拿破仑：独一无二的数学家

　　拿破仑在书房

　　撰文

　　周红* （华为硬件工程院总裁、欧洲研究院院长）

　　1797年，年仅28岁的拿破仑从11位候选人中脱颖而出，竞选成为法兰西科学院数学部院士。拿破仑对于自己能当选数学院士非常自豪，总是把这个头衔签在他的命令和文告的最前面。

　　在17世纪和18世纪，法国的大学在数学方面是不活跃的，按克莱因教授在《古今数学思想》中所说的，它们没有作出任何贡献。直到18世纪末，拿破仑建立了第一流的技术学校，集中了顶级数学家，开创了对科学家系统的训练，法国的大学才产生了为数众多的数学家。

　　作为一位数学家，拿破仑对于数学的贡献，不在于他发现了多少新的数学定理和新的数学公式，而在于他对法国科研体系和教育体制的建设、对科学家的重视和对年轻人才的培养，以及对数学和其他学科的发展与应用做出的巨大推动。

　　拿破仑认为人才的关键是教育，从1802年到1808年间，他发布了一系列法令，确立了法国精英制大学校的高等教育模式，旨在培养理论联系实际，既有知识又有应用技术的人才。实际上，目前法国最好的两所精英大学：巴黎高等师范和巴黎综合理工，就是拿破仑时代组建的。（编者注：详见《自由、激情、高效：巴黎高师何以盛产菲尔兹奖得主》）他还在军队中积极推广先进的数学方法，从三角函数到微积分方程，拿破仑对炮兵和海军军官工程师提出了很高的数学水平要求，拿破仑的大炮打到哪里，工程师的图就画到哪里。除了数学外，拿破仑也非常鼓励其他学科的发明创造，比如罐头、种牛痘、电解法制备钾和钠等，哪怕是敌对国家的发明，都得到他的积极支持与推广。

　　拿破仑几乎与同时代的法国数学家都交上了朋友，支持他们的研究，并乐意帮助经济困难的科学家们。从拉普拉斯到蒙日、傅立叶、拉格朗日、勒让德等，那个时代很多数学家得到了拿破仑的友谊和器重。拿破仑非常珍视人才，在两军交战时还尽可能保护好科学家和学生。1814年，当反法联军兵临城下，法国兵员短缺，有人提议调理工学校的学生参加战斗时，拿破仑说：“我不愿为取金蛋杀掉我的老母鸡。”这句话今天还刻在该校的梯形大教室的天花板上。

　　拿破仑对于法国的数学和科学体系做出了如此大的贡献，以至两百多年后的今天，法国数学界在全球仍然具有很强的优势。我曾去巴黎南郊伊薇特河畔比尔镇拜访世界数学圣地——法国高等科学研究院，十位数学家中竟然有七位获得了菲尔兹奖！在和这些科学家们喝咖啡的时候，我深切感受到他们对数学发自内心的热爱和自豪，真正体会到“超越人类极限，做宇宙主人”的追求。我想正因如此，在全球人均获得菲尔兹奖的比例上，法国目前还遥遥领先，接近美国的四倍、俄罗斯的五倍。

　　在走向未来信息社会和智能社会的路上，我们需要有千千万万优秀的数学家、科学家，更离不开拿破仑这样的伟大胸怀和远见卓识。

　　*本文经作者授权发表于《赛先生》。周红博士2014年当选为WWRF（ Wireless World Research Forum ）会士。http://it.sohu.com/20161002/n469523559.shtml

読んでためになりました：

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\title{\bf  Announcement 300:  New challenges on the division by zero z/0=0\\

(2016.05.22)}

\author{{\it Institute of Reproducing Kernels}\\

Kawauchi-cho, 5-1648-16,\\

Kiryu 376-0041, Japan\\

%\date{\today}

\maketitle

{\bf Abstract: } In this announcement, for its importance we would like to state the

situation on the division by zero and propose basic new challenges.

\bigskip

\section{Introduction}

%\label{sect1}

By a {\bf natural extension} of the fractions

\begin{equation}

\frac{b}{a}

\end{equation}

for any complex numbers $a$ and $b$, we found the simple and beautiful result, for any complex number $b$

\begin{equation}

\frac{b}{0}=0,

\end{equation}

incidentally in \cite{s} by the Tikhonov regularization for the Hadamard product inversions for matrices and we discussed their properties and gave several physical interpretations on the general fractions in \cite{kmsy} for the  case of real numbers.

 The division by zero has a long and mysterious story over the world (see, for example, Google site with the division by zero) with its physical viewpoints since the document of zero in India on AD 628,  however,

  Sin-Ei Takahasi (\cite{kmsy}) established a simple and decisive interpretation (1.2) by analyzing the extensions of fractions and by showing the complete characterization for the property (1.2):

 \bigskip

 {\bf  Proposition 1. }{\it Let F be a function from  ${\bf C }\times {\bf C }$  to ${\bf C }$ satisfying

$$

F (b, a)F (c, d)= F (bc, ad) 

$$

for all

$$

a, b, c, d  \in {\bf C }

$$

and

$$

F (b, a) = \frac {b}{a },  \quad   a, b  \in  {\bf C }, a \ne 0.

$$

Then, we obtain, for any $b \in {\bf C } $

$$

F (b, 0) = 0.

$$

}

 Note that the complete proof of this proposition is simply given by  2 or 3 lines.

\medskip

We thus should consider, for any complex number $b$, as  (1.2);

that is, for the mapping

\begin{equation}

w = \frac{1}{z},

\end{equation}

the image of $z=0$ is $w=0$ ({\bf should be defined}). This fact seems to be a curious one in connection with our well-established popular image for the  point at infinity on the Riemann sphere. Therefore, the division by zero will give great impacts to complex analysis and to our ideas for the space and universe.

However, the division by zero (1.2) is now clear, indeed, for the introduction of (1.2), we have several independent approaches as in:

\medskip

1) by the generalization of the fractions by the Tikhonov regularization or by the Moore-Penrose generalized inverse,

\medskip

2) by the intuitive meaning of the fractions (division) by H. Michiwaki,

\medskip

3) by the unique extension of the fractions by S. Takahasi,   as in the above,

\medskip

4) by the extension of the fundamental function $W = 1/z$ from ${\bf C} \setminus \{0\}$ into ${\bf C}$ such that $W =1/z$ is a one to one and onto mapping from $ {\bf C} \setminus \{0\} $ onto ${\bf C} \setminus \{0\}$ and the division by zero $1/0=0$ is a one to one and onto mapping extension of the function $W =1/z $ from  ${\bf C}$ onto ${\bf C}$,

\medskip

and

\medskip

5) by considering the values of functions with the mean values of functions.

\medskip

Furthermore, in (\cite{msy}) we gave the results in order to show the reality of the division by zero in our world:

\medskip

\medskip

A) a field structure  containing the division by zero --- the Yamada field ${\bf Y}$,

\medskip

B)  by the gradient of the $y$ axis on the $(x,y)$ plane --- $\tan \frac{\pi}{2} =0$,

\medskip

C) by the reflection $W =1/\overline{z}$ of $W= z$ with respect to the unit circle with center at the origin on the complex $z$ plane --- the reflection point of zero is zero,

\medskip

and

\medskip

D) by considering rotation of a right circular cone having some very interesting

phenomenon  from some practical and physical problem.

\medskip

In (\cite{mos}),  many division by zero results in Euclidean spaces are given and  the basic idea at the point at infinity should be changed. In (\cite{ms}), we gave beautiful geometrical interpretations of determinants from the viewpoint of the division by zero. The results show that the division by zero is our basic and elementary mathematics in our world.

\medskip

See  J. A. Bergstra, Y. Hirshfeld and J. V. Tucker \cite{bht} for the relationship between fields and the division by zero, and the importance of the division by zero for computer science. It seems that the relationship of the division by zero and field structures are abstract in their paper.

Meanwhile,  J. P.  Barukcic and I.  Barukcic (\cite{bb}) discussed recently the relation between the divisions $0/0$, $1/0$ and special relative theory of Einstein. However, their logic seems to be curious and their results contradict with ours.

 Furthermore,  T. S. Reis and J.A.D.W. Anderson (\cite{ra,ra2}) extend the system of the real numbers by introducing an ideal number for the division by zero $0/0$.

 Meanwhile, we should refer to up-to-date information:

{\it Riemann Hypothesis Addendum - Breakthrough

Kurt Arbenz

https://www.researchgate.net/publication/272022137 Riemann Hypothesis Addendum -   Breakthrough.}

\medskip

Here, we recall Albert Einstein's words on mathematics:

Blackholes are where God divided by zero.

I don't believe in mathematics.

George Gamow (1904-1968) Russian-born American nuclear physicist and cosmologist remarked that "it is well known to students of high school algebra" that division by zero is not valid; and Einstein admitted it as {\bf the biggest blunder of his life} [1]：

1. Gamow, G., My World Line (Viking, New York). p 44, 1970.

 For our ideas on the division by zero, see the survey style announcements 179,185,237,246,247,250 and 252 of the Institute of Reproducing Kernels (\cite{ann179,ann185,ann237,ann246,ann247,ann250,ann252,ann293}).

\section{On mathematics}

Apparently, the division by zero is a great missing in our mathematics and the result (1.2) is definitely determined as our basic mathematics, as we see from Proposition 1.  Note  its very general assumptions and  many fundamental evidences in our world in (\cite{kmsy,msy,mos}). The results will give great impacts  on Euclidean spaces, analytic geometry, calculus, differential equations, complex analysis and  physical problems. See our announcements for the details.

The mysterious history of the division by zero over one thousand years is a great shame of  mathematicians and human race on the world history, like the Ptolemaic system (geocentric theory). The division by zero will become a typical  symbol of foolish human race with long and unceasing struggles. Future people will realize this fact as a definite common sense.

We should check and fill our mathematics, globally and beautifully, from the viewpoint of the division by zero. Our mathematics will be more perfect and beautiful,  and will give great impacts to our basic ideas on the universe.

\section{Albert Einstein's biggest blunder}

The division by zero is directly related to the Einstein's theory and various

physical problems

containing the division by zero.  Now we should check the theory and the problems by the concept of the RIGHT and DEFINITE division by zero. Now is the best time since 100 years from Albert Einstein. It seems that the background knowledge is timely fruitful.

\section{Computer systems}

The above Professors listed are wishing the contributions in order to avoid the zero division trouble in computers. Now,  we should arrange  new computer systems in order not to meet the division by zero trouble in computer systems.

\section{General  ideas on the universe}

The division by zero may be related to religion,  philosophy and the ideas on the universe, and it will creat a new world. Look the new world.

\bigskip

We are standing on a new  generation and in front of the new world, as in the discovery of the Americas.

 \bigskip

\bibliographystyle{plain}

\begin{thebibliography}{10}

\bibitem{bb}

J. P.  Barukcic and I.  Barukcic, Anti Aristotle—The Division of Zero by Zero. Journal of Applied Mathematics and Physics,  {\bf 4}(2016), 749-761.

doi: 10.4236/jamp.2016.44085.

\bibitem{bht}

J. A. Bergstra, Y. Hirshfeld and J. V. Tucker,

Meadows and the equational specification of division (arXiv:0901.0823v1[math.RA] 7 Jan 2009).

\bibitem{cs}

L. P.  Castro and S. Saitoh,  Fractional functions and their representations,  Complex Anal. Oper. Theory {\bf7} (2013), no. 4, 1049-1063.

\bibitem{kmsy}

M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane,

New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$,

Int. J. Appl. Math.  {\bf 27} (2014), no 2, pp. 191-198,  DOI: 10.12732/ijam.v27i2.9.

\bibitem{ms}

T. Matsuura and S. Saitoh,

Matrices and division by zero $z/0=0$,

Linear Algebra \& Matrix Theory (ALAMT)(to appear).

\bibitem{msy}

H. Michiwaki, S. Saitoh,  and  M.Yamada,

Reality of the division by zero $z/0=0$.  IJAPM  International J. of Applied Physics and Math. {\bf 6}(2015), 1--8. http://www.ijapm.org/show-63-504-1.html

\bibitem{mos}

H.  Michiwaki, H. Okumura, and S. Saitoh,

Division by Zero $z/0 = 0$ in Euclidean Spaces.

 International Journal of Mathematics and Computation

 (in press).

\bibitem{ra}

T. S. Reis and J.A.D.W. Anderson,

Transdifferential and Transintegral Calculus,

Proceedings of the World Congress on Engineering and Computer Science 2014 Vol I

WCECS 2014, 22-24 October, 2014, San Francisco, USA

\bibitem{ra2}

T. S. Reis and J.A.D.W. Anderson,

Transreal Calculus,

IAENG  International J. of Applied Math., {\bf 45}(2015):  IJAM 45 1 06.

\bibitem{s}

S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices,  Advances in Linear Algebra \& Matrix Theory.  {\bf 4}  (2014), no. 2,  87--95. http://www.scirp.org/journal/ALAMT/

\bibitem{ttk}

S.-E. Takahasi, M. Tsukada and Y. Kobayashi,  Classification of continuous fractional binary operations on the real and complex fields,  Tokyo Journal of Mathematics,   {\bf 38}(2015), no. 2, 369-380.

\bibitem{ann179}

Announcement 179 (2014.8.30): Division by zero is clear as z/0=0 and it is fundamental in mathematics.

\bibitem{ann185}

Announcement 185 (2014.10.22): The importance of the division by zero $z/0=0$.

\bibitem{ann237}

Announcement 237 (2015.6.18):  A reality of the division by zero $z/0=0$ by  geometrical optics.

\bibitem{ann246}

Announcement 246 (2015.9.17): An interpretation of the division by zero $1/0=0$ by the gradients of lines.

\bibitem{ann247}

Announcement 247 (2015.9.22): The gradient of y-axis is zero and $\tan (\pi/2) =0$ by the division by zero $1/0=0$.

\bibitem{ann250}

Announcement 250 (2015.10.20): What are numbers? -  the Yamada field containing the division by zero $z/0=0$.

\bibitem{ann252}

Announcement 252 (2015.11.1): Circles and

curvature - an interpretation by Mr.

Hiroshi Michiwaki of the division by

zero $r/0 = 0$.

\bibitem{ann281}

Announcement 281(2016.2.1): The importance of the division by zero $z/0=0$.

\bibitem{ann282}

Announcement 282(2016.2.2): The Division by Zero $z/0=0$ on the Second Birthday.

\bibitem{ann293}

Announcement 293(2016.3.27):  Parallel lines on the Euclidean plane from the viewpoint of division by zero 1/0=0.

\end{thebibliography}

\end{document}