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Latest measurements from the AMS experiment unveil new territories in the flux of cosmicrays

Latest measurements from the AMS experiment unveil new territories in the flux of cosmicrays

Geneva 18 September 2014. The Alpha Magnetic Spectrometer (AMS1) collaboration has today presented its latest results. These are based on the analysis of 41 billion particles detected with the space-based AMS detector aboard the International Space Station. The results, presented during a seminar at CERN2, provide new insights into the nature of the mysterious excess of positrons observed in the flux of cosmic rays. The findings are published today in the journal Physical Review Letters.

Cosmic rays are particles commonly present in the universe. They consist mainly of protons and electrons, but there are also many other kinds of particles, including positrons, travelling through space. Positrons are the antimatter counterparts of electrons, with the same mass but opposite charge. The presence of some positrons in space can be explained from the collisions of cosmic rays, but this phenomenon would only produce a tiny portion of antimatter in the overall cosmic ray spectrum. Since antimatter is extremely rare in the universe, any significant excess of antimatter particles recorded in the flux of energetic cosmic rays indicates the existence of a new source of positrons. Very dense stars, such as pulsars, are potential candidates.

The AMS experiment is able to map the flux of cosmic rays with unprecedented precision and in the results published today, the collaboration presents new data at energies never before recorded. The AMS collaboration has analysed 41 billion primary cosmic ray events among which 10 million have been identified as electrons and positrons. The distribution of these events in the energy range of 0.5 to 500 GeV shows a well-measured increase of positrons from 8 GeV with no preferred incoming direction in space. The energy at which the positron fraction ceases to increase has been measured to be 275±32 GeV.

“This is the first experimental observation of the positron fraction maximum after half a century of cosmic rays experiments,” said AMS spokesperson Professor Samuel Ting. “Measurements are underway by the AMS team to determine the rate of decrease at which the positron fraction falls beyond the turning point.”

This rate of decrease after the “cut-off energy” is very important to physicists as it could be an indicator that the excess of positrons is the signature of dark matter particles annihilating into pairs of electrons and positrons. Although the current measurements could be explained by objects such as pulsars, they are also tantalizingly consistent with dark matter particles with mass of the order of 1 TeV. Different models on the nature of dark matter predict different behaviour of the positron excess above the positron fraction expected from ordinary cosmic ray collisions. Therefore, results at higher energies will be of crucial importance in the near future to evaluate if the signal is from dark matter or from a cosmic source.

"With AMS and with the LHC to restart in the near future at energies never reached before, we are living in very exciting times for particle physics as both instruments are pushing boundaries of physics,” said CERN Director-General Rolf Heuer.

AMS also reported a new observation that both the electron flux and the positron flux change their behaviour at about 30 GeV, the fluxes being significantly different from each other both in their magnitude and energy dependence. In particular, between 20 and 200 GeV, the rate of change of the positron flux is surprisingly higher than that for electrons. This is important proof that the excess seen in the positron fraction is due to a relative excess of high-energy positrons, and not the loss of high-energy electrons. This new result is very important for a better understanding of the origin of cosmic ray electrons and positrons, and may be the sign of an unknown phenomenon.

In his seminar, Professor Ting also presented some interesting new results to be published in the near future. These show that, at high energies and over a wide energy range, the combinedflux of electrons plus positrons can be described by a single constant spectral index, with no existence of structure as suspected by previous measurements of other experiments.

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, googlesite 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 inLinear 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}

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