「全力すぎる 大分県豊後高田市」Vol.19 千年のロマンに誘う“ライトアップ”イベントを続々開催

「全力すぎる　大分県豊後高田市」Vol.19
千年のロマンに誘う“ライトアップ”イベントを続々開催

　大分県豊後高田市がリリースする「全力すぎる　大分県豊後高田市」Vol.19です。
　豊後高田市の魅力満載情報を全力でお届けします。
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
【Contents】
　　■世界農業遺産の郷 田染荘 「千年のきらめき」
　　■六郷満山開山1300年　2016ライトアップイベント
　　■秋そば解禁・スタンプラリー
　　■全力発展中な豊後高田市
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

【■世界農業遺産の郷 田染荘 「千年のきらめき」】
　国の重要文化的景観に選定された昔ながらの美しい田園風景の広がる、大分県豊後高田市の田染荘小崎地区で、11月19日（土）から12月3日（土）にかけて、稲刈り後の水田を舞台にした大規模なイルミネーションイベント「千年のきらめき」を開催します。

　このイベントは田染荘では初めての企画で、あぜ道に設置した約１万個のソーラーＬＥＤライト「ペットボタル」により、期間中、毎日イルミネーションが楽しめます。
　日没後辺りが暗くなるにつれ、まるでほたるが出現するかのように、１つ１つライトが点灯していく幻想的な光景が広がります。

　≪ペットボタルについて≫
　　使用するLED装置は通称「ペットボタル」と呼ばれ、昼間の太陽光エネルギーで充電、
　暗くなると自動で発光するしくみになっています。

　≪田染荘小崎の農村景観≫
　　田染荘は、中世の時代に宇佐八幡宮の根本荘園の中でも最も重要視された荘園として
　栄えました。そのころから集落や水田の位置がほとんど変わらずに残されており、中世
　荘園村落の姿を今に残す美しい景観が高く評価され、平成２２年に「田染荘小崎の農村
　景観」として国の重要文化的景観に選定されました。また平成２５年には、豊後高田市
　を含む国東半島・宇佐地域が「世界農業遺産」に認定され、「世界農業遺産の郷」を象
　徴する場所となっています。

　　■開催期間：平成28年11月19日（土）～12月3日（土）
　　■会　　場：大分県豊後高田市田染小崎２５９６　ほたるの館周辺
　　■点灯時間：日没から３時間程度
　　　　　　　※期間中の日没時間予想　１７：２０頃～
　　　　　　　　センサー感知のため、ライトが全て点灯するまでに若干のタイムラグ
　　　　　　　　が生じます。
　　▼世界農業遺産の郷 田染荘 「千年のきらめき」
　　　http://www.city.bungotakada.oita.jp/events/detail/275


【■六郷満山開山1300年　2016ライトアップイベント】
　大分県の国東半島が誇る文化遺産「六郷満山」の開山1300年を前に、国東半島の代表的な寺社仏閣で、ライトアップイベントを開催します。ライトアップ期間中は、神楽やゴスペル、護摩など様々なイベントも開催。幻想的な夜の六郷満山をお楽しみいただけます。

　≪国東半島と六郷満山について≫
　　六郷満山とは、大分県の国東半島の６つの郷（来縄・田染・伊美・国東・武蔵・安岐）
　の谷々で発展した寺院群の総称です。
　　奈良時代から平安時代にかけて成立したものといわれる六郷満山。古来の山岳信仰に
　天台仏教や浄土思想が結びついて宇佐神宮の神仏習合がとけこみ、神と仏が一体となっ
　た霊地は、僧たちの修行の場として次々に寺院が建立されたといわれ、全盛期には、
　一大寺院群となり華やかな仏教文化（六郷満山文化）が華開きました。
　　国東半島は六郷満山文化の名残が今も色濃く伝えられており古仏・古刹に刻まれた
　千年の祈りに触れることができます。

　　▼国東半島が誇る文化遺産「六郷満山」について
　　http://www.city.bungotakada.oita.jp/page/page_03125.html

　≪六郷満山開山1300年≫
　　２年後の平成３０年に、六郷満山は開山1300年を迎えます。今年度からこの節目に向
　けた観光キャンペーンを実施するため、国東半島宇佐地域の関連６市町村(豊後高田市、
　宇佐市、国東市、杵築市、日出町、姫島村)と大分県が主体となって、「国東半島宇佐
　地域・六郷満山開山１３００年誘客キャンペーン実行委員会事務局」を組織し、各寺院
　と連携してキックオフイベントを開催します。

　■ライトアップスケジュール
　　以下の日程でライトアップとイベントが開催されます。
　　ライトアップ時間帯は、１７：３０～２１：００です。
　　※以下の日程以外はライトアップされませんのでご注意ください。
　　
　　＜霊仙寺・実相院・六所神社＞
　　１１／１２（土）　高千穂神楽と夷里神楽（18：30～） 拝観料：500円
　　１１／１３（日）　ライトアップのみ　拝観料：無料
　　＜両子寺＞
　　１１／１８（金）　ライトアップのみ　　拝観料：500円
　　１１／１９（土）　声明　（18：30～）　拝観料：1,000円
　　＜熊野磨崖仏＞
　　１１／１８（金）　ライトアップのみ　拝観料：500円
　　１１／１９（土）　ゴスペルクワイヤーJOY　拝観料：500円
　　＜文殊仙寺＞
　　１１／２５（金）　声明（17：30～）　拝観料：1,000円
　　１１／２６（土）　八千枚大護摩供（終日）　拝観料：無料
　　＜富貴寺＞
　　１２／２（金）　鈴木利枝（津軽三味線）と深田宏一（ピアノ）の共演（18：30～）
　　　　　　　　　　拝観料：300円
　　１２／３（土）　舞楽常行三昧（ぶがくじょうぎょうざんまい）（17：00～）
　　　　　　　　　　拝観料：1,000円（座席 ※100席程度）
　　　　　　　　　　拝観料：　300円（立ち見）

　▼六郷満山開山１３００年ライトアップイベント
　　http://www.city.bungotakada.oita.jp/events/detail/278


　≪【関連イベント】第１４回全国鬼サミットｉｎ国東半島≫
　　六郷満山開山1300年キックオフイベントの一環として、12月3日（土）に豊後高田市
　の中央公民館で、「第１４回全国鬼サミットｉｎ国東半島」が開催されます。
　　これは、全国鬼サミットに加入している市町村が集まり、情報交換を図るイベントで
　す。当日は歓迎アトラクションで神楽が楽しめるほか、全国各地の鬼伝説の紹介や鬼伝
　統行事の披露（豊後高田市の天念寺修正鬼会や鹿児島県曽於（そお）市の奇習鬼追い）
　さらには、秋田県から「なまはげ郷神楽」が来場し特別出演を行います。

　　■開催日時：１２月３日（土）１３：３０～
　　■会　　場：豊後高田市中央公民館
　　■入 場 料：無料


【■秋そば解禁・スタンプラリーも開催】
　豊後高田市の特産品である「そば」が収穫時期を迎え、11月5日（土）から、秋の新そばが解禁されました。豊後高田市は、「春と秋の年２回新そばを楽しめるそば処」として有名です。春そばは爽やかな風味があり、秋そばは豊かな香りと深い味わいが特徴といわれています。

　豊後高田市では、皆さまに安心して美味しいそばを楽しんでいただくために「豊後高田手打ちそば認定店」という認定店制度を創設し、認定店では挽きたて・打ちたて・茹でたての「三たて」を厳しく守っています。秋の新そばは、市内の認定店１２店舗で提供しています。
　▼認定店12店舗の詳細
　　http://www.showanomachi.com/special/gourmet.html

　≪秋そばスタンプラリー≫
　　秋そば解禁に合わせ、「秋そばスタンプラリー」を開催しています。
　　スタンプラリーでは、豊後高田手打ちそば認定店で食事をすると、一人につきひとつ
　スタンプを差し上げます。スタンプを集めると、豊後高田市内旅館の一泊二食付ペア宿
　泊券が抽選で当たります。

　　■実施期間：平成28年11月5日（土）～平成29年1月31日（火）
　　■参加店舗：豊後高田手打ちそば認定店　うち11店舗
　　■賞　　品：Ａ賞（スタンプ５個で応募可能）
　　　　　　　　豊後高田市内旅館施設の一泊二食付ペア宿泊券（抽選で６組）
　　　　　　　　Ｂ賞（スタンプ２個で応募可能）
　　　　　　　　陶芸家木下栄司氏 特製「そばちょこ」（抽選で２０名）
　　▼秋そばスタンプラリー
　　　http://www.city.bungotakada.oita.jp/events/detail/280

【■全力発展中な豊後高田市】
　豊後高田市は、平成27年3月31日に新市誕生10周年を迎え、新たなブランドコンセプトとして「全力発展中　豊後高田市」を掲げ、ブランドマークを制定しました。
このブランドマークは、まちが全力で挑んでいく「熱気や情熱」を表現しています。
豊後高田市は、市民一人ひとりが何事にも「全力」で取り組み、「夢をかたちに　未来に光り続けるまち」の実現に向けて全力で発展し続けます。http://prw.kyodonews.jp/opn/release/201611106138/

地域経済：独自・ユニーク

\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}