Not only mathematics has changed; gone were the days of tedious and strenuous methods of proofs with a paper and a pencil. Nowadays most of heavy and monotonous work in mathematics is done with computers. But as we solve larger and more complex mathematical problems with greater computational power and cleverer algorithms, the problems we cannot tackle with computers begin to stand out. In this connection, one may ask whether there is a 'mechanical' method that can be applied to any mathematical assertion in order to check the truth or falsity of that assertion.

Thus, the next intellectual thrill in mathematics will be P versus NP question: If the solution to a problem can be verified fast (i.e., in polynomial time), can it be found fast as well? Accordingly, in 2000, the Clay Math Institute named P ≟ NP as one of the seven most important open questions in mathematics (and has offered a million-dollar prize for a proof that determines whether or not P = NP).

To understand the importance of the P versus NP question just imagine for a moment a world where P = NP. In such a world, the mental effort of the mathematicians in the case of yes-or-no questions will be completely replaced by computers. Everything will be much more efficient: Transportation of all forms will be scheduled optimally to move people and goods around quicker and cheaper; manufacturers will be able to improve production to increase speed and create less waste. Learning will become easy: Near perfect vision recognition, language comprehension, translation, and all other learning tasks will be trivial.

RH wont be proved. Maybe Langlands program will be completed. More math will be computerized.

Computerized and programized.

We need maths that supports multi-parallel processing. Well, I need it!

Not only mathematics has changed; gone were the days of tedious and strenuous methods of proofs with a paper and a pencil. Nowadays most of heavy and monotonous work in mathematics is done with computers. But as we solve larger and more complex mathematical problems with greater computational power and cleverer algorithms, the problems we cannot tackle with computers begin to stand out. In this connection, one may ask whether there is a 'mechanical' method that can be applied to any mathematical assertion in order to check the truth or falsity of that assertion.

Thus, the next intellectual thrill in mathematics will be P versus NP question: If the solution to a problem can be verified fast (i.e., in polynomial time), can it be found fast as well? Accordingly, in 2000, the Clay Math Institute named P ≟ NP as one of the seven most important open questions in mathematics (and has offered a million-dollar prize for a proof that determines whether or not P = NP).

To understand the importance of the P versus NP question just imagine for a moment a world where P = NP. In such a world, the mental effort of the mathematicians in the case of yes-or-no questions will be completely replaced by computers. Everything will be much more efficient: Transportation of all forms will be scheduled optimally to move people and goods around quicker and cheaper; manufacturers will be able to improve production to increase speed and create less waste. Learning will become easy: Near perfect vision recognition, language comprehension, translation, and all other learning tasks will be trivial.

I think the in future mathematics have a very important role in all types of sector.

Mathematics has traditionally been described as the science of number and shape.The historic boundaries of mathematics have all but disappeared. So have the boundaries of its applications: no longer just the language of physics and engineering, mathematics is now an essential tool for banking, manufacturing, social science, and medicine. When viewed in this broader context, we see that mathematics is not just about number and shape but about pattern and order of all sorts.

I think there will be a grand unification of math and theoretical computer science based on the notion of type. Automated theorem proving will become commonplace. But of course the fun of sharing ideas on a table corner or on a whiteboard with a friend will remain. But social networks and webcam blackboards will become more

important.

As Sergiy Mishin wrote, in the basis of XXI century Mathematics should be the new physics and mathematics theory “Structural analysis” (http://vk.com/doc124153280_132552693?hash=1970f63e3de4fae4da&dl=1cf68455483cc35cc5)

Every era needs and develop its mathematics. Thus in the future the need for automation leads naturally to the study of man himself. Thus the basic event of mathematics in the future will be the sift of focus from nature to human. Thus mathematics and biology, mathematics and psychology, sociology will be the focus of mathematics.

Elsevier Scopus

Date: Sun, Jul 27, 2014 at 11:08 AM

Subject: Scopus releases 2013 Journal Metrics

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Elsevier

Scopus releases 2013 Journal Metrics

Dear Fethi Bin Muhammad Belgacem,

Elsevier is pleased to announce the release of the 2013 Journal Metrics based on Scopus data.

The metrics provide alternative, transparent and accurate views of the citation impact a journal makes, and are all available for free download at www.journalmetrics.com. The impact metrics are based on methodologies developed by external bibliometricians and use Scopus as the data source. Scopus is the largest citation database of peer-reviewed literature and features tools to track, analyze and visualize research output.

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Department of Mathematics,

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PAAET, Al-Aardhia, Kuwait.

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I agree @ prof Patrick Solé , @ Prof Muhammad Javaid, and Prof.Arkady Bolotin

I agree @ Prof.Sylantyev Sergiy.

I would be curious to read an english version of the "Structural analysis" tract!

 I copy here from: The Next 50 Years by Chris Budd CMath FIMA

So, where do I see our fine subject going in the next 50 years? Having said that the great driver of the last 50 years was physics and engineering, we have seen a more recent driver being biology, and now the real powerhouse behind the developments and applications of mathematics is in information and related technologies such as computer graphics, signal and image processing. Witness the explosive growth of the Internet from modest beginnings only about 20 years ago, coupled to Google and the use of social networks. We see also the profound influence of mathematics in computer animation, graphic design and virtual reality. None of these technologies would work without mathematics, and all of them were developed in part by mathematical ideas and the inspiration of many mathematicians. I see no let up in this rapid advance withideas in mathematics (such as sparse matrix theory, compression algorithms, quantum theory, network theory, complexity, nonlinear systems, Grassmann Manifolds) driving new technologies, which in turn will drive new mathematics. I suspect that mathematically driven artificial intelligence cannot be far off, and maybe a robot will pass the Turing test in the next fifty years.

For full article see;

Sift of focus, from the mathematics of nature to the mathematics of human. Computer science reforms mathematics in a revolutionary way.

The pure mathematics of 20th century will be the applied mathematics of the 21st century. Especially Category Theory will be for psychology what was Calculus for physics. 

In present traditional finite—infinite theory system, people have been creating many new “understandings” on “infinite”, “potential infinite” and “actual infinite” since Zeno’s time 2500 years ago. But it is difficult to solve those infinite related problems produced by the fundamental defects disclosed by the infinite related paradoxes since Zeno’s time, because within the present traditional finite—infinite theory system, “the infinite related problems” are strongly interlocked together with the foundation. So, though trying very hard willing to solve “some infinite related problems” with some new “understandings” within the present traditional finite—infinite theory system, but people finally discovered that nothing can be done because “everything is perfect” in present traditional finite—infinite theory system.

I know few people agree with me, but this is true. I sincerely hope that we can devote some fundamental improvements in mathematics in the 20th century.

I think that The concept of Fuzziness and Multivalued as well as  set valued functions will have its influence on new definitions of infinity startifications. In other words we will have various rankings for infinity as well as r/0  ( a favotite question of Saburo Saitoh).

More on this later. in Shaa Allah.  

I expect to develop of the division by zero as a new research topics, because the results100/0=0,0/0=0 are very simple and fundamental:

\documentclass[12pt]{article}

\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}

\numberwithin{equation}{section}

\begin{document}

\title{\bf Announcement 185　: The importance of the division by zero $z/0=0$

}

\author{{\it Institute of Reproducing Kernels}\\

Kawauchi-cho, 5-1648-16,\\

Kiryu 376-0041, Japan\\

E-mail: [email protected]\\

}

\date{November 28, 2014}

\maketitle

{\bf Abstract: } In this announcement, we shall state the importance of the division by zero $z/0=0$. The result is a definite one and it is fundamental in mathematics.

\bigskip

{\bf Introduction}

\bigskip

%\label{sect1}

By {\bf 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

Furthermore, note that Hiroshi Michiwaki with his 6 years old daughter gave the important interpretation of the division by zero $z/0=0$ by the intuitive meaning of the division, independently of the concept of the product (see \cite{ann}) .

We shall state the importance of the division by zero $z/0=0$.

\bigskip

\section{}

On AD 628, the zero was appeared in India, and the zero division $z/0=0$ was discovered on February 2, 2014, definitely with the clear definition and motivation. The uniquess and the natural interpretation were given in \cite{taka, ttk,kmsy} and \cite{ann}, respectively. Several physical interpretations of the division by zero were given in \cite{kmsy}.

\bigskip

\section{}

By the introduction of the division by zero $z/0=0$, four arithmetic operations; that is,

addition, subtraction, multiplication, and division are always possible; note that for division, we were not able to divide by zero. There was one exceptional case for the division by zero.

\section{}

For the Euclidean (B.C. 3 Century ) geometry, two non-Euclidean geometries were appered about 2 hundred years ago, and in particular, in the elliptic type non-Euclidean geometry, the point at infinity was introduced by the stereoprojection of the Euclidean plane to the sphere and the concept is a standard one in complex analysis around over one hundred years. And then we have considered as $1/0= \infty$ (\cite{ahlfors}). However, surprisingly enough, the division by zero means that $1/0=0$.

\section{}

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 a constant $G$. Of course,

\begin{equation}

\lim_{r \to +0} F =\infty,

\end{equation}

however, we obtain the important interpretation:

\begin{equation}

F = 0 = G \frac{m_1 m_2}{0}.

\end{equation}

Of course, here, we can consider the above interpretation for the mathematical formula (4.1) as the new interpretation (4.3). We can find many physical formulas with the division by zero.

\section{}

In complex analysis, linear fractional functions

$$

W = \frac{az + b}{cz + d}, \quad ad -bc \ne 0,

$$

map the extended complex plane onto the extended complex plane containing the point at infinity, one to one, conformally, beautifully. This beautiful property is changed as the beautiful formula that linear fractional functions map the whole complex plane onto the whole complex plane, one to one, however, at one point of the singular point, the linear fractional functions have strong discontinuity.

The division by zero excludes the infinity from the numbers.

\section{}

We did, essentially, not consider the division by zero, and so the property of the division by zero; that is, at the isolated singular points of analytic functions, to consider the analytic functions is a new mathematics and a new research topics, essentially.

\section{}

The impact to complex analysis is unclear, we, however, obtain a typical new theorem:

\medskip

{\bf Theorem :} {\it Any analytic function takes a definite value at an isolated singular point }{\bf with a natural meaning.} The definite value is given by the first coefficient of the regular part in the Laurent expansion around the isolated singular point.

\medskip

This will be the fundamental theorem on the division by zero in Complex Analysis and we have many applications for the Sato hyperfunction theory, generating functions theory and singular integral theory (\cite{mst}).

\section{}

In particular, the divison by zero gives new interpretations on the finite part of Hadamard

for singular integrals and the Cauchy's principal values. The division by zero will represent discontinuity properties on the universe.

\section{}

Even for middle high school students, the division by zero may be accepted as the beautiful result with great pleasures:

For the elementary function

$$

y = f(x) = \frac{1}{x},　

$$

we have $f(0) = 0$; that is, $1/0=0$.　

\section{}

We can introduce the division by zero $100/0=0,0/0=0$ with the simple and natural definition for the division by the Hiroshi Michiwachi method (\cite{ann}) in the elementary school. The division by zero will request the change of all the related books and scientific books.

\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 introduction 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

\medskip

{\bf Announcement 179} (2014.10.22):　 Division by zero is clear as z/0=0 and it is fundamental in mathematics

\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{mst}

H. Michiwaki, S. Saitoh, and M. Takagi,

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, Tokyo Journal of Mathematics (in press).

\bibitem{ann}

Announcement 179: Division by zero is clear as z/0=0 and it is fundamental in mathematics,

Institute of Reproducing Kernels, 2014.10.22.

\end{thebibliography}

\end{document}

You are right Mr. Fethi Bin Muhammad Belgacem, many people have been trying very hard with many new “ ideas and understandings” for more than 2500 years within the present traditional finite—infinite theory system for solving “the infinite related problems”. But, our science history proved that nothing can be done because “everything is so perfect” in present traditional finite—infinite theory system.

It is also true that Zeno’s Paradox of Achilles--Turtle Race is generally accepted as a perfect “defects discloser” within the present traditional finite—infinite theory system with its foundation of “potential infinite” and “actual infinite”. We can forget Zeno’s Paradox, but we can not forget Harmonious Series Paradox-------the “strict mathematical proven” modern version of Zeno’s Paradox of Achilles--Turtle Race.

Sincerer yours, Geng

Dear Mr. Fethi Bin Muhammad Belgacem, thank you for the very insightful opinion.

Sometimes we know something is “improper” in present traditional finite—infinite theory system, but we have to say “it is the only proper way to do it”, this is one of the reasons why we sometimes call science “religion”.

Looking back into our human science history, we see improvements, evolutions in many branches of science-------an important metabolism function of our human science going along with human evolution and more and more closer to nature.

Though we sometimes call science “religion” but this “religion” is different from other religions.

Sincerer yours, Geng

Dear Geng, salaam and Thank you for your kindness. Not only do I agree in principle but concur as well.  With the infinite state of knowledge, the more you know the more you dont as it opens up new tools and newer questions...!  So Matematicians and Scientists have devised various ways to value advancement (technically like complexity and NP completeness etc), and unformally like giving an estimate as to when a certain problem  will be deterministically ( just a matter of time) or stocahstically ( may or may not) be resolved.  

Speaking of Religions, in the Islamic Holy book  the Qur'an, it says, 

وَيَسْأَلُونَكَ عَنِ الرُّوحِ ۖ قُلِ الرُّوحُ مِنْ أَمْرِ رَبِّي وَمَا أُوتِيتُم مِّنَ الْعِلْمِ إِلَّا قَلِيلًا ﴿٨٥﴾ 

http://tanzil.net/#17:85

meaning

And they ask you, [O Muhammad], about the soul. Say, "The soul is of the affair of my Lord. And mankind have not been given of knowledge except a little." (85)

http://tanzil.net/#trans/en.sahih/17:85

Kindest regards to you and all readers.