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The most Useful Algebra and a New Calculus Discovered to Reshape the Giant Subject ‘Mathematics’
Region Mathematics is a new direction in Mathematics, providing a new shape to the existing super giant subject ‘Mathematics’ whatever volume of literature developed so far since the stone age of earth.
By Ranjit Biswas
Jun. 3, 2016

By ‘Mathematics’ we mean here all the branches of it. But this new subject Region Mathematics is initiated with a new Algebra called by ‘Region Algebra’ and with a new Calculus called by ‘Region Calculus’, as the beginning of a new era. However, a series of research and development work of this new subject Region Mathematics will follow with time to cater to the demand of all branches of science and engineering. Presently this new subject is at its infant stage. But this new subject will certainly cause a huge multiplicity to the volume of mathematics in very near future.

The existing huge volume of mathematics is just a part of ‘Region Mathematics’; although apparently it seems that the existing volume of mathematics has been almost sufficiently supporting the demands of the world mathematician, world scientists, world statisticians, and world engineers in their all type of mathematical works and computations. The work of ‘Region Algebra’ may apparently seem to be too simple at the first readings, because it is fact that it is simple and of very fundamental nature. Because of its very simple initial nature, the readers may have to take patience to read the materials till end, even if some of the theories/propositions happen to be unacceptable or debatable initially.

It is unearthed that none of the existing algebraic system alone like : group, ring, module, field, linear space, algebra over a field, associative algebra over a field, division algebra, etc. in general, can issue license to the world mathematicians/scientists to work smoothly and fluently; because it is observed that many of the frequently used elementary operations, rules, laws, formulas, identities, etc. are not valid in any of these existing algebraic system alone by virtue of their respective definitions and respective properties owned. This observation is surprising and unbelievable at this century where Algebra has taken almost a saturation stage, but it is fact. This important fact was hidden so far to the algebraists, and has been now unearthed in Region Algebra. With this philosophy in mind, it can be realized that all the existing classical algebraic systems are much weaker than the newly introduced algebraic system ‘region’ in terms of application potential and in-built individual caliber.

It is then observed that the existing rich calculus of Newton and Leibnitz is a particular case of Region Calculus. It is claimed that there could be requirements of new generalized types of calculus to see the universe more precisely, more appropriately, and with more confidence. If it is so then there must be a path consisting of genuine steps which will ensure whether in a certain environment a new calculus can be developed or not. If it can be developed, then the question arises on ‘how to develop’.

Calculus is one of the most important discoveries in Mathematics. The ‘very particular nature’ of Newton and Leibnitz calculus is due to the two main facts : (i) the distance between the two points x and (x+dx) is measured using a very particular metric, and (ii) the basic platform of this calculus is a very particular set which is the set R of real numbers.

It is unearthed that these two facts are two big constraints which confine the application freedom of the world mathematician, world scientists, world statisticians, and world engineers, in particular while dealing with very complex problems. Both these constraints are unlocked by introducing a new area called by ‘Region Calculus’, of which the Newton and Leibnitz calculus is just a special case.

Authors
Ranjit Biswas, Department of Computer Science & Engineering, Faculty of Engineering & Technology, JamiaHamdard University, Hamdard Nagar, New Delhi, India. (ranjitbiswas@yahoo.com)

Paper link:
http://www.sciencepublishinggroup.com/journal/paperinfo?journalid=141&doi=10.11648/j.pamj.20160503.11

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