Bayes风险的具体表达式。     

Sequential variable sampling plan for normal distribution

In this paper, a sequential variable sampling plan is studied. Suppose that the quality of an item in a batch is measured by a normally distributed random variable with a known variance, but the mean is unknown with a normal prior distribution. Then by using Bayesian approach and considering a Markov decision process, the optimality equations for the minimum total expected cost are formulated. We show that an optimal decision rule will have a control limit structure. An algorithm for a sequence of ϵ-optimal decisions is introduced. Then, the statistical procedure for conducting the sequential sampling plan is presented.     

On the empirical Bayes approach to multiple decision problems with sequential components

The empirical Bayes approach to multiple decision problems with a sequential decision problem as the component is studied. An empirical Bayesm-truncated sequential decision procedure is exhibited for general multiple decision problems. With a sequential component, an empirical Bayes sequential decision procedure selects both a stopping rule function and a terminal decision rule function for use in the component. Asymptotic results are presented for the convergence of the Bayes risk of the empirical Bayes sequential decision procedure.     

Triangular fuzzy decision-theoretic rough sets

Based on decision-theoretic rough sets (DTRS), we augment the existing model by introducing into the granular values. More specifically, we generalize a concept of the precise value of loss function to triangular fuzzy decision-theoretic rough sets (TFDTRS). Firstly, ranking the expected loss with triangular fuzzy number is analyzed. In light of Bayesian decision procedure, we calculate three thresholds and derive decision rules. The relationship between the values of the thresholds and the risk attitude index of decision maker presented in the ranking function is analyzed. With the aid of multiple attribute group decision making, we design an algorithm to determine the values of losses used in TFDTRS. It is achieved with the use of particle swarm optimization. Our study provides a solution in the aspect of determining the value of loss function of DTRS and extends its range of applications. Finally, an example is presented to elaborate on the performance of the TFDTRS model.     

基于贝叶斯信息更新的风险规避库存策略研究

在贝叶斯库存控制研究中一个著名的结论是：当缺货需求不能被观测到时,最优贝叶斯库存水平总会高于短视策略库存水平,原因是决策者需要通过多订货来获取对需求分布的认识. 这是基于风险中性的研究,然后现实中决策者都期望规避风险. 基于贝叶斯信息更新研究了风险规避背景下需求部分可观测的多周期报童问题,决策者的周期内效用函数满足独立可加性公理. 通过引入非正规化概率,研究发现,对风险规避的决策者,当其效用函数具有不变绝对风险规避特征时,最优贝叶斯库存水平也会高于短视策略库存水平. 非正规化概率简化了动态规划方程与结果的证明.     

Integrating Bayesian networks and decision trees in a sequential rule-based transportation model

Several activity-based transportation models are now becoming operational and are entering the stage of application for the modelling of travel demand. Some of these models use decision rules to support its decision-making instead of principles of utility maximization. Decision rules can be derived from different modelling approaches. In a previous study, it was shown that Bayesian networks outperform decision trees and that they are better suited to capture the complexity of the underlying decision-making. However, one of the disadvantages is that Bayesian networks are somewhat limited in terms of interpretation and efficiency when rules are derived from the network, while rules derived from decision trees in general have a simple and direct interpretation. Therefore, in this study, the idea of combining decision trees and Bayesian networks was explored in order to maintain the potential advantages of both techniques. The paper reports the findings of a methodological study that was conducted in the context of Albatross, which is a sequential rule based model of activity scheduling behaviour. To this end, the paper can be situated within the context of a series of previous publications by the authors to improve decision-making in Albatross. The results of this study suggest that integrated Bayesian networks and decision trees can be used for modelling the different choice facets of Albatross with better predictive power than CHAID decision trees. Another conclusion is that there are initial indications that the new way of integrating decision trees and Bayesian networks has produced a decision tree that is structurally more stable.     

大型群体决策问题中的简单多数规则（3个评选对象场合）

一个群体决策问题取决于两个因素,一个是群体决策的规则,另一个是投票。当选定群体决策规则时,一个群体决策问题由投票完全决定,此时,群体决策问题与投票之间一一对应。简单多数规则是个简单且被广泛采用的群体决策规则,但它有缺陷,我们可举出些群体决策问题使用简单多数规则没法从投票得到最后群体决策的结果。这里我们将给出一个简单多数规则的有趣性质,即在3个评选对象场合,使用简单多数规则没法从投票得到最后群体决策结果的n个评选人的群体决策问题的个数与所有n个评选人的群体决策问题的个数之比当评选人个数n趋向无穷时趋于零,这说明3个评选对象的大型群体决策场合,简单多数规则的缺陷不严重。     

Eliciting Electre Tri category limits for a group of decision makers

Multiple criteria sorting aims at assigning alternatives evaluated on several criteria to predefined ordered categories. In this paper, we consider a well known multiple criteria sorting method, Electre Tri, which involves three types of preference parameters: (1) category limits defining the frontiers between consecutive categories, (2) weights and majority level specifying which coalitions form a majority, and (3) veto thresholds characterizing discordance effects. We propose an elicitation procedure to infer category limits from assignment examples provided by multiple decision makers. The procedure computes a set of category limits and vetoes common to all decision makers, with variable weights for each decision maker. Hence, the method helps reaching a consensus among decision makers on the category limits and veto thresholds, whereas finding a consensus on weights is left aside. The inference procedure is based on mixed integer linear programming and performs well even for datasets corresponding to real-world decision problems. We provide an illustrative example of the use of the method and analyze the performance of the proposed algorithms.     

Fast algorithms for robust classification with Bayesian nets

We focus on a well-known classification task with expert systems based on Bayesian networks: predicting the state of a target variable given an incomplete observation of the other variables in the network, i.e., an observation of a subset of all the possible variables. To provide conclusions robust to near-ignorance about the process that prevents some of the variables from being observed, it has recently been derived a new rule, called conservative updating. With this paper we address the problem to efficiently compute the conservative updating rule for robust classification with Bayesian networks. We show first that the general problem is NP-hard, thus establishing a fundamental limit to the possibility to do robust classification efficiently. Then we define a wide subclass of Bayesian networks that does admit efficient computation. We show this by developing a new classification algorithm for such a class, which extends substantially the limits of efficient computation with respect to the previously existing algorithm. The algorithm is formulated as a variable elimination procedure, whose computation time is linear in the input size.