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Some Innovations in Mathematics, Discrete in Nature

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... C from the above sets of axioms, a new discrete space is evolved. ... Mate axiom: In a set X, for a,b X, a*b is none, one or many elements inside or outside X ... – PowerPoint PPT presentation

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Title: Some Innovations in Mathematics, Discrete in Nature


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

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

3
Mathematics- approaches
thathvamasi
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
Mathematics- approaches
thathvamasi
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
Mathematics a new definition
thathvamasi
ahambrahmasmi
  • Mathematics is a device to facilitate the
    understanding of science, the Art of Reason.

6
Mathematics is decomposed into three
Continuous, Discrete and Finite
thathvamasi
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
Mathematics - axioms
thathvamasi
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
Philosophers axioms of Finitude
thathvamasi
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
Axioms of discretum
thathvamasi
ahambrahmasmi
  • A. There exist consecutive magnitudes everywhere
  • B. There exist no least magnitude
  • C. There exist no greatest magnitude

10
Formulation of new mathematics
thathvamasi
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
Observation !
thathvamasi
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
Q is continuous
thathvamasi
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
Agenda
thathvamasi
ahambrahmasmi
  • Mathematics
  • Origin and evolution of discrete mathematics
  • Discrete derivative
  • Mate and matoid
  • Topograph
  • The future

14
Origin and evolution of discrete mathematics
thathvamasi
ahambrahmasmi
  • 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,

15
Aryabatta and p
thathvamasi
ahambrahmasmi
  • 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.

16
Aryabatta new informations
thathvamasi
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
Agenda
thathvamasi
ahambrahmasmi
  • Mathematics
  • Origin and evolution of discrete mathematics
  • Discrete derivative
  • Mate and matoid
  • Topograph
  • The future

18
Discrete derivative on the real line
thathvamasi
ahambrahmasmi
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
19
Discrete Derivative on the Complex
thathvamasi
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
Agenda
thathvamasi
ahambrahmasmi
  • Mathematics
  • Origin and evolution of discrete mathematics
  • Discrete derivative
  • Mate and matoid
  • Topograph
  • The future

21
Mate and Matoid
thathvamasi
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

22
MATE is close to nature
thathvamasi
ahambrahmasmi
  • 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.
23
Agenda
thathvamasi
ahambrahmasmi
  • Mathematics
  • Origin and evolution of discrete mathematics
  • Discrete derivative
  • Mate and matoid
  • Topograph
  • The future

24
Topograph
thathvamasi
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
Topograph and discrete models
thathvamasi
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
Agenda
thathvamasi
ahambrahmasmi
  • Mathematics
  • Origin and evolution of discrete mathematics
  • Discrete derivative
  • Mate and matoid
  • Topograph
  • The future

27
The future
thathvamasi
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
The future
thathvamasi
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
References
thathvamasi
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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