素数与量子物理的结合能带来解决黎曼猜想的新可能吗？
在寻找模式（patterns）的过程中，数学家无意中发现了素数与量子物理的联系。人们不禁要问，亚原子世界（subatomic world）是否可以帮助揭示素数那难以捉摸的本质？
作者 Marcus du Sautoy（牛津大学数学教授）
翻译墨竹
校对杨璐
黎曼（图片来源于网络）
1972年，物理学家弗里曼·戴森（Freeman Dyson）写了一篇名为《错失的机会》（MissedOpportunities）的文章。在该文中，他阐述道：如果像哥廷根这类地方的数学家曾与当时潜心研究描述电磁现象的麦克斯韦方程的物理学家进行讨论，那么可能在爱因斯坦公布其研究结果的很多年之前，相对论就被发现了。实现这一突破性成果的要素在1865年就已具备，而爱因斯坦在大约四十年后才宣布了这一结果。
令人惊讶的是，戴森居然认为科学的航船还在黑夜中摸索前行。就在他的文章发表后不久，物理学和数学之间的一次意外碰撞产生了二十世纪下半叶最为伟大的科学思想之一：量子物理和素数之间有着千丝万缕的联系。
这种与物理学意料之外的联系给了我们一个窥探数学的机会，或许它将最终揭示这些神秘数字的秘密。起初，这种联系看起来非常微弱。然而，数字42所扮演的重要角色最近甚至说服了最有力的怀疑者：亚原子世界也许是开启数学界一个最重大的未解难题的钥匙。
素数，例如17和23，是指那些只能被自身和1整除的正整数。它们是数学中最重要的对象，因为，正如古希腊人所发现的那样，它们是所有整数的基石：任何整数都可以分解成素数的乘积（例如，105 = 3 × 5 × 7）。素数是数学世界中的氢和氧，是算术中的原子。此外，它们代表了数学中最大的挑战之一。
作为一名数学家，我穷尽一生之力试图找到我身处的表观混沌之中所蕴含的模式、结构和逻辑。然而，这种模式的科学似乎是由一组数的集合所建立的，这些数之间没有任何逻辑。素数看起来更像是一组彩票号码的集合，而不是由简单的公式和规则所产生的序列。
两千年来，素数的模式问题就像一块磁铁，吸引着困惑的数学家们。波恩哈德·黎曼（Bernhard Riemann）就是其中一位。他于1859年，即达尔文发表进化论的同一年，发表了一篇具有同等革命性的论文，论述了素数的由来。黎曼是哥廷根大学的数学家，他开造了一门将为爱因斯坦的伟大突破奠定基础的几何学。然而，他的理论并不仅仅只是打开相对论的钥匙。
黎曼发现了一个几何学的大陆，其轮廓蕴藏着素数在整个数字世界中的分布方式的秘密。他意识到，可以通过 zeta 函数构建一幅景象，使得一个三维图中的波峰和波谷对应于该函数的值。Zeta 函数建立了素数和几何学之间的桥梁。通过进一步的研究，黎曼发现 zeta 函数值为零的地方（对应于波谷）蕴含着有关素数本质的关键信息。
黎曼这一发现所具备的革命性意义可以与爱因斯坦发现 E = mc2相提并论。与爱因斯坦方程中质量转化为能量不同，黎曼方程将素数转化为 zeta 函数景象中水平线（sea-level）处的点。然而，黎曼后来注意到，更加不可思议的事情发生了。当他标注了前十个零点的位置后，一个令人吃惊的模式开始出现。这些零点并不是散落各处，它们似乎分布在景象区域中的一条直线上。黎曼无法相信这仅仅只是一个巧合。他假设，所有的零点——无穷多个零点——可能都落在这条临界直线上，这就是著名的黎曼猜想（Riemann Hypothesis）。
但是，这种令人着迷的模式对素数而言意味着什么呢？如果黎曼的发现是正确的，那就意味着大自然对素数的分布是尽可能公平的。这意味着素数的行为更像是一个房间里随机的气体分子：虽然你可能不知道每个分子的确切位置，但是你可以确定不可能一个角落是真空的而另一个角落聚集着很多分子。
对于数学家而言，黎曼关于素数分布的预言是强有力的。如果这个猜测是正确的，它就意味着其他上千个定理都是成立的，其中也包括我自己的一些定理，这些定理都是以黎曼猜想的正确性为前提的。但是，经过了将近150年的努力，还是没有人能够证明所有的零点确实都落在黎曼所预言的直线上。
1972年，物理学家弗里曼·戴森和数论专家休·蒙哥马利（Hugh Montgomery）在普林斯顿高等研究院喝茶时的会面是一个机遇，它揭示了素数故事中一种令人惊叹的新关系，或许能为最终解决黎曼问题提供一条线索。他们发现，如果将黎曼临界直线上的零点和实验记录的大原子（例如铒，元素周期表中的第68个原子）的核的能级相比较，两者的分布惊人的相似。
看起来，蒙哥马利所预测的零点在黎曼临界直线上的分布模式与量子物理学家所预测的重原子的核的能级是一致的。这个关系的影响是巨大的：如果人们可以弄清楚量子物理中描述原子核结构的数学，也许同样的数学就可以用来解决黎曼猜想。
数学家是多疑的。尽管数学曾常常为物理学家服务，例如爱因斯坦，但是他们怀疑物理学是否真的能够回答数论中的困难问题。于是在1996年，普林斯顿大学的彼得·萨奈克（Peter Sarnak）向物理学家们提出挑战，请他们告诉数学家关于素数的新见解。最近，布里斯托尔大学的乔·基廷（Jon Keating）和妮娜·斯奈思（Nina Snaith）对此作出了正式回应。
有一个重要的数列叫做“黎曼 zeta 函数的矩”（the moments of the Riemann zeta function）。尽管我们知道如何抽象地去定义它，但是精确地计算该数列中的每个数却非常困难。自上世纪二十年代以来，我们已经知道前两个数是1和2。然而，直到近几年，数学家们才猜想该数列中的第三个数可能是42——它在《银河系漫游指南》（The Hitchhiker’s Guide to the Galaxy）一书中被描述为具有重要意义的数字。
确立素数与量子物理之间的联系同样具有重要意义。利用这种联系，基廷和斯奈思不仅解释了为什么生命、宇宙以及黎曼zeta函数第三矩的答案是42，而且还给出了一个预测该数列（即黎曼 zeta 函数的矩）中所有数字的公式。在这个突破性进展之前，量子物理与素数相关联的证据只来自于有趣的统计比较，但是数学家对统计学是持怀疑态度的。我们喜欢精确的事物。基廷和斯奈思运用物理学得到了一个非常精确的预测，它使得统计学在预测模式的过程中没有发挥的余地了。
现在，数学家们深信不疑了。普林斯顿一个普通房间里的那次偶然会面成就了当前素数理论最激动人心的进展之一。数学中的很多大问题，例如费马大定理（Fermat’s Last Theorem），都是在建立了与其他数学分支的联系之后才被解决的。150年来，许多数学家在解决黎曼猜想的道路上怯而止步。我们可能最终找到理解素数的工具的这一希望，已经激励更多的数学家和物理学家直面挑战。希望弥漫在空气中，我们可能离真解更近一步了。戴森也许是对的，人们错失了提前四十年发现相对论的机会；然而，如果没有数学家们喝茶讨论的机遇，谁又能知道我们还要等多久才能发现素数与量子力学的联系呢！
本文作者Marcus du Sautoy（马库斯·杜·索托伊）是牛津大学的数学教授，著有《素数的音乐》（The Music of Primes）一书，并曾担任BBC专题记录片《数学的故事》（The Story of Maths）的主讲人。
原文链接
http://seedmagazine.com/content/article/prime_numbers_get_hitched/?from=singlemessage&isappinstalled=0
延伸阅读
投稿、授权等请联系：iscientists@126.com
您可回复"年份+月份"（如201510），获取指定年月文章，或返回主页点击子菜单获取或搜索往期文章。
赛先生由百人传媒投资和创办，文小刚、刘克峰、颜宁三位国际著名科学家担任主编，告诉你正在发生的科学。上帝忘了给我们翅膀，于是，科学家带领我们飞翔。http://it.sohu.com/20161111/n472876235.shtml
ゼロ除算とも関係が・・・・・・
\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}
\numberwithin{equation}{section}
\begin{document}
\title{\bf Announcement 326: The division by zero z/0=0 - its impact to human beings through education and research\\
(2016.10.17)}
\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 to education and research on our wrong world history.
\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.
We should define $F(b,0)= b/0 =0$, in general.
\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 impact 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 and by the Moore-Penrose generalized inverse,
\medskip
2) by the intuitive meaning of the fractions (division) by H. Michiwaki - repeated subtraction method,
\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, not the point at infinity.
\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.
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 impact on Euclidean spaces, analytic geometry, calculus, differential equations, complex analysis and physical problems.
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 impact to our basic ideas on the universe.
For our ideas on the division by zero, see the survey style announcements.
\section{Basic Materials of Mathematics}
(1): First, we should declare that the divison by zero is possible in the natural and uniquley determined sense and its importance.
(2): In the elementary school, we should introduce the concept of division by the idea of repeated subtraction method by H. Michiwaki whoes method is applied in computer algorithmu and in old days for calculation of division. This method will give a simple and clear method for calculation of division and students will be happy to apply this simple method at the first stage. At this time, they will be able to understand that the division by zero is clear and trivial as $a/0=0$ for any $a$. Note that Michiwaki knows how to apply his method to the complex number field.
(3): For the introduction of the elemetary function $y= 1/x$, we should give the definition of the function at the origin $x=0$ as $y = 0$ by the division by zero idea and we should apply this definition for the occasions of its appearences, step by step, following the curriculum and the results of the division by zero.
(4): For the idea of the Euclidean space (plane), we should introduce, at the first stage, the concept of steleographic projection and the concept of the point at infinity -
one point compactification. Then, we will be able to see the whole Euclidean plane, however, by the division by zero, the point at infinity is represented by zero. We can teach the very important fact with many geometric and analytic geometry methods. These topics will give great pleasant feelings to many students.
Interesting topics are: parallel lines, what is a line? - a line contains the origin as an isolated
point for the case that the native line does not through the origin. All the lines pass the origin, our space is not the Eulcildean space and is not Aristoteles for the strong discontinuity at the point at infinity (at the origin). - Here note that an orthogonal coordinates should be fixed first for our all arguments.
(5): The inversion of the origin with respect to a circle with center the origin is the origin itself, not the point at infinity - the very classical result is wrong. We can also prove this elementary result by many elementary ways.
(6): We should change the concept of gradients; on the usual orthogonal coordinates $(x,y)$,
the gradient of the $y$ axis is zero; this is given and proved by the fundamental result
$\tan (\pi/2) =0$. The result is trivial in the definition of the Yamada field. This result is derived also from the {\bf division by zero calculus}:
\medskip
For any formal Laurent expansion around $z=a$,
\begin{equation}
f(z) = \sum_{n=-\infty}^{\infty} C_n (z - a)^n,
\end{equation}
we obtain the identity, by the division by zero
\begin{equation}
f(a) = C_0.
\end{equation}
\medskip
This fundamental result leads to the important new definition:
From the viewpoint of the division by zero, when there exists the limit, at $ x$
\begin{equation}
f^\prime(x) = \lim_{h\to 0} \frac{f(x + h) - f(x)}{h} =\infty
\end{equation}
or
\begin{equation}
f^\prime(x) = -\infty,
\end{equation}
both cases, we can write them as follows:
\begin{equation}
f^\prime(x) = 0.
\end{equation}
\medskip
For the elementary ordinary differential equation
\begin{equation}
y^\prime = \frac{dy}{dx} =\frac{1}{x}, \quad x > 0,
\end{equation}
how will be the case at the point $x = 0$? From its general solution, with a general constant $C$
\begin{equation}
y = \log x + C,
\end{equation}
we see that, by the division by zero,
\begin{equation}
y^\prime (0)= \left[ \frac{1}{x}\right]_{x=0} = 0,
\end{equation}
that will mean that the division by zero (1.2) is very natural.
In addition, note that the function $y = \log x$ has infinite order derivatives and all the values are zero at the origin, in the sense of the division by zero.
However, for the derivative of the function $y = \log x$, we have to fix the sense at the origin, clearly, because the function is not differentiable, but it has a singularity at the origin. For $x >0$, there is no problem for (2.6) and (2.7). At $x = 0$, we see that we can not consider the limit in the sense (2.3). However, for $x >0$ we have (2.6) and
\begin{equation}
\lim_{x \to +0} \left(\log x \right)^\prime = +\infty.
\end{equation}
In the usual sense, the limit is $+\infty$, but in the present case, in the sense of the division by zero, we have:
\begin{equation}
\left[ \left(\log x \right)^\prime \right]_{x=0}= 0
\end{equation}
and we will be able to understand its sense graphycally.
By the new interpretation for the derivative, we can arrange many formulas for derivatives, by the division by zero. We can modify many formulas and statements in calculus and we can apply our concept to the differential equation theory and the universe in connetion with derivatives.
(7): We shall introduce the typical division by zero calculus.
For the integral
\begin{equation}
\int x(x^{2}+1)^{a}dx=\frac{(x^{2}+1)^{a+1}}{2(a+1)}\quad(a\ne-1),
\end{equation}
we obtain, by the division by zero,
\begin{equation}
\int x(x^{2}+1)^{-1}dx=\frac{\log(x^{2}+1)}{2}.
\end{equation}
We will consider the fundamental ordinary differential equations
\begin{equation}
x^{\prime \prime}(t) =g -kx^{\prime}(t)
\end{equation}
with the initial conditions
\begin{equation}
x(0) = -h, x^{\prime}(0) =0.
\end{equation}
Then we have the solution
\begin{equation}
x(t) = \frac{g}{k}t + \frac{g(e^{-kt}- 1)}{k^2} - h.
\end{equation}
Then, for $k=0$, we obtain, immediately, by the division by zero
\begin{equation}
x(t) = \frac{1}{2}g t^2 -h.
\end{equation}
In those examples, we were able to give valuable functions for denominator zero cases. The division by zero calculus may be applied to many cases as a new fundamental calculus over l'Hôpital's rule.
(8): When we apply the division by zero to functions, we can consider, in general, many ways. For example,
for the function $z/(z-1)$, when we insert $z=1$ in numerator and denominator, we have
\begin{equation}
\left[\frac{z}{z-1}\right]_{z = 1} = \frac{1}{0} =0.
\end{equation}
However,
from the identity --
the Laurent expansion around $z=1$,
\begin{equation}
\frac{z}{z-1} = \frac{1}{z-1} + 1,
\end{equation}
we have
\begin{equation}
\left[\frac{z}{z-1}\right]_{z = 1} = 1.
\end{equation}
For analytic functions we can give uniquely determined values at isolated singular points by the values by means of the Laurent expansions as the division by zero calculus, however, the values by means of the Laurent expansions are not always reasonable. We will need to consider many interpretations for reasonable values. In many formulas in mathematics and physics, however, we can see that the division by zero calculus is reasonably valid. See \cite{kmsy,msy}.
\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.
Note that the Big Bang also may be related to the division by zero like the blackholes.
\section{Computer systems}
The above Professors listed are wishing the contributions in order to avoid the division by zero trouble in computers. Now, we should arrange new computer systems in order not to meet the division by zero trouble in computer systems.
By the division by zero calculus, we will be able to overcome troubles in Maple for specialization problems.
\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 introduced.
\bigskip
We are standing on a new generation and in front of the new world, as in the discovery of the Americas. Should we push the research and education on the division by zero?
\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 $z/0=0$, Advances in Linear Algebra
\& Matrix Theory, 6, 51-58. http://dx.doi.org/10.4236/alamt.2016.62007 http://www.scirp.org/journal/alamt
\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.
\bibitem{ann300}
Announcement 300 (2016.05.22): New challenges on the division by zero z/0=0.
\end{thebibliography}
\end{document}
0 件のコメント:
コメントを投稿