关于费马大定理

本文通过对费马提出的费马大定理的证明过程的叙述,拟向人们展示数学的一种魅力,这种魅力告诉人们一个命题的偶然发现会吸引很多人为之奋斗,在奋斗的过程中会产生许许多多的令人惊奇的成就。     

Assisting Australian indigenous resource management and sustainable utilization of species through the use of GIS and environmental modeling techniques

Information on distribution and relative abundance of species is integral to sustainable management, especially if they are to be harvested for subsistence or commerce. In northern Australia, natural landscapes are vast, centers of population few, access is difficult, and Aboriginal resource centers and communities have limited funds and infrastructure. Consequently defining distribution and relative abundance by comprehensive ground survey is difficult and expensive. This highlights the need for simple, cheap, automated methodologies to predict the distribution of species in use, or having potential for use, in commercial enterprise. The technique applied here uses a Geographic Information System (GIS) to make predictions of probability of occurrence using an inductive modeling technique based on Bayes' theorem. The study area is in the Maningrida region, central Arnhem Land, in the Northern Territory, Australia. The species examined, Cycas arnhemica and Brachychiton diversifolius, are currently being 'wild harvested' in commercial trials, involving sale of decorative plants and use as carving wood, respectively. This study involved limited and relatively simple ground surveys requiring approximately 7 days of effort for each species. The overall model performance was evaluated using Cohen's kappa statistics. The predictive ability of the model for C. arnhemica was classified as moderate and for B. diversifolius as fair. The difference in model performance can be attributed to the pattern of distribution of these species. C. arnhemica tends to occur in a clumped distribution due to relatively short distance dispersal of its large seeds and vegetative growth from long-lived rhizomes, while B. diversifolius seeds are smaller and more widely dispersed across the landscape. The output from analysis predicts trends in species distribution that are consistent with independent on-site sampling for each species and therefore should prove useful in gauging the extent of resource availability. However, some caution needs to be applied as the models tend to over predict presence which is a function of distribution patterns and of other variables operating in the landscape such as fire histories which were not included in the model due to limited availability of data.     

Zipf定则及其广延在自然资源数量计算中的应用

吉弗定则（Zipf Theorem）具有相当广泛的解释功能,因此有可能用该定则去估算自然资源的数量分布形态。本文总结了吉弗定则的普适性,并对原式实施了广延,得出了一些很有启发的结论。首先将该定则从 P_r=P₀/r 扩大到 P_r=P₀/r^b,并从b的行为中去认识自然资源数量分布的特性,把原先b=1 的吉弗方式推广到b在非1情形下的各种方式。以我国的水资源为案例作了验证,取得了满意的结果。最后指出,进一步对于参数b进行数理解析,是深入认识各类资源形态分布机制的关键所在,并发现有可能找出吉弗定则与“奇异吸引子”之间的联系。     

代际外部性与不可持续发展的根源

外部性是经济学者一直关注的问题之一，可持续发展概念的提出，事实上涉及到了外部性问题在时间上的表现，即表现为当代人行为对后代人带来的外部性。这将促使我们更加深入地思考外部性的涵义和外部性问题。以往经济学研究中常用的处理外部性行为的两个途径——庇古税和科斯定理，在面对由于主体缺失情况下的代际外部性时，表现得无能为力。本文认为，代际的外部性是不可持续发展的根源，应该成为可持续发展研究中的焦点。     

大连市水土流失控制模拟研究

从控制大连市水土流失出发,结合大连市地表总体状况,根据水土流失通用方程,利用拉格朗日中值定理分析原理,模拟计算了土壤有机质含量、地面覆盖类型、地面覆盖率和水土流失控制距离等方面对水土流失造成的影响;分别计算了不同地表条件下控制水土流失的理论经济点和理论敏感点;定量分析了不同地表作用因子对控制水土流失的影响程度.模拟结果表明:对水土流失影响最大的地表作用因子是土壤有机质含量,其次依次为地面覆盖率、地面覆盖类型和水土流失控制距离;控制水土流失最基本的地表条件是,土壤有机质含量不得低于1.78%,地面覆盖率不得低于41%,地面不得裸露,最基本的植被覆盖类型是草地,水土流失控制距离不得低于3　603 m.依据模拟结果,提出了控制大连市水土流失的措施.      

Optimal patch time allocation for time-limited foragers

The Charnov Marginal Value Theorem (MVT) predicts the optimal foraging duration of animals exploiting patches of resources. The predictions of this model have been verified for various animal species. However, the model is based on several assumptions that are likely too simplistic. One of these assumptions is that animals are living forever (i.e., infinite horizon). Using a simple dynamic programming model, we tested the importance of this assumption by analysing the optimal strategy for time-limited foragers. We found that, for time-limited foragers, optimal patch residence times should be greater than those predicted from the classic, static MVT, and the deviation should increase when foragers are approaching the end of their life. These predictions were verified for females of the parasitoid Anaphes victus (Hymenoptera: Mymaridae) exploiting egg patches of its host, the carrot weevil Listronotus oregonensis (Coleoptera: Curculionidae). As predicted by the model, females indeed remained for a longer time on host patches when they approached the end of their life. Experimental results were finally analysed with a Cox regression model to identify the patch-leaving decision rules females used to behave according to the model’s predictions.     

基于贝叶斯公式的地下水污染源识别

将贝叶斯公式与地下水二维水质对流-扩散方程相耦合,建立依靠监测井监测值的地下水污染源参数（污染源强度M、排放位置（X₀,Y₀）和排放时刻T₀）反演模型.针对监测井监测值信息量不充分或者监测值与模型参数关联性较弱的问题,提出了一种基于贝叶斯公式与信息熵的监测井优化设计方法.构建一个污染物在承压含水层中瞬时排放的算例,在确定单井监测及监测次数条件下,以监测井位置D及监测频率Dt的优化为目标,分别进行模型参数后验分布信息熵最小的单目标监测方案优化,以及信息熵最小且监测耗时最短的多目标监测方案优化.依据优化后的监测方案采用延迟拒绝自适应Metropolis算法进行污染源参数反演识别.算例研究表明：在预设定单井监测,且监测次数为5次条件下,单目标优化后的监测方案为D=（830.2,199.8）,△t=2.7,在此监测方案下,4个污染源参数M,X₀,Y₀,T₀的反演均值误差分别为19.5%、13.2%、3.4%、1.3%;多目标优化后的监测方案为D=（807.9,199.4）,△t=1.2,在此监测方案下,4个污染源参数M,X₀,Y₀,T₀的反演均值误差分别为19.9%、13.4%、3.7%、4.2%.与基于单目标优化的监测方案的反演结果相比,基于多目标优化的监测方案条件下,污染源参数的反演均值误差虽分别增加了0.4%、0.2%、0.3%、2.9%,但监测时间却显著缩短了55.6%.     

拉氏中值定理的发现与证明

给出了拉氏中值定理的发现过程和一个新的证明方法．