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                            MATHEMATICS AND MATHEMATICAL LOGIC: NEW RESEARCH
MATHEMATICS AND MATHEMATICAL LOGIC: NEW RESEARCH
CONTENTS
PREFACE
ON NUMBER’S NATURE
	ABSTRACT
	INTRODUCTION
	ETYMOLOGY OF THE ‘NUMBER’ AND ‘ARCHETYPE’
	PART I: PYTHAGORAS AND HIS DESCENDANTS ON NUMBER
		1. Pythagoras View about Number
		2. Pythagoras Vindication in Contemporary Mathematics
	PART II: NUMBER IN CONTEMPORARY MATHEMATICS
		3. Number and Standard Contemporary Mathematics
		4. Number and Logic
		5. Number and Philosophy: From Frege to Benacerraf
		6. Number and Neurophysiological and Social Aspects
		7. Number and Category Theory
			7.1. First Definition of a Category
			7.2. Existence of a ‘Natural Numbers Object’
			7.3. ‘Natural Numbers Object’ in ETCS (Elementary Theory of the Category of Sets)
		8. Number and Non-Standard View
		9. Number and Fuzziness
	PART III: JUNG AND THE CONCEPT OF ARCHETYPE
		10. ‘Chthonic’ and ‘Celestial’ Mathematics
		11. Jung’s - Pauli’s General Hypothesis of Archetypes
		12. Number as Archetype
		13. Jung and Cantor - Gödel
			13.1. Cantor’s Continuum Hypothesis
			13.2. Gödel’s Incompleteness Theorems
			13.3. Jung and Cantor’s Continuum Hypothesis
			13.4. Jung and Gödel’s Incompleteness Theorems
		14. Number and Jung’s Psychology
	PART IV: NUMBER, MONOID AND ARCHETYPE
		15. 2-Categories; Monad, Monoid, Monoidal Category, Monoid Object; theDynamical System as Functor
			15.1. 2-Categories
			15.2. Monad, Monoid, Monoidal Category, Monoid Object
			15.3. Dynamical System as Functor
				Example
		16. The Archetype of Number as Monoid and its Interpretation as the Setof Numbers
		17. A Holistic View of Mathematics
		18. Final View
	EPILOGUE
	REFERENCES
THE APPLICATION OF FUZZY LINEAR PROGRAMMING IN ENGINEERING
	ABSTRACT
	INTRODUCTION
	STRUCTURE OF THE MODEL: THEORY BACKGROUND
	PYROLYSIS OF ETHANE AS STUDY CASE
	CONCLUSION
	REFERENCES
A NEW CONSENSUS SCHEME FOR MULTICRITERIA GROUP DECISION MAKING UNDER LINGUISTIC ASSESSMENTS
	ABSTRACT
	1. INTRODUCTION
	2. MULTIPERSON MULTICRITERIA DECISION PROBLEM
	3. LINGUISTIC HIERARCHY
	4. MULTIPERSON AGGREGATION MODEL
	4. CONSENSUS SCHEME
		Guidance Procedure
	5. FUZZY DECISION APPROACHES BASED ON FUZZY PREFERENCE RELATION MODELING
	6. APPLICATION EXAMPLE
	CONCLUSION
	ACKNOWLEDGEMENTS
	REFERENCES
THE MATHEMATICAL BASIS OF PERIODICITY IN ATOMIC AND MOLECULAR SPECTROSCOPY
	INTRODUCTION
	COMBINATORIAL PERIODICITY IN MOLECULARELECTRONIC AND ATOMIC SPECTROSCOPY
	COMBINATORIAL PERIODICITY IN MOLECULAR AND NMR SPECTROSCOPIES
	PERIODICITY OF DOUBLE GROUPS AND ELECTRONIC STATES
	ACKNOWLEDGEMENT
	REFERENCES
SECOND-ORDER DEFINABILITY IN A MODEL
	Abstract
	1. Introduction
	2. A Normal Form Theorem for Formulas
	3. The Characterization of Definability in A
	4. Definable Pairing Functions
	5. Conclusion
	References
ALGEBRAIC TOPICS ON DISCRETE MATHEMATICS*
	Abstract
	1. Elemental Concepts
		1.1. Equivalence Relations
		1.2. Groups
		1.3. Permutation groups
	2. Polya’s Enumeration Method
		2.1. Action of a Group over a Set
		2.2. Orbits
		2.3. Theorem of Burnside
		2.4. Generating Function for a Group of Permutations
		2.5. Polya’s Enumeration Method
	3. Finite Fields and Latin Squares
		3.1. Polynomial Rings
		3.2. Finite Fields and Polynomials
		3.3. Latin Squares
			3.3.1. Orthogonal Array Representation
			3.3.2. Orthogonal Latin Squares
	4. Finite Geometry and Block’s Design
		4.1. Finite Affine Plane
		4.2. Finite Projective Plane
		4.3. Incomplete Block Designs
	5. Applications
		5.1. Discrete Modeling in Foundation Design
			5.1.1. General Definitions
			5.1.2. Symmetric Group Solution
		5.2. Enumeration of RNA Graphs Using Graph Theory
			5.2.1. Graphical Representation of RNA Structures
			5.2.2. Planar Tree Graph Rules
			5.2.3. Planar Dual Graph Rules
			5.2.4. Enumeration of RNA Graphs
				Enumeration of RNA tree graphs
				Enumeration of RNA dual graphs
			5.2.5. Conclusion
		5.3. Octonions and Transylvania Lottery
		5.4. Discrete Mathematics with Mathematica
			5.4.1. Basic Combinatorial
			5.4.2. Graph Theory
			5.4.3. Pölya Theory
	References
MATHEMATICAL MODELS OF QOS MANAGEMENTF OR COMMUNICATION NETWORKS*
	1 Abstract
	2 Introduction
	3 Network Architecture
	4 QoS on All-IP Networks
		4.1 Quality of Service
		4.2 Resource Allocation & Routing with QoS Guarantees
	5 Problem Definition
	6 Network Requirements
	7 Bandwidth Allocation Policies
		7.1 Maximum Throughput Allocation
		7.2 Arctan-Utility Maximizing Allocation
		7.3 Minimal Potential Delay Allocation
		7.4 Max-Min Fair Allocation
		7.5 Proportionally Fair Allocation
		7.6 (w, )-Proportionally Fair Allocation
		7.7 "-Proportionally Fair Allocation
	8 Implication
	9 Blocking Probability with Predetermined Optimal Solutions
	10 Conclusions and Future Research
		Acknowledgments.
	References
REVERSIBLE LOGIC
	Abstract
	1. Introduction
	2. Group Theory
	3. Control Gates
	4. Cosets
	5. Double Cosets
	6. Double Cosets Again
		Algorithm A
		Algorithm B
	7. Garbage Bits
	8. Experimental Prototypes
	9. Conclusion
		Acknowledgements
	Appendix A: A Remarkable Theorem from Combinatorics
	Appendix B: Optimal Syntheses
	References
IMAGINARY CUBIC OSCILLATOR AND ITS SQUARE-WELL APPROXIMATIONS IN X –AND P– REPRESENTATION
	ABSTRACT
	ACKNOWLEDGEMENTS
	1 Introduction
	2 Models in coordinate representation
	3 Matching conditions at x = ¼
	4 Energies
	5 Wave functions in the weak coupling regime
	6 Wave functions in the strong coupling regime
	7 Transition to the momentum representation
	8 Piecewise constant approximate kinetic energy
	9 The Z¡dependence of the spectrum
	10 Energies in the square-well approximation
	11 Wave functions and their zeros
	12 Outlook
	References
INDEX
                        
Document Text Contents
Page 141

In: Mathematics and Mathematical Logic: New Research
Editors: Peter Milosav and Irene Ercegovaca

ISBN 978-1-60692-862-2
c© 2010Nova Science Publishers, Inc.

Chapter 6

ALGEBRAIC TOPICS ON DISCRETE
M ATHEMATICS *

Gloria Gutiérrez Barranco, Javier Martínez,
Salvador Merino and Francisco J. Rodríguez

Departamento de Matemática Aplicada.
E.T.S.I. Telecomunicación. Universidad de Málaga.

Campus de Teatinos. 29071 Málaga. (Spain)

Abstract

Many applications of discrete mathematics for science, medicine, industry and
engineering are carried out using algebraic methods, such as group theory, polynomial
rings or finite fields.

This chapter is intended as a survey of the main algebraic topics used for devel-
oping methods in ambit of combinatorial theory. Pólya’s enumeration method, Latin
squares, patterns design or block design are examples of this usage. Others applica-
tions can be seen in Section 5, about foundation design, RNA patterns or octonions.

1. Elemental Concepts

1.1. Equivalence Relations

A binary relation on a setA is a subsetR ⊆ A×A. If (a, b) ∈ R, we say that the element
a is related tob and we will denoteaRb. Equivalence relations are binary relations that
formalize the natural process of classification of the elements in a set. Formally:

Definition 1.1. A binary relationR on a setA is an equivalence relationif it is reflexive
(aRa for all a ∈ A), symmetric (aRb impliesbRa for all a, b ∈ A), and transitive (aRb
andbRc implyaRc for all a, b, c ∈ A).

Equivalence relations are usually denoted by∼ or≡.

Definition 1.2. LetA be a set,∼ be an equivalence relation onA anda ∈ A. Theequiva-
lence classof a, denoted by[a], is the set

[a] = {b ∈ A | a ∼ b}

Page 142

130 Gloria Gutiérrez Barranco, Javier Martínez, Salvador Merino et al.

Thequotient set, denotedA/∼, is the set of all the equivalence classes ofA, that is,A/∼ ={
[a] | a ∈ A

}
.

Definition 1.3. LetA be a set. ApartitionofA is a family of subsets, namedparts, that are
disjoint by pairs and whose union isA.

Obviously the quotient setA/∼ is a partition ofA. Conversely, if a partitionP is given
onA, we can define an equivalence relation onA asx ∼ y iff there exists a part ofP which
contains bothx andy. “Equivalence relation” and “partition” are thus essentially the same
notion.

1.2. Groups

The idea of operating the elements of a set arises in a natural way. Thus, for example, with
numbers it is possible to add, to multiply, etc; with sets it is possible to join, to meet, etc.
Binary operations are the keystone of the algebraic structures studied in abstract algebra1.

Definition 1.4. A magmais a pair (A, ∗) whereA is a set and∗ is a binary operation on
A, that is, a function∗ : A× A→ A.

Binary operations are often written using infix notation such asa ∗ b, a+ b, a · b or a � b
rather than by functional notation of the formf(a, b). Sometimes they are even written just
by juxtaposition:ab.

Example 1.5. The sum of natural numbers is a binary operation. Nevertheless the sub-
traction of natural elements is not a binary operation because for example if we consider
2, 3 ∈ N it verifies that2 − 3 = −1 6∈ N.

The binary operations on a setA, do not have to verify any property in particular.
However, the fact that these binary operations verify certain properties is going to play a
very outstanding role.

Definition 1.6. Let∗ be a binary operation on a setA. It is said that:

1. ∗ is associativeif it satisfies thata ∗ (b ∗ c) = (a ∗ b) ∗ c for all a, b, c ∈ A.

2. ∗ is commutativeif it satisfies thata ∗ b = b ∗ a for all a, b ∈ A

3. e ∈ A is anidentity elementor neutral elementfor ∗ if it satisfies thata∗e = e∗a = a
for all a ∈ A.

4. a′ ∈ A is an inverse elementof a ∈ A if it satisfies that a ∗ a′ = a′ ∗ a = e being
e ∈ A the identity element.

Many binary operations of interest in the algebraic environment are commutative or
associative. Many of them also have identity elements and inverse elements.

Typical examples of associative and commutative binary operations are the addition
(+) and multiplication(·) of numbers. The product of matrixes and the composition of
functions are associative but not commutative. The subtraction(−) and the division(/) of
numbers are neither associative nor commutative.

The following result is an immediate consequence of the definition.
1All basic results about algebra can be found in several algebra books like [5].

Page 282

Index 270

tangible, 12
TCP, 191, 199
teaching, 3, 26
technological developments, ix
technology, ix, 203, 204, 227, 228, 230, 231, 232,

240
telecommunication, ix, 179, 180, 183, 198, 199,

203
telecommunication networks, 183, 198, 199
telephone, 234
telephony, 182, 183, 184
temperature, 63, 203
Tesla, 13
tetrad, 40
textbooks, 207, 255
Theory of Everything, 39
thinking, 2, 18, 27, 36
third order, x, 243, 249, 255
Thomson, 53
threat, 9
threshold, 78, 229, 231, 232, 252, 253, 254, 255,

256
tics, 199
time, ix, 2, 4, 5, 11, 13, 16, 17, 22, 23, 25, 30, 32,

35, 36, 40, 46, 50, 57, 154, 174, 179, 180, 181,
182, 183, 184, 190, 195, 196, 197, 198, 224,
244

tolerance, 58
topological, vii, 43, 170
topology, 170, 180, 183, 186, 188, 231
Topos, 19, 55
total utility, 190
traction, 130
traffic, 14, 180, 181, 182, 183, 184, 185, 186,

196, 197, 198, 199, 200, 241
traffic flow, 182, 183, 184
training, 16, 18
traits, 34
trans, 17, 31, 53, 54, 195
transfer, 62, 180, 181, 184, 186, 187
transformation, 70, 71, 257
transformations, 24, 41, 42
transistor, 227, 229, 230, 231
transistors, 227, 229
transition, 57, 58, 70, 248, 257
transitions, 96
translation, 70
transmission, 15, 164, 165, 181, 183, 191, 195,

197, 227
transparent, 30, 251
transport, 183
transportation, 47
Transylvania, 170

trees, 31, 166, 167, 168, 169, 170
Trinidad and Tobago, 57
two-dimensional, 41, 43, 108
two-dimensional space, 43

U

UMTS, 182, 200
uncertainty, viii, 2, 23, 28, 34, 58, 67, 68, 69, 71,

181, 198
uniform, 8
universal grammar, 26
universality, 16, 238
universe, 31, 68
unpredictability, 184, 185
utilitarianism, 51

V

vacuum, 229
validity, 5, 38, 68, 186
values, 15, 57, 58, 63, 72, 73, 77, 93, 97, 99, 108,

111, 112, 187, 204, 210, 251
variability, 161
variables, 33, 58, 63, 77, 115, 116, 118, 119, 120,

121, 122, 123, 126, 140, 161, 188, 189, 194,
210, 216, 224, 227, 231

variation, 180, 190, 246
vector, 43, 73, 74, 173, 187, 193, 194
vehicles, 154
vertebrates, 18
virus, 170
visible, 26, 52, 256
vision, 41
visuospatial, 12
vocabulary, 115, 116
voice, 5, 182, 184
voids, 10
VoIP, 182

W

wealth, 15, 207
web, 182, 184
wells, 252, 255, 257
West Indies, 57
Western societies, 17
winning, 174
wireless, ix, 179, 184, 198, 201
wireless networks, ix, 184, 201
wires, 15, 221, 234
wisdom, vii, 1, 3
withdrawal, 162
women, 15
wood, 177

Page 283

Index 271

World Wide Web, 175

Y

yield, 88, 105, 112, 150, 208, 220, 227, 236

Z

zeitgeist, 51
Zen, 50

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