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Some Innovations in Mathematics, Discrete in

Nature

thathvamasi

ahambrahmasmi

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

Pollachi - Director, SAHITI, COIMBATORE AND PALGHAT

Agenda

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ahambrahmasmi

- Mathematics
- Origin and evolution of discrete mathematics
- Discrete derivative
- Mate and Matoid
- Topograph
- The future

Mathematics- approaches

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- 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.

Mathematics- approaches

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- 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.

Mathematics a new definition

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- Mathematics is a device to facilitate the

understanding of science, the Art of Reason.

Mathematics is decomposed into three

Continuous, Discrete and Finite

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- 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

Mathematics - axioms

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- 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

Philosophers axioms of Finitude

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- 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.

Axioms of discretum

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- A. There exist consecutive magnitudes everywhere
- B. There exist no least magnitude
- C. There exist no greatest magnitude

Formulation of new mathematics

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- 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

Observation !

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- 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

Q is continuous

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- 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.

Agenda

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ahambrahmasmi

- Mathematics
- Origin and evolution of discrete mathematics
- Discrete derivative
- Mate and matoid
- Topograph
- The future

Origin and evolution of discrete mathematics

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- Avayavah is the terminology for derivative

(discrete) used by Aryabhatta - Avayavah is nothing but f (xh) 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,

Aryabatta and p

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- Aryabhatta considered a circle of circumference

21600 units. The corresponding radius is

calculated. This is denoted by Ma. Ma3537.738 - Ma is the parameter used in Aryabhattian

difference calculus corresponding to p. - Aryabhatta knew that the ratio of circumference

to the radius of a circle is a constant. He

calculated value of p from the above relation.

Aryabatta new informations

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- 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.

Agenda

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- Mathematics
- Origin and evolution of discrete mathematics
- Discrete derivative
- Mate and matoid
- Topograph
- The future

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(n1)-f(n-1) / 2

Calculus of finite differences D3 f f

(n1)- 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

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.

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

Agenda

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ahambrahmasmi

- Mathematics
- Origin and evolution of discrete mathematics
- Discrete derivative
- Mate and matoid
- Topograph
- The future

Mate and Matoid

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ahambrahmasmi

- Closure axiom
- In a set X, for a,b X, ab is a unique element

in X - It is a mapping (function X2 X )
- Mate axiom
- In a set X, for a,b X, ab 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

MATE is close to nature

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- Population dynamics
- Ti is a population Ti1 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.

Agenda

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ahambrahmasmi

- Mathematics
- Origin and evolution of discrete mathematics
- Discrete derivative
- Mate and matoid
- Topograph
- The future

Topograph

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- 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.

Topograph and discrete models

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- 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.

Agenda

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ahambrahmasmi

- Mathematics
- Origin and evolution of discrete mathematics
- Discrete derivative
- Mate and matoid
- Topograph
- The future

The future

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- 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.

The future

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- 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,

References

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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

Any questions ?

- Thank you

thathvamasi

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thathvamasi

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