归纳法

First the theorem is verified for the smallest admissible value of the integer.

归纳法：证明涉及到正整数变量的定理所运用的两部法。

However, there exist some paradoxes in decision making logic, such as the Allais paradox and exchange paradox in expected utility theory, backward induction paradox and Gaifmann-Koons paradox originated from Newcomb problem in interactions, the famous Arrow voting paradox in group decision-making.

决策逻辑是社会科学的基础，然而在决策逻辑中存在悖论：不确定的概然决策中的期望收益理论面临阿莱斯悖论和"交换悖论"；互动决策中存在演绎性的逆向归纳法悖论，以及存在由纽柯姆难题改造而来的盖夫曼—孔斯悖论；群体决策中存在著名的阿罗悖论，等等。

In this paper, by using the method of analogue to treat a special determinant, we come to a general conclusion, which has been proved correct by induction.

通过对一种特殊的行列式作 q -类比，得到更为一般的结论，并利用数学归纳法对该结论作出了严格的证明

In this paper,the asymptotic property for two kinds of ecological systems and oscillation for solutions of a neutral differential equations is studied by employing differential inequalities and using eigenvalue analysis,iterative algorithm,induction and comparison theorem etc.Here, the system\'s asymptotic property includes the uniform persistence,the global attractability of the solutions,the asmptotic stability of the solutions and the oscillation of solutions.

本文利用微分不等式、特征根分析法、迭代法、归纳法及比较原理等方法和理论，研究了两类生态模型解的渐近性和一类脉冲中立型微分方程解的振动性，其中包括模型的一致持久生存性，正平衡态的全局吸引性、局部与全局渐近稳定性以及解的振动性。

The Brahe theory stated that all of the planets except the earth revolved around the sun, which in turn revolved around the earth. Brahe's assistant, German astronomer Johannes Kepler (1571-1630), used induction to analyze the data collected by Brahe over a twenty-year span. Kepler supported the heliocentric theory by proposing three laws of plant revolution.

在哥白尼思想的影响下，泰革(Tycho Brahe， 1546-1601)在观察的基础上，提出了哥白尼式的地心说︰地球以外的行星都绕太阳转；但太阳及其行星都绕地球转﹗泰革的助手、德国天文学家凯普勒(Johannes Kepler， 1571-1630)用归纳法分析泰革二十年积累的数据，提出了"行星运行三大定律"，支持了日心说。

One merely needs to learn what signals to disregard as noise. Chartism is organized induction in Doyne Farmer's mode.

而所谓图表分析主义，其实就是一种按照Doyne Farmer的方式而被组织了起来的归纳法。

In the second part, it is argued that the effect of Novum Organum on the philosophy and science is just as the function of beacon. The thesis in this part systematically introduces Bacon\'s basic and extensive thoughts excluding Baconian Induction, Theory of the Idols and the Empirical Epistemology that people have thoroughly researched; moreover, the mistakes which people have made are corrected and the limitations of the viewpoints of Novum Organum are also focused on. In the third part, Bacon\'s basic and extensive scientific and technological researches are discussed. Regarded to this, the thesis finds that people only paid attention to Bacon\'s view that heat is a motion which Bacon took lots of space in Novum Organum to summarized according to Baconian Induction.

本文总共分为四大部分：首先，分析《新工具》是对亚里斯多德《工具论》的勇敢挑战，并着重论述了培根的政治和文化背景对他的新思想所造成的深刻影响；其次，系统探讨《新工具》在哲学和科学中起到的灯塔作用，本文论述了培根的基本哲学和科学思想及具体应用和反映，但是对前人已深入探讨过的培根式归纳法、四假象说和经验认识论不再论述，同时对前人的一些误读进行澄清，并分析培根的《新工具》在哲学和科学思想方面的局限；再次，对前人很少涉猎的《新工具》中的科学研究方面进行系统论述，本文发现人们基本上仅仅注意到了培根在《新工具》中大篇幅运用培根式归纳法进行举例论证的热动说，而对这本书中其他的科学研究不甚关注。

In the second part, it is argued that the effect of Novum Organum on the philosophy and science is just as the function of beacon. The thesis in this part systematically introduces Bacon"s basic and extensive thoughts excluding Baconian Induction, Theory of the Idols and the Empirical Epistemology that people have thoroughly researched; moreover, the mistakes which people have made are corrected and the limitations of the viewpoints of Novum Organum are also focused on. In the third part, Bacon"s basic and extensive scientific and technological researches are discussed. Regarded to this, the thesis finds that people only paid attention to Bacons view that heat is a motion which Bacon took lots of space in Novum Organum to summarized according to Baconian Induction.

本文总共分为四大部分：首先，分析《新工具》是对亚里斯多德《工具论》的勇敢挑战，并着重论述了培根的政治和文化背景对他的新思想所造成的深刻影响；其次，系统探讨《新工具》在哲学和科学中起到的灯塔作用，本文论述了培根的基本哲学和科学思想及具体应用和反映，但是对前人已深入探讨过的培根式归纳法、四假象说和经验认识论不再论述，同时对前人的一些误读进行澄清，并分析培根的《新工具》在哲学和科学思想方面的局限；再次，对前人很少涉猎的《新工具》中的科学研究方面进行系统论述，本文发现人们基本上仅仅注意到了培根在《新工具》中大篇幅运用培根式归纳法进行举例论证的热动说，而对这本书中其他的科学研究不甚关注。

After questionary,the circumstance of college students of science and engineering mastering the mathematical induction is obtained:many students knew how to use the mathematical induction,but they didn't know its logical basis and logical theory.

通过问卷调查，了解到理工科学生掌握数学归纳法的状况，绝大多数人会用数学归纳法，但他们不知道数学归纳法的逻辑基础和逻辑原理。

Then he gave a high appraise to logic of Moism. What he had done established a new way for the research of logical history in Chinese modern times, and pushed the research of Chinese logical history toward a new developing stage.

梁启超在逻辑——方法论方面取得的主要成就：①对比分析了归纳法与演绎法，特别是对归纳法的分析比较详尽，也基本正确；②主要将比较法作为归纳法的一个组成部分来进行分析，论述了比较法的认识作用，讨论了比较法在历史研究中的运用；③比较分析了分析法与综合法的思维过程，并结合治史和读史具体化为历史研究方法；④阐述了因果律的基本要求，与近代莱布尼茨的充足理由原则和普通逻辑的充足理由律思想相似；⑤探讨了名的分类方法，分析了前人的错误分类，提出明确划分根据的重要性。

According to the clear water experiment, aeration performance of the new equipment is good with high total oxygen transfer coefficient and oxygen utilization ratio.

曝气设备的动力效率在叶轮转速为120rpm～150rpm时取得最大值，此时氧利用率和充氧能力也具有较高值。

The environmental stability of that world - including its crushing pressures and icy darkness - means that some of its most famous inhabitants have survived for eons as evolutionary throwbacks, their bodies undergoing little change.

稳定的海底环境─包括能把人压扁的压力和冰冷的黑暗─意谓海底某些最知名的栖居生物已以演化返祖的样态活了万世，形体几无变化。