Some Innovations in Some Innovations in Mathematics, Mathematics,

Discrete in NatureDiscrete in Nature

Some Innovations in Some Innovations in Mathematics, Mathematics,

Discrete in NatureDiscrete in Nature

Dr. K. K. Velukutty,Director of MCA, STC, Pollachi

Director, SAHITI, COIMBATORE AND PALGHAT

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AgendaAgenda

• Mathematics• Origin and evolution of discrete

mathematics• Discrete derivative• Mate and Matoid• Topograph• The future

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Mathematics- approachesFor 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”.

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Mathematics- approaches…

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.

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Mathematics –a new definition

Mathematics is a device to facilitate the understanding of science, the Art of Reason.

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Mathematics is decomposed into three:

Continuous, Discrete and Finite• 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

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Mathematics - axioms

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

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Philosopher’s axioms of Finitude

a. There exist consecutive magnitudes everywhere

b. There is a magnitude smaller than any other of the kind

c. There is a magnitude greater than any other of the kind.

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Axioms of discretum

A. There exist consecutive magnitudes everywhere

B. There exist no least magnitude

C. There exist no greatest magnitude

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Formulation of new mathematics

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

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Observation !

• “ 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:

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Q is continuous• 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.

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Agenda

• Mathematics

• Origin and evolution of discrete mathematics

• Discrete derivative

• Mate and matoid

• Topograph

• The future

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Origin and evolution of discrete mathematics

• 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, …

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Aryabatta and π• 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.

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Aryabatta – new informations

• 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.

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AgendaAgenda

• Mathematics• Origin and evolution of discrete

mathematics• Discrete derivative• Mate and matoid• Topograph• The future

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Discrete derivative on the real line

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

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Discrete Derivative on the Complex • 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

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AgendaAgenda

• Mathematics• Origin and evolution of discrete

mathematics• Discrete derivative• Mate and matoid• Topograph• The future

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Mate and Matoid

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

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MATE is close to nature

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.

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AgendaAgenda

• Mathematics• Origin and evolution of discrete

mathematics• Discrete derivative• Mate and matoid• Topograph• The future

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TopographTopograph

• 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.

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Topograph and discrete modelsTopograph and discrete models

• 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.

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AgendaAgenda

• Mathematics• Origin and evolution of discrete

mathematics• Discrete derivative• Mate and matoid• Topograph• The future

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The future The future

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.

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The future …“ 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,

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References

1. K. K. Velukutty, Discrete Analysis in a Nutshell, Sahithi, 2001

2. K.K. Velukutty, Some Research Problems in Discrete Analysis, Sahithi, 2003

3. K. K. Velukutty, Geometrical and Topological Aspects in Discrete Analysis, Sahithi, 2003

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Any questions ?Any questions ?

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