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Some Innovations in Mathematics, Discrete in Nature

thathvamasi. ahambrahmasmi. Some Innovations in Mathematics, Discrete in Nature. Dr. K. K. Velukutty, Director of MCA, STC, Pollachi Director, SAHITI, COIMBATORE AND PALGHAT. thathvamasi. Agenda. ahambrahmasmi. Mathematics Origin and evolution of discrete mathematics

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Some Innovations in Mathematics, Discrete in Nature

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  1. thathvamasi ahambrahmasmi Some Innovations in Mathematics, Discrete in Nature Dr. K. K. Velukutty,Director of MCA, STC, Pollachi Director, SAHITI, COIMBATORE AND PALGHAT

  2. thathvamasi Agenda ahambrahmasmi • Mathematics • Origin and evolution of discrete mathematics • Discrete derivative • Mate and Matoid • Topograph • The future

  3. thathvamasi Mathematics- approaches ahambrahmasmi For a few centuries before, Mathematics stood and withstood for aesthetic beauty and perfection through emotional contemplation, a philosophical transaction of the mind. According to Currant and Robbins: “ Mathematics is an expression of the Human mind reflects the active will, the contemplative reason and the desire for aesthetic perfection”.

  4. thathvamasi Mathematics- approaches… ahambrahmasmi The Last century found a change. • Practicability and applicability in day to day affairs of mankind. • Mathematics is brought back to earth from heaven, indeed, it is a rebirth of Discrete Mathematics.

  5. thathvamasi Mathematics –a new definition ahambrahmasmi Mathematics is a device to facilitate the understanding of science, the Art of Reason.

  6. thathvamasi Mathematicsis decomposed into three: Continuous, Discrete and Finite ahambrahmasmi • Continuous Mathematics (Descartes, Newton & Leibnitz) anticipated the great Renaissance of science • Discrete mathematics ( Ruark, Heisenberg, Von Neumann and Margenau ) anticipated the present great IT revolution (Renaissance) • Quantum Mechanics is the forerunner of Discrete Mathematics

  7. thathvamasi Mathematics - axioms ahambrahmasmi Philosophers axioms of continuum: 1. No two magnitudes of the same kind are consecutive 2. There is no least magnitude 3. There is no greatest magnitude

  8. thathvamasi Philosopher’s axioms of Finitude ahambrahmasmi • There exist consecutive magnitudes everywhere • There is a magnitude smaller than any other of the kind • There is a magnitude greater than any other of the kind.

  9. thathvamasi Axioms of discretum ahambrahmasmi A. There exist consecutive magnitudes everywhere B. There exist no least magnitude C. There exist no greatest magnitude

  10. thathvamasi Formulation of new mathematics ahambrahmasmi Combining a , b, and C from the above sets of axioms, a new discrete space is evolved. This space spans from a finite point to infinity

  11. thathvamasi Observation ! ahambrahmasmi • “ The wonder is Q belongs to continuum. In modern terms Q is dense; Q is countable with usual integers and thus Q belongs to discretus”. - Proc UGCSNS on DA (22-24, 3, 99) • Proof follows:

  12. thathvamasi Q is continuous ahambrahmasmi • Metric Axioms: An infinite set is a discrete set if the distance between every pair of elements is finite. • If the set is not metric, one to one correspondence between the set and z+ makes the set discrete. • Any countable set may be treated as discrete if either it does not have a metric or the metric of the set is the same as usual metric of z+ • Therefore, Q cannot be discrete, but continuous But we accept Q as discrete especially due to the hypothesis of rational description for physical problems.

  13. thathvamasi Agenda ahambrahmasmi • Mathematics • Origin and evolution of discrete mathematics • Discrete derivative • Mate and matoid • Topograph • The future

  14. thathvamasi Origin and evolution of discrete mathematics ahambrahmasmi • Avayavah is the terminology for derivative (discrete) used by Aryabhatta • Avayavah is nothing but [ f (x+h) – f (x) ] / h or [f (x) – f (qx) ] / (1-q) x where h, q constants – the present day notions in discrete analysis • Aryabatta constructed a lattice to derive this derivative • The whole calculation of ancient Indian astronomy, geometry and Vastu were connected to Avayavah ( first difference – discrete derivative) of certain functions: sine, tangent, …

  15. thathvamasi Aryabatta and π ahambrahmasmi • Aryabhatta considered a circle of circumference 21600 units. The corresponding radius is calculated. This is denoted by ‘Ma’. Ma=3537.738 • Ma is the parameter used in Aryabhattian difference calculus corresponding to π. • Aryabhatta knew that the ratio of circumference to the radius of a circle is a constant. He calculated value of π from the above relation.

  16. thathvamasi Aryabatta – new informations ahambrahmasmi • Aryabhatta is identified as Vararuchi of Kerala Vikramadithya. • Every member of Panthirukulam is a mathematician. • The known 7 disciples of Aryabhatta are identified from panthirukulam. • Aryabhatta is the father of [Indian] Trigonometry. • He introduced and popularized sine function.

  17. thathvamasi Agenda ahambrahmasmi • Mathematics • Origin and evolution of discrete mathematics • Discrete derivative • Mate and matoid • Topograph • The future

  18. thathvamasi Discrete derivative on the real line ahambrahmasmi On Z discrete derivative D1 f = [ f (n) – f (n-1)] / 1 D2 f= [ f(n+1)-f(n-1)] / 2 Calculus of finite differences D3 f=[ f (n+1)- f (n) ] / 1 d1 f = [ f (x) – f (qx) ] / (1-q ) x d2 f = [ f (q-1x)-f(qx) ] / (q-1-q) x q – basic theory d3 f = [f(q-1x) – f (x)] / (q-1-1) x In general if { xn, n € Z } is the sequence of discrete space, d f = [ f (xn)- f (xn-1) ] / (xn-xn-1) or similar ones

  19. thathvamasi Discrete Derivative on the Complex ahambrahmasmi • Unlike the classical case, derivative in two directions only are made equal. • Three directions are also being attempted. Triads (4) 9C2 equalities are possible Tetrad (1) = 9 That much derivatives ! Unit Rectangle (4) Still more are there Monodiffricity of the first and second type and pre-holomorphicity

  20. thathvamasi Agenda ahambrahmasmi • Mathematics • Origin and evolution of discrete mathematics • Discrete derivative • Mate and matoid • Topograph • The future

  21. thathvamasi Mate and Matoid ahambrahmasmi Closure axiom: • In a set X, for a,b € X, a*b is a unique element in X • It is a mapping (function: X2 X ) Mate axiom: • In a set X, for a,b € X, a*b is none, one or many elements inside or outside X • It is a relation: X2 Y X • This composition is mate and a set with a mate is matoid

  22. thathvamasi MATE is close to nature ahambrahmasmi Population dynamics • Ti is a population Ti+1 is the next generation got by a single mate between every of population Biological studies. • Species - members of different species will not mate at all. Enumeration is a powerful method of discrete mathematics. Enumeration will work in such models well.

  23. thathvamasi Agenda ahambrahmasmi • Mathematics • Origin and evolution of discrete mathematics • Discrete derivative • Mate and matoid • Topograph • The future

  24. thathvamasi Topograph ahambrahmasmi • A set X is a graphoid if neighbourhood structure (interior, boundary and exterior) is assigned to every subset of the set. • A regular normal symmetric, fully ordered graphoid with union intersection property is a topograph. Topograph is in between graphoid and topology.

  25. thathvamasi Topograph and discrete models ahambrahmasmi • Topograph suits discrete models; It enriches integers and thus it enriches discrete mathematics. • It is envisaged that topograph instead of topology will suit and fit discrete circumstances of nature.

  26. thathvamasi Agenda ahambrahmasmi • Mathematics • Origin and evolution of discrete mathematics • Discrete derivative • Mate and matoid • Topograph • The future

  27. thathvamasi The future ahambrahmasmi The latest desire of the discrete analyst that the discrete analysis should have ways and means of its own to construct the analysis not depending on the continuous analysis, is stressed and made an issue of progress one step ahead. It is envisaged that this initiative will stand long and may be found established fully grown in the near future.

  28. thathvamasi The future … ahambrahmasmi “ At present research in the theory of analyticity in the discrete is steadily gaining recognition…… In fact, one may prophesize the advent of the day when the direct application of discrete analyticity will replace the discretising of many of the continuous models in classical analysis”. Berzsenyi,

  29. thathvamasi References ahambrahmasmi • K. K. Velukutty, Discrete Analysis in a Nutshell, Sahithi, 2001 • K.K. Velukutty, Some Research Problems in Discrete Analysis, Sahithi, 2003 • K. K. Velukutty, Geometrical and Topological Aspects in Discrete Analysis, Sahithi, 2003

  30. Any questions ? Thank you thathvamasi ahambrahmasmi

  31. thathvamasi ahambrahmasmi

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