既是数学大咖又是力学大神的哥廷根大师们
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在这个安静的夜晚，刚看了一篇关于哥廷根应用力学学派思想发展的综述论文，思索再三，终于敲打起键盘，写我喜欢的应用力学。应用力学的思想精髓是什么？应用力学研究什么？应用力学的发展历史是怎样的？有哪些著名的力学大师？我国的力学发展情况？我会就我个人的看法，一一回答。总之，仅代表我个人的看法，望能激起你对应用力学的兴趣。
应用力学，我个人认为是沟通自然科学与工程科学的桥梁，从生活实践中提取，归纳理论，进而形成规律，然后运用于生活，这便是其思想精髓。自然科学侧重探索自然的奥秘，工程科学侧重运用理论解决生活实践中的实际问题。理论在与实践结合的过程中，力学扮演了重要的角色，处于核心地位。
那么，应用力学的研究内容主要有哪些了？我举具体的例子说明这个问题：
力学就其历史上经典的门类，可以粗略的分为固体力学，流体力学（现在还有物理力学，生物力学等等）。在二十世纪以前，弹性理论，流体力学是理论物理的一部分。在后来的发展过程中，弹性理论主要运用于分析材料的力学性能，形成一门经典的学科，也就是固体力学。现在，我们大学开设的工科力学基础课程就包括，材料力学，弹性力学，结构力学，还有建筑系的建筑力学，可以说都是固体力学的一部分。固体力学就完成了力学在工程实践中的应用，但同时和物理保持着深刻的联系。流体力学在当时也是理论物理的一部分，至今在某种程度上也可以说是理论物理的一部分。流体力学的分支学科包括：空气动力学，多相流体力学，燃烧学，湍流与流动稳定性等等。可以这样说，流体力学既是基础科学又是应用科学。
空气动力学主要应用于航空航天领域，解决飞行器飞行过程中的具体问题，包括：激波，边界层减阻，音障突破等。多相流体力学主要研究汽液固三相物质之间的相互作用。简单的例子就是，气泡在液体中的生成和溃灭现象；以及海洋中，海水与固体建筑物，船只等的相互耦合作用。燃烧学的应用非常广，比如在发动机领域，燃烧室的燃烧现象是一个涉及面广且困难的问题。燃烧现象涉及到复杂的热化学反应，复杂的湍流运动，还有各种未知的环境影响，综合来说，燃烧学是一个与流体力学结合紧密的学科，了解燃烧现象，对很多基础学科都有本质上的进步。湍流与流动稳定性领域，主要研究流体如何从层流状态过渡到湍流状态，以及流体在湍流状态后的一切动力学特征。这是一个非常困难的领域，世界各国都有很多科学家致力于湍流的研究。从哥廷根大学的应用力学研究所开始，普朗特曾致力于湍流的研究，提出了著名的“混合长度理论”；普朗特的学生，冯?卡门也曾孜孜不倦的研究湍流，写下了著名论文《湍流的力学相似原理》。这些领域的应用，都说明了流体力学既是一门重要的应用学科，同时又是一门重要的物理科学，从而也论证了，为什么力学是联系自然科学和工程科学的桥梁。
应用力学的发展历史是怎样的了？我才疏学浅，仅就我所了解的范围斗胆谈论这个问题，恳请有识之士批评指正我，并改正，补充和完善相应的内容。在哥廷根应用力学研究所建立以前，我所知道的力学大师有泊松，欧拉，达朗贝尔，阿佩尔，拉格朗日，拉普拉斯。我们能发现，其实我所列举的力学大师更准确的说，应该称为数学家。没错，我曾在费米的传记中看到一段话，在当时的意大利，力学都是由数学系的老师授课的，也就是说，力学是作为数学系的一部分，这个现象在当时很多国家都是类似的。
在哥廷根应用力学研究所建立以后，力学作为一门独立的学科逐渐开始从数学，物理中分化出来。在二十世纪初叶，哥廷根是世界的数学中心，有着深厚的数学底蕴。高斯，黎曼，希尔伯特，克莱因都是鼎鼎有名的数学大师。同时，在哥廷根大学还有一个传统，就是既要在纯粹数学领域深入研究下去，另外还要把数学应用于生活，以及其他科学领域。从高斯起，哥廷根大学就坚持着这个传统。所以，我们会发现，高斯既是一位数学家，又是一位在物理学领域颇有建树的物理学家。
在希尔伯特和克莱因作为哥廷根数学领袖的时候（二十世纪初），希尔伯特更侧重纯粹数学的研究，但也支持相关的物理学研究（曾支持波恩建立物质结构研究所，海森堡曾在这里学习）。与此同时，克莱因较侧重应用数学的研究，建立了哥廷根应用力学研究所，邀请著名科学家普朗特带领应用力学研究所。两位数学大师在各自的信仰下，使数学在各个方面蓬勃的发展。
应用力学研究所成立以后，力学进一步蓬勃的发展。当时的哥廷根，力学主要在固体力学和流体力学方面得到了充足的进步。后来应用力学研究所又继续分为流体力学研究所，由普朗特主持；以及空气动力学研究所，由贝茨任主任。此后，应用力学研究所有着许多成果：托儿明研究了非定常边界层的稳定性；尼姑拉兹在管道的阻力方面做了一系列的开创性实验；贝茨研究了翼型阻力；阿克莱特研究了超声速流相似律和吸气边界层等等。
应用力学研究所培养了一个时代的力学大师，冯?卡门，铁摩辛柯都在应用力学研究所学习过。学成的冯?卡门更是培养了一个时代的力学家。我们国家的钱学森先生，钱伟长先生，郭永怀先生都师从于冯?卡门。
总之，从哥廷根应用力学研究所建立开始，力学作为真正的科学开始在自然科学和工程科学间建立起沟通的桥梁。有哪些著名的力学大师了？
当然，我所列举的力学的大师，可能存在我个人的偏爱。
普朗特，当之无愧的力学大师，开创了哥廷根应用力学研究的先锋，并使工程科学的思想开始生根，发芽。
泰勒尔，英国著名的流体力学家，湍流统计学派的代表。
冯?卡门，继续把哥廷根应用力学的工程科学思想发扬光大，把空气动力学应用于航空航天，取得了相当大的成就，著名的卡门涡街应该是人人皆知。并培养了钱学森，钱伟长，郭永怀等我们国家一个时代的科学家。
巴彻勒，英国著名的流体力学家，师从泰勒尔，在局地均匀各向同性湍流中取得了很大的成就，同时开创了悬浮流体力学的研究，也是剑桥大学应用数学与理论物理系的创始系主任。我曾看过我国著名物理学家温景嵩老师对巴彻勒老师的回忆，深深感动于巴彻勒老师的品质。我对我自己的要求是，一定会去剑桥大学，去看看巴彻勒老师工作过的地方，巴彻勒老师也是我前进的目标。
科尔莫果洛夫，前苏联著名的数学家，力学家。提出的湍流“负5/3律”，至今都是，湍流领域的重要成果。
铁摩辛柯，著名工程力学家，被誉为“现代工程力学之父”。
周培源，我们国家著名物理学家，力学家，在湍流统计领域取得了巨大成就，对我国的力学事业做出了巨大贡献。
还有很多力学大师。我个人认为，前辈们的品质，纯真好奇心的求知，值得我们用心学习。
我国的力学发展情况是怎样的？我国力学学科的发展，我认为有两个平行的方面。
第一个方面是周培源教授带领下的湍流研究，使我国的湍流研究在国际上有一定地位。第二个方面是钱学森先生，钱伟长先生，郭永怀先生在二十世纪五十年代回国后所做的贡献。钱学森先生，郭永怀先生创建了中科院力学研究所。中科院力学研究所为我国的火箭发动机，风洞研究做出了巨大贡献。今年，我国发射了首颗微重力研究卫星，力学所是主要负责研究所。随后，钱学森先生还在中科大成立了近代力学系。钱伟长先生在上海建立了应用数学和力学研究所，进一步发扬工程科学思想，把数学理论应用于生活实践。
简单的叙述，也就到此为止了，这些都是我平时看的点滴，肯定有叙述不当或错误的地方，望亲爱的读者能批评指正，更希望能激起你对力学的兴趣。我了，也得踏踏实实努力，继续在力学考研的道路上前进啦，共勉。
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\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}
\numberwithin{equation}{section}
\begin{document}
\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}
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