ビッグバン理論の見直し迫る新たな観測
2014年09月11日
いて座矮小楕円銀河に属する球状星団M54（メシエ54）の最新の観測結果から、天の川銀河の外でもやはり、ビッグバン理論から予測されるリチウムの量と実際の値には開きがあることが分かった。いて座矮小楕円銀河に属する球状星団M54（メシエ54）の最新の観測結果から、天の川銀河の外でもやはり、ビッグバン理論から予測されるリチウムの量と実際の値には開きがあることが分かった。 地球から約8万光年の距離にある星団でも、どうやら私たちの銀河と同様に、金属元素のリチウムの量が理論上の値よりも大幅に少ないらしいことが、9月10日に発表された最新の研究で明らかになった。このリチウム量の不足から考えられる可能性は、これまでの天体物理学研究ではビッグバンを十分に説明できていないか、恒星のはたらきを十分に説明できていないかのいずれかであると、論文の著者らは示している。ただし、今回の発見は、ビッグバンの概念そのものを覆すものではない。
「この（リチウム量の）問題に関する最も極端な説明は、ビッグバン理論が不完全であるということだ。そこまで極端にならずに、この問題を説明する方法は見つかっていない」と、イリノイ大学アーバナ・シャンペーン校の理論物理学者ブライアン・フィールズ（Brian Fields）氏は言う。フィールズ氏は今回の観測には参加していない。
宇宙についての基本的な理論では、宇宙は誕生後の数分間、ちょうど原子炉のような働きをして、最も軽い3つの元素である水素、ヘリウム、リチウムを生成したとされている。これより重い、酸素や窒素や炭素やケイ素などの元素は、それ以降に、恒星の核の中か、強力な超新星爆発において作られたという。最初期の核反応に関するこうした理解に基づいて、最も軽い3つの元素がどれだけ生成されたかが理論上予測された、とカリフォルニア大学サンタクルーズ校（UCSC）の物理学者ジョエル・プリマック（Joel Primack）氏は説明する。なお、プリマック氏も今回の観測には参加していない。全体的に見て、これらの予測は正確に当たっていた。「宇宙学の最大の成功の1つだ」とプリマック氏は言う。だがそれも、リチウムには当てはまらない。「リチウムについては、この予測は私たちが実際に恒星で確認できた数値より、約3倍も多かった」とプリマック氏は言う。
◆別の銀河系ではどうなる？
リチウムの量が予測より少ないことは、フランソワ・スピート（Francois Spite）氏とモニク・スピート（Monique Spite）氏の計測によって、1982年に初めて明らかになった。この2人は夫婦でともに天文学者である。「その後も多くの人が計測し直して、やはり同じ結果を得ている」とプリマック氏は言う。ただし、天の川銀河の外の恒星におけるリチウムの量は、これまで計測されてこなかった。初めてそれを行ったのが、ボローニャ大学のアレッシオ・ムッチャレッリ（Alessio Mucciarelli）氏らによる今回の研究である。「ほかの銀河系でもこの問題が同じであるなら、ローカルな問題ではなく全宇宙的な問題なのだと確認できるだろう」とムッチャレッリ氏は言う。
チームが観測対象に選んだのは、いて座矮小楕円銀河に属する球状星団M54（メシエ54）だ。「これらの星は非常に暗いので、正直なところ、うまく行くかどうかも定かでなかった」とムッチャレッリ氏は言う。最終的には、チリにあるヨーロッパ南天天文台（ESO）の超大型望遠鏡VLTを用いた30時間の観測によって、天の川銀河の外にある恒星でも、やはりリチウム量は予測より少ないとの確証を得た。
◆謎は深まるばかり
リチウム量の不足の説明として考えられるものの1つは、恒星にはもともと現在より多くのリチウムが含まれていたが、核反応によって破壊されたという説だ。「そんなに突飛な考えではないが、詳細まで確認するのは難しい」とフィールズ氏は言う。より可能性の高い説明として、フィールズ氏とプリマック氏がともに認めているのは、ビッグバン直後の最初の数分間に、これまでの研究では明らかになっていない何らかのエネルギー放出があって、リチウムの生成が抑制された、というものだ。もしそうだとすれば、リチウムははじめ、くずのような形で誕生し、それがやがて崩壊して暗黒物質（ダークマター）になった可能性がある。
プリマック氏は、もしそうであれば「このリチウムの問題から窺えることは、恒星に関する些細な事柄などではなく、ダークマターに関する根本的な事柄なのかもしれない」と言う。
ダークマターの性質は、宇宙学の分野で今なお謎とされている主要な問題の1つなので、プリマック氏の仮説の通りなら、これは本当に大きな話になってくる。
球状星団M54のリチウム量に関する今回の論文は、「Monthly Notices of the Royal Astronomical Society」誌のオンライン版に9月10日付けで掲載された。
Michael D. Lemonick for National Geographic News
「ビッグバン理論の見直し迫る新たな観測」（拡大写真付きの記事）
Announcement 179: Division by zero is clear as z/0=0 and it is fundamental in mathematics
\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}
\numberwithin{equation}{section}
\begin{document}
\title{\bf Announcement 179: Division by zero is clear as z/0=0 and it is fundamental in mathematics\\
}
\author{{\it Institute of Reproducing Kernels}\\
Kawauchi-cho, 5-1648-16,\\
Kiryu 376-0041, Japan\\
E-mail: kbdmm360@yahoo.co.jp\\
}
\date{\today}
\maketitle
{\bf Abstract: } In this announcement, we shall introduce the zero division $z/0=0$. The result is a definite one and it is fundamental in mathematics.
\bigskip
\section{Introduction}
%\label{sect1}
By a natural extension of the fractions
\begin{equation}
\frac{b}{a}
\end{equation}
for any complex numbers $a$ and $b$, we, recently, found the surprising 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 result is a very special case for general fractional functions in \cite{cs}.
The division by zero has a long and mysterious story over the world (see, for example, google site with division by zero) with its physical viewpoints since the document of zero in India on AD 628, however,
Sin-Ei, Takahasi (\cite{taka}) (see also \cite{kmsy}) established a simple and decisive interpretation (1.2) by analyzing some full extensions of fractions and by showing the complete characterization for the property (1.2). His result will show that our mathematics says that the result (1.2) should be accepted as a natural one:
\bigskip
{\bf Proposition. }{\it Let F be a function from ${\bf C }\times {\bf C }$ to ${\bf C }$ such that
$$
F (b, a)F (c, d)= F (bc, ad)
$$
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 } $
$$
F (b, 0) = 0.
$$
}
\medskip
\section{What are the fractions $ b/a$?}
For many mathematicians, the division $b/a$ will be considered as the inverse of product;
that is, the fraction
\begin{equation}
\frac{b}{a}
\end{equation}
is defined as the solution of the equation
\begin{equation}
a\cdot x= b.
\end{equation}
The idea and the equation (2.2) show that the division by zero is impossible, with a strong conclusion. Meanwhile, the problem has been a long and old question:
As a typical example of the division by zero, we shall recall the fundamental law by Newton:
\begin{equation}
F = G \frac{m_1 m_2}{r^2}
\end{equation}
for two masses $m_1, m_2$ with a distance $r$ and for a constant $G$. Of course,
\begin{equation}
\lim_{r \to +0} F =\infty,
\end{equation}
however, in our fraction
\begin{equation}
F = G \frac{m_1 m_2}{0} = 0.
\end{equation}
\medskip
Now, we shall introduce an another approach. The division $b/a$ may be defined {\bf independently of the product}. Indeed, in Japan, the division $b/a$ ; $b$ {\bf raru} $a$ ({\bf jozan}) is defined as how many $a$ exists in $b$, this idea comes from subtraction $a$ repeatedly. (Meanwhile, product comes from addition).
In Japanese language for "division", there exists such a concept independently of product.
H. Michiwaki and his 6 years old girl said for the result $ 100/0=0$ that the result is clear, from the meaning of the fractions independently the concept of product and they said:
$100/0=0$ does not mean that $100= 0 \times 0$. Meanwhile, many mathematicians had a confusion for the result.
Her understanding is reasonable and may be acceptable:
$100/2=50 \quad$ will mean that we divide 100 by 2, then each will have 50.
$100/10=10 \quad$ will mean that we divide 100 by10, then each will have 10.
$100/0=0 \quad$ will mean that we do not divide 100, and then nobody will have at all and so 0.
Furthermore, she said then the rest is 100; that is, mathematically;
$$
100 = 0\cdot 0 + 100.
$$
Now, all the mathematicians may accept the division by zero $100/0=0$ with natural feelings as a trivial one?
\medskip
For simplicity, we shall consider the numbers on non-negative real numbers. We wish to define the division (or fraction) $b/a$ following the usual procedure for its calculation, however, we have to take care for the division by zero:
The first principle, for example, for $100/2 $ we shall consider it as follows:
$$
100-2-2-2-,...,-2.
$$
How may times can we subtract $2$? At this case, it is 50 times and so, the fraction is $50$.
The second case, for example, for $3/2$ we shall consider it as follows:
$$
3 - 2 = 1
$$
and the rest (remainder) is $1$, and for the rest $1$, we multiple $10$,
then we consider similarly as follows:
$$
10-2-2-2-2-2=0.
$$
Therefore $10/2=5$ and so we define as follows:
$$
\frac{3}{2} =1 + 0.5 = 1.5.
$$
By these procedures, for $a \ne 0$ we can define the fraction $b/a$, usually. Here we do not need the concept of product. Except the zero division, all the results for fractions are valid and accepted.
Now, we shall consider the zero division, for example, $100/0$. Since
$$
100 - 0 = 100,
$$
that is, by the subtraction $100 - 0$, 100 does not decrease, so we can not say we subtract any from $100$. Therefore, the subtract number should be understood as zero; that is,
$$
\frac{100}{0} = 0.
$$
We can understand this: the division by $0$ means that it does not divide $100$ and so, the result is $0$.
Similarly, we can see that
$$
\frac{0}{0} =0.
$$
As a conclusion, we should define the zero divison as, for any $b$
$$
\frac{b}{0} =0.
$$
See \cite{kmsy} for the details.
\medskip
\section{In complex analysis}
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 $w=0$. 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.
However, we shall recall the elementary function
\begin{equation}
W(z) = \exp \frac{1}{z}
\end{equation}
$$
= 1 + \frac{1}{1! z} + \frac{1}{2! z^2} + \frac{1}{3! z^3} + \cdot \cdot \cdot .
$$
The function has an essential singularity around the origin. When we consider (1.2), meanwhile, surprisingly enough, we have:
\begin{equation}
W(0) = 1.
\end{equation}
{\bf The point at infinity is not a number} and so we will not be able to consider the function (3.2) at the zero point $z = 0$, meanwhile, we can consider the value $1$ as in (3.3) at the zero point $z = 0$. How do we consider these situations?
In the famous standard textbook on Complex Analysis, L. V. Ahlfors (\cite{ahlfors}) introduced the point at infinity as a number and the Riemann sphere model as well known, however, our interpretation will be suitable as a number. We will not be able to accept the point at infinity as a number.
As a typical result, we can derive the surprising result: {\it At an isolated singular point of an analytic function, it takes a definite value }{\bf with a natural meaning.} As the important applications for this result, the extension formula of functions with analytic parameters may be obtained and singular integrals may be interpretated with the division by zero, naturally (\cite{msty}).
\bigskip
\section{Conclusion}
The division by zero $b/0=0$ is possible and the result is naturally determined, uniquely.
The result does not contradict with the present mathematics - however, in complex analysis, we need only to change a little presentation for the pole; not essentially, because we did not consider the division by zero, essentially.
The common understanding that the division by zero is impossible should be changed with many text books and mathematical science books. The definition of the fractions may be introduced by {\it the method of Michiwaki} in the elementary school, even.
Should we teach the beautiful fact, widely?:
For the elementary graph of the fundamental function
$$
y = f(x) = \frac{1}{x},
$$
$$
f(0) = 0.
$$
The result is applicable widely and will give a new understanding for the universe ({\bf Announcement 166}).
\medskip
If the division by zero $b/0=0$ is not introduced, then it seems that mathematics is incomplete in a sense, and by the intoduction of the division by zero, mathematics will become complete in a sense and perfectly beautiful.
\bigskip
section{Remarks}
For the procedure of the developing of the division by zero and for some general ideas on the division by zero, we presented the following announcements in Japanese:
\medskip
{\bf Announcement 148} (2014.2.12): $100/0=0, 0/0=0$ -- by a natural extension of fractions -- A wish of the God
\medskip
{\bf Announcement 154} (2014.4.22): A new world: division by zero, a curious world, a new idea
\medskip
{\bf Announcement 157} (2014.5.8): We wish to know the idea of the God for the division by zero; why the infinity and zero point are coincident?
\medskip
{\bf Announcement 161} (2014.5.30): Learning from the division by zero, sprits of mathematics and of looking for the truth
\medskip
{\bf Announcement 163} (2014.6.17): The division by zero, an extremely pleasant mathematics - shall we look for the pleasant division by zero: a proposal for a fun club looking for the division by zero.
\medskip
{\bf Announcement 166} (2014.6.29): New general ideas for the universe from the viewpoint of the division by zero
\medskip
{\bf Announcement 171} (2014.7.30): The meanings of product and division -- The division by zero is trivial from the own sense of the division independently of the concept of product
\medskip
{\bf Announcement 176} (2014.8.9): Should be changed the education of the division by zero
\bigskip
\bibliographystyle{plain}
\begin{thebibliography}{10}
\bibitem{ahlfors}
L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Company, 1966.
\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}
S. Koshiba, H. Michiwaki, S. Saitoh and M. Yamane,
An interpretation of the division by zero z/0=0 without the concept of product
(note).
\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. Vol. 27, No 2 (2014), pp. 191-198, DOI: 10.12732/ijam.v27i2.9.
\bibitem{msty}
H. Michiwaki, S. Saitoh, M. Takagi and M. Yamada,
A new concept for the point at infinity and the division by zero z/0=0
(note).
\bibitem{s}
S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. Vol.4 No.2 (2014), 87-95.http://www.scirp.org/journal/ALAMT/
\bibitem{taka}
S.-E. Takahasi,
{On the identities $100/0=0$ and $ 0/0=0$}
(note).
\bibitem{ttk}
S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classification of continuous fractional binary operators on the real and complex fields. (submitted)
\end{thebibliography}
\end{document}
0 件のコメント:
コメントを投稿