雅克·夏尔·弗朗索瓦·施图姆Charles Sturm

雅克·夏尔·弗朗索瓦·施图姆的父亲是Jean-Henri施图姆，其家族从斯特拉斯堡来到日内瓦定居，大约在施图姆出生前50年。Jean-Henri施图姆是一位算术教师，娶了Jeanne-Louise-Henriette Gremay。施图姆的父母给了他良好的教育，在学校他表现出极大的前途，特别是在希腊和拉丁诗歌方面，他有非凡的才能。

施图姆来自一个新教家庭，为了学习德语，他参加了当地的路德教会，那里用该语言布道。当施图姆十六岁时，他的父亲去世，他改变了学术研究的方向，离开人文学科，开始学习数学。1821年，他在日内瓦科学院由SimonLhuilier教授数学，Lhuilier立即认出了施图姆的数学天才。然而，Lhuilier当时已年过七十，接近退休，所以是他的继任者Jean-Jacques Schaub启发了施图姆。Schaub不仅教施图姆数学，还在科学院为他提供经济支持。施图姆的父亲去世后，家庭陷入相当大的经济困难，所以经济援助让施图姆能够继续他的教育。

在科学院施图姆最好的朋友是Daniel Colladon，这段友谊对施图姆早期的研究生涯产生了显著影响。离开科学院后，施图姆被任命为位于日内瓦附近科佩城堡的斯塔尔夫人的小儿子的家庭教师。他于1823年5月上任，发现这给了他大量空闲时间致力于自己的研究。他很好地利用了自己的时间，开始撰写关于几何学的文章，这些文章发表在约瑟夫·热尔岗的Annales de mathématiques pures et appliquées上。1823年底之前，这家人从城堡搬到了巴黎，在那里待了六个月，施图姆作为家庭教师自然随行。

在巴黎，他通过这个家庭被引入了科学界。施图姆写信给他的朋友Colladon（见[1]）：-

至于M 弗朗索瓦·阿拉戈，我曾两三次置身于他每周四邀请到他家的科学家群体中，在那里我见到了顶尖科学家皮埃尔·西蒙·拉普拉斯、西莫恩·德尼·泊松、约瑟夫·傅里叶、Gay-Lussac、安德烈-马里·安培等……我经常参加每周一举行的研究所会议。

这对施图姆来说显然是一个极其幸运的机会。尽管他在1824年5月回到了城堡，但六个月后他又离开了，投身于科学研究。Paris Academy设定了一个关于水的压缩性的奖项题目，施图姆和他的朋友Colladon决定在日内瓦湖上开始实验，目的是提交参赛作品。实验并不十分成功，因为它们没有产生预期的结果，而且Colladon在进行实验时手部受了重伤。

1825年12月，施图姆和Colladon前往巴黎学习数学和物理课程，并收集更多仪器以重复他们的实验。施图姆在巴黎建立的人脉证明是有用的，因为他曾在弗朗索瓦·阿拉戈家中住了一段时间，担任其儿子的家庭教师。他还获准使用安德烈-马里·安培的实验室。这段时间对施图姆来说非常富有成果，他听了安德烈-马里·安培、Gay-Lussac、奥古斯丁·路易·柯西和西尔维斯特·佛朗索瓦·拉克鲁瓦的讲座。约瑟夫·傅里叶为施图姆和Colladon都建议了研究项目，他认识到Colladon本质上是一位物理学家，而施图姆是一位数学家。

尽管他们完成了为Académie des Sciences大奖赛准备的论文，但他们没有获奖；事实上，没有一份提交的作品被认为足够好，同一题目被再次设立。此时，施图姆和Colladon都在为约瑟夫·傅里叶做助手。Colladon在日内瓦湖上做了进一步的实验，在修改了他们的联合论文后，他们成功赢得了奖项。奖金的数额足以让施图姆和Colladon在巴黎继续他们的研究。这一点标志着他们成功合作的结束，两人开始了不同的研究项目。施图姆在数学物理方面的理论工作涉及焦散曲线以及圆锥曲线的极点和极线。

施图姆最著名的论文之一Mémoire sur la résolution des équations numériquesⓉ（关于解数值方程的学位论文）于1829年发表。它考虑了确定方程在给定区间上实根个数的问题。这个问题是一个著名的问题，有着悠久的历史，曾被勒内·笛卡儿、米歇尔·罗尔、约瑟夫·拉格朗日和约瑟夫·傅里叶考虑过。第一个给出完整解的是奥古斯丁·路易·柯西，但他的方法繁琐且不实用。施图姆凭借他的论文获得了名声，该论文利用约瑟夫·傅里叶的思想，给出了一个简单的解。夏尔·埃尔米特写道：-

施图姆的定理有幸立即成为经典，并在教学中占据了一个它将永远保持的位置。他的证明只利用了最基本的考虑，是简洁和优雅的罕见例子。

奇怪的是，尽管该定理很快成为经典，但它不久就被 relegated 到历史中，并且与夏尔·埃尔米特所相信的相反，从教科书中消失了。正如标题所示，[5]考察了施图姆代数定理历史上的两个事件。作者描述了阿尔弗雷德·塔斯基如何在1940年表明施图姆的证明方法可用于数学逻辑中，以证明初等代数和几何的完备性。1829年的论文并不是施图姆关于这些代数方程工作的最后成果，在[12]中Sinaceur:-

...试图确定奥古斯丁·路易·柯西和Ch-F 施图姆从1829年到大约1840年关于代数方程根的研究之间的相互影响。

当时，对于外国人和新教徒来说，巴黎不是一个容易获得职位的地方，尽管他因1829年的论文而闻名，但他并未被任命。1830年七月革命改变了政治气候，此后弗朗索瓦·阿拉戈成功让施图姆被任命为罗林学院的数学教授。他于1833年成为法国公民，并于1836年当选为Académie des Sciences。这些年间，他发表了一些关于微分方程的重要成果。

施图姆对获得西莫恩·德尼·泊松热理论中出现的特定微分方程的结果产生了兴趣。约瑟夫·刘维尔也在研究从热理论导出的微分方程。施图姆和约瑟夫·刘维尔在1836-1837年关于微分方程的论文涉及函数的级数展开，今天被称为施图姆-约瑟夫·刘维尔问题，是二阶微分方程中的一个eigenvalue问题。

他于1838年起在巴黎的巴黎综合理工学院工作，1840年成为分析与力学教授。同年，他接替西莫恩·德尼·泊松担任巴黎理学院力学讲席。在大约十年间，他讲授了出色的课程，但他希望给学生提供尽可能好的课程，这意味着他花费大量时间准备关于微分与积分学以及理性力学的讲义。这些讲义成为广泛使用的教材Cours d'analyse de l'École Polytechnique 2 Vol（1857-63）和Cours de mécanique de l'École Polytechnique 2 VolⓉ（巴黎综合理工学院力学教程，第2卷）（1861），两者均在他去世后出版。

他用于研究的时间现在有限，但他仍做出了重要贡献，从事无穷小几何、射影几何以及曲线与曲面的微分几何的研究。他还在几何光学方面做了重要工作。

从1851年起，他的健康开始恶化，尽管他勇敢地试图克服问题并重返教学（他设法坚持了一段时间），但他在长期患病后去世。

Charles-François Sturm's father was Jean-Henri Sturm whose family came from Strasbourg to settle in Geneva about 50 years before Charles-François's birth. Jean-Henri Sturm was a teacher of arithmetic who had married Jeanne-Louise-Henriette Gremay. Charles-François's parents gave him a good education and at school he showed great promise, particularly in Greek and Latin poetry for which he had a remarkable talent.

Sturm came from a Protestant family and, in order to learn German, he attended the local Lutheran church where sermons were preached in that language. When Sturm was sixteen years old his father died and he changed tack in his academic studies, leaving the humanities and taking up the study of mathematics. He was taught mathematics at Geneva Academy by Simon Lhuilier in 1821 and immediately Lhuilier recognised the mathematical genius in Sturm. However, Lhuilier was over seventy years of age and close to retiring at this time so it was his successor Jean-Jacques Schaub who inspired Sturm. Schaub did more than teach Sturm mathematics for he supported him financially at the Academy. Sturm's family had been left in considerable financial difficulties on the death of his father so the financial assistance allowed Sturm to continue with his education.

At the Academy Sturm's best friend was Daniel Colladon and the friendship would have a marked influence on Sturm's early research career. After leaving the Academy, Sturm was appointed as a tutor to the youngest son of Mme de Staël at the Châteaux de Coppet close to Geneva. He took up his appointment in May 1823 and found that it left him plenty of free time to devote to his own studies. He used his time well and began to write articles on geometry which were published in Gergonne's Annales de mathématiques pures et appliquées. Before the end of 1823 the family moved from the château to spend six months in Paris and Sturm, as tutor, naturally accompanied them.

In Paris he was introduced into the scientific circles by the family. Sturm wrote to his friend Colladon (see [1]):-

As for M Arago, I have two or three times been among the group of scientists he invites to his house every Thursday, and there I have seen the leading scientists, Laplace, Poisson, Fourier, Gay-Lussac, Ampère, etc. ... I often attend the meetings of the Institute that take place every Monday.

This was clearly an extremely fortunate opportunity for Sturm. Although he returned to the château in May 1824 he left after six further months to devote himself to scientific research. The Paris Academy had set a prize topic on the compressibility of water and Sturm, with his friend Colladon, decided to begin experiments on Lake Geneva with the aim of putting in an entry for the prize. The experiments were not a great success since they did not yield the expected results and Colladon received a serious injury to his hand while conducting the experiments.

In December 1825 Sturm and Colladon went to Paris to take courses in mathematics and physics and also to collect further instruments to repeat their experiments. The Paris contacts that Sturm had made proved useful for he lived at Arago's house for a while as tutor to his son. He was also given the use of Ampère's laboratory. The time was very fruitful for Sturm who attended lectures by Ampère, Gay-Lussac, Cauchy, and Lacroix. Fourier suggested projects for both Sturm and Colladon, recognising that Colladon was essentially a physicist while Sturm was a mathematician.

Despite completing their paper for the Grand Prix of the Académie des Sciences they did not win the prize; in fact none of the submissions was deemed good enough and the same topic was set again. By this time Sturm and Colladon were both working as assistants to Fourier. Colladon made further experiments on Lake Geneva and after revising their joint memoir they successfully won the prize. The value of the prize was enough to allow Sturm and Colladon to continue their research in Paris. This point marked the end of their successful collaboration and the two embarked on different research projects. Sturm's theoretical work in mathematical physics involved the study of caustic curves, and poles and polars of conic sections.

One of Sturm's most famous papers Mémoire sur la résolution des équations numériques Ⓣ was published in 1829. It considered the problem of determining the number of real roots of an equation on a given interval. The problem was a famous one with a long history having been considered by Descartes, Rolle, Lagrange and Fourier. The first to give a complete solution was Cauchy but his method was cumbersome and impractical. Sturm achieved fame with his paper which, using ideas of Fourier, gave a simple solution. Hermite wrote:-

Sturm's theorem had the good fortune of immediately becoming a classic and of finding a place in teaching that it will hold forever. His demonstration, which utilises only the most elementary considerations, is a rare example of simplicity and elegance.

Strangely although the theorem quickly became a classic it was soon relegated to history and, contrary to what Hermite believed, vanished from textbooks. As the title indicated, two events in the history of the algebraic theorem of Sturm are examined in [5]. The author describes how Tarski showed in 1940 that Sturm's method of proof could be used in mathematical logic to prove the completeness of elementary algebra and geometry. The 1829 paper was not the last of Sturm's work on this algebraic equations and in [12] Sinaceur:-

... seeks to determine the mutual influence between A-L Cauchy's and Ch-F Sturm's research from 1829 to around 1840 on the roots of algebraic equations.

Paris was not an easy place for a foreigner and Protestant to obtain a post at this time and, despite his fame from the 1829 paper, he was not appointed. The revolution of July 1830 changed the political climate and after this Arago succeeded in getting Sturm appointed as professor of mathematics in the Collège Rollin. He became a French citizen in 1833 and was elected to the Académie des Sciences in 1836. These were the years during which he published some important results on differential equations.

Sturm became interested in obtaining results on specific differential equations which occurred in Poisson's theory of heat. Liouville was also working on differential equations derived from the theory of heat. Papers of 1836-1837 by Sturm and Liouville on differential equations involved expansions of functions in series and is today well-known as the Sturm-Liouville problem, an eigenvalue problem in second order differential equations.

He worked at the École Polytechnique in Paris from 1838 where he became a professor of analysis and mechanics in 1840. In the same year he succeeded Poisson in the chair of mechanics in the Faculté des Sciences, Paris. For around ten years he gave excellent lectures but his wish to give his students the best possible courses meant that he gave a great deal of his time to preparing his lecture courses on differential and integral calculus and on rational mechanics. These courses became the widely used texts Cours d'analyse de l'École Polytechnique 2 Vol. (1857-63) and Cours de mécanique de l'École Polytechnique 2 Vol Ⓣ (1861) both published posthumously.

His time for research was now limited but he still made important contributions undertaking research on infinitesimal geometry, projective geometry and the differential geometry of curves and surfaces. He also did important work on geometrical optics.

From 1851 his health began to fail and despite brave attempts to overcome the problem and return to teaching (which he managed to do for a while) he died after a long illness.