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Some Innovations in
Mathematics,
Discrete in Nature
Dr. K. K. Velukutty,
Director of MCA, STC, Pollachi
Director, SAHITI, COIMBATORE AND PALGHAT
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Agenda
• Mathematics
• Origin and evolution of discrete
mathematics
• Discrete derivative
• Mate and Matoid
• Topograph
• The future
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Mathematics- approaches
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”.
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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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Agenda
• 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
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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
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• Unlike the classical case, derivative in two directions only are made equal.
• Three directions are also being attempted.
9C2 equalities are possible
Triads (4)
Tetrad (1)
Unit Rectangle (4)
=9
That much derivatives !
Still more are there
Monodiffricity of the first and second type and pre-holomorphicity
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Agenda
• Mathematics
• Origin and evolution of discrete
mathematics
• Discrete derivative
• Mate and matoid
• Topograph
• The future
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Mate and Matoid
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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
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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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Agenda
• Mathematics
• Origin and evolution of discrete
mathematics
• Discrete derivative
• Mate and matoid
• Topograph
• The future
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Topograph
• 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 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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Agenda
• Mathematics
• Origin and evolution of discrete
mathematics
• Discrete derivative
• Mate and matoid
• Topograph
• The future
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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
Any questions ?
Thank you
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