Welcome to my Computer Technology Page
Concepts of Computer Technology
All the research papers here are the byproducts of teaching at
University of Pennsylvania, Drexel University,
Indian Institute of Science, and Jawaharlal Nehru University.
Simplifying the basic concepts of computer
science gave ample scope for investigating the foundations. To
view these papers, your browser should be
setup to read PDF files.
1. A Note on Inductive Probability
Karl Popper's contention that probabilistic induction is
impossible is contradicted with an example.
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2. A GraphTheoretic Proof of Arrow's Dictator Theorem
A simple and short proof of Dictator Theorem is given. Loosely
stated, the theorem says that democracies
are not possible.
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3. An Axiomatic Definition of Shannon's Entropy
A definition of information, which forms the basis of the current
information technology, is given in terms of
two axioms.
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4. A Graphic Illustration of RogersRamanujan Identities
Apart from illustrating the RRIdentities using graph theory, two
new forms of the identities are derived.
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5. Ackermann Functions and Transfinite Ordinals
An important part of Cantor's set theory, which forms the
foundations of mathematics, is the concept of
transfinite ordinals. A systematic way of writing the sequence of
ordinals is given.
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6. NuMachine and NuAlgebra
An intuitive alternative to Turing Machine, the machine which
defines computation, is given in terms of
a directed graph.
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7. Sentient Arithmetic and Godel's Theorems
Elementary Arithmetic of Godel is extended with three more
derivation rules, and the Incompleteness
Theorems are proved without using any metalanguage.
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8. Justification of the Continuum Hypothesis
Intuitive arguments are given to suggest that Continuum
Hypothesis should be accepted.
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9. Hall’s Theorem and Compound Matrices
The powerful concept of compound matrices is used to prove Hall’s
Theorem in a few steps.
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10. Matchings and Pfaffians
Pfaffians are used to enumerate the matchings in undirected graphs.
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11. Matrices with Elements from a Division Ring
The entire set of paths in a graph are enumerated by inverting a
matrix with elements from a division ring.
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12. Structured Labels and Directed Paths
The entire set of paths in a graph are enumerated by
inverting a matrix with elements from a field.
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13. BoyceCodd Normal Form Decomposition
An algorithm is given to decompose a relation into BCNF.
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14. Axiomatic Derivation of the Continuum Hypothesis
Continuum Hypothesis is derived from an axiom called Axiom of
Monotonicity.
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15. Real Set Theory
A set theory is defined in which Generalized Continuum Hypothesis
and Axiom of Choice are theorems.
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16. Intuitive Set Theory
A set theory is defined in which Skolem Paradox does not arise.
Also, there are no sets which are not Lebesgue measurable.
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17. Two Axioms to Extend ZermeloFraenkel Theory
Axiom of Monotonicity is used along with ZermeloFraenkel
set theory to derive Generalized Continuum Hypothesis.
Axiom of Fusion is used to investigate the cardinality
of the set of points in a unit interval.
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18. Generic Dependencies and Database Design
A generalization of the functional dependency called generic dependency
is defined. Use of generic dependencies in database design is illustrated.
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19. Visualization of Intuitive Set Theory
Intuitive Set Theory is defined as the theory we get when axioms of
Monotonicity and Fusion are added to ZF theory. Cardinals in the theory
are visualized using illustrations.
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20. Compound Matrices and Three Celebrated Theorems
Laplace's Theorem, BinetCauchy Theorem, and Jacobi's Theorem are
stated in terms of compound matrices and illustrated with examples.
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21. Shannon's Communication Channels and Word Spaces
Communication channels are represented by labelled graphs and analyzed
using the concept of word spaces. A slight generalization of the notion of
regular expressions is used to represent channel signals.
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22. Theory of Search Engines
Using PerronFrobenius theorem, four stochastic matrices are defined,
that can be used to rank the pages and hyperlinks in the Web.
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23. White Hole, Black Whole, and The Book
Physical and intellectual spaces are visualized making use of concepts
from Intuitive Set Theory. A book containing all the proofs of mathematics
is called The Book.
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24. The Essence of Intuitive Set Theory
The ideas which motivated the defintion of intuitive set theory
are explained. Only a passing acquaintance with the transfinite
cardinals of Cantor is assumed on the part of the reader.
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25. Generalized Continuum Hypothesis and the Method of Fusing
The method of fusing explains the basis for the formulation of the
axioms of monotonicity and fusion, the two axioms which define intuitive
set theory.
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26. Generalized Continuum Hypothesis and the Axiom of Combinatorial Sets
Axiom of Combinatorial Sets is defined and used to derive generalized continuum
hypothesis.
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27. Definition of Intuitive Set Theory
Two axioms which define intuitive set theory, Axiom of Combinatorial
Sets and Axiom of Infinitesimals, are stated. Generalized continuum
hypothesis is derived from the first axiom, and the infinitesimal is
visualized from using the latter.
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28. Arrow's Paradox and the Fractional Voting System
It is shown that the fractional voting system can be used to
cirumvent Arrow's paradox. The paradox states that fair elections
are not possible with the present voting systems.
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29. Two Open Problems and a Conjecture in Mathematical Logic
The open problems attempt to extend ZermeloFraenkel set theory and
the conjecture suggests an extension of Godel's incompleteness theorems.
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30. Twenty Problems of Information Technology
Out of the many significant and pressing problems of information technology,
twenty are listed.
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31. Teaching Generalized Pythagoras Theorem
Generalized Pythagoras theorem is discussed in terms of matrices.
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32. Teaching Generalized Continuum Hypothesis
Generalized Continuum Hypothesis is derived from a
simple axiom called Axiom of Combinatorial Sets.
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33. Logsets and ZF Theory
Logset, the inverse of the powerset operation, is introduced
into set theory.
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34. The Mathematical Universe in a Nutshell
The mathematical universe discussed here gives models of
possible structures our physical universe can have.
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35. Geometrical Equivalents of Goldbach Conjecture and Fermat Like Theorem
Five geometrical eqivalents of Goldbach conjecture are given,
calling one of them Fermat Like Theorem.
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36. Uncertainty Principle of Phantom Mechanics
This paper shows that not only the particles
of physics, but even a mathematical point in motion is governed
by the uncertainty principle.
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37. InformationTheoretic Equivalent of Riemann Hypothesis
Riemann Hypothesis is viewed as a statement about the capacity of a communication channel as defined by Shannon.
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38. Electrical Equivalent of Riemann Hypothesis
Riemann Hypothesis is viewed as a statement about the power dissipated in an electrical network.
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39. Derivation of Continuum Hypothesis from Axiom of Combinatorial Sets
Continuum Hypothesis is derived from an axiom called Axiom of Combinatorial Sets.
The derivation is simple enough to be understood by any novice, who has a passing
acquintance with cardinals of Cantor.
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40. Intuitive Set Theory: A Tutorial
Ackermann functions are used recursively to define the transfinite
cardinals of Cantor. Axiom of Monotonicity is defined and used to derive
Continuum Hypothesis. Axiom of Fusion is defined and used to
split the unit interval into infinitesimals.
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The web surfer is forewarned that many of the concepts discussed
here are new, and hence they should be accepted only after careful
consideration.
The web surfer is reminded that in ancient societies, the most gruesome
punishment was reserved for those who polluted the village water. In
today's global village the most gruesome punishment should be
reserved for those who pollute the Internet. On my part, I have
written all the papers here, as terse as possible, and all the
cencepts as clear as I could, even when there was a possibility of
their being wrong.
For a subset of the concepts above, concerned with the foundations
of computer science,
click here.
For a subset of the concepts above, concerned with the foundations
of mathematics,
click here.
K. K. Nambiar
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