Mathematics and Mathematical Logic

MATHEMATICS RESEARCH DEVELOPMENTS SERIES

MATHEMATICS AND MATHEMATICAL LOGIC: NEW RESEARCH

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MATHEMATICS RESEARCH DEVELOPMENTS SERIES Boundary Properties and Applications of the Differentiated Poisson Integral for Different Domains Sergo Topuria 2009. ISBN 978-1-60692-704-5 Quasi-Invariant and Pseudo-Differentiable Measures in Banach Spaces Sergey Ludkovsky 2009. ISBN 978-1-60692-734-2 Operator Splittings and their Applications Istvan Farago and Agnes Havasiy 2009. ISBN 978-1-60741-776-7

Measure of Non-Compactness for Integral Operators in Weighted Lebesgue Spaces Alexander Meskhi 2009. ISBN: 978-1-60692-886-8 Mathematics and Mathematical Logic: New Research Peter Milosav and Irene Ercegovaca (Editors) 2010. ISBN: 978-1-60692-862-2

MATHEMATICS RESEARCH DEVELOPMENTS SERIES

MATHEMATICS AND MATHEMATICAL LOGIC: NEW RESEARCH

PETER MILOSAV AND

IRENE ERCEGOVACA EDITORS

Nova Science Publishers, Inc. New York

Copyright © 2010 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Mathematics and mathematical logic : new research / [edited by] Peter Milosav and Irene Ercegovaca. p. cm. Includes bibliographical references and index. ISBN 978-1-61470-218-4 (eBook) 1. Logic, Symbolic and mathematical. I. Milosav, Peter. II. Ercegovaca, Irene. QA9.M3475 2009 510--dc22 2009012375

Published by Nova Science Publishers, Inc. Ô  New York

CONTENTS Preface

vii

Chapter 1

On Number’s Nature Dimitris Gavalas

Chapter 2

The Application of Fuzzy Linear Programming in Engineering C. Riverol

Chapter 3

A New Consensus Scheme for Multicriteria Group Decision Making under Linguistic Assessments P. Bernardes, P. Ekel and R. Parreiras

67

The Mathematical Basis of Periodicity in Atomic and Molecular Spectroscopy K. Balasubramanian

87

Chapter 4

1 57

Chapter 5

Second Order Definability in a Model George Weaver

115

Chapter 6

Algebraic Topics on Discrete Mathematics Gloria Gutiérrez Barranco, JavierMartínez, SalvadorMerino and Francisco J. Rodríguez

129

Chapter 7

Mathematical models of QoS Management for Communication Networks Chia-Hung Wang and Hsing Luh

Chapter 8

Reversible Logic Alexis DeVos

Chapter 9

Imaginary Cubic Oscillator and Its Square-well Approximations in x– and p– Representation Miloslav Znojil

Index

179 203

243 261

PREFACE A computational model is a mathematical structure which is constructed in an easy way and is useful in performing certain computations on the structure. This book discusses how the idea of "computational model" has been formalized by connecting the domain theory with the theory of metric spaces and related topological spaces. The nature of number is also examined, from a synthetic, holistic and interdisciplinary point-of-view, which is mostly mathematical but also encompasses respective psychological, neuro-physiological and philosophical views. Reversible logic circuits are discussed, which are beneficial to both classical and quantum computer design. Three experimental prototypes are used to illustrate how, in the near future, reversible computers will outperform conventional computers, in terms of power dissipation and heat generation. In quantum mechanics and field theory, Schroedinger equation for a single particle in one-dimensional imaginary potential represents one of the most popular schematic non-Hermitian models. Via a solvable square-well approximation technique, this book examines the problems in both the coordinate and momentum representations. The optimization problem is also looked at, which in traditional designs, is stated in precise mathematical terms but in real life, are stated in vague and linguistic terms. Chapter 1 - This chapter deals with the question regarding the essential nature of number. It is about ideas, which commence from the Pythagoreans and through the Platonists end to the modern era. Here the authors meet on the one hand Cantor, Gödel and their descendents and on the other Jung and his students. The distance, between the number’s well-known concept as cardinality and means of numbering and the one of dynamical psychic factor, which is sustained in this article, is enormous. What really happens with number’s issue? This question concerns us in a work, which poses more problems than those that solves. In the present text, the authors’ goal is to investigate the nature of number from a synthetic, holistic, interdisciplinary aspect, which mostly uses mathematical, but also the respective psychological, neuro-physiological and philosophical view. The spread of information also, which is taken as evidence of progress and knowledge, can be viewed as an effort to compensate and counterbalance the loss of the collective memory and the natural wisdom. For this reason the recourse to ancient knowledge and to modern research about the collective memory, that is the unconscious and its structural elements -Jung’s archetypes-, is obligatory. The structure of this chapter goes as follows: After the Introduction and Etymology of the Concepts ‘Number’ and ‘Archetype’ it encompasses four parts.

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In Part I: Pythagoras and His Descendants on Number, the authors discuss Pythagoras View about Number and His Vindication in Contemporary Mathematics. In Part II: Number in Contemporary Mathematics, the authors present number’s concept in relation to Standard Contemporary Mathematics, Logic, Category Theory, Philosophy, Neurophysiological and Social Aspects, Fuzziness, and Non-standard View. In Part III: Jung and the Concept of Archetype, the authors refer to subjects as ‘Chthonic’ and ‘Celestial’ Mathematics, Jung’s - Pauli’s General Hypothesis of Archetypes, Number as Archetype, Jung and Cantor - Gödel, and Number and Jung’s Psychology. In Part IV: Number, Monoid and Archetype, the authors discuss the issues 2-categories; Monad, Monoid, Monoidal Category, Monoid Object; The Dynamical System as Functor, The Archetype of Number as Monoid and its Interpretation as the Set of Numbers, and A Holistic View of Mathematics. The chapter comes to an end with the Final View and the Epilogue. Chapter 2 - As will be seen, the mathematical apparatus of the fuzzy theory of fuzzy sets provides a natural basis for the theory of the possibility as well described [1]. Viewed in this perspective, a fuzzy restriction may be interpreted as a possibility distribution with its membership function playing the role of possibility distribution function. This principle will be described in this chapter. Chapter 3 - This work proposes a new consensus scheme for group decision making, which allows one to obtain a consistent collective opinion, from information provided by each expert in terms of multigranular fuzzy estimates. It is based on a linguistic hierarchical model with multigranular sets of linguistic terms, and the choice of the most suitable set is a prerogative of each expert. From the human viewpoint, using such model is advantageous, since it permits each expert to utilize linguistic terms that reflect more adequately the uncertainty level intrinsic to his evaluation. From the operational viewpoint, the advantage of using such model lies in the fact that it allows one to express the linguistic information in a unique domain, without information losses, during the discussion process. Such consensus scheme is applied in the analysis of a multicriteria decision problem, generated with the use of the Balanced Scorecard methodology for enterprise strategy planning. Three techniques for multicriteria analysis, based on fuzzy preference relation modeling, are considered. They permit the evaluation, comparison, selection, prioritization, and/or ordering of alternatives with the use of both quantitative and qualitative estimates. With the availability of different techniques, the most appropriate one can be chosen, considering possible sources of information and its uncertainty. Chapter 4 - This chapter applies combinatorial and group-theoretical relationships to the study of periodicity in atomic and molecular spectroscopy. The relationship between combinatorics and both atomic and molecular energy levels must be intimate since the energy levels arise from the combinatorics of the electronic or nuclear spin configurations or the rotational or vibrational energy levels of molecules. Over the years the authors have done considerable work on the use of combinatorial and group-theoretical methods for molecular spectroscopy. The role of group theory is evident since the classification of electronic and molecular levels has to be made according to the irreducible representations of the molecular symmetry group of the molecule under consideration. Combinatorics plays a vital role in the enumeration of electronic, nuclear, rotational and vibrational energy levels and wave functions. As can be seen from other chapters in this book, the whole Periodic Table of the elements has a mathematical group-theoretical basis since the electronic shells have their origin in group theory.

Preface

ix

Indeed, this concept can even be generalized to other particles beyond electrons such as bosons or other fermions that exhibit more spin configurations than just the bi-spin orientations of electrons. Chapter 5 - The back and forth characterization of equivalence of interpretations for finitary (and some infinitary) second-order languages introduced in Weaver and Penev [2005] is applied to obtain a condition necessary and sufficient for an attribute of an interpretation to be definable in that interpretation by a second-order formula (either finitary or infinitary). This condition is applied to obtain some "reduction" theorems for the second-order theories of those infinite interpretations having pairing functions that are definable by two simple classes of second-order formulas. Chapter 6 – 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 developing methods in ambit of combinatorial theory. Pólya's enumeration method, Latin squares, patterns design or block design are examples of this usage. Others applications can be seen in Section 5, about foundation design, RNA patterns or octonions. Chapter 7 - Network design and network synthesis have been the classical optimization problems in telecommunication for a long time. In the recent past, there have been many technological developments such as digitization of information, optical networks, Internet, and wireless networks. These developments have led to a series of new optimization problems. In communication networks, a number of requirements can be identified. For instance, the network operator requires a good earning capacity, and the network users require reliable communication. The degree to which these and other requirements are fulfilled, can in many cases be deduced from how the network resources are distributed. In contrast to the Public Switched Telephony Network (PSTN), services offered in a data communication network vary significantly in terms of the required bandwidth. This adds a new challenging dimension into design of the networks, since inevitably, questions of fair medium sharing, quality guarantees, delays, etc. become crucial. Different methods of obtaining fair resource sharing are derived and investigated, for example, Max-Min Fairness, Proportional Fairness, etc. The authors focus on fairness on backbone networks possessing certain desired fairness properties. Chapter 8 - Reversible logic circuits are beneficial to both classical and quantum computer design. Present-day logic building-blocks (like OR gates and NAND gates) are logically irreversible and therefore cannot be used for designing reversible computers. Thus reversible computation needs an appropriate design methodology. In contrast to conventional digital logic circuits, reversible logic circuits (of a same logic width w) form a mathematical group. The reversible circuits of width w form a group isomorphic to the symmetric group S2w. Its Young subgroups allow systematic and efficient synthesis of an arbitrary reversible circuit. The author can choose a left coset, a right coset, or a double coset approach. The optimal design is reminiscent of the so-called banyan networks of telecommunication. As an illustration, three experimental prototypes (in c-MOS chip technology) of reversible computing devices are presented. Special care has been taken to avoid as much as possible the appearance of garbage bits. The examples illustrate how, in a near future, reversible com-

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puters will outperform conventional computers, in terms of power dissipation and heat generation. Chapter 9 - Schrödinger equation with imaginary PT symmetric potential V (x) = i x3 is studied using the numerical discretization methods in both the coordinate and momentum representations. In the former case our results confirm that the model generates an in¯nite number of bound states with real energies. In the latter case the di®erential equation is of the third order and a square-well, solvable approximation of kinetic energy is recommended and discussed. One finds that in the strong-coupling limit, the exact PT symmetric solutions converge to their Hermitian predecessors.

In: Mathematics and Mathematical Logic: New Research ISBN 978-1-60692-862-2 Editors: Peter Milosav and Irene Ercegovaca © 2010 Nova Science Publishers, Inc.

Chapter 1

ON NUMBER’S NATURE Dimitris Gavalas* Varvakeios Experimental School Athens, Greece

Where is the wisdom we have lost in knowledge? Where is the knowledge we have lost in information? T. S. Eliot, Choruses from the ‘Rock’, I, 15-16.

ABSTRACT This chapter deals with the question regarding the essential nature of number. It is about ideas, which commence from the Pythagoreans and through the Platonists end to the modern era. Here we meet on the one hand Cantor, Gödel and their descendents and on the other Jung and his students. The distance, between the number’s well-known concept as cardinality and means of numbering and the one of dynamical psychic factor, which is sustained in this article, is enormous. What really happens with number’s issue? This question concerns us in a work, which poses more problems than those that solves. In the present text, our goal is to investigate the nature of number from a synthetic, holistic, interdisciplinary aspect, which mostly uses mathematical, but also the respective psychological, neuro-physiological and philosophical view. The spread of information also, which is taken as evidence of progress and knowledge, can be viewed as an effort to compensate and counterbalance the loss of the collective memory and the natural wisdom. For this reason the recourse to ancient knowledge and to modern research about the collective memory, that is the unconscious and its structural elements -Jung’s archetypes-, is obligatory. The structure of this chapter goes as follows: After the Introduction and Etymology of the Concepts ‘Number’ and ‘Archetype’ it encompasses four parts.

* Folois 6 112 56 Athens, Greece [email protected]

2

Dimitris Gavalas In Part I: Pythagoras and His Descendants on Number, we discuss Pythagoras View about Number and His Vindication in Contemporary Mathematics. In Part II: Number in Contemporary Mathematics, we present number’s concept in relation to Standard Contemporary Mathematics, Logic, Category Theory, Philosophy, Neurophysiological and Social Aspects, Fuzziness, and Non-standard View. In Part III: Jung and the Concept of Archetype, we refer to subjects as ‘Chthonic’ and ‘Celestial’ Mathematics, Jung’s - Pauli’s General Hypothesis of Archetypes, Number as Archetype, Jung and Cantor - Gödel, and Number and Jung’s Psychology. In Part IV: Number, Monoid and Archetype, we discuss the issues 2-categories; Monad, Monoid, Monoidal Category, Monoid Object; The Dynamical System as Functor, The Archetype of Number as Monoid and its Interpretation as the Set of Numbers, and A Holistic View of Mathematics. The chapter comes to an end with the Final View and the Epilogue.

INTRODUCTION From a cognitive point of view, the reason to discuss the subject of number is the fundamental changes in its mental representation. Indeed, after the failure of the effort to limit number in a frame of Logic -that is to limit it only to its logical aspect-, nowadays inside the cognitive realm in order to have a complete image for it, is emerged the need to see it from all of its aspects, logical and non-logical. Gödel set aside the Logic of Frege - Russell - Hilbert and posed the matter of uncertainty for Mathematics, using its own means. Other mathematicians also, like Weyl, talk about the ‘abysmal’ character of numbers. Afterwards, the introduction of methods of Fuzzy Logic and Non-standard Analysis leads to corresponding views about number. In a society, and especially in a scientific community, where is considered officially that the dominating function is the rational thinking, even though this is just a myth, a little attention is paid to the ‘other’ side of things. Jung stresses that no matter how much beautiful and perfect finds man his logic, this is just one of his psychological functions and man is covered only by one side. From all the other sides though, man is surrounded by the irrational, the collective, the unconscious, the chance. Therefore, next to the rational exists the irrational and in much more proportion. When logic and the time linear sequence of events stops being applied, then our rationalism does not function anymore and in order to understand what is happening we need other functions and methods.

ETYMOLOGY OF THE ‘NUMBER’ AND ‘ARCHETYPE’ In Gk. the word for number, αριθμός (arithmos), is etymologized from the verb which means: Attach, adject, seam, collect, osculate, put together, fit, piece, conjoin, unite, integrate, and the like. So, αριθμός (arithmos) means: Amplitude, deal, crowd, multitude, oodles, pack, plurality, quantum, throng, lot, number, amount, quantity, count, enumeration, measurement, reckoning, numeration, calculation, computation, account, sign of value, of order, of position, of integration and so on. From αριθμός (arithmos) comes the term Arithmetic. According to another view, αριθμός (arithmos) means: Best/ optimum psyche, best movement, animate time. This view agrees entirely to Xenocrates who says that “psyche is

On Number’s Nature

3

number which moves itself” (Heinze, 1965, Fragm. 61). Aristotle explains that “because psyche is both kinetic and knowing, so somebody united these two and said that psyche is number which moves itself” (De an.). In L. we have the words numero (verb), numerus (noun), from which comes the word number (verb and noun): The sum, total, count, or aggregate of a collection of units, or the like. Both, Gk. and L words are related to rhythm. In Mathematics and Science number is a basic concept, and a mode of thought: Number is the basis of science. Also, the term αρχέτυπον (archetype) means: Original pattern from which copies are made, from L. archetypum, which comes from Gk. archetypon ‘pattern, model, firstmoulded’ from arche- ‘first’ + typos ‘model, type, blow, mark of a blow’. So, from the Gk. for ‘original pattern’, archetype is a basic model from which copies are made; therefore a prototype. In general terms, the abstract idea of a class of things which represents the most typical and essential characteristics shared by the class, thus a paradigm or exemplar. Jungian psychology speeks about ‘pervasive idea or image from the collective unconscious’.

PART I: PYTHAGORAS AND HIS DESCENDANTS ON NUMBER 1. Pythagoras View about Number “Number is the within of all things”, Pythagoras.

Pythagoras considers that number lies behind all things and facts, beginning from musical harmony and finishing to the planets orbits. This aspect leads him to the belief that everything is number. Iamblichus (1922; Waterfield, 1988a, 1988b) writes about the Pythagorean view on number. According to him, the Pythagorean philosophy bases mostly on the mathematical semantics of numbers and their relationships. The study of number entails not only the material phenomena but mostly the spiritual and mental aspects, the symbolic meaning of each separate number as well as its mystical properties. All these views about number do not limit to the relationships of the material phenomena, instead number has mental, cosmological, and ethical applications. This means that the Pythagorean entailment on number is not just a mathematical science according to the modern view, but study of the hidden wisdom of number, which contains truths that are being taught on the initiatory rites of antiquity. In Metaphysics Aristotle describes the teaching of Pythagoreans: They thought they found in numbers, many resemblances to things which are and become; thus such and such an attribute of numbers is justice, another is soul and mind, another is opportunity, and so on; and again they saw in numbers the attributes and ratios of the musical scales. Since, then, all other things seemed in their whole nature to be assimilated to numbers, while numbers seemed to be the first things in the whole of nature, they supposed the elements of numbers to be the elements of all things, and the whole heaven to be a musical scale and a number. According to O’ Meara (1990), the Pythagorean idea that number is the key to understanding reality inspired Neoplatonist philosophers in Late Antiquity to develop theories in Physics and Metaphysics based on mathematical models. He examines this theme, describing first the Pythagorean interests of Platonists in the second and third centuries and

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then Iamblichus’s program to Pythagoreanize Platonism in the fourth century in his work On Pythagoreanism -whose unity of conception is shown and parts of which are reconstructed for the first time. The impact of Iamblichus’s program is examined as regards Hierocles of Alexandria and Syrianus and Proclus in Athens: Their conceptions of the figure of Pythagoras and of Mathematics and its relation to Physics and Metaphysics are examined and compared with those of Iamblichus. This provides insight into Iamblichus’s contribution to the evolution of Neoplatonism, to the revival of interest in Mathematics, and to the development of a philosophy of Mathematics and a mathematizing Physics and Metaphysics. So, for Philolaus the nature of number is ‘gnomonical, magisterial and pedagogical’: If one focuses and reflects on number as symbol, then one can realize concepts that lie beyond the limits of the usual human consciousness. According to Aristotle, the Pythagoreans sustain that entities pattern upon number. Syrianus also characterizes number as eidetic, that is its meaning is demonstrative of the kind, the quality and not only the quantity of the things and entities. Proclus helps us to realize the relationship between number and the rest Pythagorean philosophy. By his sayings it is clear that the mystical meaning of number is closely related to the Pythagorean mystical revelation. This meaning is revealed to the initiates at rites, where the relations and lows that refer to the divine aspect of number for the principles and things are being explained to them. Many writers of the antiquity compare the Pythagorean meaning of number with that of Orpheus who seems to be the first that introduces this mystical theory. Syrianus says that the Pythagoreans accepted the theory about number from the Orphics. They also extended it up to the sensual material phenomena beginning from the spiritual and the mental aspects. For the non-sensual that is the spiritual, mental, mystical and esoteric meaning of number, we have many fragments from the writers of antiquity. However, a complete and systematic study is not rescued nowadays except the one attributed to Iamblichus. In his work the mathematical properties of the first ten natural numbers are developed and are referred the many appellative names for each one of them. Both from the mathematical properties and from the appellative names we can figure out the way and the spirit under which the Pythagoreans dealt with number. We can also compare this spirit with the modern approach and ask ourselves, whether except from ours is there another one lost and forgotten spirit, which is the other side of the coin that must co-exist with the current one. Therefore, for the Pythagoreans, number is the structural element both for the material world and for the soul and pre-existed of them and indeed the world and the soul were created in pattern upon the number. The Pythagoreans deepen in the number’s concept, considering that the number’s principles are all entities principles. The natural elements are inadequate for the world’s interpretation; in contrary to the number that contains mental power. The elements consisting number are also all entities’ elements and the heaven is governed by harmony and number. With Pythagoras a new multi-leveled world is created, since the sensual is reduced to the conceptual, while the conceptual in its turn is reduced to the ideal. This view is further developed by Plato. With the Pythagoreans, number acquires transcendental power, since by its existence derive and become thinkable the physical magnitudes. In Plato also, number tends to replace idea. Indeed, Plato is inducted to the Hen (One) and he approaches the idea to the number, reducing the many to the One. So, if the world’s nature is mathematical according to Pythago-

On Number’s Nature

5

ras and Plato, this happens in virtue of number mental power, which ensures ontological validity. “Number is the highest degree of knowledge; it is knowledge itself”, Plato says.

2. Pythagoras Vindication in Contemporary Mathematics Eves (1983, Lecture 32), about Pythagoras vindication in contemporary Mathematics, says that, if we could hear Pythagoras’s voice about the end of the 19th century, it would say: I have been told you over 2000 years ago: everything depends on natural numbers. This is so because after an extremely important line of discoveries, mathematicians showed that Mathematics is consistent, if the system of natural numbers is consistent. Indeed, the structure of Mathematics is a huge inverted pyramid leaning its peak on the system of natural numbers. Analysis, mainly on its first steps, displayed many problems. Under every element in it were deep properties of the system of the real numbers that demanded immediate understanding. Weierstrass sustained a program according to which the system of real numbers itself should first be developed logically and afterwards Analysis’s concepts to be defined according to the system of real numbers. This program is known as the ‘Arithmetization of Analysis’ and was realized at the end of the 19th century. The success of this program had great impact. Since Analysis bases on the system of the real numbers we conclude that it is consistent only if the system of the real numbers is consistent. Euclidean Geometry, after its arithmetization by Descartes, can also base on the system of the real numbers; therefore it is consistent only if the system of the reals is consistent. Still, various geometries are consistent if Euclidean Geometry is consistent. The same happens also with Algebra. All these lead to the conclusion that the majority of Mathematics is consistent if the system of the reals is also consistent. Here lies the immense importance of the system of the reals for the foundation of Mathematics. Finally, was adopted the view that we can start from an initial set of axioms more basic than the one of the system of the reals and genetically, that is by stating the appropriate definitions and by applying clever techniques, to reach the system of the reals. This forces the consistency of the system of the reals -and therefore of the majority of Mathematics- to depend on the consistency of this initial and more basic set of axioms. This was succeeded and the system of the reals came along only from apposite definitions and appropriate techniques, without further assumptions, from a set of axioms for the much more simple and basic system of the natural numbers. This success gave to the mathematicians an important sense of security about the consistency of the majority of Mathematics. This derives from the fact that the system of the natural numbers has intuitive simplicity, and clarity and has been vastly used for a longer period of time without presenting any internal contradictions and does not present the complexity of other mathematical systems. Mainly, it is inherent in the humans and natural, because it is an archetype. Therefore, beginning from the naturals following various techniques we can construct the integers and then from the integers the rational numbers. Subsequently, we can introduce the irrationals basing on the rationales and finally we can define the reals. The final step is the hardest, but is realized with many ways, like Dedekind cuts etc. This development is represented as follows: N → Z → Q → R → C, since from the reals we can have the complex numbers. Hence, on the system of the naturals bases the construction of Mathematics and its consistency. Pythagoras’s view that literally everything in the world depends on the naturals

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was vindicated and is expressed by Kronecker as follows: ‘God has made the naturals, all the rest is man’s project. Their divine nature makes them so simple, but their deepest substance always escapes from us’.

PART II: NUMBER IN CONTEMPORARY MATHEMATICS 3. Number and Standard Contemporary Mathematics The natural numbers are: (i) A sequence of marks/ signs, e.g. |, ||, |||, ||||, |||||, …, for use in counting or labeling. Any other sequence of marks/ signs would do as well, provided the sequence always provides a way to write down the next (the successor) mark/ sign. This is the case with decimal notation 0, 1, 2, …, 10, 11, 12, …, 100, …, 200, … . (ii) Finite sets, with equinumerous sets regarded as equal numbers, while successor means ‘adjoin one more element’. (iii) Cardinal-equivalence classes of finite sets. (iv) Ordinal-equivalence classes of finite ordered sets, while successor means ‘adjoin one new element, to come after all the others’. (v) Finite sets of sets, linearly ordered by the membership relation as follows: 0=∅, 1={0}, 2={0, 1}, …, ω={0, 1, 2, …}. In each of these cases, the natural numbers satisfy the Peano postulates. From this multiplicity of answers to the question ‘what is a natural number’, we must conclude that there is no answer to the question. There are various alternative concrete descriptions depending on the sort of counting intended or on the prior assumptions. In each case, the description provides ‘numbers’ which do satisfy the Peano postulates. Hence we conclude that one does not define what a natural number ‘is’ by itself. Instead, one defines the system of all natural numbers, with successor operation s(n)=n+1. Then N is any such system which satisfies the Peano postulates. This means that there are many such systems within Set Theory, but that they are all isomorphic. The postulates themselves are, by no means, unique. The Peano postulates, for example, may be replaced by the Recursion Theorem as an axiom. Here, as in other axiomatic descriptions of mathematical objects, there are a variety of choices for lists of axioms. Number Theory, like other subjects in Mathematics, is not the study of a unique model nor yet the examination of a unique axiomatic system, is it rather a study of the form exemplified by the various models and specified in the axioms. Perhaps the simplest aspect we have for the natural numbers is that, there is a minimum number 0, that each number n has an immediate successor s(n) and that, if we commence with the minimum 0 and we construct consecutively the successor of each number ad infinitum, 0, s(0)=1, s(1)=2, s(2)=3,… we enumerate all natural numbers. This aspect is expressed by Peano axioms on the structured set . The “Theorem of Existence and Uniqueness of Natural Numbers’’ tells us that there is at least one system of natural numbers. If there are two systems of natural numbers, then there is

On Number’s Nature

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exactly one isomorphism between them that is all systems of natural numbers are isomorphic by twos. Thus we determine a particular system of natural numbers, the members of which are called natural numbers. A choice for this system could be the following: N={∅, {∅}, {{∅}}, …} and s: n → s(n)={n}. Another choice is Ν={0, 1, 2, 3, …} and s(n)=n+1, that is to accept that there is indeed the set of ‘true’ natural numbers, where each of them is not a set but a number, and the successor is not the function s: n → s(n)={n} but the function which correlates to every number n its successor n+1. The classical theory of Zermelo, allows for instance such non-sets -as the ‘true’ natural numbers- as urelements and it only insists on the premise that the system of the natural numbers must satisfy Peano’s axioms. Therefore, we construct the natural numbers beginning with 0 or ∅ and infinitely repeating the operation of the successor. To summarize: The natural numbers start out from elementary operations of counting, listing and comparing. They then develop into effective tools for calculation. The rules for calculation are formal and can be organized as the consequences of simple systems of postulates. The consequences of these postulates include the remarkably varied and rich properties studied in Number Theory, properties by no means apparent in the original processes of counting and listing. (Mac Lane, 1986) From a philosophical point of view, the common aspect of mathematicians and philosophers of science was, and up to a point still is, that it is really feasible to construct the set of natural numbers from nothing and only with Logic. But all the relevant efforts lean against the fallacious ‘General Principle of Inclusion’ and lead to contradictions. Logic, by its nature, seems that cannot prove the existence of anything, which means that it cannot have ontological demands. This is a critical point for the continuation of our aspects. Since it is not Logic, and the resulting typical theories that lead to the answer about number’s existence, then which is?

4. Number and Logic Work in Logic has provided several developments bearing on numbers. In retrospect, the development of the concept of number prompts the question of ‘What number is?’ This can be interpreted as the question of whether number is Platonic object, mental construction, or nothing more than mystified numeral. Alternatively, in virtue of the plethora of kinds of numbers, we can interpret the question as asking what makes entities of certain kinds, but not others, numbers. Beyond a rather vague characterization, it seems difficult to give a general one of number. The most fundamental numbers, the naturals, measure size or order, are subject to distinctive operations and can be the roots of equations. Each of these central features has played a role in generating new kinds of numbers. The result is a collection of entities which are related by family resemblance (Wittgenstein, 1974), though the boundaries of the family seem somewhat arbitrary and vague. It is difficult to see why, for example, complex numbers are called numbers, but not vectors or numerical matrices; both of these share the central features of natural numbers. This conclusion is reinforced by the work of Conway’s (1976). He gives a transfinite recursive construction that generalizes both constructions, the Dedekind’s of the reals and the von Neumann’s of ordinals. Essentially, a number is any pair , such that all the mem-

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bers of L and R are numbers, and every member of R is greater than or equal to every member of L. The ‘≥’ and the arithmetical operations are also defined in a natural recursive manner. The construction generates virtually all the numbers, including infinitesimals, but excluding the complex numbers and if cardinals are to be identified with initial ordinals, a nonuniform definition of arithmetical operations is necessary. Moreover, the construction generates many novel numbers, for example, numbers obtained by applying the full range of realnumber operations to infinite numbers, which make no sense on the usual understanding. Moreover, a simple generalization of the construction -dropping the ordering condition on L and R-, produces even more number-like objects. Just conceivably, a unifying account of number might eventually be found, but in the meantime the emergence of new kinds of numbers seems likely. There are non-standard inconsistent models of Arithmetic which contain inconsistent numbers -natural numbers with inconsistent properties. These have some notable applications; for example, some of them can be shown to provide solutions for arbitrary sets of simultaneous linear equations. Just as the existence of non-standard models of Analysis made infinitesimals legitimate, so might these legitimize the notion of an inconsistent number. Besides, there are three points of particular importance: (i) The proof by Gödel in 1931 that the Peano axioms, and all other consistent axiom systems for Arithmetic, are incomplete, in the sense that there are truths of Arithmetic that cannot be proved from the axioms -at least if the underlying Logic is firstorder (Gödel’s Theorems). The axioms are complete if the underlying Logic is second-order and the Induction Principle is formulated as a second-order axiom and not just a first-order schema; but second-order Logic is not itself axiomatizable. This raises profound questions about the nature of both numbers and our knowledge about them. (ii) The paradoxes surrounding transfinite numbers. The dominant view is that they are embedded in ZF Set Theory. According to this, there is no totality of all ordinals, all sets or other large collections, and so the question of their size does not arise. Although this account provides enough Set Theory for most Mathematics -though not all: Category Theory appears to require large sets of just this kind-, it can hardly be said to be conceptually adequate. Standard Logic, for example, defines the sense of a quantifier in terms of the domain (totality) over which it ranges. It is therefore unclear what the sense of the quantifiers of ZF is, if, as it claims, there is no such totality. (iii) Robinson’s Non-standard Analysis. As was proved by Löwenheim-Skolem, firstorder theories of number have non-standard models. In particular, any theory of the reals has such models. Robinson shows that in all of these models, there are non-zero numbers that are smaller than any real number: Infinitesimals. Using these, he demonstrates that the reasoning of the Infinitesimal Calculus -which is much more intuitive than limit reasoning-, can be interpreted in a perfectly consistent manner. Hence, infinitesimals have been rehabilitated as perfectly good numbers (Priest, 1998). It is regarded that natural numbers should obey the Induction Principle, but this exhibits a form of circularity known as ‘impredicativity’: The statement of the principle involves quantification over properties of numbers, but to understand this quantification we must assume a

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prior grasp of the number concept, which it was our intention to define. It is nowadays a commonplace to draw a distinction between impredicative definitions and impredicative specifications: The first are illegitimate, while the second are not. The conclusion in this case is that the Induction Principle on its own does not provide a non-circular route to an understanding of the natural number concept. It is therefore needed an independent argument. Five main strategies have been attempted, which are the well known: Intuitionism, Platonism, Formalism, Logicism, and Empiricism.

5. Number and Philosophy: From Frege to Benacerraf Frege (1884/ 1980, 1893-1903/ 1967) considered the introduction of numbers via numerically definite quantifiers. What led him to reject it was a problem facing not just this but any strategy involving implicit definitions; such definitions do not fully determine the identity conditions of the objects they attempt to introduce. This is known as the ‘Julius Caesar problem’ because Frege posed it by asking ‘how one could tell from the implicit definitions whether or not Julius Caesar is a number’. This question creates a problem for the strategy of explaining the number-words’ substantive use in terms of their adjectival use. The problem bites only when we attempt to move beyond elementary Arithmetic and quantify over numbers. Another logicist strategy is to derive Arithmetic from the ‘numerical equivalence’, that is, the contextual principle -called ‘Hume’s principle’- that the number of Fs is the same as the number of Gs if and only if the Fs and the Gs can be correlated 1-1. Frege sketched a construction of the natural numbers and a proof that they satisfy Peano’s axioms, assuming only second-order Logic and the numerical equivalence. He nevertheless rejected the strategy of basing Arithmetic on this equivalence because the ‘Julius Caesar problem’ applies to it as much as to the previous contextual strategy. Interest in the strategy has been revived by Wright (1983), who argues that the ‘Julius Caesar problem’ can be solved by appeal to an independently plausible ‘sortal inclusion principle’ to the effect that objects must be of different sorts if the content of their identity conditions is sufficiently different. The proposal Frege settled on instead was to define the number of Fs explicitly as the class of all concepts equinumerous with F. He showed that on this definition of number Arithmetic can be deduced from what he took to be logical principles. However, one of his allegedly logical principles was inconsistent, and subsequent attempts -most notably by Whitehead and Russell- to repair Frege’s system have had to appeal to principles even their advocates have baulked at calling logical. Frege’s solution to the ‘Julius Caesar problem’ -the underdetermination of the objects of Arithmetic by the principles about them to which we are committed- is in any case under threat from the opposite problem: Any non-arithmetical determination of the objects which solves the ‘Julius Caesar problem’ will give numbers extra properties which, since they do not flow from the principles governing numbers, must be arbitrary and hence spurious. This problem was mentioned by Dedekind (Ewald, 1996), but it has come to prominence more recently through a much-cited paper by Benacerraf (1965). Two ways of dealing with it have been advanced under the generic label of ‘structuralism’. Dedekind’s way -that we are capable, once we have given a logical construction of one model of Peano’s axioms, of abstracting away from its particular features to gain a conception of a model without those features- ap-

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peals to a mental process (abstraction) that many have found mysterious. Benacerraf’s way that Arithmetic should be seen as the study not of one particular model of Peano’s axioms but of the structure which all such models have in common- is in danger of relapsing into the axiomatic formalism. The well-known Russell’s paradox suggests to Dummett (1991) that the concept ‘set’ is what he calls ‘indefinitely extensible’. This means that any attempt to regard the objects falling under the concept as forming a definite totality leads inevitably to the realization that there are other objects not in the totality which we are nevertheless forced to admit as falling under the concept. Dummett holds that the presence of ‘indefinitely extensible’ concepts is a characteristic feature of Mathematics which should lead us to espouse for it the anti-realism which his more general meaning-theoretic arguments make room for. He recommends that we abandon the law of the excluded middle and espouse the Mathematics of Intuitionism but not Brouwer’s solipsistic conception of its objects. Dummett thinks that in this way we can retain Frege’s logicist insight that numbers are abstract objects truths about which embody deductive subroutines whose application to the world is validated by Logic alone. Just as Brouwer’s Intuitionism has been accused of an instability which reduces it to strict Finitism, not everyone is persuaded that Dummett’s position does not collapse into the ultra-intuitionism of middle-period Wittgenstein, according to which the meaning of an arithmetical generalization is identical with its proof. On this view Goldbach’s conjecture does not in the present state of knowledge have any meaning at all. About Frege’s work one can see the following contemporary studies: Boolos, 1986, 1987; Heck, 1993; Maddy, 1992; Zalta, 1998, 1999. Benacerraf (1965), on the other hand, discusses the classical problem for set-theoretic foundations -that Arithmetic has no unique set-theoretic representation- on the line of thought of Dedekind’s (1888/ 1963), Hilbert’s (1900), Weyl’s (1927/ 1949) and Bernays’ (1950/ 1976). This argument was influential in shaping later work. It is the inspiration of the position known as Structuralism -the view that mathematical objects are essentially positions in structures and have no important additional internal composition or nature (Resnik, 1981, 1982; Shapiro, 1983, 1989). The version of questions with regard to numbers, which is discussed more often in the modern philosophical bibliography, emanates from this work of Benacerraf’s. The essay of Benacerraf’s does not aspire it explains what precisely the numbers are, but is focused on exposing the limits of various opinions on numbers, explaining what they are not. Concretely, Benacerraf concludes that numbers are not sets, as most mathematicians believe. In the end in any case, where he presents his own opinion for the question, he adopts the aspect that numbers do not exist at all. With this view, for which he offers some justification, avoids the difficulties of various theories for the numbers. Moreover, this view is mathematical acceptable, after the Mathematics do not require to be the numbers certain concrete objects, but simply some structure exists. It is possible for one to conclude that, after the numbers are a clean mathematical affair, then the opinion that these do not exist as separate entities it should be also philosophically acceptable. (Potter, 1998; Heijenoort, 1967).

6. Number and Neurophysiological and Social Aspects As the reader can find in Campbell (2005), some basic questions are posed concerning the domain of mathematical cognition: How does the mind represent number and make

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mathematical calculations? What underlies the cognitive development of numerical and mathematical abilities? What factors affect the learning of numerical concepts and procedures? What are the biological bases of number knowledge? Do humans and other animals share similar numerical representations and processes? What underlies numerical and mathematical disabilities and disorders, and what is the prognosis for rehabilitation? These questions are the domain of mathematical cognition, the field of research concerned with the cognitive and neurological processes that underlie numerical and mathematical abilities. Mathematical cognition research intersects a wide array of disciplines including cognitive development, neurological development, computational science, cognitive and educational psychology, animal cognition, cognitive and clinical neuropsychology, neuroscience, and cognitive science. Already since 1998, Ramachandran & Blakeslee wonder whether each of us has a centre for numbers and arithmetical operations in the left angular gyrus. This region is in a way necessary for arithmetical calculations, and weirdly is not necessary for the understanding of arithmetical concepts that rule such calculations. And they continue: We don’t know yet how this arithmetic circuit works in the angular gyrus, but at least we know where to search. The fact that we execute the arithmetical operations without special effort, allows us to conclude that this ability is ‘hard-wired’ by birth. But in reality, this ability was acquired after the introduction of basic concepts as zero and the line of numbers. It has also been sustained that there is in the brain the line of numbers, a kind of graphical, scalar representation of numbers and that each point of it consists of a group of neurons which point to particular mathematical value. But are there proofs that such a line exists in the brain? When healthy individuals are asked which of two given numbers is greater, the subjects need more time to answer when the numbers are close than being far. Undoubtedly, there is independent mechanism for the arithmetical operations and this mechanism bases on the angular gyrus of the left brain hemisphere. Rickard in the UCSD, using fMRI (functional magnetic resonance imaging) technique, showed that the region of arithmetical operations is not entirely on the left angular gyrus but also a little in front. This fact does not affect the basic assumption and it is matter of time to prove the existence of line of numbers with the aid of a modern imaging technique. About relevant issues see also Dehaene (1997a). ‘Number Sense’ is a short-hand for our ability to quickly understand, approximate, and manipulate numerical quantities. Dehaene’s hypothesis is that number sense rests on cerebral circuits that have evolved specifically for the purpose of representing basic arithmetic knowledge. Four lines of evidence suggesting that number sense constitutes a domainspecific, biologically-determined ability: (i) The presence of evolutionary precursors of arithmetic in animals; (ii) The early emergence of arithmetic competence in infants independently of other abilities, including language; (iii) The existence of a homology between the animal, infant, and human adult abilities for number processing; and (iv) The existence of a dedicated cerebral substrate. In adults of all cultures, lesions to the inferior parietal region can specifically impair number sense while leaving the knowledge of other cognitive domains intact. Furthermore, this region is demonstrably activated during number processing. Dehaene postulates that

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higher-level cultural developments in arithmetic emerge through the establishment of linkages between this core analogical representation -the ‘number line’- and other verbal and visual representations of number notations. The neural and cognitive organization of those representations can explain why some mathematical concepts are intuitive, while others are so difficult to grasp. Thus, the ultimate foundations of mathematics rest on core representations that have been internalized in our brains through evolution (Dehaene, 2001). In the last decade new research promotes our knowledge regarding the relationship Mathematics-brain. Specifically, Göbel et al. (2001) sustain that in order to investigate the hemispheric organization of a language-independent spatial representation of number magnitude in the human brain they applied focal repetitive Transcranial Magnetic Stimulation (rTMS) to the right or left angular gyrus while subjects performed a number comparison task with numbers between 31 and 99. Repetitive TMS over the angular gyrus disrupted performance of a visuospatial search task, and rTMS at the same site disrupted organization of the putative “number line.” In some cases the pattern of disruption caused by angular gyrus rTMS suggested that this area normally mediates a spatial representation of number. The effect of angular gyrus rTMS on the number line task was specific. rTMS had no disruptive effect when delivered over another parietal region, the supramarginal gyrus, in either the left or the right hemisphere. Besides, Dehaene (2003) claims that the recent discovery of number neurons allows for a dissection of the neuronal implementation of number representation. Recently, some researchers demonstrate a neural correlate of Weber’s law, and thus resolve a classical debate in psychophysics: The mental number line seems to be logarithmic rather than linear. In another interesting article, Dehaene et al. (2004), refer that recent studies in human neuroimaging, primate neurophysiology, and developmental neuropsychology indicate that the human ability for arithmetic has a tangible cerebral substrate. The human intraparietal sulcus (IPS) is systematically activated in all number tasks and could host a central amodal representation of quantity. Areas of the precentral and inferior prefrontal cortex also activate when subjects engage in mental calculation. A monkey analogue of these parieto-frontal regions has recently been identified, and a neuronal population code for number has been characterized. Finally, pathologies of this system, leading to acalculia in adults or to developmental dyscalculia in children, are beginning to be understood, thus paving the way for brainoriented intervention studies. More specifically, regarding negative numbers and their mental representation, Fischer & Rottmann (2005), refer that the cognitive representation of negative numbers has recently been studied with magnitude classification and variable standard. Results suggested that negative numbers are cognitively represented on a “mental number line”. Conflicting with this observation they report here that response biases for parity classification of negative numbers are related to absolute digit magnitude only. A control experiment with magnitude classification relative to a fixed standard shows that our result does not reflect mere inattention to the minus sign. Together, the available evidence suggests that human process negative numbers less automatically compared to positive numbers. The IPS is thought to play a role in mental functions, including processing symbolic numerical information. According to Cantlon et al. (2006), adult humans, infants, pre-school children, and non-human animals appear to share a system of approximate numerical processing for non-symbolic stimuli such as arrays of dots or sequences of tones. Behavioral studies of adult humans implicate a link between these non-symbolic numerical abilities and sym-

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bolic numerical processing -e.g., similar distance effects in accuracy and reaction-time for arrays of dots and Arabic numerals. However, neuroimaging studies have remained inconclusive on the neural basis of this link. The IPS is known to respond selectively to symbolic numerical stimuli such as Arabic numerals. Recent studies, however, have arrived at conflicting conclusions regarding the role of the IPS in processing non-symbolic, numerosity arrays in adulthood, and very little is known about the brain basis of numerical processing early in development. Addressing the question of whether there is an early-developing neural basis for abstract numerical processing is essential for understanding the cognitive origins of our uniquely human capacity for math and science. Using functional Magnetic Resonance Imaging (fMRI) at 4-Tesla and an event-related fMRI adaptation paradigm, the researchers found that adults showed a greater IPS response to visual arrays that deviated from standard stimuli in their number of elements, than to stimuli that deviated in local element shape. These results support previous claims that there is a neurophysiological link between non-symbolic and symbolic numerical processing in adulthood. In parallel, they tested 4 years-old children with the same fMRI adaptation paradigm as adults to determine whether the neural locus of nonsymbolic numerical activity in adults shows continuity in function over development. They found that the IPS responded to numerical deviants similarly in 4 years-old children and adults. To our knowledge, this is the first evidence that the neural locus of adult numerical cognition takes form early in development, prior to sophisticated symbolic numerical experience. More broadly, this is also, to our knowledge, the first cognitive fMRI study to test healthy children as young as 4 years-old, providing new insights into the neurophysiology of human cognitive development. Finally, according to Lakoff & Núñez (2000), the only mathematical ideas that human beings can have are ideas that the human brain allows. We know a lot about what human ideas are like from research in Cognitive Science. Most ideas are unconscious, and that is no less true of the mathematical ones. Abstract ideas, for the most part, arise via conceptual metaphor -a mechanism for projecting embodied that is, sensory-motor reasoning to abstract reasoning. They argue that conceptual metaphor plays a central, defining role in mathematical ideas within the cognitive unconscious -from Arithmetic and Algebra to sets and Logic to infinity in all of its forms: transfinite numbers, points at infinity, infinitesimals, and so on. Even the real numbers, the imaginary numbers, Trigonometry, and Calculus are based on metaphorical ideas coming out of the way we function in the everyday physical world. This work is about mathematical ideas, about what Mathematics means -and why. The authors believe that understanding the metaphors implicit in Mathematics will make Mathematics make more sense. Moreover, understanding mathematical ideas and how they arise from our bodies and brains will make it clear that the brain’s Mathematics is Mathematics, the only Mathematics we know or can know. Let us see now another aspect: For Hersh, Mathematics has existence or reality only as part of human culture. Despite its seeming timelessness and infallibility, it is a social - cultural - historic phenomenon. The questions he posses -What are numbers? What are infinite sets? What is the meaning and nature of Mathematics?- are answered in this framework: It is neither physical nor mental, it is social; it is part of culture and of history. It is like law, like religion, like all those other things which are very real, but only as part of collective human consciousness. To the question “What kind of a thing is a number?” Hersh says that we can think of two basic answers -either it is out there some place, like material objects; or it is inside, a thought

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in somebody’s mind. Philosophers have defended one or the other of those two answers, which are both completely wrong. A number is not a thing out there; there is not any place that it is, or any thing that it is. Neither is it just a thought, because after all, two and two is four, whether you know it or not. Frege, made quite an issue of the fact that mathematicians didn’t know the meaning of One. What is One? Nobody could answer coherently. Of course Frege answered, but his answer was no better, or even worse, than the previous ones; and so it has continued to this very day, strange and incredible as it is. We know all about so much Mathematics, but we don’t know what it really is. Of course when we say, “What is a number?” it applies just as well to a triangle, or a circle, or a differentiable function, or a selfadjoint operator. We know a lot about it, but what is it? What kind of a thing is it? When we say that a mathematical thing, object, entity, is completely external, independent of human thought or action, or else internal, a thought in our mind -we are not just saying something about numbers, but about existence- that there are only two kinds of existence. Everything is either internal or external; and given that choice, that polarity or dichotomy, numbers do not fit -that’s why it is a puzzle. The question is made difficult by a false presupposition, that there are only two kinds of things around. But if we are real, and ask what there is around, for instance there is the traffic ticket we have to pay, there is the news on the TV, etc. -none of these things are just thoughts in our mind, and none of them is external to human thought or activity. They are a different kind of reality, that’s the trouble. This kind of reality has been excluded from Metaphysics and Ontology, even though it is well-known -the sciences of Anthropology and Sociology deal with it. But when we become philosophical, somehow this third answer is overlooked or rejected. Mathematics is neither physical nor mental, it is social, part of culture, part of history, and it is like all those real things which are real only as part of collective human consciousness. Being part of society and culture, it is both internal and external: Internal to society and culture as a whole, external to the individual, who has to learn it from books in school. That is what Mathematics is. But for some Platonic mathematicians, that proposition is so outrageous that it takes a lot of effort even to begin to consider it. Hersh calls it ‘humanistic philosophy of Mathematics’. He uses the term ‘humanism’ because it is saying that Mathematics is something human. There is no Mathematics without people. Many people think that numbers are there whether or not any people know about them; this is confusion. ‘Humanistic philosophy of Mathematics’ faces Mathematics as part of human culture and history. This philosophy lands Mathematics, makes it psychologically accessible and increases the possibility for someone to learn it, because it is exactly one of the things people do. He names his view ‘social conceptualism’ too, because Mathematics consists of concepts, but not individually held concepts; socially held concepts. This view sounds like an anthropic principle of Mathematics. One can say “There are nine planets; there were nine planets before there were any people. That means there was the number nine, before we had any people”. We do see mathematical things, like small numbers, in physical reality; and that seems to contradict the idea that numbers are social entities. We use number words in two different ways: as nouns and adjectives. This is an important observation. We say nine apples, nine is an adjective. If it is an objective fact that there are nine apples on the table, that is just as objective as the fact that the apples are red, or that they are ripe, or anything else about them, that is a fact. And there is really no special difficulty about that. Things become difficult when we switch unconsciously, and carelessly, between this real-world adjective interpretation of mathematical

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words like nine, and the pure abstraction. That is not really the same nine. Although there is of course a correlation and a connection. But the number nine as an abstract object, as part of a number system, is a human possession, a human creation; it does not exist without us. The possible existence of collections of nine objects is a physical thing, which certainly exists without us. The two kinds of nine are different. Like we can say a plate is round, an objective fact, but the conception of roundness, mathematical roundness, is something else (Hersh, 1997). According to Ernest (1995), the Philosophy of Mathematics is in the middle of a Kuhnian revolution. For over two thousand years, Mathematics has been dominated by an absolutist paradigm, which views it as a body of infallible and objective truth, far removed from the affairs and values of humanity. Currently, this is being challenged by a growing number of philosophers and mathematicians, including Lakatos, Davis and Hersh, Tymoczko. Instead, they are affirming that Mathematics is fallible, changing and, like any other body of knowledge, the product of human inventiveness. How Mathematics is viewed is significant on many levels, but nowhere more so than in education and society. For if Mathematics is a body of infallible, objective knowledge then it can bear no social responsibility. Thus, the underparticipation of sectors of the population, such as women; the sense of cultural alienation from Mathematics felt by many groups of students; the relationship of Mathematics to human affairs, such as the transmission of social and political values; its role in the distribution of wealth and power; none of these issues are relevant to Mathematics. On the other hand, if it is acknowledged that Mathematics is a fallible social construct, then it is a process of inquiry and coming to know, a continually expanding field of human creation and invention, not a finished product. Such a dynamic view of Mathematics needs to include the empowerment of learners to create their own mathematical knowledge; Mathematics can be reshaped, at least in school, to give all groups more access to its concepts, and to the wealth and power its knowledge brings; the social contexts of the uses and practices of Mathematics can no longer be legitimately pushed aside, the implicit values of Mathematics need to be squarely faced. When Mathematics is seen in this way, it needs to be studied in living contexts which are meaningful and relevant to the learners, including their languages, cultures and everyday lives, as well as their school based experiences. This view of Mathematics provides a rationale, as well as a foundation, for multicultural and girl-friendly approaches to Mathematics. Overall, Mathematics becomes responsible for its uses and consequences in education and society. Next Dehaene wonders “What is a number?” Dehaene, as a neurophysiologist studying how the human brain wires itself to do Mathematics, answers that number is a parameter of our physical environment which is extracted and processed by dedicated cerebral networks just like color, which is a subjective property entirely made up by brain area V4. Indeed, he shows how animals and infants have a largely innate intuition about numerical quantities and their properties. Recent experimental evidence suggests that: (i) The human baby is born with innate mechanisms for individuating objects and for extracting the numerosity of small sets. (ii) This ‘number sense’ is also present in animals, and hence that it is independent of language and has a long evolutionary past. (iii) In children, numerical estimation, comparison, counting, simple addition and subtraction all emerge spontaneously without much explicit instruction. (iv) The inferior parietal region of both cerebral hemispheres hosts neuronal circuits dedicated to the mental manipulation of numerical quantities, and that a lesion to that

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area leads to a loss of ‘number sense’, including not knowing what is 3-1, or what number falls between 2 and 4. This inner feeling of quantity serves as a foundation for the later ‘construction of number’ through mathematical axiomatizations. Yet as a basic category of experience provided by a dedicated brain circuit, number is as undefinable as color, space, movement, happiness, or beauty. Dehaene agrees with Hersh that Platonism -the view that mathematical facts are abstract and independent of human existence and knowledge- is not a tenable position. His neurobiological interpretation is that Platonism is a cognitive illusion that imposes itself upon so many great mathematicians because with training, their brains develop a vivid, seemingly real, internal image of mathematical objects. Presumably, one can only become a mathematical genius if one has an outstanding capacity for forming vivid mental representations of abstract mathematical concepts -mental images that soon turn into an illusion, eclipsing the human origins of mathematical objects and endowing them with the semblance of an independent existence. Mathematics is indeed a product of the human mind and brain, and as such it is indeed a very human enterprise, fallible, revisable, and highly dependent on the limits and abilities of our cerebral equipment. Does that mean, however, that mathematics is a purely social activity? The trouble with labelling Mathematics as ‘social’ or ‘humanistic’, and with comparing it to art and religion, is that this view completely fails to capture what is so special about Mathematics -first, its universality, and second, its effectiveness. A mathematician can go to any place in the world and, given enough time, can convince anyone about mathematical truths -3 is a prime number, or that the 3rd decimal of Pi is a 1 or that ‘Fermat’s Last Theorem’ is true. The point is universal agreement is often easily reached about what constitutes a mathematical fact. This makes an unqualified relativistic, social, Lakatosian, or post-modernist view of Mathematics totally untenable. Relativists notwithstanding, the value of Pi does not vary from culture to culture, nor does each culture have its own different ‘mathematical universality’, which is one and only. The other key difference between Mathematics and other cultural objects is its effectiveness. This was, and still is, a subject of awe and wonder for physicists like Wigner and Einstein. “How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?” Einstein asked in 1921. This is bound to remain forever a mystery as long as you adhere to a strong relativistic position, which asserts that Mathematics is the result of the arbitrary cultural choices of mathematical ‘churches’. For that matter, indeed, the effectiveness of Mathematics is also not easy to explain if you believe, as Hersh seems to do, that mathematicians pursue their work for the sole purpose of its abstract beauty. Dehaene’s solution to both of these riddles appeals to evolution -of the brain and of Mathematics. In his opinion, mathematical objects are universal and effective, first, because our biological brains have evolved to progressive internalize universal regularities of the external world -such as the fact that one object plus another object usually makes two objects, and second, because our cultural mathematical constructions have also evolved to fit the physical world. If mathematicians throughout the world converge on the same set of mathematical truths, it is because they all have a similar cerebral organization that: (i) Lets them categorize the world into similar objects -numbers, sets, functions, projections, etc. (ii) Forces to find over and over again the same solutions to the same problems. One can remind of the

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reinvention of place-value number notation in 4 different cultures widely separated in space and time -Chinese, Babylonian, Maya and Indian. The common structure of our brains can explain why there is trans-cultural convergence in Mathematics despite different social perspectives. In that respect, mathematical objects, while they are human constructions, are radically different from other cultural constructions of Western societies such as religion, Nouveau roman, symphony orchestras etc. Pies come in all sorts, Greek, American, or French; but hopefully there’ll always be only one number pi (π) (Dehaene, 1997b). Dehaene claims that number is like color. Because we live in a world full of discrete and movable objects, it is very useful for us to be able to extract number. This can help us to track predators or to select the best foraging grounds, to mention only very obvious examples. This is why evolution has endowed our brains and those of many animal species with simple numerical mechanisms. In animals, these mechanisms are very limited: they are approximate, their representation becomes coarser for increasingly large numbers, and they involve only the simplest arithmetic operations. Humans have also had the remarkable good fortune to develop abilities for language and for symbolic notation. This has enabled us to develop exact mental representations for large numbers, as well as algorithms for precise calculations. Mathematics, or at least Arithmetic and Number Theory, is a pyramid of increasingly more abstract mental constructions based solely on (i) our ability for symbolic notation, and (ii) our nonverbal ability to represent and understand numerical quantities. He argues that many of the difficulties that children face when learning Mathematics and which may turn into fullblown adult ‘innumeracy’ stem from the architecture of our primate brain, which has not evolved for the purpose of doing Mathematics. It is his view that the human brain does not work like a computer and that the physical world is not based on Mathematics -rather Mathematics evolved to explain the physical world the way that the eye evolved to provide sight. What are numbers, really? If we grant that we are all born with a rudimentary number sense that is engraved in the very architecture of our brains by evolution, then clearly numbers should be viewed as a construction of our brains. However, contrary to many social constructs such as art and religion, number and arithmetic are not arbitrary mental constructions. Rather, they are tightly adapted to the external world. Whence this adaptation? The puzzle about the adequacy of our mathematical constructions for the external world loses some of its mystery when one considers two facts: (i) The basic elements on which our mathematical constructions are based, such as numbers, sets, space, and so on, have been rooted in the architecture of our brains by a long evolutionary process. Evolution has incorporated in our minds/ brains structures that are essential to survival and hence to veridical perception of the external world. At the scale we live in, number is essential because we live in a world made of movable, denumerable objects. Things might have been very different if we lived in a purely fluid world, or at an atomic scale, and hence we have to concur with other mathematicians such as Poincare, Delbruck, or Hersh in thinking that other life forms could have had Mathematics very different from our own. (ii) Our Mathematics has seen another evolution, a much faster one: a cultural evolution. Mathematical objects have been generated at will in the minds of mathematicians of the past thirty centuries (this is what we call "pure mathematics"). But then they have

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Dimitris Gavalas been selected for their usefulness in solving real world problems, for instance in Physics. Hence, many of our current mathematical tools are well adapted to the outside world, precisely because they were selected as a function of this fit.

Many mathematicians are Platonists. They think that the Universe is made of mathematical stuff, and that the job of mathematicians is merely to discover it. Dehaene strongly denies this point of view. This does not mean, however, that he is a ‘social constructivist’. On the contrary he believes that mathematical constructions transcend specific human cultures. In his view, however, this is because all human cultures have the same brain architecture that ‘resonates’ to the same mathematical tunes. The value of Pi does not change with culture. Furthermore, he is in no way denying that the external world provides a lot of structure, which gets incorporated into our Mathematics. He only objects to calling the structure of the Universe ‘mathematical’. We develop mathematical models of the world, but these are only models, and they are never fully adequate. Planets do not move in ellipses -elliptic trajectories are a good, but far from perfect approximation. Matter is not made of atoms, electrons, or quarks -all these are good models, indeed, very good ones-, but ones that are bound to require revision some day. A lot of conceptual difficulties could be clarified if mathematicians and theoretical physicists paid more attention to the basic distinction between model and reality, a concept familiar to biologists (Dehaene, 1997c). Finally, according to Cybernetics, number is a conceptual scheme, an abstraction of the second level from specific numbers. The abstraction procedure to recognize specific numbers is counting. Counting is based on the ability to divide the surrounding world up into distinct objects. This ability emerged quite far back in the course of evolution; the vertebrates appear to have it in the same degree as humans do. The use of specific numbers is a natural integrated description complementary to the differential description by recognizing distinct objects. This ability would certainly be advantageous for higher animals in the struggle for existence. And cybernetic apparatus for counting could be very simple -incomparably simpler than for recognition of separate objects in pictures. Yet nature, for some reason, did not give our brain this ability. The numbers we can directly recognize are small, up to five or six at best -though it can be somewhat extended by training. Thus the number 2 is a neuronal concept, but 20 and 200 are not. We can use them only through counting, creating artificial representations in the material external to the brain. The material may be, and was historically, fingers and toes, then pebbles, notches etc., and finally sophisticated signs on paper and electronic states of computer circuitry. For theoretical purposes the best is still the ancient-style representation where a chosen symbol, say 'I' stands for one object. Thus 2 is 'II', and 5 is 'IIIII' (Principia Cybernetica). From the above report, it is obvious that there is a vivid developing research concerning the neurological and social basis of the human mathematical ability. It is also obvious that this research has not yet arrived at definite conclusions. When this happens we will be able to explain in detail both how the human brain and society process the mathematical and especially numerical notions and magnitudes.

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7. Number and Category Theory 7.1. First Definition of a Category A category consists of a collection of ‘objects’ and a collection of ‘morphisms’. Every morphism f has a ‘source/ domain’ object and a ‘target/ co-domain’ object. If the source of f is X and its target is Y, we write f: X → Y. In addition, we have: (1) Given a morphism f: X → Y and a morphism g: Y → Z, there is a morphism gf: X → Z, which we call the ‘composite’ of f and g. (2) Composition is associative: (fg)h = f(gh). (3) For each object X there is morphism 1Χ: X → X, called the ‘identity’ of X. For any f: X → Y we have 1X f = f 1Y = f. We should visualize the composite of f: X → Y and g: Y → Z as follows: X⎯f→Y ⏐ gf g ↓ Z. This is a first definition of a category.

7.2. Existence of a ‘Natural Numbers Object’ Category and Topos Theory is based only on objects and arrows/ morphisms between them. So, for an arbitrary Topos E the Axiom of Infinity states that there is an object N of E with arrows 0 και s: 1 ⎯0→ Ν ⎯s→ Ν (1) such that for any object X of E with arrows x and f: 1 ⎯x→ X ⎯f→ X there is a unique arrow h which makes the following diagram commute: 1 ⎯0→ Ν ⎯s→ h 1 ⎯x→

X ⎯f→

Ν h

(2). X

The object N is then called a natural numbers object (N.N.O.) for E. For such a N.N.O. N the definition states that the diagram (1) is universal among diagrams of the form 1 ⎯x→ X ⎯f→ X. It readily follows that N, together with the arrows 0 and s, is unique up to isomorphism. We can thus speak of the N.N.O. of a Topos E, if there is one. In Set the usual set of all natural numbers N={0, 1, 2, …} has the required universal property for a N.N.O. as in (2), where the arrow 0 sends the one element of 1={0} to 0∈N, while s is the usual successor function n → n+1. Given a set X, an element x∈X and a function f: X → X, the arrow h uniquely provided by (2) thus satisfies h(0)=x, h(n+1)=f(h(n)) (3). In other words, h is de-

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fined from x and f by recursion or as one often says by induction. (Mac Lane, 1986; Mac Lane & Moerdijk, 1992).

7.3. ‘Natural Numbers Object’ in ETCS (Elementary Theory of the Category of Sets) Since Dedekind raised the issue for modern Mathematics, some claim that mathematical objects should have only relevant properties. So, natural numbers should have only arithmetic properties. Every ZF set has non-arithmetic properties given by its particular elements. So, on this view numbers cannot be ZF sets. But ETCS models of Arithmetic have no irrelevant properties. So, this view could say the numbers form an ETCS set (Benacerraf, 1965). Specifically, ‘recursion data’ on a set X means an element x of X and a function f: X → X. We define a ‘natural number object’ to be a set N with recursion data 0 and s: N → N, called zero and successor, such that: For any recursion data x, f on any X there is a unique function h: N → X with h(0) = x, h(sn) = f(h(n)), for all n∈N. Provably in ETCS there are infinitely many different natural number objects, but all are isomorphic. None has any properties but the shared ones which follow from the definition (McLarty, 1994). Benacerraf and McLarty show the structuralism of ETCS.

8. Number and Non-Standard View The existence of non-standard models for mathematical structures implies from Löewenheim-Skolem Theorems. It is also known from Gödel’s Theorems that the intuitively clear system of the natural numbers can not be fully characterized axiomatically. Therefore, the axioms about N do not conceive the intuitive depth of the natural numbers; consequently is given the chance to various intuitive aspects about N to coexist having simultaneously their axiomatic analogue. It is generally accepted for instance that the smaller the natural numbers are the more familiar man is with them; conversely the bigger they become the less accessible and difficult to be understandable. Let us consider a natural model in order to better comprehend the fuzzy observable nature of the predicate ‘standard’. Let’s assume that we have a series of cards:

0

1

2

… 3 …

… n …

We paint the first card white using a specific quantity of white color. Next, we add a tiny bit of black color, so as we cannot discern, with our observing abilities, cards 1 and 2 in relation to color. We continue in this way until we have a completely black card and we continue to paint in black all the following cards. Hence, the predicate ‘standard’ determines the white cards that are literally the cards which site before the completely black card, which in their turn can mean our familiarization with the corresponding natural numbers. Therefore, when we say ‘standard natural’ we mean here ‘white number’ and when we say ‘non-standard natural’ we mean ‘black number’ and non-accessible. We must underline here that there is not ‘maximum’ white card neither ‘minimum’ black card. We can consider also that ‘standard natural’ means ‘graduated finite white number’ and ‘non-standard natural’ means ‘graduated

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infinite black number’. So the natural numbers can be presented as a tape where the white color becomes gradually black: standard

0

1

non-standard

2

3

4 ...

n ... fuzzy borders

We make the following assumptions: (i) 0 is standard natural number. (ii) For every n∈Ν, n is standard natural implies n+1 is standard. (iii) There is a non-standard natural ν∈Ν. (iv) The Principle of Induction also holds: If φ is any logical expression, then (φ(0) and φ(n) → φ(n+1) for every standard n∈Ν), implies that formula φ also holds for every standard natural (Nelson, 1977, 1988). We have to note that (i) and (ii) indeed hold in the colorful tape: 0 is white and if a card is white, then its successor is also white, since observably are indiscriminate. Also from (ii) is obvious that there is no minimum non-standard that is ‘minimum’ black card, neither maximum standard that is ‘maximum’ white card. Hence it is obvious that standards are separated by non-standards with fuzzy limits and therefore standards cannot be set of ZFC, but essentially are a fuzzy set or differently a class or semiset, according to Vopenka’s (1979) terminology. The local non-standard regard of N has as consequence the local non-standard regard of many structures and particularly of R. A basic ambiguity is imported thus in Mathematics, which is covered behind the initial not-definable predicate ‘standard’. Same observations that are made for N are also in effect for R, that is to say the axioms of R are not capable to conceive the intuitive perception we might have for this.

9. Number and Fuzziness According to Kosko (1994), everything is a matter of degree; the same holds for numbers, which are the hallmark of precision. The question then is: Can we fuzz these up too? Are fuzzy numbers possible? The old answer is no; how could numbers be fuzzy? Numbers are ‘pure forms’ and they either are or are not, they belong all or none to the well-known sets of numbers. Every whole number is either even or odd, no in between, no middle ground, no gray cases. Mathematicians and scientists have devised Mathematics that seems to have escaped the sloppiness of everyday fuzzy sets and thoughts. Where could fuzziness hide in the black-and-white world of Mathematics? If we consider number zero (0) we can find it at a glance as a spike on a number line. The spike means the number zero belongs 100% to the

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set, say, ZERO and no other number belongs to it. Every number is either in the ZERO set or out of it; all or none. In this set sense the number zero alone belongs to the set ZERO. But what about numbers close to, almost, nearly zero? These numbers are fuzzy numbers, they define a spectrum of numbers near zero and some belong more in the set than others. The closer a small number to zero is, the more it belongs to the fuzzy set of small numbers. The number 1 is closer to 0 than the number 2 is, and 2 is closer than 3 is, and so on. The number 0 belongs 100% to the set ZERO but close numbers may belong only 80% or 50% or 10%. We might draw the fuzzy number zero as a triangle centered at the exact number 0. If we draw the triangle narrow enough, we get back the spike of classical Mathematics. That is another surprise: Mathematics as we know it is but a special case of fuzzy Mathematics, a special limiting case -the degenerate case of black-and-white extremes in a mathematical world of grays. We can add and subtract triangles just as we add and subtract spikes/ numbers. We can also draw the fuzzy number ZERO in infinitely many ways; each one can draw it differently just as each one thinks differently of HOUSE, SMALL, SMART, FAIR, NICE or CLEAN. There are as many ways to draw the fuzzy number ZERO as there are numbers.

Exact/ Non-fuzzy Number Zero (0) and Inexact/ Fuzzy Number Zero (-2, 0, 2).

What can we do with fuzzy numbers? We can reason with them; we do it all the time. Fuzzy Logic means reasoning with fuzzy numbers and sets. The knowledge or intelligence comes from associating fuzzy events of everyday. For now we have proved the point: Numbers are fuzzy too. We work with fuzzy numbers all the time and, if numbers are fuzzy, then everything is and indeed is. So, according to the above view, the consideration of a number can be double: Classical/ exact or fuzzy/ inexact. If we take the horizontal axis of the real numbers, then we can add the vertical axis up to 1 and transform the exact representation of the number to fuzzy representation using a membership function, for example triangular, trapezoid, conoid etc. Consequently, a fuzzy number can be represented by a triangle, the apex of which lies exactly above the number in one unit height (1) and its basis lies on the horizontal axis. The classical/ exact eight (8), for example, has a basis limited to a point, number 8, and therefore is represented by a vertical arrow sited on point 8 and of one unit height. Hence, while exact 8

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can be considered as a unique triad of the form (8,8,8), the fuzzy 8 has the possibility for many triangles such as: basis’ interval (7,9) or (6,9) according to data and then the fuzzy numbers are correspondingly (7,8,9) and (6,8,9). The ordered triad which represents a fuzzy number has as its centre the apex of the triangle and therefore the corresponding exact number. The first number shows the left point of the basis and the third one shows the right point. For example, the (3,5,7) is the fuzzy number 5 with basis the interval (3,7). In conclusion, we would say that a fuzzy number is a quantity the value of which is not accurate, as it happens with usual numbers, but can be considered as a function with domain the set of the real numbers and co-domain the interval of the reals between 0 and 1. To each numerical value in the interval is attributed a particular membership degree, where 0 represents the minimum possible grade and 1 the maximum. Fuzzy numbers on many aspects represent the natural world more realistic than the usual numbers. Fuzzy numbers are used in Statistics, Programming and Science. The concept of ‘fuzzy number’ considers the fact that natural phenomena have a degree of uncertainty. A way then to represent uncertainty is to use fuzzy numbers and sets. In some cases, a number can be known only approximately or inexact even though its exact value is known. Such paradigms are well-known in Programming.

PART III: JUNG AND THE CONCEPT OF ARCHETYPE 10. ‘Chthonic’ and ‘Celestial’ Mathematics There is a common sense of antithesis between, on the one hand, the ‘Superior World’ of ideal mathematical objects, archetypal ideas, pure relationships, qualities and the like (‘Celestial’ Mathematics) and, on the other hand, the current ‘inferior world’ of material objects, quantity, amplitude, application, counting (‘Chthonic’ Mathematics). This is an important and basic antithesis. This basic antithesis looks impressive and may be not so random. If we also base on this sense of antithesis, we can say that there are some conclusions and some messages are transmitted by the nature of this situation. Indeed, if we consider as ‘horizontal plane’ the usual human empirical consciousness, which generally conceives objects in motion into a time-spatial continuum and deals with them and their governing laws, then this plane is crossed by a ‘vertical plane’ and the only thing we can say about this order is that it is, on the one hand, something mathematically abstract and, on the other hand, legend, sacred and mysterious. If we face the number as revelation/ discovery and not as invention for measuring, then because its nature is ‘mythological’ it belongs to the region of ‘divine’ forms and it is as archetypal as they do. But unlikely to them, it is also real, in the sense that we meet it on the region of human experience as quantity. Therefore, it constitutes the bridge and the intersection between the natural, real, outside world and the ideal, imaginary, inner world. This means that it participates in both worlds. It is a part of the material and a part of the psychic. It does not only count, neither is it quantitative and static. It also denotes the quality and it is a dynamical schema and for this reason it is an entity, which is in part revelation and in part invention. This exact double nature of the number makes it symbolic and a vehicle of mental and psychic processes.

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11. Jung’s - Pauli’s General Hypothesis of Archetypes Jung wrote that number is the key to the mystery, since it is just as much discovered as it is invented. It is a quantity as well as a meaning. He understood number to be the most primitive element of order in the human mind and defined number psychologically as an archetype of order which has became conscious. Von Franz (1974) undertook an exhaustive investigation of number archetype acting as dynamic organizing principle in both psyche and matter. She published her findings in Number and Time and her work represents a significant extension of the General Hypothesis of Archetypes of Jung-Pauli. Indeed, according to Jung’s - Pauli’s General Hypothesis of Archetypes (Card, 1991, 1993, 1995, 1996; Cohen, 1975; Jung 1957-1977; Jung & Pauli, 1955; Meier, 1992, 2001; Nagy, 1991; Stevens, 1982), the frames of mind and matter -psyche and nature- are complementary views of the same transcendental reality which they call ‘unus mundus’. The archetypes function as fundamental dynamical patterns, the various representations of which characterize all procedures either mental or physical. In the psychic realm the archetypes organize the ideas and the images, while in the physical one they organize the structure and the transformations of matter and energy and also are responsible for the order. The simultaneous action of the archetypes both in the psychic and in the physical realm is responsible for the cases of synchronistic phenomena, that is phenomena which are linked acausally, but have the same or similar meaning for the observer (Peat, 1987). Later on, von Franz’s (1974, 1992) work on the number archetype cleared up and extended Jung’s - Pauli’s General Hypothesis of Archetypes as follows: All mental and physical phenomena are complementary aspects of the same unitary, transcendental reality. At the basis of all physical and mental phenomena there exist certain fundamental dynamical forms or patterns of behavior which may be called number archetype and they are manifestations of the later. Any specific process, physical or mental, is a particular representation of certain of this archetype. In particular, the number archetype provides the basis for all possible symbolic expression. Therefore, it is possible that a neutral language/ hyper-code constructed from abstract symbolic representations of the number archetype may provide highly unified, although not unique, descriptions of all mental and physical phenomena. This view leads to the potential of arithmetization/ mathematization of the totality of psychophysical phenomena, which are the study matter of Cognitive Science (Warner & Szubka, 1995). The parallel, also, between von Franz’s search for primal archetype inherent in numbers and Chomsky’s search for linguistic universals has been examined by the physicist Card (2000). Von Franz found a quaternary of order archetypes connected immediately with the first four numbers. The representations of these archetypes are very often dynamical in nature. She condenses their characteristics as follows: Numbers become then typical dynamical patterns, for which we have to say the following: One (1) comprises totality. Two (2) divides, repeats and generates symmetries. Three (3) gives center to the symmetry and introduces linear succession. Four (4) functions as a stabilizer by returning to one and it presents four aspects creating limits. The fourth order archetype is interesting since its various representations concern feedback and self- reference. Self- reference is a repeating subject in the development of Set Theory and Symbolic Logic that led to Gödel’s Incompleteness Theorem. We have to note here that it plays an important role in the chaotic dynamics of non-linear systems. Eventually, the well-known mathematician and author Ian Stewart does not exclude the fact of being ar-

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chetypal patterns of numbers, a common set of ideas in many minds for which our intellect is predisposed to work and thanks to them to function. Since archetypes precondition all existence, they are manifest in the spiritual achievements of science, art and religion, as well as in the organization of organic and inorganic matter. The archetype thus provides a basis for a common understanding of data derived from all sciences and human activities -not least because of its implications for epistemology.

12. Number as Archetype “Number is an archetype of order that has become conscious”, Jung

In the platonic dialogue Menon, Socrates addresses questions to slave Menon and he shows thus, beginning from a particular square, how is constructed another one having the double surface. Menon recognizes that an obvious construction is not correct and afterwards he recognizes the correct one. Then Socrates results to the following conclusion: “The knowledge he has now either he acquired it some time in the past or he had it all along. If he had it all along, he also knew all these all along, but if he acquired it some time in the past, it is not possible he acquired it in his present life. Did someone teach him Geometry? Because he will do the same he just did both for all Geometry and for each other discipline of knowledge. But did anyone really teach him all these?” The answer is ‘no’ because Menon is an illiterate slave. This experiment shows two things: on the one hand that there are three stages in order for someone to reach knowledge. Firstly, knowledge is unconscious. Next, conjecture and opinion rouse up with questions and through the dialectic approach. Finally conjecture and opinion are transformed into knowledge through the understanding of the relation that exists between the cognitive object and the idea/ form from which it comes. On the other hand that we know truths we never learned externally, that is from education and experience. This knowledge is a pattern/ model of the universal truths which we can conceive and acknowledge. Finally, there is a higher hierarchical level of the absolute knowledge and truth, fountain-head of the absolute knowledge and truth, fountain-head of the knowledge of Good (Hill, 1992). The modern research detects the existence of innate models, universal dynamical schemes, which are well-known as archetypes and are different from the platonic ideas/ forms. Indeed, what Plato called idea, is a model of ultimate perfection only with the positive/ bright meaning, while archetype is bipolar and realizes both the positive and the negative/ shadowy side, because it is not a pattern of perfection, but of wholeness. In addition, it is accompanied by emotional charge. Therefore a kind of knowledge based on innate principles mostly, not acquired or resulting from the external experience, is the system of archetypes. The study of archetypes and their structure, like the study of numbers, can lead to better understanding of the typical properties of mind. The archetypes are structures, models in the mental realm, norms and limitations, patterns of mental and psychic behavior in the unconscious, giving the possibility to unconscious messages to be sent to the conscious mind with the form of images. The nature of these messages is symbolic. The archetypes correspond to a universal grammar of mind. They predispose it towards particular kinds of universal knowledge, assuring that some issues are universal, like myths, numbers, language structures, ideas/ forms.

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The archetypes provide the potential and the circumstances for the production of ideas and fantasy. They are not found in the mind as result of information, practice or teaching, but instead are a pan-human legacy. An archetype is not immediately known, but becomes known only through the emergence of the archetypal image, presented spontaneously. Because the archetypal information has a universal basis, out of the particular space-time, it is independent on specific experience. Our coincidence with our conscious and the rational rejects, repulses or represses archetypes and their images, but these always return with one form or another. However, this form is always inside a frame and not completely free, i.e. it is not free of being any form. Archetypes in general have their own special logic and structure and these special elements illustrate the structure of mind. The above mentioned issues lead us to think that the opposition between the ‘Superior World’ of Pure Mathematics and the ‘Inferior World’ of Applied Mathematics is not absolute. These two Worlds are seemingly incompatible, because the connecting bridge is not missing. Between them, as common factor, stands a medium, the number, the reality of which is valid in both Worlds, because it is exactly an archetype in its essence, in its ultimate reality. Number belongs to both Worlds real and fantastic; it is visible and invisible, quantitative and qualitative. The fact that number participates and characterizes the nature of the medium form and that appears as an interceder between the basic opposition and all the others, is a very important fact. Because we immediately conclude that the symbol realizing the intermediation, the conjunction, the wholeness, the unity, the order, the identity, the synthesis, the reconciliation can be generally expressed by a mathematical fashion. The meaning of this symbol necessarily presupposes an excess which does not lead to some metaphysical hypothesis. It is just a borderline meaning and the fact that exists something beyond this cognitive limit is proved by the spontaneous appearance and observation of archetypes and more clearly by number. The latter, on the one side of the limit is quantity and on the other side autonomous entity capable of qualitative denotations expressed in a priori structures of order, regularity and eurhythmy. With the existence of a factor, mediating between the two phenomenally incompatible Worlds, we know for sure today that the one World takes through this exact factor, properties of the other World. With each other’s help the two Worlds interact continuously and even more the ultimate reality bases on a common substratum, still unknown, having properties simultaneously from the two Worlds which gives us the chance to build a new model of the World, closer to the idea of the ‘unus mundus’. It seems that the method best fits to the nature of order and chance is arithmetic. From ancient years people used numbers to express the meaning of order, as well as coincidences with meaning, the ones that can be interpreted. Also if we denude a set of objects from all of its properties and its characteristics, that is we consider an abstract set, what finally remains is its multitude, that is its number, a fact the denotes at least that number is a magnitude initial and irreducible. The succession of natural numbers seems to be something more than a simple accumulation of identity monads. It includes the set of Mathematics and all that are going to be discovered in this field. That is why number is in a sense non-predictable entity. Number helps more than anything else to the ordering of the chaos of phenomena. It is the predefined tool for the creation of order or to become understandable some already existing, but still unknown smooth arrangement. May be it is the most primitive element of order of the human mind. The fact that numbers have an archetypal foundation, it makes obvious that it is not so arbitrary to define number as an archetype of order which became conscious by man. Even the

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intuitionist/ constructivist Brower said that Mathematics refer to becoming and not to being, a view completely platonic. It is a spiritual action, which is in direct relation with the possibility of conceptual construction of all mathematical entities from the natural numbers. People believe in general that numbers were invented or were the result of thinking and for this reason they are nothing else but concepts of quantities, the content of which was attributed by the human intellect. But it is equally possible the numbers to be discovered. In this case they are not just concepts, but something more: autonomous entities which contain, in a way something more than mere quantities. Contrary to the concepts, they do not base in a hypothesis, they are un-hypothetical according to Plato, but to the property of identity, i.e. they are themselves, something that cannot be expressed by a conscious intellectual invention. Under these circumstances it is possible to attribute to them easily some properties which are not yet been discovered. We tend towards the synthetic view that numbers both were invented and discovered and therefore have an autonomy analogous to that of the archetype. Consequently, they preexisted of consciousness and that is why they determine it than being determined by it. Also, the archetype as an a priori ideal form both was invented and discovered. It was discovered because nobody knew its unconscious, autonomous existence and was invented because its presence was the effect of analogous human psychophysical structures. It seems though, that natural numbers have archetypal character. Some numbers and combinations of numbers relate to or act on particular archetypes. The opposite is real also, as the research of Jung, von Franz etc assure. The set of natural numbers is an archetype of order, which is illustrated in consciousness, helping it to place the chaos of phenomena into some sort of ordering diagram. The system of natural numbers is a form of a priori knowledge, extremely capable of organizing the random nature of experience. Exactly like the native language it is being learned easily, because different kinds of numerical systems are less accessible. They were not named natural numbers without special reason. Number which is used for counting and the quantitative relations is something more than we generally believe. It is simultaneously an entity of the same order with the mythological elements that is why the Pythagoreans thought it sacred. But when we use it only for practical purposes, we do not seem to realize that aspect. The archetypal structures and numbers as such, are not static forms but dynamic elements. We perceive the special power of archetypes and numbers when we have the opportunity to appreciate the appeal, that is the emotion through the unconscious or the awe or the fear that exert on us. They seem fateful and in this case we can think as example numerophobia, stress for Mathematics, superstitions, arithmomancy etc. It is characteristic the issue of number 13, which is taboo for many people, it has unfavorable character and the superstitious feels awe for this number. Archetypes are images and simultaneously emotions and we can talk about archetypes only when these two aspects are presented simultaneously. When the image is charged with emotional intensity it acquires mysterious, ‘divine’ dimension, it becomes dynamic and it necessarily has impacts on the person. The fact that number seems to be initial and innate concept for man, which is accompanied by emotion, is interpreted by Gauss sayings: “It is curious that all people who study seriously the science of Number Theory, are dominated by a kind of passion for it.” Cantor also confesses: “When I think and study infinite, it always follows a real pleasure, in which I succumb pleasantly as I see that the concept of the integer number is split in two concepts and we ascend to the infinite. The one is the concept of power (cardinality) and the other the concept of counting.’’

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Even those contents that are considered the most certain of the consciousness are surrounded by the shadow of uncertainty and vagueness. And the most austere mathematical concept, that we believe it contains only those elements we attribute to it and is well defined and limited, it is something more than we believe it is. It is simply a psychic fact and as such it is partially unknown just like number. Of course these faint and implicate differentiations are being abolished by being considered needless, ignorable, not worth a fig or they do not have relation to our needs and their usual applications. Even though we face exact definitions and clear analysis, we discover sometime the most unexpected differentiations, not only with the pure mental concept of the term, but with the value that is attributed to number as well as with its application. The fact that these differentiations exist displays exactly what was originally sustained. Weyl believed about natural numbers and used to say that he could not understand how something so simple, constructed by the human mind, could contain something abysmal. He should ask himself first, whether the human mind really constructed them. He feels that he controls and leads completely the phenomenon, but this is not true. We claim that what happens is exactly the opposite that is numbers as archetypal dynamic processes control us at least up to a point. Weyl by the term ‘abysmal’ means that irrational factor, which is contained in all numbers, and which we cannot conceive and explain with rational terms. It is very interesting here to see what a psychologist of the Jungian school, von Franz (1980), exactly says about this issue: I want to read you in detail what the well-known mathematician, Hermann Weyl, says in his book Philosophy of Mathematics and Natural Science. You know that until about 1930 the great and passionate occupation of most mathematicians was the discussion of the fundamentals. They hoped, as has been the fashion nowadays, to re-discuss the fundamentals of all science. But the famous German mathematician, David Hilbert, created a new construction of the whole building of Mathematics, so to speak, and hoped that this would contain no internal contradictions. There would be a few basic axioms on which one could build up all branches of Mathematics: Topology, Geometry, Algebra, and so on; it was to be a big building with solid foundations in a few axioms. That was in 1926, and Hilbert was even bold enough to say: “I think that with my theory the discussion of fundamentals has been forever removed from Mathematics.”

Then in 1931 came another very famous mathematician, Kurt Gödel, who took a few of those basic axioms and showed that one could reach complete contradictions with them: Starting from the same axioms, one could prove something and its complete opposite. In other words, he showed that the basic axioms contain an irrational factor which could not be eliminated. Nowadays in Mathematics one must not say that obviously this is so-andso and that therefore that and that are also so, but: “I assume that it is so-and-so, and if so then that and that follow”. The axioms must be presented as assumptions, or must be postulated, after which a logical deduction can be made, but one cannot infer that what has been assumed or postulated could not be contradicted or doubted as an absolute truth. In order to make such assumptions, Mathematics is generally formulated in such terms as: “It is self-evident” or “It is reasonable to think” -that is how mathematicians posit an axiom nowadays, and from there they build up. From then on there is no contradiction, only one conclusion is possible, but in “it is reasonable to assume” that is where the dog lies buried, as we say. Gödel showed that, and thus threw over the whole thing. Strangely enough that

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did not reopen the discussion of fundamentals. From then on, as Weyl says, nobody touched that problem, they just felt awkward and scratched behind their ears and said, “Don't let’s discuss fundamentals, there’s nothing doing: it is reasonable to assume, we cannot go beyond that” and there the situation rests today. Weyl, however, went through a very interesting development. At first he was very much attracted by the physicist Werner Heisenberg. He was very much of a Pythagorean and was attracted by the numinosity and irrationality of natural integers. Then he became fascinated by David Hilbert, and in the middle of his life had a period during which he became more and more attracted by Hilbertian logic and dropped the problem of numbers, treating them, erroneously as I think, as simply posited quantities. He says, for instance, that natural integers are just as though one took a stick and made a row of marks, which one then named conventionally; there was nothing more behind them, they were simply posited by the human mind and there was nothing mysterious about them; it was “reasonable and self-evident” that one could do that. But at the end of his life he added -only to the German edition of his book on the philosophy of Mathematics, and shortly before his death- this passage: “The beautiful hope we had of freeing the world of the discussion of fundamentals was destroyed by Kurt Gödel in 1931 and the ultimate basis and real meaning of Mathematics are still an open problem. Perhaps one makes Mathematics as one does music and it is just one of man’s creative activities, and though the idea of an existing completely transcendental world is the basic principle of all formalism, each mathematical formalism has at every step the characteristics of being incomplete (which means that every mathematical theory is consistent in itself but is incomplete, at the borders are questions which are not self-evident, are not clear, and are incomplete) in so far as there are always problems, even of a simple arithmetical nature, which can be formulated in the frame of a formalism, but which cannot be decided by deduction within the formalism itself.”

That is put in a mathematician’s complicated way; put simply, it means that I daresay it is self-evident, by which I posit something irrational, because it is not self-evident. Now one could make an uroboros movement and say: “But from my deduction I can reprove my beginning”. You cannot! You cannot from the deductive formalism afterwards deduce a proof, except by a tautology, which naturally is not allowed, even in Mathematics. “We are therefore not surprised that in an isolated phenomenal existence a piece of nature surprises us by its irrationality and that one cannot analyze it completely. As we have seen, Physics therefore projects everything which exists onto the background of possibility or probability”.

That is important because it sums up in one word what modern science does. In other words, any fragment of phenomenal existence, let us say these spectacles, contains something irrational which one cannot exhaust in physical analysis. Why the electrons of these millions and millions of atoms of which my spectacles consist are in this place and not in another, I cannot explain; therefore through Physics, when it comes to a single event in nature, there is no completely valid explanation. The single event is always irrational, but in Physics one proceeds by projecting this onto the background of a possible, i.e., one makes a matrix. For instance, in these spectacles there

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are so many atoms and so many particles of them, and so on, and out of a whole group one can make a mathematical formula in which one could even count the particles -not 1, 2, 3, 4, 5, but by projecting onto the background of what is possible. That is why these matrices are nowadays used in engineering and so on, because one can cope with the uncountable; they provide an instrument with which to cope with the things which cannot be counted singly. Weyl says: “It is not surprising that any bit of nature we may choose (these spectacles or anything) has an ultimate irrational factor which we cannot and never will explain and that we can only describe it, as in physics, by projecting it onto the background of the possible”.

But then he continues: “But it is very surprising that something which the human mind has created itself, namely the series of whole natural integers (I told you that he has this erroneous idea that the human mind created 1, 2, 3, 4, 5, by making dots), and which is so absolutely simple and transparent to the constructive spirit, also contains an aspect of something abysmal which we cannot grasp”.

That is the confession of one of the most remarkable -because one of the most philosophically oriented- modern mathematicians, Hermann Weyl. We can naturally say that we do not believe what he believed, namely that the natural integers simply represent the naming of posited dots, therefore to us it is not surprising that natural integers are abysmal and beyond our grasp. He believed that, and that is why he could not understand. It is incredible that it should be so, but it is so; in other words, because the natural integers have something irrational (he called it abysmal) the fundamentals of Mathematics are not solid, because the whole of Mathematics is ultimately based on the givenness of the series of natural integers. Now precisely because numbers are irrational and abysmal -to quote Weyl- they are a good instrument with which to grasp something irrational. If one uses numbers to grasp the irrational, one uses irrational means to get hold of something irrational, and that is the basis of divination. They took those irrational, abysmal numbers which nobody has so far understood, and tried to guess reality or their connection with reality (von Franz, 1980). This other factor is illustrated very well on the mythological aspects of number, which are depicted among the others to the anaglyphs of ancient peoples, like the Mayas about 700 B.C, when the numerical subdivisions of time were personalized by gods. Here characteristic is the fact that Kant reduces the concept of number to that of time, because the numeration is in immediate relation with some time evolution. Also, according to the known, the pyramid with the spots presents the Pythagorean ‘tetraktyn’ (quaternary number). It consists of the most used natural numbers 1,2,3,4 which form the sum 10. 4 and 10 were for the Pythagoreans essentially sacred numbers. The archetype, as well as number, is not just a word or concept, it is literally a part of life itself, an undivided part of the vivid entity and this happens through emotion. While we do not perceive the special emotional tone of the archetype, we can intellectually prove that it has a meaning or that it has not. But the archetype starts living only when we try patiently to

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discover why and how it has meaning for the particular vivid person. While we consider the archetype as a mere image or idea, the ‘divine’ power of which we never felt, we will talk without knowing for which thing we talk and the words we use are empty and of no value. The archetype gets vivid only when we try to understand its ‘divine’ existence and its relation to the person. Only that moment do we start to understand that the word archetype means some particular things and that everything depends on the way these are connected to us, in other words on our stance towards it. The field Jung considered as the most fertile for future research was that of the basic mathematical axioms, the first mathematical intuitions, paying special attention to the idea of the infinite series of the natural numbers and the continuum in Geometry. As Arendt (1958) says, contemporary Mathematics becomes the science that studies the structure of the human spirit. In this line of thought it was also discovered that our representations are ‘ordered’ before we even realize it. Van der Waerden (1975), who refers many paradigms of elementary mathematical intuitions that come from the unconscious, concluded that the unconscious is capable not only of taking in but also of combining and judging. The judgment of the unconscious is intuitive, but when the circumstances allow it is perfectly certain. Among the many initial mathematical intuitions or a priori ideas, the natural numbers are the most interesting ones. These numbers are not only useful for counting or executing numerical operations, but also for centuries were the unique means man had in order to decipher the meaning of the various divination techniques and oracles. Of such kind were astrology, numerology, and geomancy etc, which base on the numerical calculation and were interpreted by Jung in terms of synchronicity that is a-causal coincidence bearing meaning to the observer. Besides, the natural numbers are certainly archetypal representations, because we are obliged to bring them in our mind in a particular way. No one for example denies that two (2) is the only and first number which means a couple, even if one has not thought of it consciously. In other words numbers are not concepts invented consciously by man just to execute calculations. It is obvious that they are spontaneous, autonomous creations of the unconscious, just like the rest archetypal symbols. The natural numbers though, are also properties of the external objects: We can assure and calculate that here are two stones and three trees. When we denude the objects from their properties, what is left is the common property of being together many of them that is a particular magnitude. But the same numbers are undoubtedly part of our psychic composition; they are abstract concepts we can study without reference to external objects. Therefore numbers are presented to us as a concrete association between the two fields: nature and psyche. Therefore number is revealing: it expresses the total World and translates logically the hidden relations of its elements, on its nature is irrational, is reduced to the wholeness, but when it is analyzed and developed it reveals universal relations, movements and positions. In the universe, number is greater than the natural magnitudes, because the latter are expressed by number. Number de-materializes World and its powers, reduces the empirical thought to pure contemplate, to noesis, it reaches the human intellect in its origins, since number belongs neither to time nor to space, since monad, contains everything and is contained in everything. According to the Pythagoreans, we thus reach the depths of the myth; we touch what can be called mystery of the mysteries. It is characteristic the observation that human intellect places everything in a finite World. Mind can assert truths, which the human intellect cannot confirm (Gödel, undecidable propositions). Intellect is the capability of counting and numbering, mind is the realizing of number itself.

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13. Jung and Cantor - Gödel 13.1. Cantor’s Continuum Hypothesis Cantor develops Set Theory introducing the cardinal number, card A, of a set A. The cardinal numbers are linearly ordered. Cantor proves that the set R of real numbers is not denumerable; this means that card N < card R. He then raises the question “Is there any cardinal number properly between these two?” The statement that there is no such is Cantor’s Continuum Hypothesis (CH). R is commonly called ‘the continuum’. For ZF Set Theory many sentences are independent of the axioms and so remain undecided and undecideable; one such is the CH: There is no cardinal number between the cardinal of the natural numbers and that of the reals. It turns out that CH can be neither proved nor disproved in ZFC: It is independent of ZFC. The second half of this result is due to Gödel: CH is consistent with ZFC; that is its negation cannot be proved from ZFC. Subsequently Cohen establishes the other half of the independence of CH: it cannot be proved from ZFC. CH is thus an example of an interesting statement about sets which is independent of the axioms of ZFC; similarly the Axiom of Choice (AC) is independent of ZFC. Cantor uses for the cardinal number of N, the symbol ℵ0 (aleph-naught) -the countable infinity, the infinity of the natural numbers-, the symbol 2ℵ0 for the power-set of ℵ0, and the symbol ℵ1 for the continuum infinity, the infinity of the real numbers. Since the cardinals are comparable by twos, CH is expressed by the successive cardinal; then CH says, in mathematical terms, that 2ℵ0 =ℵ1. This says that the continuum is the very next transfinite number after the infinity of the natural numbers. Gödel points out just how far this is from being proved. But nothing has been proved so far about the question what the power of the continuum is. In other words, there might be one other infinity in between, or a million, or a countably infinite number, or even an uncountably infinite number of infinities in between the natural numbers and the continuum. Cohen believes that there probably is no limit on how much such infinity that there are and suggested the likelihood that the CH was obviously false, and he conjectured that 2ℵ0 might well turn out to be larger than any transfinite aleph. In his view, the continuum is an incredibly rich set one produced by a bold new axiom which could never be approached by any piecemeal process of construction (Bell, 1986; Gödel, 1947; Hofstadter, 1982; Mac Lane, 1986). 13.2. Gödel’s Incompleteness Theorems Studying several views of this issue (Gödel, 1931; Nagel & Newman, 1973; Mac Lane, 1986; Uspensky, 1987), we can say that in 1931 Gödel proves two ‘incompleteness’ theorems. Gödel considers a formal theory T which contains Arithmetic and shows how to construct a sentence G such that neither G nor not-G could be proved within the system T. Such a G is then undecideable and its existence is the Gödel’s First Incompleteness Theorem. Gödel also shows that we can formulate within the system T a sentence con(T) which interpreted states that ‘T is consistent’. Then he shows this sentence could not be proved within the system T; in other words, no such system T is strong enough to establish its own consistency. It is the Gödel’s Second Incompleteness Theorem. Besides the worth of these two theorems, it is of particular significance the way which Gödel follows to prove these theorems. The proofs depend on a coding -arithmetized symbols-, by ‘Gödel numbers’ which translate statements about proofs of T to statements about

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numbers within T, i.e. into the language of Arithmetic. Since T contains ordinary Arithmetic, it has constants such as 0, 0΄=1, 0΄΄ = 2 etc. for numerals, variables x, y, z, … for numbers, equality, a successor function and symbols for operations, as well as, the apparatus of the first order predicate calculus. By the use of ‘Chinese Remainder Theorem’, in T we can define the prime number and prove in T the ‘Fundamental Theorem of Arithmetic’: Every natural number is uniquely a product of primes. By explicitly enumerating formulas and proofs in T in some code, we can replace statements about proofs by statements about the corresponding numbers, their codes. Here is an example of a code: ¬ 2

∨ 3

∃ 5

0 7

΄ 11

= 13

+ 17

* 19

x 23

y 29

z 31

Each formula is a finite list of symbols, so is coded by a list of numbers n1, n2, n3, …: This can be replaced by a single number m = 2n1 3n2 5n3 … Conversely, give a number m, we can factor it into primes and so read off the corresponding list of basic symbols. An analogous procedure is followed just for the proofs in T. The above process achieves to establish an 1-1 mapping between all the formulas and the proofs on the one hand and a definite subset of the natural numbers -in particular the set of prime numbers P- on the other hand. What characterizes not only Gödel’s work but also the new era which initiates with it, is the arithmetizing of Meta-mathematics that is the correspondence of logical relations to pure arithmetical relations. With this arithmetizing, it is finally achieved a reduction of Meta-mathematics, as it happened before with Mathematics, to the natural numbers. Therefore, what Gödel achieves, is to properly map all propositions of Meta-mathematics which refer to structure properties of the logico-mathematical system in use containing all natural numbers, to the system itself. The method of arithmetizing Meta-mathematics that Gödel follows is like that of the Analytic Geometry where the coordinates correspond to the geometric figures and the latter are expressed by arithmetical relations. By Gödel’s work, was also made clear that the noncontradictory of a logical-mathematical system which contains all natural numbers is not expected to be proved with the means of the system itself. This fact is parallel to the descriptive image where one who is being drawn cannot be saved by dragging own hair out of the sea. What he needs is someone else to drag him out; the same holds for the proof of noncontradictory, where we must use proofing means out of the system. Jung uses an analogous argument, when he sustains that the reply to the question ‘How does the system hummer-nail work?’ cannot be found into the system, but outside of it, at the hand which holds the hummer and hits the nail. But according to Gödel himself, the problem can be solved with some appropriate philosophical realism, analogous to that of the platonic archetypal ideas. It is important to note here that when Gödel shows that no axiomatic system with enough axioms -like that of Arithmetic- can prove the true or false of each proposition expressed inside it, he actually shows that there is inherent uncertainty in Mathematics and therefore we cannot confine ourselves only to the study of its clear and definite characteristics. Weyl, finally, presents the consequences of these theorems, saying that the arithmetization method leads to the aspect that the natural numbers, along with their Arithmetic, consist such a broad field, so that any formalized theory can be mapped inside it. This astonishing property of the number, acknowledged by Pythagoras and Plato, is being used by Gödel for

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the meta-mathematical study of a given mathematical formalism. All these show that our view about natural numbers is still located in initial stages. After that we have to study the interesting work of Robertson (1995), who connects Jung’s view about number with these of Cantor’s and Gödel’s. Drawing on the initial concept advanced by Pythagoras, i.e that reality is number, and Plato’s later theory, i.e. that a world of perfect ideas exists which transcends the world we live in, the author sketches the scientific advances made from the Renaissance through the 20th century. He explains Descartes’ Geometry, the Calculus of Newton and Leibniz, Cantor’s Theory of Infinite Sets, Freud’s Theory of Personality, Jung’s model of the psyche, and Gödel’s Incompleteness Theorems. Utilizing the above mentioned background material, Robertson then explains how Jung and Gödel both posited that a Platonic world of ideas -archetypes- exist beyond physical reality and the inner world of the psyche. And his well reasoned conclusion is that these archetypes are Pythagoras’ simple counting numbers. This is a work for anyone interested in Mathematics, Psychology, Philosophy, and archetypes.

13.3. Jung and Cantor’s Continuum Hypothesis Through his study of his patients’ ‘number dreams’, Jung came to believe that the smaller natural numbers are symbols in much the same sense that the people and events of our dreams are symbols of personified collective character traits and behavioral situations. The integers seemed to correspond to progressive stages of development within the psyche. In brief, one corresponds to a stage of non-differentiation; two to polarity or opposition; three to movement toward resolution; four to stability, wholeness, as in a quaternity, or a mandala, which is most commonly four-sided. Jung went beyond this limited model, and took a brilliant leap toward generalization of these discoveries: he speculated that number itself -as expressed most basically in the small integers- was the most primitive archetype of order. There is something peculiar, one might even say mysterious, about numbers. If a group of objects is deprived of every single one of its properties or characteristics, there still remains, at the end, its number, which seems to indicate that the number is something irreducible, which helps more than anything else to bring order into the chaos of appearances. It may well be the most primitive element of order in the human mind. We can define number psychologically as an archetype of order which has become conscious (Jung, CW8). Since the natural numbers were each true symbols of order, the implication was that the development of Mathematics reflected the progressive development of order within the psyche. Even more than that, Jung felt that number might be the primary archetype of order of the unus mundus itself; i.e., the most basic building blocks of either psyche or matter are the integers. Number is a very ancient archetype that seems to predate humanity itself. Dantzig (1954) mentions a number of examples of animals and even insects which seem to possess a number sense. The same does Dehaene (1999) more than forty years later. They say that animals, insects, infants have a sense of “one”, “two”, “three”, and “many”. Interestingly, anthropologists found that the numeric systems of some African, South American, Oceanic and Australian cultures had the same limitations -e.g., the Australian aborigines only had numbers for “one” through “six”, and “many”. The archetypal quality of the smaller numbers is so ancient that it predates humanity itself, and is carried in the heritage of creatures even as primitive as insects. Because human beings are capable of counting, we imagine that is how numbers were arrived at. But when

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crows can recognize “one”, “two”, “three” and “many”, few of us would argue they arrived at these numeric relationships by counting per se. Instead there must be pattern recognition, a “primordial image”, to use Jung’s earlier formulation of “symbol”, which corresponds to the smaller integers. In other words, we have an innate sense of what “one” and “two” and “three” mean. Now, if we conceive numbers as having been discovered, and not merely invented as an instrument for counting, then on account of their mythological nature they belong to the realm of godlike human and animal figures and are just as archetypal as they are (Jung, CW10). As civilization developed, there was a need for ever larger numbers. This need puts a strain on any system of separate and distinct symbols. Even among great mathematicians, it is the rare genius for whom virtually all numbers come to possess true symbolic stature. One such was Ramanujan, who recognized that number 1729 “It is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways”. [1729 = 123+13 = 103+93] (Hardy, 1992). For most of us, however, this archetypal pattern recognition is unlikely to extend past the smaller counting numbers. Since, at this stage, the recognition of number is the recognition of a primordial image or pattern, there is as yet little if any distinction between Arithmetic and Geometry. As soon as Arithmetic and Geometry split and go different directions, it becomes much less clear that Jung is necessarily right in his guess that all Mathematics emerges from the smaller counting numbers. Geometry by its various nature deals with continuous lines, figures and planes, while Arithmetic develops out of ever grander extensions of the discrete counting numbers. At the time when Jung was developing these ideas, he was corresponding with Pauli. Pauli, inspired by Jung, was searching for a neutral language which could underlie both the physical and psychological worlds (Pauli, 1955; Card, 1991). Pauli recognized that the issue came to a head when the development of Arithmetic reached the point where it was forced to deal with infinite quantities. If, therefore, a more general concept of archetype is used today, then it should be understood in such a way that included within it is the mathematical primal intuition which expresses itself, among other ways, in Arithmetic, in the idea of the infinite series of integers, and in Geometry, in the idea of the continuum (Card, 1993). At that point, Jung’s speculation becomes identical with Cantor’s Continuum Hypothesis; i.e., are there any infinity that lie between the infinity of the integers and the infinity of the geometric continuum? Cantor’s Continuum Hypothesis is the question: How many points are there on a straight line in Euclidean space? An equivalent question is: How many different sets of integers do there exist? This question, of course, could arise only after the concept of number had been extended to infinite sets (Gödel, 1947). Though Arithmetic and Geometry inhabit separate realms, it is inordinately productive when either turns to the other for a new way of thinking about a problem. Back and forth goes the cross-fertilization between the two fields. This is because at their extremes -dealing with the discrete counting numbers and the geometric continuum respectively- Arithmetic and Geometry seem like very different fields. There is, however, a great fuzzy area where they overlap, since reality itself is more fuzzy than not. Similarly, though at their extremes mind and matter seem totally distinct, the boundaries between the two are fuzzy. There is little blackand-white in the real world. Between the extremes there is a grey spectrum where everything is matter of grade (Kosko, 1994). The CH marks the point where the boundaries of Arithmetic and Geometry are very hazy indeed. There is dispute that the power-set of the countable numbers is the same size as the

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continuum; the CH says that there is no infinity that lies between the two. Starting with the small natural numbers, we build up ineluctably to the countable infinity of all integers; but taking all the combinations of those numbers is a very different thing than merely accumulating. On the human level, we certainly understand how much more complex it is to deal with relationships than with things. We can think of countable infinity as a natural infinity, as the limit of what we encounter in dealing with the things of the world. The uncountable infinity of the continuum is the limit of what we encounter dealing with the relationships in the world. The higher infinities beyond the continuum are all power-sets of the continuum, hence relationships between relationships, etc. Though Cantor remained convinced that the CH was true, others were less sure from the beginning. Cantor’s CH assumed the identifiability of two concepts that were intrinsically different and of no comparable orders of magnitude. The two ideas were inherently antithetical: The nature of the continuum, regarded as the collection of all infinite sequences of rational numbers, was something totally different from the infinity of natural numbers. Gödel had already proved that any system at least as rich as Arithmetic contains undecidable mathematical truths; he guessed that the CH might be just such an undecidable proposition. There are -assuming the consistency of the axioms- a priori three possibilities for Cantor’s conjecture: It may be (i) demonstrable, (ii) disprovable, or (iii) undecidable. The third alternative is the most likely. To seek a proof for it is, at present, perhaps the most promising way of attacking the problem. One result along these lines has been obtained already, namely that Cantor’s conjecture is not disprovable from the axioms of Set Theory, provided that these axioms are consistent. The result was given by Gödel by his own proof in 1940 that is, if a modified Set Theory, which does not include the CH is consistent, then it will remain consistent if the CH is added as an additional axiom. However, this was only half of what was needed to prove that the CH was undecidable within Set Theory. In order to prove the other half, Gödel needed to show that if a modified Set Theory, which does not include the CH is consistent, then it will still be consistent if the CH is assumed to be false. Though Gödel made some progress toward a solution, he was never able to prove it, as it was simply too complex to be resolved with the mathematical tools available at the time. Finally, in 1963, Cohen was able to prove the second half of the problem, thus showing that within Set Theory, the CH was undecidable. So, Gödel’s incompleteness proof had demonstrated that every logical system contains essentially undecidable propositions. Now Gödel - Cohen had shown that the CH was undecidable within Set Theory. Since by this time Mathematics and Set Theory were inseparable for most mathematicians, this was a haunting proof. According to Cohen (1963, 1964, and 1966) and Gödel (1958) we can present the evolution of this ‘story’ as follows: Gödel had already conjectured that even if the CH was undecidable within Set Theory as then constituted, within some extended Set Theory it would eventually be resolved in some way. The undecidability of CH from the axioms being assumed today can only mean that these axioms do not contain a complete description of the reality. Gödel was confident that Set Theory is an important positive step for Mathematics. He believed that it necessarily describes a ‘well-determined reality’, perhaps the mathematical reality which underlies the reality in which we actually exist -perhaps Jung’s unus mundus; in any case, some ‘well-determined reality’. Gödel is confident that the CH must be either true or false within that particular reality. Since he and Cohen together proved that the CH is nei-

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ther true nor false, but undecidable within Set Theory as it is currently constituted, then Gödel can only assume that is because the current set of axioms are insufficient to fully define that reality; i.e., that there are further axioms yet to be discovered. Whereas Gödel’s Incompleteness Theorems seemed to ring the death knell for the development of Mathematics, Gödel has faith that Mathematics is inexhaustible. He considered that Set Theory is the right direction for Mathematics, and that within an extended Set Theory, the CH is resolvable, and finally that any axioms powerful enough to enable the CH to be resolved will necessarily be powerful enough to lead Mathematics in new directions. So far, Gödel’s view has not been justified. Sixty years have passed since 1947, and no extension of Set Theory has yet been discovered in which the CH can be resolved. Perhaps it is truly an undecidable proposition, or perhaps Mathematics is waiting for another Cantor, or Gödel. While we wait, we have to know that both Gödel and Cohen feel that eventually the CH will be resolved. That is where things rest at this point. Jung guessed that the natural numbers are the primary archetype of order in the unus mundus, and that all Mathematics develops out of the natural numbers. Cantor would have agreed with him. Jung’s whole psychology developed out of an attempt to deal with the problems presented by the fact that the world outside is somehow contained within our minds/ brains, while at the same we and our minds/ brains are obviously contained within the world. His Depth Psychology explored the world within, especially as it found expression in the dreams, myths, and other unconscious expressions of human beings. At each level of the psyche, he discovered structure and order. As he went deeper and deeper, it is only natural that eventually he arrived at a level where that structure and order had little or no human qualities attached to it. That was the level of number as archetype. (Robertson, 1995).

13.4. Jung and Gödel’s Incompleteness Theorems We can connect the works of Jung and of Gödel. Jung’s view is that number itself is archetypal: “It may be the most primitive element of order in the human mind. We can define number psychologically as an archetype of order which has become conscious”. Jung feels that number might be the primary archetype of order of the unus mundus itself: i.e., the most basic building blocks of either psyche or matter are the integers. If the Archetypal Hypothesis that has emerged from the work of Jung - Pauli - von Franz holds that the most basic building blocks of psyche and matter are natural numbers, then the properties of natural numbers must be sufficient to account for the nature of the geometrical continuum. In other words, Cantor’s CH must indeed be true. Here at last Jung meets Gödel. Gödel takes up the proof of Cantor’s CH subsequent to his proof of the Incompleteness Theorems, and he comes to suspect that within the axiom system of standard Set Theory, the CH is in fact one such formally undecidable proposition, the possibility of which existence had been established by the Incompleteness Theorems. Gödel was able to complete half of the proof of the undecidability of the CH, and Cohen completed the remainder of the proof in 1963. If the CH is formally undecidable within standard Set Theory, then the possibility of fully describing the properties of the continuum on the basis of infinite sequences of natural numbers is thrown into doubt. Gödel believed that an extension to the axiom system of Set Theory could be constructed in which the validity of the CH could be determined, but no such extended theory has been discovered in nearly sixty years. As a consequence, the prospects

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for finding representations of the unus mundus based upon the integer nature of the number archetype is called into question. After all, isn’t the attempt to reduce the world to the archetypes of the small natural numbers much the same as the attempt of Russell - Whitehead to reduce Mathematics to Logic, the attempt of physicists to reduce the physical world to first atoms, then subatomic particles, then most recently to quarks? Isn’t it more likely that the world is richer than we can ever hope to comprehend? Jung thought that there was a unitary reality -the unus mundus- that underlay both psyche and matter, and speculated that the primary archetypes of this unitary reality were the simple counting numbers. In this case, each number is, itself, a true symbol: undefinable and inexhaustible -a much less reductionistic stance than the hope that all reality can be reduced to Logic. But even so, doesn’t it seem unlikely that we will ever find any lowest level to reality? The Archetypal Hypothesis is a starting point to explore the magic and wonder of the world, not an end point to circumscribe its possibilities. This argument raises important issues for understanding the Archetypal Hypothesis and its implications. If the number archetype is to be understood to be completely equivalent to the natural numbers, then the evocation of Gödel’s and Cohen’s work on the CH is valid without qualification. The Archetypal Hypothesis would then be seen to be a recycled version of Pythagoreanism, brought to the dust by the undecidability of the CH, in much the same way as the original dream of the Pythagoreans foundered with the discovery of the irrational numbers. The crucial issue, then, is whether the number archetype is simply the natural numbers. Jung does not express himself clearly on this point, and even von Franz is not altogether consistent in her discussion of the relationship of number archetype to the natural numbers. The problem is the failure to distinguish between the concept of the archetype-as-such and the representations of the archetype; the images and ideas that are the specific realizations of the archetype, each pointing toward the existence of the abstract archetype-as-such but none being strictly equivalent to it. In the context of number archetype, this problem emerges as the failure to distinguish between the number archetype-as-such and a specific representation of the number archetype such as the natural numbers. Thus the number archetype is not the natural numbers; the natural numbers are only one specific representation of the number archetype. As with specific representations of any archetype, the natural numbers are symbols with an unlimited potential for expression, but their properties do not exhaust all of the possibilities for representation implicit in the number archetype-as-such. The application of the undecidability of the CH to one representation of the number archetype, namely the natural numbers, does not, in fact, lead to a conundrum for the Archetypal Hypothesis. Rather, this result is in concordance with the above mentioned fact about representations of archetypes: No representation is a complete representation. Apart from what can be expressed by any specific representation of an archetype, there are other aspects of the archetype that are valid but unrepresented. Here, it seems, Jung’s empirical findings resonate with Gödel’s Incompleteness Theorems. From this perspective, the answer to the question about the attempt to reduce the world only to the archetypes of the small natural numbers much the same as the attempt of Russell - Whitehead to reduce Mathematics to Logic, is simply “no”. It is an important distinguishing feature of any prospective archetypal science that it entertains no dreams of a ‘Final Theory’ and has no presumptions of being a ‘Theory of Everything’ in the sense that contemporary Physics presently imagines. An archetypal science would be, by its nature, a self-reflective science that is not only aware of its own epistemo-

On Number’s Nature

39

logical capacities and limitations but actually incorporates them -e.g. archetypes- in its representations of phenomena. Given these considerations, the question naturally arises: If natural numbers are simply one representation of the number archetype, what other representations might there be? Finding such representations would be a prerequisite for any future development of science from an archetypal perspective. Some hints in this direction can be found in the work of von Franz, and nowadays of Robertson and Card. Jung and Gödel, each one in his own way, try to open our mind to the ‘other’ reality, which exists in inner and outer world. These two views of the world, external and internal, have common basis and are expressed by a common factor, which is number. We are still very early in our understanding of the archetypal nature of reality. Plato’s aspect of an ideal world is an early attempt at describing the archetypal nature of reality, and it lacks so much. Plato’s ideal world lies, like other early views of the humans, totally separate from us and our experience. Science proceeded by exploring the outer rather, not the inner world. Mathematics plays a role in that process; without Mathematics there is no science. Gödel’s Incompleteness Theorems demonstrated once and for all that Mathematics is bigger than Logic; so are science and all human endeavors. But, if Logic is insufficient, perhaps Mathematics, as is develops out of the simple archetypal counting numbers, is enough. Is it really so surprising that the CH proves intractable?

14. Number and Jung’s Psychology Jung (CW6) claims that in order to live in space-time and in the relations of the things of the world, in order to communicate and to exchange information with the environment, we need psychological functions, of which here are the four basic types: Sensation, Thinking, Feeling and Intuition. Jung places counter-diametrically on a circle these four basic psychological functions and names the centre of the circle, where all these functions meet, with the Greek classical term pemptusia (quintessence). Rucker (1988) draws on Jung’s view to explain why Mathematics has developed in five ways: Number, Logic, Space, Infinity and Information. These five ways are related to the five above-mentioned psychological functions, where here he regards Information as Quintessence; as Quintessence unifies all the psychological functions, similarly Information unifies all the branches of Mathematics. Then he makes the following correlation with Mathematics: Sensation ≈ Number, Thinking ≈ Logic, Feeling ≈ Space, Intuition ≈ Infinity and Quintessence ≈ Information. The interesting, for this work, is that he explains the correlation between Sensation and Number: Sensation informs us that ‘something’ exists; this ‘something’ is the result of a distinction which leads us to number. The world of Sensation and that of number is the world of the discrete. So, distinctions among objects can be made leads to our perception of discreteness; discreteness leads, in turn, to number. If we examine through time the basic concept of Mathematics in each era, then we observe that this is number according to the Middle Ages, space according to Renaissance, Logic according to Industrial Revolution and infinity according to Modern Times. Nevertheless, with computers’ progress, we are already following the new era in Mathematics, the fifth in a row, in which the dominant concept is the one of information. The conception of the

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world is based on the exchange of information and on communication with it. Whenever we are learning, we are sending and receiving transmissions, generating and absorbing information; we are communicating (Gavalas, 1999). Finally, for Jung different numbers make up various kinds of patterns. Certain archetypes are associated with some of the smaller numbers: 1. 2. 3. 4. 5.

monad: unity dyad: opposition triad: thesis-antithesis-synthesis tetrad: balance of quaternity quintad: a step further out.

Robertson (1989) sustains that as number evolved, so did the relationship between the ego and the Self, which is, according to Jung, the totality of the human personality: (i) Natural numbers coincided with the period of ‘participation mystique’, when man could only experience himself through his projections on nature. (ii) Zero appeared at the birth of the Christian era, as the ego began to emerge as a thing in itself. As zero gradually led to the explicit formulation of infinity, the ego unsuccessfully tried to swallow the Self. (iii) Infinity gave way to self-reference in the twentieth century, as order began to crumble into chaos everywhere around us. The ego, unable to swallow the Self, gave way to despair. God was dead; chaos reigned. But even chaos was revealed as possessing a self-referential order. The relationship between ego and Self is not one in which ego rules Self or Self rules ego. Rather, the Self is at one and the same time both the goal of the ego, the process by which the ego attains that goal. The ego is both the Self’s expression in the world of limits and the process of evolution of the Self. Sensing the Self as something irrational, as an indefinable existent, to which the ego is neither opposed nor subjected, but merely attached, and about which it revolves much as the earth revolves around the sun -thus we come to the goal of individuation. (Jung, CW7). Number, as the archetype of order, is in the process of finding a new symbol with which to clothe itself. A new order is trying to emerge from chaos. When that happens, it will correspond to a new vision of the Self, as it did when natural numbers ruled, when zero emerged, when infinity found concrete expression. At the end of this Part we must underline the fact that the initial work of Jung and the other of Jung - Pauli opened a completely new way for an essential consideration of numbers’ nature. The work of these pioneers was continued by von Franz; today this way is followed mainly by Card and Robertson and also by others (Gough & Shacklett, 1993). In the next Part IV we present our own view which is closer to Mathematics and expresses the previous aspects in a mathematical way through the contemporary Category Theory.

On Number’s Nature

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PART IV: NUMBER, MONOID AND ARCHETYPE 15. 2-Categories; Monad, Monoid, Monoidal Category, Monoid Object; the Dynamical System as Functor 15.1. 2-Categories Firstly, by the term double category we mean that the set of morphisms of a category has two different compositions which satisfy the ‘exchange law’, as we shall see. By the term 2category, which is a brevity for the term two-dimensional category, we also mean a double category in which every identity morphism is such for both compositions. There are also ncategories. Indeed, sets are zero-dimensional in that they only consist of objects and there is no way to go from one object to another within a set. Nonetheless, we can go from one set to another using a function. So the category of all sets, the Set, is one-dimensional because it has both, objects and morphisms/ arrows/ functions between objects. In general, categories are onedimensional in this sense. But this in turn makes the collection of all categories, the Cat, into a two-dimensional structure, a 2-category having objects, morphisms between objects/ categories which we call functors, and 2-morphisms between morphisms/ functors which we call natural transformations. This process never stops and the collection of all n-categories is an (n+1)-category until ω-categories. In 7.1 we discourse about category, but a 2-category also consists of a collection of 2morphisms. Every 2-morphism T has a ‘source’ morphism f and a ‘target’ morphism g. If the source of T is f and its target is g, we write T: f ⇒ g. If T: f ⇒ g, we require that f and g have the same source and the same target; for example, f: X → Y and g: X → Y. We should visualize T as follows: ⎯f→ X T⇓ Y. ⎯g→ In addition, we have: (1) Given a 2-morphism S: f ⇒ g and a 2-morphism T: g ⇒ h, there is a 2-morphism ST: f ⇒ h, which we call the ‘vertical composite’ of S and T. (2) Vertical composition is associative: (ST)U = S(TU). (3) For each morphism f there is a 2-morphism 1f: f ⇒ f, called the ‘identity’ of f. For any T: f ⇒ g we have 1f T = T 1g = T. It is obvious that these are just like the previous 3 rules for a category. We draw the vertical composite of S: f ⇒ g and T: g ⇒ h like this:

X

⎯f→ S⇓ ⎯ g → Y. T⇓ ⎯h→

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Dimitris Gavalas We also require that we can ‘horizontally’ compose 2-morphisms as follows:

X

⎯f→ S⇓ ⎯g→

Y

⎯ f΄→ T⇓ ⎯ g΄ →

Z.

So we also demand: (1) Given morphisms f, g: X → Y and f', g': Y → Z and 2-morphisms S: f ⇒ g and T: f' ⇒ g', there is a 2-morphism ST: ff' ⇒ gg', which we call the ‘horizontal composite’ of S and T. (2) Horizontal composition is associative: (ST)U = S(TU). (3) The identities for vertical composition are also the identities for horizontal composition. That is, given f, g: X → Y and Τ: f ⇒ g, we have 11x T = T 11y = T. Finally, we demand the ‘exchange law’ relating horizontal and vertical composition: (ST)(S'T') = (SS')(TT'). This makes the following 2-morphism unambiguous:

X

⎯f→ S⇓ ⎯g→ T⇓ ⎯h→

Y

⎯ f΄→ S΄ ⇓ ⎯ g΄ → T΄ ⇓ ⎯ h΄ →

Z.

We can think of it either as the result of first doing two vertical composites, and then one horizontal composite, or as the result of first doing two horizontal composites, and then one vertical composite. Like the archetype of category is the category of sets and functions, that is Set, so the archetype of 2-category is the category with objects the small categories and morphisms the functors, that is Cat, where 2-morphisms are the natural transformations. Here it also holds: The set of all natural transformations is the set of morphisms of two different categories supplied with two different compositions which satisfy the exchange law and the identity morphism is such for both compositions. In case of Cat the objects for horizontal composition are categories, while for the vertical ones are functors. From all the above we see 2-category can be considered as abstract two-dimensional space where the axioms tell us how we pose together 2-morphisms in these two-dimensions. So we can pose them horizontally, that is the one next to the other, or vertically, the one above the other. Note that all is meant by ‘archetype of category’ for Set is that this is a familiar category and that, if we start from Set and various categories of structures built using sets, e.g. groups, rings, vector and topological spaces, we can then abstract the notion of category and thus obtain Cat. In the same sense Cat is the ‘archetype of 2-category’ and so on.

15.2. Monad, Monoid, Monoidal Category, Monoid Object The 2-category is a framework where we can present the general concepts of ‘multiplication’ or ‘combination’. Multiplication is a function Μ: SxS → S: (a, b) → M(a, b) = ab, where

On Number’s Nature

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we take two elements a and b from a set S, and multiply them to get a new one M(a,b)=ab of the set S. We can visualize this as follows: X

S

S M

X

X S

where this triangular shape takes two ‘inputs’ from the two slanted edges labelled S, and spits out one ‘output’ from the horizontal edge labelled S on the bottom. It is clear from the geometry here that M is something 2-dimensional, hence, a 2-morphism, and that S is 1dimensional, hence, a morphism from X to itself. Here X, being 0-dimensional, is an object. We can take the ‘dual’ diagram like this:

which illustrates more vividly how M is the process of two copies of S getting squashed down into one copy. This sort of diagram is called a ‘string diagram’ and it is literally the Poincare dual of the earlier picture, meaning that stuff that was k-dimensional is now drawn as (2-k)-dimensional. The 0-dimensional object X is now the 2-dimensional ‘background’. The essence of multiplication can be described generally in a situation where we have a 2-category with an object X in it, a morphism s: X → X and a 2-morphism M: ss ⇒ s. We are often interested in situations like this where the multiplication M is associative and there is a multiplicative unit. The first means that the composite 1s M M M1s M sss ⇒ ss ⇒ s equals sss ⇒ ss ⇒ s, where 1s: s ⇒ s is the identity 2-morphism from s to itself.

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Dimitris Gavalas The second means that there is a 2-morphism I: 1X → s for which I1s M s = 1Xs ⇒ ss ⇒ s equals 1s, and so does 1 sI M s = s1X ⇒ ss ⇒ s.

If we have a 2-category with stuff in it satisfying these rules, we say we have a Monad in that 2-category. As an example of a Monad we can consider the original example where s is a set and M is a function. Then the 2-category has only one object X, the morphisms of this 2-category are sets, the composing morphisms corresponds to taking the Cartesian product of sets and the 2morphisms of this 2-category are functions between sets. In this case the Monad is formulated as follows: The multiplicative unit 1X corresponds to the one-element set 1, s is a set, the 2morphism I: 1X → s is a function from the one-element set 1 to s, which picks out a special ‘element’ of s. The 2-morphism M: ss ⇒ s is the multiplication operation and is associative while the special element of s is the multiplicative unit that is, it serves as the left and right identity for multiplication. So we have a set with an associative multiplication and a unit for this multiplication. That is what is called a Monoid. So a Monoid is a special sort of Monad. Finally, a 2-category with just one object is called a Monoidal category. For example, Set is a Monoidal category where we can multiply objects, i.e. sets, with the Cartesian product. If we consider the 2-category with just one object, forget the object and correspond the morphisms to objects and the 2-morphisms to morphisms, then we have got a category where we can compose/ multiply objects, because they were secretly morphisms from X to itself. So we do this degradation: 2-morphisms → morphisms morphisms → objects object X → we forget it. This is a very important point for our view in the following. Because a Monoidal category is a 2-category with one object, we can talk about Monads in any Monoidal category. These are usually called Monoid Objects, because they are like a Monoid living in the category in question. The category Cat having (small) categories as objects and functors as morphisms becomes a Monoidal category with the Cartesian product of categories as the way to multiply objects and a Monoid Object in this is a Monoidal category. A Monoid Object is defined in a Monoidal category, but a Monoidal category is itself a kind of Monoid Object. If a category is a Monoidal category with one object X, then the set hom(X, X), of all morphisms from X to X, is a set with an associative binary product, namely composition, and a unit element, namely 1Χ. Conversely, if we have a Monoid S in the traditional sense, we can easily cook up a category with one object X and hom(X, X) = S.

On Number’s Nature

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We can see morphisms as paths between objects, 2-morphisms as paths between paths and so on. We can think of morphism as a ‘process’ which transforms or projects an object into another. The neutral morphism corresponds to the ‘process of doing nothing at all’. We can compose processes h and g, ‘do h and then g’, and get the product gh. Crucially, every process g can be ‘undone’ using its inverse g-1. We tend to think of this ability to ‘undo’ any process as a key aspect of symmetry and for a full understanding of symmetry we should really study Monoids. So, a Monoid is like a group, but the symmetries no longer need be invertible; a category is like a Monoid, but the symmetries no longer need to be composable. In contrast to a set, which consists of a static collection of objects, a category consists not only of objects but also of morphisms which can be viewed as ‘processes’ transforming one object into another. Similarly, in a 2-category, the 2-morphisms can be regarded as ‘processes between processes’, and so on. Coming to an end: A category with one object is a Monoid and the classical definition of a Monoid is this: A set S with an associative binary product and a unit element 1 such that a1 = 1a = a for all a in S. Monoids abound in Mathematics and they are, in a sense, the most archetypal algebraic structures.

15.3. Dynamical System as Functor Let it be N a Monoid, so that it only has one object denoted *, the maps are natural numbers, the composition is addition (or multiplication), the identity of *, 1*, is the 0 (or the 1). A functor, arrow between categories, Ν → Set, means that we interpret the * as a set S and that every map n: * → * in N is interpreted as an endomap of the set S, gn: S → S in such a way that g0 = 1S and gn+m = gngm (or gnm = gngm ) Example Consider as S a set of numbers and define gn: gn(x) = n+x. The classical example of interpretation of a Monoid in Set is that the object of the Monoid * is interpreted as the set of maps of the Monoid (natural numbers) and in this way always get a functor from the Monoid N to the category of sets Set. Of course, there are lots of functors other than the classical one from N to Set. Consider a set X together with an endomap α, , then we have a dynamical system. Let interpret * as X and send each map n of N (natural number) to the composite of α with itself n times, that is n → αn, and 0 to the identity of X, i.e. 0 → 1Χ. In this way we get a functor H from N to sets, H: N → Set, which satisfies: (i) Η(*) = Χ, (ii) Η(n) = αn, (iii) Η(0) = 1Χ, (iv) H(n+m) = H(m)H(n), because the composition of α with itself is associative. So, when we specify a ‘set with an endomap’ , we obtain a functorial interpretation of N in Set. This means that another name and way of being for the category of dynamical systems is SetN. Indeed, this is the category of dynamical systems of discrete-time. So, a discrete-time dynamical system is just a functor from Monoid N to Set. In analogous way, we can have a continuous-time dynamical system, if replace natural numbers with real numbers, i.e. we allow in the Monoid all real numbers as maps. Then, to give a functor from the Mon-

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Dimitris Gavalas

oid R to Set, is to give a set X with an endomap αt, , for every real number t. We must ensure, of course, that α0= 1Χ and αs+t= αsαt. We can think of X as the set of ‘states’ of a system which, if it is in the state x at a certain time, then t units of time later it will be in the state αt(x) (Baez, 1996; Lawvere & Schanuel, 2005).

16. The Archetype of Number as Monoid and its Interpretation as the Set of Numbers There is only one archetype for number, which belongs to the existential subsoil of humans. One cannot refer to his own number archetype, but only to the one and only number archetype. This is the principle according to which Mathematics -and not only- is formulated. Man is carrier of this archetype and contains an image or idea or model in his non-conscious mind and, perhaps, brain. This archetype is inaccessible and unmodified, but man can capture its manifestations. These last ones and their consequences are the only things that we know, since at bottom we do not know the nature of the archetype itself. This point is not a paradox if we consider that neither for energy we know something, only its manifestations and its consequences. We simply can say that all these come from the region of hyper-personal factors which lie not on the surface of the consciousness, but on the deepest ends of man. Because this region is much older than man and is the presupposition of his life, we call it ‘divine’ in contrary to the human. Let’s not forget that the initial meaning of Mathematics and especially of number and of geometrical figures was divine and magical. Therefore, Mathematics as such is psychogenic that is spontaneous psychic fact, product of non-conscious procedures. This is independent on the fact that these first ‘divine’ hints are elaborated and are used consciously by the human factor Number is one of the most important archetypes and may be the womb of all the others. It is an archetype of order which has become conscious. Wherever exists order also exists structure, which is expressed numerically or geometrically and therefore Mathematics comes in. Hence the archetype of number is one but the imprints are many. The question is: Which one of these many imprints of number -that is numbers as we know and use them- are really different between them? We said before that the natural numbers produce all known numbers and the majority of Mathematics: N → Z → Q → R → C. But as we know not all natural numbers are of the same kind. Instead, there are the prime and the composite natural numbers and according to the Fundamental Theorem of Arithmetic every composite is analyzed uniquely in a product of prime numbers. Therefore, the primes are adequate to produce all natural numbers, which produce all the others and the biggest part, if not all, of Mathematics. If we denote with P the set of all primes, then we have: Ρ → N → Z → Q → R → C. This means that the primes are the atoms of mathematics and play an analogous role with Chemistry elements: They produce all numbers. Since the number archetype is unique, we can consider it as the one and only object of a Monoidal category and to represent it as ⊗. All arrows in this category are endomaps of this unique object. There are a lot of arrows in this category all arrows are from ⊗ to ⊗, since ⊗ is the unique object. As arrows, we can consider all natural numbers and therefore 0, 1 etc. are arrows from ⊗ to ⊗. Thus, we can write:

On Number’s Nature

47

... As an arrow composition, we can choose the multiplication of natural numbers, that is the composition of two arrows/ numbers is their product nom=nm. Because there is only one object, every pair of arrows must be composible and indeed the composition/ multiplication satisfies this convention. If 1⊗ is identical then must 1⊗οn = n = no1⊗ for every n. Therefore must be 1⊗ = 1. Such a category is called Monoidal, if it contains just one element. Next, we can interpret this category to the well-known and most familiar category of sets Set. If we name Α this Monoidal category, with composition of arrows the multiplication, then Α → Set: ⊗ → N, the unique object ⊗ of Α is interpreted as the set of natural numbers and that is why every arrow in Α -a natural number- is interpreted as an arrow from the set of naturals to itself: fn: N → N: x → fn(x) = nx, with f1 = 1N and fnοfm = fnm. All these show that the objects correspond to objects, arrows correspond to arrows, the composition holds and the identity corresponds to identity. Therefore holds the structure of the category. Such a transportation of one category to another, which retails the structure, is called functor. Of course a functor must retail domain and co-domain but especially here this happens automatically, since all arrows have the same domain and co-domain. The conventional paradigm of interpretation of a Monoidal category to the category of sets Set is the following: The object of the Monoidal category is interpreted as the set of arrows of the Monoidal category and in this way we always take a functor from the Monoidal category to the category of sets Set. Therefore, the object ⊗, which is the archetype of number, is interpreted as the set of the naturals N. This means that the imprints of the archetype are the natural numbers. But furthermore we can consider as arrows not natural numbers but rationals and to have in this way the inverses that is invertible arrows. And even further we can consider as arrows all real numbers and in the same way, as above, to interpret the object/ archetype as the set of real numbers. In this way we take all numbers, from naturals to the reals. We can interpret the archetype of number as the set of the reals. May be it is possible to consider as 2-morphisms the complex numbers or any other allowed procedure between numbers:

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Dimitris Gavalas

17. A Holistic View of Mathematics How can we, using the archetype view, integrate the partial philosophical views for Mathematics? The following figure helps towards this direction:

Archetypal Aspect *: Archetype-as-such 1: Archetypal Images 2: Archetypal Ideas 3: Archetypal Effect 4: Projection to the outer World.

Correspondence to Mathematics Mathematical Archetypes, especially natural numbers and geometrical continuum Intuitionism Platonism Formalism - Logicism Applied Mathematics, Mathematics as Science, Empiricism.

The figure, which comes from Jung’s view, and the correspondence, which I think is natural, show that the different views on Mathematics are just the different levels of realizing/ understanding archetypes and all together belong to the same wholeness. Therefore the different views are not necessary to be considered as competitive or as contradictions but as complementary; that depends on which level of the archetype one realizes according to one’s nature and consciousness.

On Number’s Nature

49

Goodman (1979, 1990 and 1991) for instance and others (Atiyah et al., 1994; Jaffe & Quinn, 1993), conceive and deal with level 4 that is the projection to the outer world. This shows that they have not realized their projections and are not interested in finding from where they come and what exactly they mean. Pythagoras instead is placed in the center that is he works out the essence of archetype. Formalists and logicians are affected by archetype’s action, but do not examine the subject further and therefore they need no interpretations; the syntactic part suffices for them and they do not search for the meaning, which in this way remains hidden. In my view, the course from the centre to the circumference, faces all aspects and illuminates all elements, while simultaneously unifies all philosophical views of Mathematics. Thus a particular mathematical issue should be firstly reached by its center, the archetype, and tracked till its applications and only then is it complete. For this reason I insist on the archetypal view of Mathematics, because all the others have been done in a way or another and up to a point, while the archetypal view has been abandoned.

18. Final View We have a 2-category with just one object that is a Monoidal category in which holds: Object → Number Archetype Morphisms → Numbers 2-morphisms → ? But, as we mentioned before, we degrade this situation and we consider morphisms/ numbers as objects, 2-morphisms as morphisms/ order ‘ 0 , k=1,…,m;

m

∑λ k =1

k

= 1. With the construction of μ T ( X i , X j ) , it is possible to obtain nd

the membership function μ T ( X i ) of the set of nondominated alternatives, following a procedure similar to the one described above for the global relation μ G ( X i , X j ) , which involves Equations (11)-(13). Finally, the nondominance level can be obtained by performing nd

nd

the intersection between μ R ( X i ) and μ T ( X i ) : nd μWnd ( X i ) = min{μ nd R ( X i ), μ T ( X i )} ,

(16)

and the following set of fuzzy nondominated solutions can be met:

X nd = { X ind ∈ X | μ Wnd ( X ind ) = sup μ Wnd ( X i )} .

(17)

X i ∈X

The second procedure has a lexicographic character. It consists in a step-by-step introduction of criteria for comparing alternatives. In this case, a sequence X1, X2,…, Xm, such that

X ⊇ X 1 ⊇ ... ⊇ X m , is obtained with the following expressions:

μ nd Rk ( X j ) = infk −1{1 − μ Pk ( X j , X i )} = 1 − sup μ Pk ( X j , X i ) , k = 1,..., m , X i ∈X

X i ∈ X k −1

(18)

A New Consensus Scheme for Multicriteria Group Decision Making… nd k X k = { X ind k ∈ X k −1 | μ nd ) = sup μ nd Rk ( X i R k ( X i )} .

77 (19)

X j ∈ X k −1

In the third procedure, the use of (12) permits one to construct the membership functions of the set of nondominated alternatives for each fuzzy preference relation. The membership nd

functions μ Rk ( X i ) , k=1,…,m play a role identical to membership functions replacing objective functions Fk ( X ) , k=1,…,m in analyzing < X , M > models [7]. Therefore, in order to obtain X

nd

, it is possible to construct:

μ nd ( X i ) = min μ nd Rk ( X i ) . 1≤ k ≤ m

(20)

If it is necessary to differentiate the importance of each preference relation, it is possible to rewrite (20) as follows: λk μ nd ( X i ) = min[μ nd Rk ( X i )] . 1≤ k ≤ m

(21)

It should be mentioned that the use of (21) does not require the normalization of

λk ,

k=1,…,m as it is required to apply (17).

6. APPLICATION EXAMPLE The enterprise's board of directors, which includes five members (e1,…,e5), is to plan the development of large projects (strategy initiatives) for the following five years. Four possible projects (X1, X2, X3, and X4) have been marked. It is necessary to compare these projects to select the most important of them, as well as order them from the point of view of their importance, taking into account four criteria (categories) suggested by the Balanced Scorecard methodology (it should be noted that all of them are of the maximization type): c1) financial perspective, c2) the customer satisfaction, c3) internal business process perspective, c4) learning and growth perspective. First, the specialists are asked to give their opinion relative to each project in terms of fuzzy estimates, using the linguistic variables from the linguistic hierarchy shown in Figure 1. The semantic values (trapezoidal fuzzy sets) corresponding to the labels from this hierarchy are given in Table 1. The professionals involved are considered of the same importance, except for e1, whose opinions are judged more important. Therefore, the parameters wz are set as: w1=0.3 and wz=0.175, z=2,…,5. The parameter minconsensus is specified as 0.85 by the moderator. It is interesting to observe that the proposed consensus scheme allows the discussion process to be divided into several sections (more specifically m × n sections), being each

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P. Bernardes, P. Ekel and R. Parreiras

section related to a specific alternative and a specific criterion. Only the sections regarding the criterion c1 and alternative X1 are exposed here. At the first cycle, as shown in Table 2, the consensus level is lower than the threshold parameter minconsensus specified by the moderator. In this way, as the expert e5 is the most discordant one, he is invited to update his opinion. The data concerning the second cycle is shown in Table 3. As it can be seen, the expert e5 made only a small change in his evaluation and remained as the most discordant member of the group. Consequently, he is invited to review his opinion again. Table 1. Fuzzy numbers associated with the linguistic terms of the linguistic hierarchy Level t=1

t=2

t=3

Granularity 3

Label

l

5

9

Fuzzy number (0, 0, 0.5, 5)

1 0

l11

(0, 4.5, 5.5, 10)

l21

(5, 9.5, 10, 10)

l02

(0, 0, 0.5, 2.5)

l12

(0, 2, 3, 5)

l 22

(2.5, 4.5, 5.5, 7.5)

l32

(5, 7, 8, 10)

l 42

(7.5, 9.5, 10, 10)

l03

(0, 0, 0.5, 1.25)

l13

(0, 0.75, 1.75, 2)

l23

(1.25, 2, 3, 3.75)

l33

(2.5, 3.25, 4.25, 5)

l43

(3.75, 4.5, 5.5, 6.25)

l53

(5, 5.75, 6.75, 7.5)

l63

(6.25, 7, 8, 8.75)

l73

(7.5, 8.25, 9.25, 10)

l83

(8.75, 9.5, 10, 10)

Table 2. Fuzzy estimates given by each expert at cycle=1 e1

e2

e3

e4

e5

2 0

3 2

l13

l

3 4

2 3

0.16

0.18

Opinion

l

Discordance Consensus level

0.27 0.79

l

0.05

l

0.40

Group 2.42

A New Consensus Scheme for Multicriteria Group Decision Making…

79

Table 3. Fuzzy estimates given by each expert at cycle=2 e1

e2

e3

e4

e5

2 0

3 2

l13

l

3 4

3 6

0.16

0.18

Opinion

l

Discordance Consensus level

0.27 0.79

l

0.04

l

Group 2.42

0.40

Table 4 presents the data collected at the third cycle of the discussion process. Although e5 remained as the most discordant expert, as he had been already invited twice to review his opinion, now e1 is invited (he is the second most discordant expert). It can be seen in Table 5 that, at the fourth cycle, the consensus level finally achieves a satisfactory value. Therefore, the procedure is interrupted and the output estimate is given by Δ(2.74) = (l33 ,−0.26) . Table 4. Fuzzy estimates given by each expert at cycle=3 e1

e2

e3

e4

e5

2 0

3 2

l13

l

3 4

3 5

0.15

0.19

Opinion

l

Discordance Consensus level

0.26 0.82

l

0.03

l

Group 2.31

0.30

Table 5. Fuzzy estimates given by each expert at cycle=4 e1

e2

e3

e4

e5

2 1

3 2

l13

l

3 4

3 5

0.19

0.14

Opinion

l

l

Discordance Consensus level

0.08 0.85

0.08

l

Group 2.74

0.25

Table 6. Collective evaluations of each alternative, considering each criterion c1

c2

c3

c4

3 3

2 3

2 4

l12

l 42

l63

l32

l73

l33

l 23

l 43

l13

l53

l73

l 42

X1

l

X2

l32

X3 X4

l

l

The collective opinions related to each alternative, obtained with the use of the consensus scheme, are exposed in Table 6 and illustrated in Figures 2-5. Having calculated the collective estimates, the following comparison matrices, corresponding to the collective nonstrict fuzzy preference relations, are obtained for each criterion, using (9) and (10):

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P. Bernardes, P. Ekel and R. Parreiras

⎡1 ⎢1 R1 = ⎢ ⎢1 ⎢ ⎣0

0

1⎤ 0.63 ⎡ 1 ⎢ ⎥ 1 0.91 1⎥ , 1 1 R2 = ⎢ ⎢ 0 1 1 1⎥ 0 ⎢ ⎥ 0 0 1⎦ ⎣0.91 0

⎡ 1 ⎢0.45 R3 = ⎢ ⎢ 0 ⎢ ⎣ 0.91

0

1 1

1 1⎤ 1 1⎥⎥ , 1 0⎥ ⎥ 1 1⎦

1 ⎤ 0.45 0 ⎤ ⎡1 0 ⎢ ⎥ 1 1 0.83⎥ , 1 1 1 0.63⎥⎥ . ⎢ R4 = ⎢1 0.45 0 1 0 ⎥ 1 0 ⎥ ⎢ ⎥ ⎥ 1 1 1 ⎦ 1 1 ⎦ ⎣1 1

Figure 2. Collective fuzzy estimates related to c1.

Figure 3. Collective fuzzy estimates related to c2.

A New Consensus Scheme for Multicriteria Group Decision Making…

Figure 4. Collective fuzzy estimates related to c3.

Figure 5. Collective fuzzy estimates related to c4.

81

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P. Bernardes, P. Ekel and R. Parreiras

The strict preference relations Pk, k=1,…,4 can be obtained by applying (11) to each nonstrict preference relation Rk:

0 ⎡0 ⎢1 0 P1 = ⎢ ⎢1 0.09 ⎢ 0 ⎣0

0 0 0 0

1⎤ ⎡ 0 ⎥ ⎢0.37 1⎥ , P2 = ⎢ ⎢ 0 1⎥ ⎥ ⎢ 0⎦ ⎣ 0

⎡0 0.55 ⎢0 0 P3 = ⎢ ⎢0 0 ⎢ ⎣0 0.17

1 0.09⎤ 0 0 ⎡ 0 ⎢ 1 1 0 ⎥⎥ 0 0.55 , P4 = ⎢ ⎢0.55 0 0 ⎥ 0 0 ⎥ ⎢ 1 0 ⎦ 0.37 1 ⎣ 1

0 0 0 0

1 0.09⎤ 1 1 ⎥⎥ , 0 0 ⎥ ⎥ 1 0 ⎦ 0⎤ 0⎥⎥ . 0⎥ ⎥ 0⎦

Let us consider the application of the first procedure for analyzing models. The aggregation of R1, R2, R3, and R4 into R, using (14), results in

⎡ 1 ⎢0.45 R=⎢ ⎢ 0 ⎢ ⎣ 0

0 0 0 ⎤ 1 0.91 0.63⎥⎥ 0 1 0 ⎥ ⎥ 0 0 1 ⎦

(22)

The strict fuzzy preference relation is derived from (22), using Equation (11):

⎡ 0 ⎢0.45 P=⎢ ⎢ 0 ⎢ ⎣ 0

0 0 0 ⎤ 0 0.91 0.63⎥⎥ . 0 0 0 ⎥ ⎥ 0 0 0 ⎦

(23)

The nondominance level of each alternative, calculated with (12), is given by

μ

nd Rk

= [0.55 1 0.09 0.37] . Thus, according to the first procedure for analyzing multic-

riteria problems, the alternatives can be ranked from best to worst as X 2 ; X 1 ; X 4 ; X 3 . The second procedure requires the ranking of criteria from the most important to the least important one. Supposing that they can be ranked as c1 ; c2 ; c3 ; c4 , at the first step, the nondominance

μ

nd R1

level

of

each

alternative

is

calculated

applying

(18)

to

P1 :

= [0 0.91 1 0] . As alternatives X1 and X4 are undistinguishable, the elements of P2

related to X1 and X4, can be analyzed as follows:

A New Consensus Scheme for Multicriteria Group Decision Making…

83

⎡0 0.09⎤ P2′ = ⎢ . 0 ⎥⎦ ⎣0 The nondominance level of X1 and X4 is given by: μ R2 = [1 0.91]. Therefore, the use nd

of the second procedure provides the ranking: X 3 ; X 2 ; X 1 ; X 4 . In the third procedure, having calculated the strict preference relations Pk, k=1,..,4, the use of (12) provides the nondominance level of each alternative for each criterion:

μ nd 0.91 1 0] , R1 = [0

μ nd 0] , R 2 = [0.63 1 0

μ nd 0 0.91] R3 = [1 0.45

and

μ nd 0.63 0 1] . The global nondominance level of each alternative is obtained on R4 = [0 the basis of (20):

μ nd = [0 0.45 0 0] . Hence, according to the third procedure, the alternatives X1, X3, and X4 are undistinguishable and we have X 2 ; X 1 ~ X 3 ~ X 4 .

CONCLUSION The present work introduced a consensus scheme for analyzing decision problems, which allows aggregating the opinions of multiple experts into a collective opinion. Each expert is invited to evaluate the alternatives in terms of multi-granular linguistic terms that are organized as a linguistic hierarchy structure. The proposed consensus scheme has some positive aspects: it has an intuitive appeal and involves simple calculus; it admits a computational component for regulating the discussion process; it can be coupled to different procedures for analyzing multicriteria problems. However, it is important to mention that loss of information happens at the final step of the proposed consensus scheme, when the output data is generated (more specifically, when the a parameter, which reflects the difference of information between bc and its nearest integer value, is eliminated). In order to demonstrate the usefulness of the presented consensus scheme, three different decision procedures based on analyzing models are utilized for solving a hypothetical enterprise strategy planning problem, generated with the use of the Balanced Scorecard methodology, by a group of experts.

ACKNOWLEDGEMENTS This research is supported by the National Council for Scientific and Technological Development of Brazil (CNPq) - grant 302406/2005-0 and the State of Minas Gerais Research Foundation (FAPEMIG) - grant TEC 00140/07.

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REFERENCES [1]

[2] [3] [4]

[5]

J. Lu, G. Zhang, D. Ruan, and F. Wu, Multi-objective Group Decision Making: Methods, Software and Applications with Fuzzy Set Techniques, Imperial College Press, London, 2007. H. J. Zimmermann, Fuzzy Set Theory and Its Application, Kluwer Academic, Boston, 1990. W. Pedrycz and F. Gomide, An Introduction to Fuzzy Sets: Analysis and Design, MIT Press, Cambridge, 1998. F. Herrera and L. Matínez, A model based on linguistic 2-tuples for dealing with multigranular hierarchical linguistic contexts in multi-expert decision-making, IEEE Transactions on Systems, Man and Cybernetics – Part B: Cybernetics, vol. 31, 2001, pp. 227234. F. Herrera and E. Herrera-Viedma, Linguistic decision analysis: steps for solving decision problems under linguistic information, Fuzzy Sets and Systems, vol. 115, 2000, pp. 67-82.

[6] R. S. Kaplan and D. Norton, The Balanced Scorecard: Translating Strategy into Action, Harvard Business School, Boston, 1996. [7] P. Ya. Ekel, M. R. Silva, F. Schuffner Neto, and R. M. Palhares, Fuzzy preference modeling and its application to multiobjective decision making, Computers and Mathematics with Applications, vol. 52, 2006, pp. 179-196. [8]

[9]

[10]

[11]

[12]

[13]

[14] [15]

P. Ya. Ekel and F. H. Schuffner Neto, Algorithms of discrete optimization and their application to problems with fuzzy coefficients, Information Sciences, vol. 176, 2006, pp. 2846-2868. P. Ya. Ekel, M. Menezes, and F. Schuffner Neto, Decision making in fuzzy environment and its application to power engineering problems, Nonlinear Analysis: Hybrid Systems, vol. 1, 2007, pp. 527-536. R. C. Berredo, P. Ya. Ekel, E. A. Galperin, and A. S. Sant'anna, Fuzzy preference modeling and its management applications, in Proceedings of the International Conference on Industrial Logistics, Montevideo, 2005, pp. 41-50. P. Bernardes, P. Ekel, J. Kotlarewski, and R. Parreiras, Fuzzy set based multicriteria Decision making and its applications, in Progress on Nonlinear Analysis, Nova Science Publisher, Hauppauge, 2008, pp. 247-272. R. C. Botter and P. Ya. Ekel, Fuzzy preference relations and their naval engineering applications, in Proceedings of the XIX Congress of the Panamerican Institute of Naval Engineering, Guayaquil, 2005, Paper 7-8. E. Herrera-Viedma, F. Herrera, F. Chiclana, A consensus model for multiperson decision making with different preference structures, IEEE Transactions on Systems, Man and Cybernetics – Part A: Systems and Humans, vol. 32, 2002, pp. 394-402. H. M. Hsu and C.T. Chen, Aggregation of fuzzy opinions under group decision making, Fuzzy Sets and Systems, vol. 79, 1996, pp. 279-285. C. Lu, J. Lan, and Z. Wang, Aggregation of fuzzy opinions under group decisionmaking based on similarity and distance, Journal of Systems Science and Complexity, vol. 19, 2006, pp. 63-71.

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[16] Z. Xu, Group decision making based on multiple types of linguistic preference relations, Information Sciences, vol. 178, 2008, pp. 452-467. [17] H. S. Lee, Optimal consensus of fuzzy opinions under group decision making environment, Fuzzy Sets and Systems, vol. 132, 2002, pp. 303–315. [18] D. Ben-Arieh and Z. Chen, Linguistic-labels aggregation and consensus measure for autocratic decision making using group recommendations, IEEE Transactions on Systems, Man and Cybernetics - Part A: Systems and Humans, vol. 36, 2006, pp. 558-568. [19] J. Kacprzyk and M. Fedrizzi, A “soft“ measure of consensus in the setting of partial (fuzzy) preferences, European Journal of Operational Research, vol. 34, 1988, pp. 316325. [20] J. Fodor, and M. Roubens, Fuzzy Preference Modelling and Multicriteria Decision Support, Kluwer Publishers, Boston, 1994. [21] S. A. Orlovski, Decision making with a fuzzy preference relation, Fuzzy Sets and Systems, vol. 1, 1978, pp. 155-167.

In: Mathematics and Mathematical Logic: New Research ISBN 978-1-60692-862-2 Editors: Peter Milosav and Irene Ercegovaca © 2010 Nova Science Publishers, Inc.

Chapter 4

THE MATHEMATICAL BASIS OF PERIODICITY IN ATOMIC AND MOLECULAR SPECTROSCOPY ∗

K. Balasubramanian Center for Image Processing and Integrated Computing, University of California Davis, Livermore, CA, USA Chemistry and Applied Material Science Directorate, Lawrence Livermore National Laboratory, University of California, Livermore, CA, USA Glenn T. Seaborg Center, Lawrence Berkeley Laboratory, University of California, Berkeley, CA, USA

INTRODUCTION This chapter applies combinatorial and group-theoretical relationships to the study of periodicity in atomic and molecular spectroscopy. The relationship between combinatorics and both atomic and molecular energy levels must be intimate since the energy levels arise from the combinatorics of the electronic or nuclear spin configurations or the rotational or vibrational energy levels of molecules. Over the years we have done considerable work on the use of combinatorial and group-theoretical methods for molecular spectroscopy [1–15]. The role of group theory [1–40] is evident since the classification of electronic and molecular levels has to be made according to the irreducible representations of the molecular symmetry group of the molecule under consideration. Combinatorics plays a vital role in the enumeration of electronic, nuclear, rotational and vibrational energy levels and wave functions. As can be seen from other chapters in this book, the whole Periodic Table of the elements has a mathematical group-theoretical basis since the electronic shells have their origin in group theory. Indeed, this concept can even be generalized to other particles beyond electrons such as bosons or other fermions that exhibit more spin configurations than just the bi-spin orientations of electrons. ∗

A version of this chapter was also published in The Mathematics of the Periodic Table, edited by D.H. Rouvray and R.B. King published by Nova Science Publishers, Inc. It was submitted for appropriate modifications in an effort to encourage wider dissemination of research.

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It has been shown that Einstein’s special theory of relativity is quite important for classifying the energy levels of very heavy atoms and molecules that contain very heavy atoms [41–48]. This is because to keep balance with the increased electrostatic attraction in heavier nuclei having a large number of protons, the core electrons of such heavy atoms must move with considerably faster average speeds. We have shown, for example, that the averaged speed of the 1s electron of heavier atoms such as gold is about 60% of the speed of light. Consequently, ordinary quantum mechanics does not hold, and one needs to invoke relativistic quantum mechanics to deal with such heavy atoms and with molecules containing very heavy atoms. We have defined relativistic effects as the difference in the observable properties of electrons as a consequence of using the correct speed of light compared to the classical infinite speed. Mathematically, the introduction of relativity results in a double group symmetry owing to the spin-orbit coupling term [41], which is a relativistic term in the Hamiltonian. This is a natural symmetry consequence of the LS spin-orbit operator, which changes sign upon rotation by 360º. Thus, the periodicity of the identity operation, which is normally envisaged as a rotation through 360º, is no longer the identity operation of the group. This is illustrated in Figure 1 with a Möbius strip, which exemplifies the double group symmetry. As one completes a 360º rotation along the Möbius surface there is a sign change since one goes from the inside of the surface to the outside. This requires the introduction of a new operation R in the normal point group of a molecule that corresponds to the rotation by 360º which is not equal to E, the identity operation. Hence we have to make use of the double group and doublevalued representations in both atomic and molecular spectroscopy.

Figure 1. A Möbius strip exemplifying the double group relativistic periodicity. The introduction of spin-orbit coupling into the relativistic Hamiltonian changes the periodicity of the normal point group symmetry into a double group symmetry, as rotation through 360° is not the identity operation. Note that the Möbius strip changes sign in this operation. Generalization of this to other complex phases results in Berry’s phase, where rotation through 360° may yield exp(2πi/n), thus resulting in other kinds of periodicity.

The Mathematical Basis of Periodicity in Atomic and Molecular Spectroscopy

89

The double group consists of twice the number of operations as the normal point group but is not a simple direct product of the normal point group and another group. This is a consequence of the fact that only some of the conjugacy classes of the normal point group generate new conjugacy classes upon multiplication by the operation R. The other conjugacy classes, which are called two-sided operations, such as the C2 rotations, double in order instead of generating new classes. This is because such operations when multiplied by R become equivalent to the operation, and thus the new operations belong to the same conjugacy class as the corresponding old operations. This feature complicates double group theory and the resulting periodicity of the double group. A characteristic feature is the generation of even-dimensional double-valued representations that characterize half-integral quantum numbers. We shall discuss this in one of the ensuing sections. In this chapter we shall consider the mathematical basis of atomic periodicity and spectroscopy with the use of group theory and combinatorics. We shall also consider the combinatorics of unitary groups and Young diagrams and their connections to the electronic spin functions. We shall also discuss molecular periodicity by considering the combinatorial basis of molecular electronic states. We describe the double groups and the periodicity arising from the classification of states in the double group. We expound on the combinatorics and periodicity pertinent to the rotational levels, nuclear spin functions, and rovibronic levels of molecules and give some examples.

COMBINATORIAL PERIODICITY IN MOLECULAR ELECTRONIC AND ATOMIC SPECTROSCOPY As might be expected, the classification of atomic states and thus the Periodic Table of the elements are based on combinatorial and group-theoretical considerations. An interesting related combinatorial problem has to do with the graphical unitary group approach to manyelectron configuration and correlation problems [4, 49]. The associated fermionic algebras involving Young diagrams and the symmetric permutation group approach have been discussed previously [4, 49]. It would also be interesting to consider cases of similar enumerations for other bosons or even fermions that are more than spin 1⁄2 particles. Such cases, while possibly not applicable to electronic systems, are applicable to nuclear spin species, and would have considerable group-theoretical value. There are several applications of group theory to atomic states. An early application by Curl and Kilpatrick [16] showed that the Schur functions of the symmetric groups Sn can be used as generating functions for atomic term symbols. The method involves replacement of cycle index polynomial terms by the various mA and ms symbol powers for generating functions of the atomic states that transform according to the irreducible representations of the Sn group. From this the authors were able to establish a periodic connection to the combinatorial enumeration of atomic term symbols even for complicated cases such as those for f7 shells. Balasubramanian [4] has established the connection between the graphical unitary group approach for electronic configurations and the Schur function algebra of the Sn groups, which play an important role in the Periodic Table in terms of the classification of various spin multiplets and the term symbols of electronic states. We shall consider this in some detail before enumerating the atomic states. The electronic states arising from the many-electron configura-

90

K. Balasubramanian

tions have certain periodicities and patterns as enumerated by the Schur functions of the symmetric groups Sn. The representation theory of the symmetric group is well known [17, 20, 50], and we will not repeat it here. The irreducible representations of Sn may be characterized by Young diagrams for the various partitions of the integer n, denoted by [n]. The states of many particles (including bosons and fermions) that possess multiple spin orientations can be represented by generalized Young Tableaus (GYTs). For example, Figure 2 shows all of the possible GYTs for the partitions of six occupied by six particles that have three spin orientations (for example, a spin-1 particle such as the bosonic deuterium nucleus) with the possibility that two have the first kind of spin orientation, two have second kind and the last two particles have the third kind. We have denoted this [122232] shape as shown in Figure 2. 1

1

1

2

2 3

2 3

1

3

3

2

1

1

2

2

1

1

3

1 3

1

2 2

1

1

2

3

3

3

2

1 1

3

2 3

2

2

3

3

2

1

2 3

3

2 2

1

1

2

3

1 1

2 3

1 1

3

2

3

2

3

3

2

1 3

Figure 2. Generalized Young Tableaus (GYTs) for the partition of six for a spin 1 boson (e.g., deuterium) corresponding to the spin distribution of two particles with the first spin orientation, two with the second orientation, and two with the third or [122232] shape.

As can be seen from Figure 2, the GYTs have numbers in any column in strictly ascending order while the numbers in any row must be in non-decreasing order. These tableaus represent the nuclear spin functions that transform according to the particular irreducible representation that the diagram represents. It is interesting to note that for a spin-1 particle such GYTs can have at the most three rows and, likewise for electrons, which are spin 1⁄2 particles, the GYTs can have at the most two rows. In general for a spin-j particle there can be at most only 2j + 1 rows in the GYTs.

The Mathematical Basis of Periodicity in Atomic and Molecular Spectroscopy

91

The enumeration of the GYTs for the various shapes of the spin distributions is a fundamental problem that is common to electronic and nuclear structures. In the context of manyelectron spin functions, the graphical unitary group approach requires the enumeration of Gel’fand states which are the GYTs containing two rows. The results also have some interesting periodicity trends in the mathematical sense. These GYTs and the associated spin multiplets and spin states can be enumerated by polynomials called the Schur functions of the symmetric group Sn. The Schur function corresponding to a partition λ of n is denoted by {λ} and is defined in the following way: 1 {λ} = n!

∑χλ(g)s1b s2 b …snbn 1

(1)

2

g∈G

where χλ(g) is value of the character for g in the group G = Sn corresponding to the irreducible representation [λ] of the group Sn. To illustrate this, the Schur function corresponding to the partition 4 + 1 + 1 is given by the Schur function, {6;4,1,1}, shown below: {6;4,1,1} =

1 120

[10s16 + 30 s14s2 + 40 s13s3 – 90 s12s22 – 120 s1s1s3

– 30 s23 + 40 s32 + 120 s6]

(2)

The Schur function is the generator for the GYTs, and is obtained by replacing every sk in the Schur function or S-function by ∑λik. The coefficient of a typical term λ1a1 λ2a1… λmam in i

the generating function thus obtained yields the number of GYTs with the shape [1a12a1…mam ]. The GYT generators are so powerful that they also enumerate the atomic states when applied to electronic spin functions which are GYTs with only two rows. As an illustration of GYT generation, let us consider the partition 2 + 1 for three particles. Let the particle under consideration be a spin-1 boson, which has three spin orientations that we depict symbolically as λ1, λ2 and λ3. The S-function in this case is given as: 1

{3;2,1} = 6 [2 s13 – 2 s3]

(3)

The GF of the GYTs for a spin 1 particle is thus given as: {λ1,λ2,λ3;2,1} = λ12λ2 + λ1λ22 + λ2λ32 + λ22λ3 + λ1λ32+ λ12λ3 + 2 λ1λ2λ3

(4)

The above generating function thus obtained from the S-function generates all of the GYTs shown below in Figure 3 for all possible spin distributions or shapes for the partition 2 + 1. The enumeration technique can be applied to GYTs of any shape belonging to any particle with any spin shape and spin distribution. The method is not restricted to just spin 1 or spin 1⁄2 particles.

92

K. Balasubramanian

1

1

1

3

2

2

2

1

2

3

2 2

1

3

3

1

1

3

3

1 2

3

2

3

Figure 3. All possible GYTs corresponding to the partition 2 + 1 as enumerated by the S-function {λ1,λ2,λ3;2,1}.

We can use the above method to generate all of the possible spin states for a manyelectron system or all of the possible atomic spectral energy levels for a given open-shell electronic configuration. First, we illustrate the method for obtaining all of the possible electronic spin states. The GYTs for electrons may contain at most two rows since there are only two possible distinct spin orientations for an electron (α and β) and thus there cannot be more than two rows. Accordingly, only certain partitions are allowed for an electron. This means that the GYTs can be formed only by the integers 1 and 2. Each spin distribution or spin shape then contains representations that are sums of the GYTs with the appropriate shape. For example for a system of six electrons with five spins up and one spin down there are exactly two GYTs as shown in Figure 4.

1

1

1

1

1

2

1

1

1

1

1

2 Figure 4. The GYTs for six electrons with five spin ups and one spin down.

Figure 5. The many-electron spin multiplets for an even number of electrons; there are exactly N cells and at most two rows for the spin functions.

The [1a12a1] GYTs enumerate states with a total spin quantum number Mz = (a1 – a2)/2. Consequently, once the GYTs are sorted out according to their total Mz val-

The Mathematical Basis of Periodicity in Atomic and Molecular Spectroscopy

93

ues we obtain the spin multiplets for the many-electron systems. A neat set of periodic spin multiplets are obtained for such many-electronic systems. These are shown in Figures 5 and 6, respectively, for even and odd numbers of electrons. This important result was made possible by use of the periodic S-functions.. The method of S-functions is powerful and general in that it can be applied to more than electrons. Thus, the same method can be applied to other particles that have integral spins such as the deuterium nuclear spin functions or to the cases with multinomial spin distributions. In such cases the diagrams become more complex with many more rows depending on the particles. For a spin 1 particle the diagrams have three rows at most. For a spin j particle the diagrams will have up to (2j + 1) rows yielding an array of complex spin multiplets. Next we demonstrate the periodic power of the S-function for enumerating the possible electronic states of an atom [16] which are well known as atomic term symbols in atomic spectroscopy. The method is completely analogous to generating the GYTs and manyelectron spin multiplets demonstrated above. The only difference is that we obtain a generating function for the different ML projections and spin projections and the total function must comply with the Pauli Exclusion Principle, as electrons are fermions. The method can be applied from simple cases, e. g., main group open-shells such as p2, p3, to more complex lanthanides and actinides that have fm open-shell f-electrons (Table 1). For example, consider the most complex half-filled 5f7 shells. The possible electronic states sorted according to the total spin and total angular momentum in compliance with Pauli’s Principle are given by: 2

S(2), 2P(5), 2D(7), 2F(10), 2G(10), 2H(9), 2I(9), 2J(7), 2K(4),2M(2), 2N, 2O S(2), 4P(2), 4D(6), 4F(5), 4G(7), 4H(5), 4I(5), 4J(3), 4K(3),4M, 4N 6 6 S, P, 6D, 6F, 6G, 6H, 6I 8 S 4

Table 1. All possible atomic term symbols for all actinides and lanthanides. Shell f1/f13 f2/f12 f3/f11 f4/f10

f5/f9

f6/f8

f7

States F 1 1 1 1 3 3 3 S D G I P F H 2 2 P D(2) 2F(2) 2G(2) 2H(2) 2I 2J 2K 4S 4D 4F 4G 4I 1 S(2), 1D(4), 1F(1), 1G(4), 1H(2), 1I(3), 1J, 1K(2), 1M, 3 P(3) 3D(2) 3F(4) 3G(3) 3H(4) 3I(2) 3J(2) 3K 3L 5 5 5 5 5 S D F G I 2 P(4), 2D(5), 2F(7), 2G(6), 2H(7), 2I(5), 2J(5), 2K(3),2L(2), 2M 2N 4 4 S P(2) 4D(3) 4F(4) 4G(4) 4H(3) 4I(3) 4J(2) 4K 4L 6 6 6 P F H 1 S(4), 1P. 1D(6), 1F(4), 1G(8), 1H(4), 1I(7), 1J(3), 1K(4), 1L (2), 1M(2), 1O 3 P(6) 3D(5) 3F(9) 3G(7) 3H(9) 3I(6) 3J(6) 3K(3) 3L(3) 3M 3N 5 5 5 S P D(3) 5F(2) 5G(3) 5H(2) 5I(2) 5J 5K 7 F 2 S(2), 2P(5), 2D(7), 2F(10), 2G(10), 2H(9), 2I(9), 2J(7), 2K(5) 2L(4),2M(2), 2 N, 2O 4 S(2), 4P(2), 4D(6), 4F(5), 4G(7), 4H(5), 4I(5), 4J(3), 4K(3),4M, 4N 6 6 S, P, 6D, 6F, 6G, 6H, 6I 8 S 2

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Figure 6. The many-electron spin multiplets for an odd number of electrons; there are exactly N cells and at most two rows for the spin functions.

The mathematical aspect of periodicity in atomic states is dependent on the orbital angular momentum of the electrons and spins as exemplified by the S-function generator used above for the generation of atomic term symbols. Yet another aspects of periodicity involves the molecular electronic states. The electronic configurations themselves consist of two parts, namely the spin part that was generated using the S-functions and space types that can also be generated using multinomial generators. In certain cases, as shown by the author, the orbital degeneracy can bring out additional symmetry. A space type can be imagined as a distribution of electrons in boxes such that a permutation of electrons within a box does not generate a new space type and the boxes themselves can be permuted if the orbitals are degenerate. Such groups are called wreath product groups. Balasubramanian [7] used this group theory combined with combinatorial multinomial generating functions to generate electronic space types. This can be illustrated for the benzene delocalized orbital π electrons. The periodic generating function for the number of space types of an n-orbital electronic configuration is given by: F = (1 + w + w2)n

(5)

where the coefficient of wm gives the number of space types with m electrons distributed among these n orbitals. For the case of benzene with six π electrons distributed among six orbitals we seek the coefficient of w6 with n = 6 in the above generating function. This is given by: 6 6 6 1 + ⎛⎝3⎞⎠ + ⎛⎝ 2 2 2 ⎞⎠ + ⎛⎝ 4 1 1 ⎞⎠ = 141

(6)

These 141 space types of benzene enumerated here are divided into equivalence classes of space types according to the symmetry equivalence from the wreath product groups in-

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95

duced by orbital degeneracy. As is well known, the six π orbitals of benzene are divided into 1 + 2 + 2 + 1 equivalence classes of orbitals. Thus, switching of the orbitals in the second and third set leads to equivalences, and the electrons can themselves be switched in each orbital. The result is a direct product of wreath product groups as shown below: S2 × S2[S2] × S2[S2] × S2

(7)

The cycle index polynomial of the totally symmetric representation of the above group generates the equivalence classes of the space types from the well-known Pólya Theorem [1, 52–60]. Consequently, the cycle index and the generating functions for the case of benzene are as follows. P=

F=

2

{ (s

1 4 8 1

}

+ 2 s12s2 + 3 s22 + 2 s4)

2

(8)

1 2 {2 (1 + w + w2)2} 28

(9)

The coefficient of w6 in the above generating function can be seen to be 58, which suggests that for benzene 141 space types are divided into 58 equivalence classes. Table 2 gives the number of equivalence classes of the space types for the various atoms that exhibit equivalence among the p orbitals. Table 2. Equivalence classes of the space types for the electronic configurations of atoms that have degenerate p orbitals. System He Li Be B C N O F

Total No Space Types 45 156 414 882 1554 23-4 2907 3139

Equivalence Classes 17 42 86 148 223 295 349 368

In summary, we have shown that the electronic configurations of molecules and atoms can be simplified using the mathematical periodicity of the spin functions and space types. The former case was accomplished using the S-functions of the symmetric permutation groups Sn while the latter case was simplified using the wreath product configuration symmetry groups.

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COMBINATORIAL PERIODICITY IN MOLECULAR AND NMR SPECTROSCOPIES The concept of mathematical periodicity as described by the orbit structure of a permutation finds important applications in molecular and nuclear spin spectrsocopies. The orbit structure of a permutation comprises several cycles such that in each cycle a set of nuclei is visited followed by a return to the starting point. The cyclic structure of the permutation (12345)(678)(9,10) is illustrated in Figure 7. The orbit structure in Figure 7 determines the nuclear spin statistical weights of the rotational levels of molecules. Thus, from the periodic orbit structure in Figure 7, one determines a polynomial s5s3s2, because we have one orbit of length 5, one orbit of length 3 and one orbit of length 2. The periodicity and the length of the period associated with each such orbit then determine a generating function for the nuclear spin statistical weights of the energy levels. This concept can also be used in NMR and ESR spectrsocopies where the periodicity and the length of the orbits determine the NMR spin energy levels and thus the NMR spectra associated with the molecules. Although in ordinary NMR only Zeeman-allowed transitions are observed and thus only those transitions with changes of a single spin, multiple quantum NMR offers a powerful tool to probe into transitions involving multiple spin quantum numbers. Thus, all NMR interaction energy levels can be probed.

Figure 7. Periodic orbit structure for the permutation (12345)(678)(9,10).

We shall start with an application of permutational periodic structure in molecular spectroscopy. Indeed, the rotational energy levels of a molecule themselves have periodicities based on their point groups. We illustrate this with an icosahedral cluster, namely N20 [37] and C60 [29–33] systems. Consider the highly energetic regular dodecahedral N20 cluster [37], which exhibits icosahedral symmetry analogous to that in the fullerene C20. Since 14N is a spin 1 particle it exhibits an interesting generating function and nuclear spin species distribu-

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97

tion. The generalized character cycle indices for all of the irreducible representations for the N20 cluster with Ih symmetry are shown in Table 3. These were constructed using the orbit structures of permutations as demonstrated in Figure 7. The cycle indices for the various irreducible representations were obtained by multiplying the periodic orbit structures of each permutation by the corresponding character values. Note that the resulting polynomials are the same for the T1g and T2g representations and likewise the T1u and T2u representations since the orbit structures multiplied by their character values become identical owing to accidental degeneracy. We have used our generalization [1, 5–6, 56] of Pólya’s Theorem for all characters to seek generating functions for the nuclear spin species of 14N. Note that since the 14N nucleus is a spin 1 particle, we replace every xk in the cycle index in Table 3 by λk + μk + νk where the symbols λ, μ and ν stand for –1, 0 and 1 spin projections of the spin 1 14N nucleus. The resulting generating functions for the nuclear spin functions are shown in Table 4. The generating functions shown in Table 4 have two parts, one consisting of coefficients and the other of the trinomial λiμjνk. We do not show k since k = 20 – i + j) and it can thus be deduced from the values of i and j. To illustrate how the generating functions in Table 4 are obtained, let us consider the T1g or the T2g representation. From Table 3 we obtain the GCCI for this representation and we make the substitution given by GFx = PGx (xk → λk + µk + νk)

(10)

The above substitution yields the following expression: 1

GFT1g = 120 [3(λ + μ + ν)20 + 12(λ5 + μ5 + ν5)4 – 12(λ2 + μ2 + ν2)10 + 12(λ10 + μ10

(11)

+ ν10)2 – 15(λ + μ + ν)4(λ2 + μ2 + ν2)8] When Equation (8) is simplified into a trinomial it has several terms with coefficients for each term. Table 4 shows the coefficients and powers of λ and μ for the term. The power of ν is simply 20 – (i +j) and is thus not shown. The actual computations of the generating functions for N20 (and for C60 discussed subsequently) were carried out using computer code in quadruple precision developed by Balasubramanian [8, 9]. It is important to employ a quadruple precision arithmetic especially for C60, as the coefficients grow astronomically and thus any lower precision results in errors. An interesting consequence of the periodicity is that the g and u representations differ in some of their coefficients so significantly that one can say that there is inversion contrast in combinatorics. For example, the coefficient of the term λ9μ6ν5 in Table 4 for the Ag representation is 647706 while the corresponding coefficient for the Au representation is 645606. Moreover, the first non-zero coefficient for the Au representation is for the (18,2,0) partition, which means that at least 18 colors of one kind and two colors of another kind are needed to induce chirality in the binomial distribution. A purely trinomial term has two chiral colorings for the lowest order term, i. e., the (18,1,1) term in Table 4 has a coefficient of two for Au.

98

K. Balasubramanian Table 3. The GCCIs for the dodecahedral N20 cluster. N20 Order Ag Au T1g=T2g T1u=T2u Gg Gu Hg Hu

120 1 1 1 3 3 4 4 5 5

54 24 1 1 1 ⁄2 1 ⁄2 –1 –1 0 0

1236 20 1 1 0 0 1 1 –1 –1

210 15 1 1 –1 –1 0 0 1 1

210 1 1 –1 3 –3 4 –4 5 –5

Figure 8. Nuclear frequency spin spectrum for the Ag representation of N20.

102 24 1 –1 1 ⁄2 1 – ⁄2 –1 1 0 0

263 20 1 –1 0 0 1 –1 –1 1

1428 15 1 –1 –1 1 0 0 1 –1

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99

The coefficients thus enumerated in Table 4 can be sorted according to their total MF values where the term l has the projection –1, m has the projection 0, and v has the projection +1. Thus the term λiμjνk in Table 4 represents a total nuclear spin quantum number MF of (–i + k). When these coefficients are sorted according to their total MF values, they separate into nuclear spin multiplets with MF values ranging typically from –I, –I+1, –I+2,….0,….I–2, I–1, I. Such a multiplet would represent a nuclear spin multiplet with a multiplicity of 2I + 1. In this way for each irreducible representation the nuclear spin multiplets are separated according to their multiplicities and the results are shown in Table 5 for N20. Table 4. Generating functions for the dodecahedral N20 cluster.

100

K. Balasubramanian Table 4. (Continued)

As can be seen from Table 5, the frequencies of the spin multiplets corresponding to the g and u representations differ even for the singlet spin states. For example, the 1Ag state has a frequency of 113035 while the 1Au state has a frequency of 112444. There is a similar difference in the triplet state and most of the spin multiplets. This means that the parity can be contrasted even in low spin nuclear states. The corresponding rovibronic levels will also be populated with appreciable differences in the populations. From the nuclear spin multiplets we can also obtain the total nuclear spin statistical weights by the use of the Pauli Principle. Since 14 N nuclei are bosons, the overall wavefunction, which is a product of the rovibronic wavefunction and nuclear spin function, must be symmetric or must transform as the Ag irreducible representation. The frequency of each representation is obtained by adding the product of 2S + 1 and the frequency. The results are shown as a footnote in Table 5. On the basis of this, the frequencies shown in this footnote are themselves the nuclear spin statistical weights for N20 (see Figure 8).

The Mathematical Basis of Periodicity in Atomic and Molecular Spectroscopy Table 5. Nuclear spin species for the N20 cluster

101

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K. Balasubramanian

Other irreducible representations have similar spectra comparable to that in Figure 8, except that the intensities of the peaks vary. The Ag representation is particularly important as it gives the number of lines in multiple quantum NMR spectra. The frequencies of other irreducible representations determine the intensities of the lines in the spectra. The multiple quantum NMR spectra usually contain structural information for the (n – 2) quantum as this value exhibits dipolar couplings that contrast the structure. For the present case n – 2 corresponds to the spin multiplet 2S + 1 = 37. These multiplets have the frequencies 5, 2, 2, 7, 11, 1, 5, 5, 6, for the Ag, T1g, T2g, Gg, Hg, Au, T1u, T2u, Gu, and Hu representations, respectively. Thus, the dodecahedral N20 cluster exhibits interesting mathematical periodicity in spectroscopic terms. Next we consider the C60 cluster [29–33] as another example that demonstrates mathematical periodicity and its applications. The GCCIs of C60 are constructed analogously to those of N20 discussed above. The fact that C60 has 60 vertices would of course divide the permutation of 60 vertices into various periodic orbits. The nuclear spin species thus obtained using the GCCIs are shown in Table 6. As seen from Table 6, the frequencies grow astronomically as expected. This is because of the combinatorial explosion of the coefficients in the generating functions even though these functions are binomials. The binomial expansion is due to the fact that 13C60 is comprised of 13C nuclei, which exhibit only two spin orientations, as they are spin 1⁄2 particles. The same is true of C60H60, as protons are spin 1⁄2 particles and 12C has no nuclear spin. Again a major contrast is that the g and u representations have different frequencies due to the difference in the periodicity of the permutation multiplied by the character value for these representations. This feature manifests itself as contrasting frequencies for the g and u irreducible representations. We note that earlier work had an error in the spin statistical weights of C60 [31] primarily owing to the arithmetical precision but this was subsequently corrected [30, 32]. The relative differences between the g and u parities are especially significant for high-spin nuclear multiplets. For example, for the 2S + 1 = 57 spin multiplet of 13C6, the frequencies of the Ag, T1g, T2g, Gg, Hg, Au, T1u, T2u, Gu, and Hu representations are 22, 36, 36, 58, 80, 14, 42, 42, 56, and 70, respectively. Similarly for 2S + 1 = 55 the frequency of the Ag representation is 280 while it is 260 for Au. Consequently, the contrast in the g and u spin populations can be seen experimentally if high-spin nuclear states can be excited. Table 6. Nuclear spin multiplets for 13C60 or C60H60. Frequency of the irreducible representation Ag: 9607679885269312 Spin multiplets and their frequencies for Ag: 2S+1 Frequency 2S+1 1 31791575566072 3 7 150988619146706 9 13 105558807981090 15 19 31605175642230 21 25 4481735502630 27 31 298734989924 33 37 8805633300 39 43 101874363 45 49 372752 51 55 280 57 61 1

Frequency 2S+1 Frequency 89413728633564 5 13095954950748 149756091280506 11 13219208028055 76925432220000 17 5141513084676 17892025439775 23 933143835273 1980110898945 29 80345370985 101492436960 35 3139590568 2227563126 41 50512570 18110340 47 280174 41528 53 388 22 59

The Mathematical Basis of Periodicity in Atomic and Molecular Spectroscopy Table 6. (Continued) Frequency of the irreducible representation T1g: 28823036970926496 Spin multiplets and their frequencies for T1g : 2S+1 Frequency 2S+1 Frequency 2S+1 Frequency 1 95374646372040 3 268241251090167 5 39287856402727 7 452965902231668 9 449268197030424 11 39657626655407 13 316676363633175 15 230776308338940 17 15424535176554 19 94815530686980 21 53676052490265 23 2799431557098 25 13445194549380 27 5940332333550 29 241035603798 31 896204629630 33 304475471640 35 9418755979 37 26416344630 39 6682635360 41 151524170 43 305608974 45 54304371 47 840285 49 1114158 51 124257 53 1123 55 804 57 36 59 61 0

Frequency of the irreducible representation Gg: 38430716856193728 Spin multiplets and their frequencies for Gg: 2S+1 1 7 13 19 25 31 37 43 49 55 61

Frequency 2S+1 127166221937640 3 603954521378374 9 422235171614265 15 126420706329465 21 17926930052010 27 1194939619444 33 35221977930 39 407483337 45 1486916 51 1084 57 0

Frequency 2S+1 Frequency 357654979723731 5 52383811353475 599024288311326 11 52876834683423 307701740558940 17 20566048261230 71568077929785 23 3732575392371 7920443232495 29 321380974795 405967908600 35 12558346548 8910198522 41 202036737 72414711 47 1120460 165779 53 1512 58 59

Frequency of the irreducible representation Hg 48038396740938240 Spin multiplets and their frequencies for Hg: 2S+1 1 7 13 19 25 31 37 43 49 55 61

Frequency 2S+1 158957797411208 3 754943140441100 9 527793979532265 15 158025881932935 21 22408665535200 27 1493674601616 33 44027608785 39 509357130 45 1859568 51 1354 57 0

Frequency 2S+1 Frequency 447068708357295 5 65479766312622 748780379591832 11 66096042717787 384627172778940 17 25707561349782 89460103369560 23 4665719229588 9900554131440 29 401726346555 507460345560 35 15697937361 11137761648 41 252549365 90525051 47 1400644 207307 53 1902 80 59

103

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K. Balasubramanian

Table 6. (Continued) Frequency of the irreducible representation Au: 9607678793631424 Spin multiplets and their frequencies for Au: 2S+1 Frequency 2S+1 1 31791571643468 3 7 150988613640506 9 13 105558798039270 15 19 31605170531130 21 25 4481732871390 27 31 298734348764 33 37 8805495420 39 43 101861196 45 49 371694 51 55 260 57 61 0

Frequency 2S+1 Frequency 89413727296344 5 13095954114986 149756080818726 11 13219207292373 76925425313100 17 5141512318638 17892020535870 23 933143526111 1980109351620 29 80345252581 101491992360 35 3139568730 2227502850 41 50509098 18103410 47 279955 41266 53 377 14 59

Frequency of the irreducible representation T1u: 28823037990981216 Spin multiplets and their frequencies for T1u: 2S+1 Frequency 2S+1 Frequency 2S+1 Frequency 1 95374639953380 3 268241262122232 5 39287856269005 7 452965915721858 9 449268199508214 11 39657627967718 13 316676367808710 15 230776318887660 17 15424535578410 19 94815537801090 21 53676055391160 23 2799431961645 25 13445196226770 27 5940334271070 29 241035683183 31 896205406510 33 304475780520 35 9418781778 37 26416442910 39 6682705140 41 151526692 43 305623968 45 54309474 47 840531 49 1114942 51 124548 53 1132 55 826 57 42 59 61 0

Frequency of the irreducible representation Gu: 38430716784610624 Spin multiplets and their frequencies for Gu: 2S+1 1 7 13 19 25 31 37 43 49 55 61

Frequency 2S+1 127166211596396 3 603954529362364 9 422235165847980 15 126420708332460 21 17926929098160 27 1194939755164 33 35221938330 39 407485164 45 1486642 51 1086 57 0

Frequency 2S+1 Frequency 357654989418576 5 52383810383991 599024280327336 11 52876835260051 307701744200760 17 20566047897048 71568075926790 23 3732575487756 7920443622690 29 321380935775 405967772880 35 12558350508 8910208020 41 202035787 72412884 47 1120487 165808 53 1509 56 59

The Mathematical Basis of Periodicity in Atomic and Molecular Spectroscopy

105

Table 6. (Continued) Frequency of the irreducible representation Hu: 48038395577718272 Spin multiplets and their frequencies for Hu 2S+1 1 7 13 19 25 31 37 43 49 55 61

Frequency 2S+1 158957783147612 3 754943142918890 9 527793963824370 15 158025878824830 21 22408661950230 27 1493674096176 33 44027431350 39 509345790 45 1858246 51 1336 57

Frequency 2S+1 Frequency 447068716714920 5 65479764507375 748780361146062 11 66096042558712 384627169513860 17 25707560219562 89460096462660 23 4665719015799 9900552974310 29 401726189132 507459765240 35 15697919478 11137710870 41 252544942 90516294 47 1400452 207074 53 1887 70 59 0

Table 7 shows the correlation of the rotational levels for C60 from J = 0 to 30 with the corresponding weights only in the rotational subgroup I. Note that for purposes of comparing with experimental results one must use the nuclear spin frequencies given in Table 6, but the statistical weights in Table 7 in factored form yield the orders of magnitude. All levels in Table 7 are of g symmetry since the J states can correlate into g levels. (1) The irreducible representations for J > 31 are given by q[A + 3T1 + 3T2 +4G + 5H] + Γ(r), where q is the quotient obtained by dividing J by 30 and r is the remainder. Γ(r) is the set of irreducible representations spanned by J = r listed in this Table (see text for further discussion). Note that since nuclear spin statistical weights are the same for g and u symmetries, we do not show g or u. (2) f = 19 215 358 678 900 736 for C60H60; f = 706 519 304 586 988 199 183 738 259 for C60D60. Each correlation in Table 7 was obtained using the mathematical method of subduction. As can be seen from Table 7, we have a very interesting periodicity among rotational levels. The correlations for the rotational levels with J > 31 have a periodic relation to the levels with J < 30. This is another mathematical manifestation of periodicity. The relations for all J > 30 are as follows: D(J)↓Ih = q(D(30)↓Ih – A) + (D(r) ↓Ih), q = [J/30], r = J – 30[J/30]

(12)

where the function within square brackets is the greatest integer contained in the brackets and thus q and r are quotients and remainders obtained by dividing J by 30. The term D(30) stands for the subduced representations for J = 30 that are displayed in Table 7. To illustrate this, the J = 195 rotational level contains the following representations: D(195)↓Ih = 6(Ag + 3T1g + 3T2g + 4Gg + 5Hg) + (Ag + 2T1g + 2T2g + 2Gg + 2Hg)≡ 7Ag + 20T1g + 20T2g + 26Gg + 32Hg

(13)

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K. Balasubramanian Table 7. Correlations of the rotational levels of C60: the nuclear spin statistical weights J = 0 to 30

The above concept of the periodicity of the rotational levels of C60 is illustrated in Figure 9. It is worthy of note that the nuclear spin statistical weights of the rotational levels vary approximately as (2J + 1) due to large nuclear spin statistical weights.

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107

J=0 J=1

J=30 * * * . * *

J=2

.J=3 Figure 9. Periodicity of the rotational levels of buckminsterfullerene, C60.

PERIODICITY OF DOUBLE GROUPS AND ELECTRONIC STATES The concept of the double group [17, 24–27, 42, 51] is required when the normal periodicity resulting from rotation through 360° breaks down, as demonstrated for the Möbius strip. This happens when half-integral states are considered. For example, the rovibronic states of openshell systems with an odd number of open-shell electrons exhibit half-integral spin states due to an odd number of open-shell electrons and thus we need a new concept of periodicity. This is also the case when spin-orbit coupling is introduced into the Hamiltonian. This difficulty was circumvented by Bethe through the concept of the double group. He introduced a new operation called R that changes the sign for the rotation through 360° for half-integral states and yet retains the same symmetry for the integral states, as shown above for the C60 integral rotational levels. Since the periodicity and the group structure are quite different for the double group, we provide a few examples of double group character tables and correlation tables. Most of the character tables appear in books such as those of Hamermesh [17] or Altmann and Herzig [24] for the double groups. Balasubramanian [51] developed the character table for the icosahedral double group denoted by Ih2 that is shown in Table 8. Note that the operation R introduces a few new conjugacy classes for the Ih2 double group while other conjugacy classes just double in their orders. This is a consequence of the fact that certain operations are called two-valued operations and these operations when multiplied by R become equivalent, and thus belong to the same conjugacy class. However, other operations, such as C5 and RC5, become inequivalent, and thus belong to different conjugacy classes. The new irreducible representations in the double group are called two-valued representations and they are always even dimensional and correspond to half-integral representations. The number of such representations equals the number of new conjugacy classes, as demonstrated in Table 8. These are called E1g(1/2), Gg(3/2), Ig(5/2), E2g(7/2), with the corresponding u representations. The correlation table for the half-integral states of the Ih2 double group is shown in Table 9. Note that the corresponding table for the integral representations has already been discussed for C60 (Table 7). As can be seen from Table 9, the half-integral spin or rovibronic

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K. Balasubramanian

states all correlate only into double-valued representations, which are all even dimensional. As a result, the representation corresponding to 1⁄2 is a degenerate two-dimensional irreducible representation. The quartet state with s = 3⁄2 is also four-fold degenerate and s = 5⁄2 is likewise the six-fold degenerate I representation in the double group. The first case which splits into two irreducible representations is the s = 7⁄2 case. The periodicity is reduced in the double-valued representation to half as s = 31⁄2 is related to s = 1⁄2 by periodicity. All higher s values are obtained using a periodic relation as shown in Table 9. Table 8. Character table for the Ih2 double group.

The Mathematical Basis of Periodicity in Atomic and Molecular Spectroscopy Table 8. (Continued)

109

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K. Balasubramanian

Table 9. Periodic correlation table for the half-integral states of the Ih2 double group. s

1 3 5 7 9

2 2 2 2

2 11 2 13 2 15 2 17 2 19 2 21 2 23 2 25 2 27 2 29 2 31 2

Irreducible Representationsa E1g′(1 2) Gg′(3 2) Ig′(5 2) E2g′(7 2) + Ig′(5 2) Gg′(3 2) + Ig′(5 2) E1g′(1 2) + Gg′(3 2) + Ig′(5 2) E1g′(1 2) + Gg′(3 2) + Ig′ (5 2) + E2g′(7 2) Gg′ (3 2) + 2 Ig′(5 2) G′g(3 2) + 2 Ig′(5 2) + E2g′(7 2) E1g′(1 2) + Gg′(3 2) + 2 Ig′(5 2)+ E2g′(7 2) E1g′(1 2) + 2 Gg′(3 2) + 2 Ig′(5 2) E1g′(1 2) + 2 Gg′(3 2) + 2 Ig′(5 2) + E2g′(7 2) E1g′(1 2) + Gg′(3 2) + 3 Ig′(5 2) + E2g′(7 2) 2 Gg′(3 2) + 3 Ig′(5 2) + E2g′(7 2) E1g′(1 2) + 2 Gg′(3 2) + 3 Ig′(5 2) + E2g′(7 2) 2 E1g′(1 2) + 2 Gg′(3 2) + 3 Ig′(5 2) + E2g′(7 2)

a

Ds = q{E1g′(1 2) + 2 Gg′(3 2) + 3 Ig′(5 2) + E2g′(7 2)} + Ds′, 2s + 1 q = 30 , s′ = s – 15q, if s > 31 2

We have also collected the correlation tables [42] for the octahedral double group Oh2 in Table 10, the correlation table for the Td2 in Table 11, and the correlation table for the D6h2 in Table 12. These correlation tables all demonstrate interesting mathematical periodicity for the rotational or rovibronic levels. The octahedral integral rotational levels exhibit a period of 12 analogous to that for the tetrahedral group. However, the half-integral spin states or rovibronic states exhibit a period of six both in the octahedral and tetrahedral double groups. The D6h2 double group exhibits a different periodic trend as seen from Table 12. The periodicity of six is same for both the half-valued and integral representations. Thus, the periodicity trends exhibited by the double groups are quite interesting. These correlation tables are quite valuable in obtaining the rovibronic levels of molecules with both an odd and even number of electrons. It is important to obtain the overall rovibronic correlation as opposed to individual rotational correlations owing to the fact that the total wavefunction may become a half-integral representation, especially for systems with an odd number of electrons. Furthermore, for molecules containing very heavy atoms spin-orbit effects become quite significant, and thus the coupling of the spin with orbital angular momentum splits the electronic states into spin-orbit states. The exact manner in which these states are split by spin-orbit coupling is given by the double group correlation tables shown here.

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Table 10. Periodic correlation table for the half-integral states of the Oh2 double group. Irreducible Representations in the Oh2 Groupa A1g T1g Eg + T2g A2g + T1g + T2g A1g + Eg + T1g + T2g Eg + 2T1g + T2 + T2g

s 0 1 2 3 4 5 6 ⁄2

A1g + A2g + Eg + T1g + 2T2g

1

E1g′(1⁄2) Gg′(3⁄2) E2g′(5⁄2) + Gg′(3⁄2) E1g′(1⁄2) + E2g′(5⁄2) + Gg′(3⁄2) E1g′(1⁄2) + 2 Gg′(3⁄2) E1g′(1⁄2) + E2g′(5⁄2) + 2 Gg′(3⁄2) E1g′(1⁄2) + E2g′(5⁄2) + 2 Gg′(3⁄2) + terms of s′ but interchange E1g′(1⁄2) with E2g′(5⁄2) 2n {(E1g′(1⁄2) + E2g′(5⁄2) + 2 Gg′(3⁄2)} + terms for s′

3

⁄2 ⁄2 7 ⁄2 9 ⁄2 11 ⁄2 6 + s′ 12n + s′ 5

a

Terms for other integral s values are found using the formula: D(12n+s′) = Ds′ + n(A1g + A2g + 2 Eg + 3 T1g + 3 T2g), s′ < 12.

Table 11. Periodic correlation table for the half-integral states of the Td2 group. s 0 1 2 3 4 5 6

Irreducible Representationsa A1 T1 E + T2 A2 + T1 + T2 A1 + E + T1 + T2 E + 2 T1 + T2 A1 + A2 + E + T1 + 2 T2

7 8 9 10 11 12 13 14 15 1 ⁄2 3 ⁄2 5 ⁄2 7 ⁄2 9 ⁄2 11 ⁄2 6 + s′ 2n + s′

A2 + E + 2 T1 + 2 T2 A1 + 2E + 2 T1 + 2 T2 A1 + A2 + E + 3 T1 + 2 T2 A1 + A2 + 2E + 2 T1 + 3 T2 A2 + 2E + 3 T1 + 3 T2 2 A1 + A2 + 2 E + 3 T1 + 3 T2 A1 + A2 + 2 E + 4 T1 + 3 T2 A1 + A2 + 3 E + 3 T1 + 4 T1 A1 + 2 A2 + 2 E + 4 T1 + 4 T1 E1/2 G3/2 G3/2 + E5/2 E1/2 + G3/2 + E5/2 E1/2 + 2 G3/2 E1/2 + E5/2+ 2 G3/2 E1/2 + E5/2 + 2 G3/2 + terms for s′, but interchange E1/2 and E5/2 2n(E1/2 + E5/2)

a

Other integral spin states are correlated using the formula

D(12n+s'′ = Ds′ + n(A1 + A2 + 2 E + 3T1 + 3T2), s′ < 12.

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The concept of periodicity can be extended to cases beyond the double groups. Such cases would involve Berry’s phase where a rotation through 360° would yield a complex number, exp(2πi/n), for an integer n > 2. The symmetry exhibited by such systems could be quite intriguing. It is hoped that this chapter will stimulate future investigations into Berry’s phase. Table 12. Periodic correlation table for the half-integral states of the D6h2 double group s 0 1 2 3 4 5 6 1

⁄2 ⁄2 5 ⁄2 7 ⁄2 9 ⁄2 11 ⁄2 3

6n + s′

Irreducible Representationsa A1g A2g + E1g A1g + E1g + E2g A2g + B1g + B2g + E1g + E2g A1g + B1g + B2g + E1g + 2 E2g A2g + B1g + B2g + 2 E1g + 2 E2g 2 A1g + A2g + B1g + B2g + 2 E1g + 2 E2g E1g′(1⁄2) E1g′(1⁄2) + E3g′(3⁄2) E1g′(1⁄2) + E2g′(5⁄2) + E3g′(3⁄2) E1g′(1⁄2) + 2 E2g′(5⁄2) + E3g′(3⁄2) E1g′(1⁄2) + 2 E2g′(5⁄2) + 2 E3g′(3⁄2) 2 E1g′(1⁄2) + 2 E2g′(5⁄2) + 2 E3g′(3⁄2) 6n{E1g′(1⁄2) + E2g′(5⁄2) + E3g′(3⁄2)} + terms of s′, s′ < 6

a

Terms for other integral s values may be found by using

D(12n+s′) = Ds′ + n(A1g + A2g + B1g + B2g + 2 E1g + 2 E2g), s′ < 6

ACKNOWLEDGEMENT This research was performed under the auspices of the US Department of Energy by the University of California, LLNL under contract number W-7405-Eng-48 while the work at UC Davis was supported by the National Science Foundation.

REFERENCES [1] [2] [3] [4] [5] [6] [7] [8]

K. Balasubramanian, Chem. Rev., 85, 599 (1985). K. Balasubramanian, J. Chem. Phys., 72, 665 (1980). K. Balasubramanian, J. Chem. Phys., 73, 3321 (1980). K. Balasubramanian, Theor. Chim. Acta, 59, 237 (1981). K. Balasubramanian, J. Chem. Phys., 74, 6824 (1981). K. Balasubramanian, J. Chem. Phys., 75, 4572 (1981). K. Balasubramanian, Int. J. Quant. Chem., 20, 1255 (1981). K. Balasubramanian, J. Comput. Chem., 3, 69 (1982).

The Mathematical Basis of Periodicity in Atomic and Molecular Spectroscopy [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

[26] [27]

[28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40]

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K. Balasubramanian, J. Comput. Chem., 3, 75 (1982). K. Balasubramanian, H. Strauss, and K. S. Pitzer, J. Mol. Spectrosc., 93, 447 (1982). K. Balasubramanian, J. Phys. Chem., 86, 4668 (1982). K. Balasubramanian, J. Chem. Phys, 78, 6358 (1983). K. Balasubramanian, J. Chem. Phys, 78, 6369 (1983). K. Balasubramanian, Group Theory of Non-rigid Molecules and its Applications,” Elsevier Publishing Co., 23, 149–168 (1983). K. Balasubramanian, Theor. Chim. Acta, 78, 31 (1990). R. F. Curl, Jr., and J. E. Kilpatrick, Amer. J. Phys. 28, 357 (1960) M. Hamermesh, Group Theory and its Physical Applications, Addison Wesley, Reading MA, 1964 F. A. Cotton, Chemical Applications of Group Theory, Wiley Interscience, New York, NY, 1971 P. R. Bunker, “Molecular Symmetry and Spectroscopy,Academic Press, New York, NY, 1979 D. E. Littlewood, Theory of Group Characters and Matrix Representations of Groups, , Oxford, New York, NY, 1958 H. Weyl, Theory of Groups and Quantum Mechanics, Dover Publications, New York, NY, 1950 M. Tinkham, Group Theory and Quantum Mechanics, McGraw-Hill, New York, NY 1964 B. R. Judd, Operator Techniques in Atomic Spectroscopy, Princeton University Press, Princeton,, NJ 1998 S L Altmann and P Herzig, Point-Group Theory Tables, Clarendon Press, Oxford, 1994. P. Pyykkö and H. Toivonen, Tables of Representation and Rotation Matrices for The Relativistic Irreducible Representations of 38 Point Groups, Acta Academiae Aboensis, Ser B, 43, 1 (1983) H.T. Toivonen and P. Pyykkö, Int. J. Quant. Chem., 11, 697 (1977) H.T. Toivonen and P. Pyykkö, Relativistic Molecular Orbitals and Representation Matrices for the Double Groups T and Th, Department of Physical Chemistry, Åbo Akademi, Finland, Report No. B79 (1977), 11 pp. K. Balasubramanian, J. Mag. Res., 91, 45 (1991). K. Balasubramanian, Chem. Phys. Lett., 183, 292 (1991). K. Balasubramanian, Chem. Phys. Lett., 200, 649 (1992). W. G. Harter and T. C. Reimer, Chem. Phys. Lett., 194, 230 (1992). W. G. Harter and T. C. Reimer, Chem. Phys. Lett., 198, 429E (1992). W. G. Harter and D. E. Weeks, J. Chem. Phys., 90, 4727 (1989). K. Balasubramanian, J. Chem. Phys., 95, 8273 (1991). K. Balasubramanian and T. R. Dyke, J. Phys. Chem., 88, 4688 (1984). K. Balasubramanian, J. Mol. Spectroscopy 157, 254 (1993). K. Balasubramanian, Chem. Phys. Lett., 202, 271(1993). K. Balasubramanian, J. Phys. Chem., 97, 8736 (1993) K. Balasubramanian, Mol. Phys., 80, 655 (1993). K. Balasubramanian, J. Chem. Phys., in press.

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[41] K. Balasubramanian, Relativistic Effects in Chemistry, Part B: Applications, Wileyinterscience, New York, NY, p. 527, 1997. [42] K. Balasubramanian, Relativistic Effects in Chemistry, Part A: Theory and Techniques, Wiley-Interscience, New York, NY, p. 301, 1997. [43] K. S. Pitzer, Accts. Chem. Res., 12, 271 (1979). [44] P. Pyykkö and J. P. Desclaux, Accts. Chem. Res., 12, 276 (1979). [45] P. Pyykkö, Adv. Quant. Chem., ll, 353 (1978). [46] K. Balasubramanian, J. Phys. Chem., 93, 6585 (1989). [47] P. Pyykkö Ed. Proceedings of the Symposium on Relativistic Effects in Quantum Chemistry; Int. J. Quantum Chem., 25 (1984). [48] P. Pyykkö, Relativistic Theory of Atoms and Molecules, Springer Verlag: Berlin and New York, Part I 1986 Part II 1993, Part 3 2000. For comprehensive list of references up to 2002 see http://www.csc.fi/rtam/. [49] J. Paldus, Theoretical Chemistry: Advances and Perspectives, H. Eyring and D. J. Henderson, Eds, Academic Press, New York, NY, 1976 [50] X. Y. Liu and Balasubramanian, J. Comput. Chem., 10, 417 (1989). [51] K. Balasubramanian, Chem. Phys. Lett., 260, 476 (1996). [52] G. Pólya, Acta Math, 68, 145 (1937) [53] K. Balasubramanian, Theor. Chim. Acta, 53,129 (1979) [54] G. Pólya and R.C. Read, Combinatorial Enumeration of Groups, Graphs, and Chemical Compounds, Springer, New York, NY, 1987. [55] J. H. Redfield, Amer. J. Math. 49, 433 (1927). [56] K. Balasubramanian, J. Math. Chem. 14, 113 (1993). [57] D. H. Rouvray, Chem. Soc. Rev. 3, 355 (1974). [58] D. H. Rouvray, Endeavour, 34, 28 (1975) [59] A.T. Balaban, Chemical Applications of Graph Theory, Academic Press, New York, NY, 1976. [60] R. Read, in “Graph Theory and Applications”, Y. Alavi et al. eds., Lecture Notes in Mathematics, 303, 243 (1972) Springer, 1972.

In: Mathematics and Mathematical Logic: New Research ISBN 978-1-60692-862-2 c 2010 Nova Science Publishers, Inc. Editors: Peter Milosav and Irene Ercegovaca

Chapter 5

S ECOND - ORDER D EFINABILITY IN A M ODEL George Weaver∗ Park Science Center, Bryn Mawr College Bryn Mawr, PA, USA

Abstract The back and forth characterization of equivalence of interpretations for finitary (and some infinitary) second-order languages introduced in Weaver and Penev [2005] is applied to obtain a condition necessary and sufficient for an attribute of an interpretation to be definable in that interpretation by a second-order formula (either finitary or infinitary). This condition is applied to obtain some ”reduction” theorems for the second-order theories of those infinite interpretations having pairing functions that are definable by two simple classes of second-order formulas.

Key words and phrases: second-order logic, infinitary logic, definability 1991 Mathematics Subject Classification: 03B15, 03C85

1.

Introduction

Weaver and Penev ([2005]) extended the Fra¨ıss´e back and forth characterization of elementary equivalence to interpretations in the “standard” semantics of those second-order languages without functional variables whose non-logical vocabulary is finite and excludes functional constants. This characterization fails when the non-logical vocabulary is infinite. However, there is a class of infinitary second-order languages for which the characterization holds even when the non-logical vocabulary is infinite. Weaver and Penev presented several applications of this characterization for both classes of languages. Another application is presented here: a necessary and sufficient condition for definability in an interpretation. K is a set of non-logical constants, excluding functional constants. When K is finite, LK is the (finitary) second-order language without functional variables whose non-logical ∗

E-mail address: [email protected]

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vocabulary is K. When K is infinite, L K is the infinitary second-order language without functional variables whose non-logical vocabulary is K. As in Weaver and Penev [2005] (page 504 and following), the set of formulas in this infinitary language includes the (finitary) atomic formulas over K and is closed under finitary quantifications and usual truth functional combinations, and conjunctions and disjunctions of sets of formulas of bounded quantifier rank. While the logical vocabulary of these languages includes the identity sign, the only equations in the languages are between individual symbols (i.e. individual constants or individual variables). An interpretation of type K is an ordered pair A=(A, fA ), where A is a nonempty set (the domain of A) and fA is a function defined on K in the usual way. Let B be a subset of A and φ(x) be a formula whose one and only free variable is the individual variable x. φ(x) defines B in A (or B is defined by φ(x) in A ) provided for all a in A, a∈B iff a satisfies φ(x) in A (A |= φ(x)[a]). B is definable in A iff there is φ(x), as above, such that φ(x) defines B in A. Let m be a positive integer ≥2, and ρ be an m-ary relation on A. φ(x1...xm ) is a formula in LK whose free variables are exactly the individual variables x 1,..., xm , where these variables are all different, and where for all i and j between 1 and m, if i 1. The ring Zm is obtained as quotient ring from the ideal mZ in the integer ring. Given two rings (A, +, ·) and (B, +, ·), a ring homomorphism is an application f : A → B such that f (a + b) = f (a) + f (b) and f (ab) = f (a)f (b), for all a, b ∈ A. Direct consequences of this definition are the following properties: 1. f (0) = 0. 2. For all a ∈ A, f (−a) = −f (a). 3. If A0 is a subring of A then f (A0) is a subring of B. 4. If B 0 is a subring of B then f −1 (B 0 ) is a subring of A. 5. If f is an isomorphism then f −1 is also an isomorphism. As all ring is a group with the addition, we keep the definition of kernel. That is, ker(f ) = f −1 (0) = {x ∈ A | f (x) = 0} Proposition 3.12. Let (A, +, ·) and (B, +, ·) be rings and f : A → B be an homomorphism. 1. f (A) is a subring of B. 2. ker f is an ideal of A. Theorem 3.13. Let (A, +, ·) and (B, +, ·) be rings and f : A → B be an homomorphism. The rings A/ ker f and f (A) are isomorphic.

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Definition 3.14. A polynomial in a ring (R, +, ·) is a formal expression: m

p(x) = a0 + a1 x + · · · + am x

=

m X

ai xi

i=0

where the coefficients a0, a1, . . . , am are elements of R, m is a natural number and, if m > 0, am 6= 0. We say that m is the degree of this polynomial and write ∂(p(x)) = m. Two polynomials are considered to be equal if the coefficients of each power of x are equal. Hereinafter, if no confusion arises we will write p to denote a polynomial p(x). Polynomials in R can be added by simply adding the corresponding coefficients and multiplied using the distributive law and the rules xa = ax, for all element a ∈ R, and xk xl = xk+l for all natural numbers k and l. Every polynomial p = a0 + a1 x + · · · + am xm in a ring R define a function (not necessarily homomorphism) f : A → A with f (a) = a0 +a1 a+· · ·+am am . The elements that belong to f −1 (0) are named roots (or zeros) of p(x). That is, r ∈ R is a root of p if a0 + a1 r + · · · am rm = 0. It is easy to prove that the set of polynomials with coefficients in a ring R with the addition and the multiplication is also a ring. This ring is named the polynomial ring over R and is denoted by R[x]. Formally, the addition operation is defined as follows:  max{m,n} m n  ai + bi , 0 ≤ i ≤ min{m, n}; X X X ai xi + bi xi = ci xi where ci = , m < i ≤ n; bi  i=0 i=0 i=0 , n < i ≤ m. ai and the multiplication operation is defined as follows: m X i=0

ai xi ·

n X i=0

bixi =

m+n X

X

i=0

r+s=i

ar bs

!

xi

Proposition 3.15. Let R be a ring. 1. If R is commutative then R[x] is commutative. 2. If R is a unitary ring then R[x] is a unitary ring. 3. R[x] is an integral domain if and only if R is an integral domain. Example 3.16. The polynomial ring Z6 [x] is commutative and unitary. However, it is not a integral domain because, for example, 3x + 3 and 2x + 2 are zero divisors. Note that both 3x + 3 and 2x + 2 are polynomials with degree 1 but, surprising, the degree of its product is 0. Theorem 3.17. Let R be a commutative unitary ring. It is an integral domain if and only if, for all p, q ∈ R[x] − {0}, ∂(pq) = ∂(p) + ∂(q)

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3.2. Finite Fields and Polynomials A field is a non-trivial commutative unitary ring (K, +, ·) in which every non-zero element a ∈ K has inverse element. That is, (K, +) and (K − {0}, ·)9 are abelian groups and · distributes over +. The following theorem is well-known. Theorem 3.18. 1. Any field is an integral domain. 2. Any finite integral domain is a field. Example 3.19. It is well-known that (Zp, +, ·) is a field if an only if p is prime. In this section we will define new fields builded in analogous mode as the Zp fields are designed. That is, we will define the divisibility relation in a polynomial ring, the congruence relations and the quotient ring. From now on, we will study polynomial rings K[x] in which K is a field and therefore K[x] is an integral domain. However, K[x] is not necessarily a field. 10 This allows us introduce the concept of divisor. A polynomial p1 ∈ K[x] − {0} is said to be divisor of another polynomial p2 ∈ K[x] if there exists q ∈ K[x] − {0} such that p1 q = p2 . Theorem 3.20 (of division). Let K be a field and p1 , p2 ∈ K[x] with s 6= 0. There exist unique q, r ∈ K(x) such that p2 = p1 q + r and ∂(r) < ∂(p1). In this case, q is the quotient and r is named rest of the division. Corollary 3.21. Let K be a field, a ∈ K and p ∈ K[x]. The polynomial x − a is divisor of p if and only if a is a root of p. These results and Theorem 3.17 yield to the following theorem. Theorem 3.22. Let K be a field and p ∈ K[x]. The number of roots of p is lower or equal to the degree of p. Remark 3.23. Note that the previous result is not true in every polynomial ring. For example, in Z6 [x], the polynomial 2 + 3x + x2 has four roots: 1, 2, 4 and 5. A polynomial p ∈ K[x] is said to be prime (or unreductable) if every divisor of p is itself or it has degree 0. In other case, we say that p is reductable. As a direct consequence of this definition we have the following proposition. Proposition 3.24. Let K be a field. 1. Any polynomial with degree 1 is prime. 2. A polynomial with degree 2 or 3 is reductable if and only if it has a root. Example 3.25. The polynomial 1 + x2 is prime in R[x] but it is reductable in C[x]. 9 10

Hence, K ∗ = K − {0}. For example, R[x] is not a field.

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Example 3.26. In Z2 [x], • 1 + x + x3 is prime because neither 0 nor 1 are roots. • 1 + x + x2 + x3 is reductable because 1 is a root. In fact, 1 + x + x2 + x3 = (1 + x)(1 + x2 ). Similarly to examples 3.9 and 3.11, given p ∈ K[x], the subset pK[x] = {pq | q ∈ K[x]} is an ideal of K[x] and, therefore, K[x]/pK[x] is also a ring. K[x]/p denote this quotient ring. Theorem 3.27. Let K be a field and p a prime polynomial. Then K[x]/p is a field. Moreover, if K is finite, then |K[x]/p| = |K|∂(p) Example 3.28. In R[x], 1 + x is a prime polynomial and, therefore, R[x]/(1 + x) is a field. By Theorem 3.20, for all polynomial p there exist polynomials q and r such that p = (1 + x)q + r and ∂(r) < ∂(1 + x) = 1. So, R[x]/(1 + x) = {[a] | a ∈ R} In consequence, R[x]/(1 + x) is isomorphic to R. From now on, we omit the symbols [ and ] in the notation of equivalence classes. Example 3.29. The prime polynomial 1 + x + x2 in Z2 give us the following finite field. There exist 4 equivalence classes, because the division between 1 + x + x2 can give 4 possible rests: 0, 1, x and 1 + x. Operations in this field are shown in the following tables: 0 1 x 1+x + 0 0 1 x 1+x 1 1 0 1+x x x x 1+x 0 1 1+x 1+x x 1 0

· 0 1 x 1+x

0 1 x 1+x 0 0 0 0 0 1 x 1+x 0 x 1+x 1 0 1+x 1 x

The following theorem characterize every finite field. Theorem 3.30. For all finite field K there exist a prime number p ∈ Z+ and a prime polynomial q ∈ Zp [x] such that Zp [x]/q is isomorphic to K. As a consequence of the theorems 3.27 and 3.30, any finite field has pn elements, where p is a prime number and n is a positive integer. Any two finite fields with the same number of elements are isomorphic. Hence, we can say that there is an “unique” field of pn elements named Galois’s fields of order pn . F(pn) denotes this field. 11 11

The previous theorems are consequence of the work of the French mathematician Evariste Galois (1811– 1832) about the non-existence of formulae to solve general polynomials equations with degree greater to 4.

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Figure 1. Dürer’s Melancolia I and magic square under the bell.

3.3. Latin Squares The adjective “Latin” comes from Leonard Euler (1707–1783), who used Latin characters as symbols in opposition to Greek squares in which Greek characters are used. Euler used both kinds of squares to define Graeco-Latin squares used to construct magic squares. Those squares are arrangements of n2 numbers in a square, such that all rows, all columns, and both diagonals sum the same constant named “magic number”. More information can be seen in [8]. Figure 1 shows one of the most famous copperplates entitled “Melancolia I" made by the painter and mathematician Albert Dürer. In the right upper part of the plate, under the bell, we can see the following magic square that could be one of the first magic square seen by Europeans. The magic number 34 can be obtained in many different ways. 16 3 2 13 5 10 11 8 9 6 7 12 4 15 14 1 Definition 3.31. A Latin square is an n × n array filled with n different symbols, usually 1, 2, . . .n, in such a way that each symbol occurs exactly once in each row and exactly once in each column. Here we present two examples: 1 4 2 3

2 3 1 4

3 2 4 1

4 1 3 2

a b d c

b c a d

d a c b

c d b a

Algebraic Topics on Discrete Mathematics 1 2 3 3 1 2 2 3 1

1 3 2 2 1 3 3 2 1

153

1 2 3 2 3 1 3 1 2

Figure 2. Conjugated of example 3.33

Example 3.32. For every n ≥ 2 the table of addition for Zn is a Latin square. Observe that these tables fit to definition 3.31 when we replace all 0 by n. A Latin square is said to be normalized if its first row is in natural order. For example, the first Latin square above is normalized because its first row is 1, 2, 3, 4. We can normalize any Latin square permuting the name of the symbols conveniently. For example, the second Latin square showed above is not normalized, but permuting c and d we obtain the next normalized Latin square: a b c d b d a c c a d b d c b a In general, if we permute the rows, permute the columns, and permute the names of the symbols of a Latin square, we obtain a new Latin square said to be isotopic to the first. Isotopism is an equivalence relation, so the set of all Latin squares is divided into subsets, named “isotopy classes”. 3.3.1. Orthogonal Array Representation Every entry of a Latin square can be written as a triplet (r, c, s), where r is the row, c is the column, and s is the symbol. Then we obtain a set of n2 triplets named the orthogonal array representation of the square. For example, the orthogonal array representation of the first Latin square displayed above is T = {(1, 1, 1), (1, 2, 2), (1, 3, 3), (1, 4, 4), (2, 1, 4), (2, 2, 3), (2, 3, 2), (2, 4, 1), (3, 1, 2), (3, 2, 1), (3, 3, 4), (3, 4, 3), (4, 1, 3), (4, 2, 4), (4, 3, 1), (4, 4, 2)} The definition of Latin square can be written in terms of orthogonal arrays as n2 triplets of the form (r, c, s), where 1 ≤ r, c, s ≤ n and all of the possible pairs (r, c), (r, s) and (c, s) are different. The orthogonal array representation shows that rows, columns and symbols play similar roles, as will be made clear below. If we systematically reorder the three items in each triplet, another orthogonal array (and, thus, another Latin square) is obtained. For example, we can replace each triplet (r, c, s) by (c, r, s) which corresponds to the transposed square (reflecting about its main diagonal), or we could replace each triplet (r, c, s) by (c, s, r), which is a more complicated operation. Altogether there are 6 possibilities including the identity, giving us six Latin squares (not necessary different) named the conjugates of the original square.

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Example 3.33. The normalized Latin square defined by the orthogonal array {(1, 1, 1), (1, 2, 2), (1, 3, 3), (2, 1, 3), (2, 2, 1), (2, 3, 2), (3, 1, 2), (3, 2, 3), (3, 3, 1)} defines a conjugates class with 3 elements that appear in figure 2 Finally, we can combine these two equivalence operations: two Latin squares are said to be paratopic if one of them is isotopic to a conjugate of the other. This is again an equivalence relation, with the equivalence classes named species, or paratopy classes. Each specie contains up to 6 isotopy classes. Numbers of Latin squares. In [20] we can see the number of Latin squares (or rectangles) depending on the numbers of rows (or rows and columns). Computations have been made until 10 × 10 squares and estimated for squares with n = 11, 12, 13, 14, 15. Example 3.34. Complete Sudoku puzzles are 9 × 9 Latin squares, with an additional constraint: they are divided in nine 3 sub-squares and each one of them contains the nine symbols. The name “Sudoku” is the Japanese abbreviation of “Suji wa dokushin ni kagiru”, meaning “the digits must occur only once”

Figure 3. An incomplete Sudoku puzzle.

3.3.2. Orthogonal Latin Squares In 1782 Euler (see [9]) proposed the following question known as problem of the 36 officers: How many arrangements of 36 officers of six different ranks and regiments can be made in a 6 × 6 square such that each of six ranks and each of six regiments are represented once in each row and column? Euler conjectured that there was no solution, and introduced mutually orthogonal Latin squares to decide the conjecture. G. Tarry proved it with an exhaustive search of all Latin squares of order 6 in 1901, [26]. In 1984 a more elegant proof was given by Stinson in [25]. Orthogonal Latin squares solves some problems of optimization. Example 3.35. Design a way to distribute the tests so that each one of three drivers proves three different vehicles in the smaller time. Superposing the two following tables, we are able to solve the problem in three turns.

Algebraic Topics on Discrete Mathematics Drivers 1 2 3 2 3 1 3 1 2

Vehicles 1 2 3 3 1 2 2 3 1

→

155

Schedule Turn 1: (1, 1) (2, 2) (3, 3) Turn 2: (2, 3) (3, 1) (1, 2) Turn 3: (3, 2) (1, 3) (2, 1)

Definition 3.36. Two n × n Latin squares, T1 = (aij ) and T2 = (bij ), are orthogonal if and only if the n2 ordered pairs (aij , bij ) are all different. We say that T1, T2, . . . , Tk are mutually orthogonal if and only if any two squares Tr , Ts are orthogonal. It is easy to prove that the orthogonality of T1 and T2 is equivalent to the orthogonality of the normalized squares T1∗ and T2∗ . Therefore, from now on, we will talk about normalized Latin squares. We saw above the confirmed Euler’s conjecture about there not exist mutually orthogonal 6×6 Latin square. Euler also conjectured that this problem could not be solved for n×n Latin squares when n ≡ 2 (mod 4). But, nevertheless, in 1960 works of Bose, Shrikhande and Parker proved that this conjecture is false (see [23], [3]), and interest shifted to finding how many pairs of mutually orthogonal Latin squares there are for a given n. Theorem 3.37. For n ∈ Z+ , n > 2, the number of n×n Latin squares mutually orthogonal is at most n − 1. Proof. Let be T1, T2, . . ., Tm mutually orthogonal and normalized Latin squares. We represent Tk = (akij ). So, ak1j = j for k = 1, 2, . . .m. Now, ak21 6= 1 and different for all k = 1, . . . , m, since, on the contrary, they would not be orthogonal. This proves than m is smaller than n. In general, it is difficult to construct orthogonal Latin squares. But when the order is a power of prime number, finite fields give us a method to obtain the maximum number of mutually orthogonal Latin squares. Theorem 3.38. Let be n > 2 and n = pm , where p is a prime number. Then there are n − 1 mutually orthogonal Latin squares. Proof. Consider the only finite field F(n) = {c1, c2, . . . , cn } (Galois’s field), where c1 = 1 and cn = 0. For k = 1, 2, . . . , n − 1, we define the square Tk = (akij ) being akij = ck ci + cj . Then, we can prove: 1. Each Tk is a Latin square. All elements in the same row are different. akir = akis ⇒ ck ci + cr = ck ci + cs ⇒ cr = cs ⇒ r = s All elements in the same column are different. akrj = aksj ⇒ ck cr + cj = ck cs + cj ⇒ ck cr = ck cs and, as ck has inverse, then cr = cs ⇒ r = s.

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0

2. When k 6= k0 , then Tk , Tk0 are mutually orthogonal. Suppose (akij , akij ) = (akrs , akrs) then ck ci + cj = ck cr + cs ck0 ci + cj = ck0 cr + cs and reducing, (ck − ck0 )ci = (ck − ck0 )cr . Hence, as ck 6= ck0 , we have ci = cr and hence i = r. Using this with any of the previous equations, we obtain that cj = cs and j = s. According to the previous results we can normalize these Latin squares. Example 3.39. For n = 5, using the Galois’s field F(5) = Z5 = {1, 2, 3, 4, 0}, and the above theorem, we obtain the following four squares: 2 3 4 0 1

3 4 0 1 2

4 0 1 2 3

0 1 2 3 4

1 2 3 4 0

3 0 2 4 1

4 1 3 0 2

0 2 4 1 3

1 3 0 2 4

2 4 1 3 0

4 2 0 3 1

0 3 1 4 2

1 4 2 0 3

2 0 3 1 4

3 1 4 2 0

0 4 3 2 1

1 0 4 3 2

2 1 0 4 3

3 2 1 0 4

4 3 2 1 0

3 4 5 1 2

4 5 1 2 3

5 1 2 3 4

1 3 5 2 4

2 4 1 3 5

3 5 2 4 1

4 1 3 5 2

5 2 4 1 3

1 4 2 5 3

2 5 3 1 4

3 1 4 2 5

4 2 5 3 1

5 3 1 4 2

1 5 4 3 2

2 1 5 4 3

3 2 1 5 4

4 3 2 1 5

5 4 3 2 1

normalizing, 1 2 3 4 5

2 3 4 5 1

Example 3.40. With the Galois’s field F(4) = {1, x, x + 1, 0}, we obtain three mutually orthogonal squares. The first of them is: 0 x+1 x 1 x+1 0 1 x x 1 0 x+1 1 x x+1 0 To construct a pair of orthogonal Latin squares of any odd order, we can use the group Z2m+1 . Let T1 = (i + j) be the addition table for the integers modulo n = 2m + 1 and let T2 = (2i + j), entries taken modulo n. But the number of mutually orthogonal Latin squares can be less than n − 1. Magic squares. We are going to finish by applying some kind of orthogonal Latin squares, those n × n squares which contain all numbers from 1 to n (or equivalently, 0 to n − 1) in such all rows, columns, and both diagonals. Those squares are also magic, and by juxtaposing produce new magic squares.

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Example 3.41. The other two Latin (and magic) squares that are let to calculate in the previous example 3.40 are x+1 0 1 x x 1 0 x+1 0 x+1 x 1 1 x x+1 0

and

x 1 0 x+1 0 x+1 x 1 x+1 0 1 x 1 x x+1 0

Replacing x by 2 and juxtaposing we obtain a square formed by pairs of numbers in {0, 1, 2, 3}. Converted to base 10, we have 14 1 4 11 8 7 2 13 3 12 9 6 5 10 15 0 Obviously, other substitutions give other magic squares.

4.

Finite Geometry and Block’s Design

A finite geometry is any geometric system that has only a finite number of points. These geometries were introduced by Gino Fano ([10]) at final XIX century. The usual Euclidean geometry is not finite, because any line contains infinitely many points, in fact the same number of points as there are in the real line R. A finite geometry can be any finite dimension, but for ours proposes we will describe only classic 2-dimensional (or plane) finite geometries. In this section, mainly we present two kinds of finite plane geometry: affine and projective. In an affine geometry, the normal sense of parallel lines applies. In a projective plane, by contrast, any two lines intersect at a unique point, and so parallel lines do not exist. Both geometries may be described by fairly simple axioms.

4.1. Finite Affine Plane Definition 4.1. We name affine plane to a pair (P, R), where P is a nonempty set whose elements are named points, and R is a nonempty collection of subsets of P whose elements are named lines, such that: AP1 Given any two distinct points, there is exactly one line that contains both points. AP2 Given a line r and a point P , there exists exactly one line r0 containing P such that r = r0 or r ∩ r0 = ∅ (Euclid’s fifth postulate). AP3 There exists a set of four points, no three of which belong to the same line. The last axiom ensures that the geometry is not trivial, while the first two specify the nature of the geometry. The first question is the possibility to build affine plane geometries with any (finite) number of elements. Axiom AP3 indicates that is necessary at least four elements.

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P1

P2

P4

P3

Figure 4. The affine plane of order 2. Example 4.2. The simplest affine plane contains only four points; it is named the affine plane of order 2. Since no three points are collinear, any pair of points determines a unique 4
line, and so this plane contains 2 = 6 lines. Above, in Figure 7, we can see a graphical representation 12. By means of finite fields we can build affine planes. For n = pm , where p is a prime number and m is a positive integer, we consider the Galois’s field F = F(n), and define an affine plane AP (n) = (P, R), were the set of points is P = {(x, y) | x, y ∈ F} and the set of lines R is defined in this way: 1. For each a ∈ F, “vertical” lines ra = {(a, y) | y ∈ F}, which can be represented by equation x = a. 2. For each a, b ∈ F, lines ra,b = {(x, ax + b) | x ∈ F}, which can be represented by equation y = ax + b. Therefore, AP (n) contains n2 points and n2 + n lines. Lemma 4.3. Each line r contains n points and each point is contained in n + 1 lines. Proof. Each line contains n points is evident. Now, let (x0 , y0) be a point. This point is in the line rx0 and also in the n lines ra,(y0−ax0 ) por each a ∈ F. Theorem 4.4. If F(n) is a field with n = pm elements, then AP (n) is an affine plane (named affine plane of order n) with n2 points and n2 + n lines. Proof. AP (n) verifies the three axioms: AP1 Let P0 = (x0, y0 ), P1 = (x1, y1 ) be two different points. There are three possibilities: 1. x0 = x1 , then P0 , P1 ∈ rx0 . 2. y0 = y1 , then P0 , P1 ∈ r0,y0 . 3. x0 6= x1 and y0 6= y1 , then P0 and P1 are contained in the line defined by equation (x − x0)(x1 − x0 )−1 = (y − y0 )(y1 − y0 )−1 12

Observe that the affine plane of order 2 could be seen like the complete graph K4 .

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The line that contains P0 , P1 is the unique one. Suppose P0 , P1 ∈ ra,b and P0 , P1 ∈ rc,d (other situations similar), = ax0 + b = cx0 + d y0 = ax1 + b = cx1 + d y1 y1 − y0 = a(x1 − x0 ) = c(x1 − x0) Therefore a = c and hence b = d. AP2 Lemma 4.3 stays that there are n points P1 , P2, . . . , Pn in r. If P is not in r, we consider n lines r1, r2, . . . , rn determinate by P and Pi . But lemma 4.3 says also that P is in n + 1 lines, then exists a line r0 different all ri and therefore r0 ∩ r = ∅. AP3 The four points (0, 0), (0, 1), (1, 0) and (1, 1) verify whatever three are not collinear. That proves the theorem. Example 4.5. Affine plane of order 2 defined by F(2) = Z2 = {0, 1} has 4 points P1 = (0, 0), P2 = (0, 1), P1 = (1, 0) and P1 = (1, 1) and 6 lines. Figure 7 in Example 4.2 represents this plane. We say that lines r and r0 are parallel if either r = r0 or else r ∩ r0 = ∅. The relation “be parallel to” is an equivalence relation, this takes us the following definition. Definition 4.6 (Parallel classes). Given AP (n) of order n we can define a partition in R of n + 1 classes which contain mutually parallel lines. Every class is represented by any line in the class, for example, vertical lines class is [r0]. Example 4.7. Following figure represents all parallel classes in the affine plane of order 3. 2

2

2

2

1

1

1

1

0

1

[r0]

2

0

1

[r0,0]

2

0

1

2

[r1,0]

0

1

2

[r2,0]

Figure 5. Parallel classes for AP (3).

4.2. Finite Projective Plane In the beginning of this section we saw that affine geometry differs from projective geometry because there not exist parallel lines. Every pair of different lines intersect at an unique point. A projective plane geometry can be define using following axioms.

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Definition 4.8. We name projective plane to a pair (P 0 , R0), where P 0 is a nonempty set whose elements are named points, and R0 is a nonempty collection of subsets of P 0 whose elements are named lines, such that: PP1 Given any two distinct points, there is exactly one line that contains both points. PP2 The intersection of any two distinct lines contains exactly one point. PP3 There exists a set of four points, no three of which belong to the same line. The natural way to build a projective plane is through affine plane. We considerer AP (n) = (P, R) and redefine this points in the projective plane like P ⊆ F(n) × F(n) × {0, 1} in this way: P = {(x, y, 1) | x ∈ F, y ∈ F} (4) Moreover, every parallel class defines a new point “at infinity” in projective plane in this way: • The “vertical” lines class [r0] defines the point (0, 1, 0). • The parallel class [ra,0] defines the point (1, a, 0) Then the n2 points in P according to (4) and the n + 1 points “at infinity”define the n2 + n + 1 points in projective 13 plane. P 0 = P ∪ {(0, 1, 0)} ∪ {(1, a, 0) | a ∈ F} The lines in R together the line “at infinity”r∞ form the projective lines in R0 , in other words, R0 is formed by following lines: • Line “at infinity” r∞ = {(x, y, 0) ∈ P 0 }, also represented by z = 0. • “Vertical” lines ra = {(az, y, z) ∈ P 0 }, also represented by x = az. • The rest of lines ra,b = {(x, ax + bz, z) ∈ P 0 }, also represented by y = ax + bz. Theorem 4.9. If F(n) is a field with n = pq elements, then there exists a projective plane (named projective plane of order n) with n2 + n + 1 points and n2 + n + 1 lines. Example 4.10. The smallest projective plane (of order 2), also known as Fano’s plane, has 7 points and lines. Remark 4.11. The inverse process is available. If we have a projective plane and we delete a line and all the points incident with that line then the remaining structure is an affine plane. The deleted line is obvious the line “at infinity”. Therefore, it is well-established that both affine and projective planes of order n exist when n is a prime power. It is conjectured that no finite planes exist with orders that are not prime powers, although this statement has not been proved. The best result to date is the celebrated BruckRyser theorem ([4]), which was published in 1949, which states: 13

Observe that the point (0, 0, 0) never is in a projective plane.

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161

001

011

101 111

100

110

010

Figure 6. Diagram of the projective plane of order 2. Theorem 4.12 (Bruck-Ryser). If n is a positive integer n ≡ 1 or 2 (mod 4) and n is not equal to the sum of two integer squares, then n does not occur as the order of a finite plane. The smallest integer that is not a prime power and not covered by the Bruck-Ryser theorem is 10; it is of the form 4k + 2, but it is equal to the sum of squares 12 + 32. Using sophisticated techniques and computer analysis, it has been shown that 10 is not the order of a finite plane, [17]. The next smallest number to consider is 12, for which neither a positive nor a negative result has been proved.

4.3. Incomplete Block Designs Originally, statistics deals with variability and the planning of environmental or ecological investigations, the items used to control variability are grouping (blocking) into homogeneous subgroups (blocks) for measurement of related variables and use of covariance. There are many ways of blocking the experimental units in a comparative experiment with v varieties (also named treatments). If the sample of experimental units is from a homogeneous population, then no blocking is required and a completely randomized experiment design of the v varieties randomly allotted to the rv experimental units where the replicate number (sample size) is r. If homogeneous blocks of size v are available to accommodate all v varieties, a randomized complete block design (all v varieties in each block, not necessarily an equal number of times) is used. In complete block design, every treatment is allocated to every block. In other words, every combination of treatments and conditions (blocks) is tested. For example, an agricultural experiment is aimed at finding the effect of 3 fertilizers (A,B,C) for 5 types of soil (1...5). There are 15 plots at the disposal of the researcher - 3 plots for each type of soil. Here fertilizers (A,B,C) are “treatments”, types of soil (1...5) are “blocks”; the “design” is the chosen distribution of the 3 fertilizers over the 15 available plots. In a complete block design, every possible combination of fertilizer and soil-type is used at least once. In many biological and industrial experiments the number of experimental units and their groupings into blocks are such that the experimenter cannot influence them without sacrifice of experimental purpose or material. In this cases, the block size is constant and exceeds the number of treatments; the cost of experimental units is high and this cost is principally associated with blocks as a whole; and a balanced design is required. That is, in many situations, the block size, k, is less than v and an incomplete block design (not

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all varieties in each of the blocks) is used. Yates [28] first described incomplete block experiment designs. Example 4.13. A researcher is carrying out a study of the effectiveness of four different skin creams for the treatment of a certain skin disease. He has eighty subjects and plans to divide them into 4 treatment groups of twenty subjects each. Using a randomized blocks design, the subjects are assessed and put in blocks of four according to how severe their skin condition is; the four most severe cases are the first block, the next four most severe cases are the second block, and so on to the twentieth block. The four members of each block are then randomly assigned, one to each of the four treatment groups. Definition 4.14. Let V be a set with v elements (varieties). A finite collection ∆ = {Bi }bi=1 of nonempty subsets in V is a incomplete block design, or (v, b, r, k)-design when it verifies: 1. Every Bi contains a fixed-number k of varieties, k < v. 2. Every variety x ∈ V replies r(≤ b) times, in other words, x is contained in exactly r different blocks Bi ∈ ∆. It is easy to see than vr = bk.14 If V = {vj }vj=1 , we can define the b × v incidence matrix N = (nij ) defined:  1 if vj ∈ Bi nij = 0 other case. Finally, it is usual to define the v×v symmetric matrix Λ = (λx,y ), named concurrence matrix, being λx,y the number of blocks where the varieties x, y appear together. This matrix “balance” the block design. Obvious λx,x = r and, when x 6= y and λx,y = λ is a constant, then we say that the design is balanced. Example 4.15. In the twentieth century, the most frequent treatment used in the strawberry growing is the Methyl Bromide. It is an agricultural fumigant widely used because it is a powerful agent for the control of pathogens and bad herbs. Moreover, it increases the productivity because it reduces the effect known as stress of the land. However, it is considered as one of the causes of the hole in the ozone layer. On November of 2002, in the “IV Meeting of Montreal Protocol” (Copenhagen) its progressive withdrawal was agreed, until its total disuse in the developed countries. One research team is studding a new treatment to replace the Methyl Bromide. Specifically, they are interested in testing its effect in 6 varieties of strawberry:: Camarosa, Andana, Chandler, Cartuno, Sophie and Tudla. For this experiment, the research team dispose of 3 fields with 4 plots. They use the small design of v = 6, b = 3 and k = 4 defined by the incidence matrix   1 1 1 1 0 0 N = 1 1 0 0 1 1 0 0 1 1 1 1 14

The trick for remembering this formula is using letters of the words varieties and blocks, so varieties↔blocks.

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is a (6, 3, 2, 4)-design not balanced. The concurrence matrix is   2 2 1 1 1 1 2 2 1 1 1 1   1 1 2 2 1 1  Λ= 1 1 2 2 1 1   1 1 1 1 2 2 1 1 1 1 2 2

Definition 4.16. Let V be a set with v elements (varieties). A finite collection ∆ = {Bi }bi=1 of nonempty subsets in V is a balanced incomplete block design, or (v, b, r, k, λ)-design when it verifies: 1. ∆ is a (v, b, r, k)-design. 2. Every pair x, y of varieties appears together in exactly λ(≤ b) blocks Bi . Then the five parameters have the following relations Theorem 4.17. Every balanced incomplete block design (v, b, r, k, λ)-design verifies: 1. vr = bk. 2. λ(v − 1) = r(k − 1).
Proof. 1. was commented above. To proof 2. we count the number of ones in the v2 × b matrix A = (a{x,y},i), {x, y} ⊆ V , 1 ≤ i ≤ b defined  1 if {x, y} ⊆ Bi a{x,y},i = 0 other case.
Each row contains λ ones, then the number of ones in A is λ v2 . Each column contains

k k 2 ones, then the total number of ones is b 2 . Therefore λ

k(k − 1) k−1 v(v − 1) =b = vr 2 2 2

This proves the theorem. Example 4.18. When n = pm (power of prime number) then 1. The affine plane AP (n) defines a (n2 , n2 + n, n + 1, n, 1)-design. The points are the varieties and the lines are the blocks. 2. Also the projective plane is a balanced incomplete block design. In this case there are the same number of varieties than blocks. In other words the projective plane is a (n2 + n + 1, n2 + n + 1, n + 1, n + 1, 1)-design.

5.

Applications

In this section we introduce some applications about different results of discrete mathematic displayed in the previous sections.

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P1

P2

P4

P3

Figure 7. The affine plane of order 2.

5.1. Discrete Modeling in Foundation Design Developed by Gloria Gutiérrez Barranco, Javier Martínez del Castillo, Salvador Merino Córdoba, Francisco J. Rodríguez Sánchez from Department of Applied Mathematics. University of Málaga (Spain). The study that we present arises as an application of Discrete Mathematics to the resolution of problems of circular arrangement. In particular, the original problem was the following one: It is wanted to construct an electrical line of transmission supported by metallic towers with their four legs anchored to the ground. The legs can be manufactured with different heights. The problem was the determination of how many foundation drawings are necessary to lay the tower foundations according to the circumstances in the orography.

Figure 8. Positions diagram for tower foundation.

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165

5.1.1. General Definitions We begin with some concepts that we will use in the rest of the paper. • We call chain to any sequence of elements, repeated or not, placed in successive positions. • We call length of a chain to the number of positions that it has. • We say that a chain is closed when it is considered that the successor of the last position is the first one. • We call orbit of n elements taken from k in k to the set of circularly indistinguishable closed chains of length k generated by elements of n different types. The cardinal of the set of such orbits is denoted by Onk . The problem previously exposed for the electrical transmission towers would be the calculation of On4 . We are going to solve it using algebraic procedures. 5.1.2. Symmetric Group Solution If we have elements of n different types we can generate nk chains of length k. We will consider them as closed chains. The set of such chains is named C and we represent by SC the symmetric group defined by C, this is SC = {π : C → C | π is one-to-one } We can emphasize that any permutation in SC defines the set of the invariant elements for such permutation Inv(π) = {x ∈ C | π(x) = x} that is a subgroup of C. Any group G that acts on the set C defines an equivalence relation: x RG y ⇐⇒ exists π ∈ G such that π(x) = y Thus, we can construct the quotient set C/RG The resolution of the original problem, that is to say, the calculation of the number of orbits, is reduced to calculate el cardinal of this quotient set, where G is the group of the rotations that act on the nk chains in C. The well-known Burnside’s lemma establishs that the number of equivalence classes in C/RG is given by: 1 X |Inv(π)| (5) |G| π∈G

2πi radians. k We denote hgii the subset of G generate by the rotation gi and we denote ind(gi) the index of such element, that is, k ind(gi) = |hgii| Concretely, G = {g0, g1, . . . , gk−1}, where gi means the rotation of

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

as it is known ind(gi ) = gcd(k, i) Then |Inv(gi)| = nind(gi ) , since if a chain is invariant for gi then it has ind(gi) “free" elements (these elements can be anyone of the n possible types) and the others are “forced" ones. 1 #

1

#+

$!

"+

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#

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G2

G3

)

G4

G6

Figure 9. Invariant conditions for k = 12

Using the lemma of Burnside (5) we can calculate the number of orbits and solve the original problem. Onk =

1 X ind(gi ) n k gi ∈G

Example 5.1. Using combinatory methods we have:

On4

=



n 1



+4



n 2



+9



n 3



+6



n 4



(6)

By means algebraic methods, to calculate the number of different orbits for k = 4, we have 1 On4 = (n4 + n + n2 + n) 4 that is the same value obtained using the formula (6).

5.2. Enumeration of RNA Graphs Using Graph Theory Developed by Hin Hark Gan, Samuela Pasquali and Tamar Schlick from Howard Hughes Medical Institute and Department of Physics. University of New York (USA). Understanding the structural repertoire of RNA is crucial for RNA genomics research. Yet current methods for finding novel RNAs are limited to small or known RNA families. To expand known RNA structural motifs, we develop a two-dimensional graphical representation approach for describing and estimating the size of RNA’s secondary structural repertoire, including naturally occurring and other possible RNA motifs. These estimation are based in tree enumeration theorems of Cayley for labeled trees and Harary-Prins for unlabeled trees (applications of Pölya method)

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167

Figure 10. (Left) Dual graph representations of existing RNAs. (Right) Comparing dual (column F) and digraph (G) representations of hypothetical RNA secondary structures (E). 5.2.1. Graphical Representation of RNA Structures Unlike graphs for chemical structures, where atoms are vertices and bonds are edges, our RNA graphs are RNA secondary topologies where a vertex or an edge can represent multiple nucleotide bases or base pairs, which are themselves composed of multiple atoms and bonds. To allow graphical representation of complex RNA secondary topologies, we state below rules for defining RNA graphs and provide justifications for these rules. The rules specify how to represent RNA loops, bulges, junctions and stems as vertices or edges in a graph. Essentially, the tree and dual graph rules simplify RNA secondary motifs to allow their representation as mathematical graphs; the "RNA graphs" specify the skeletal connectivity of the secondary motifs. We use tree graphs to represent RNA trees and dual graphs to represent any RNA secondary structures, including trees and pseudoknots, since pseudoknots cannot be represented as trees. Still, the tree representation is advantageous because of its intuitive appeal and the existence of applicable tree enumeration theorems, especially those by Cayley and by Harary and Prins. 5.2.2. Planar Tree Graph Rules To represent RNA trees as planar graphs, we use the following rules to assign edges and vertices. T1. A nucleotide bulge, hairpin loop or internal loop is considered a vertex when there is more than one unmatched nucleotide or non-complementary base pair. The special case of the GU wobble base pair is regarded as a complementary base pair.

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Figure 11. Schematic graphical representations for three RNA secondary topologies computed using Zuker’s MFOLD algorithm. T2. The 3 and 5 ends of a helical stem are considered a vertex. T3. An RNA stem is considered an edge; we define an RNA stem to have two or more complementary base pairs. T4. An RNA junction is a vertex. 5.2.3. Planar Dual Graph Rules To represent trees, pseudoknots and other RNA secondary topologies as planar graphs, we use the following general rules. D1. A vertex represents a double-stranded helical stem. D2. An edge represents a single strand that may occur in segments connecting the

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secondary elements (e.g., bulges, loops, junctions and stems). D3. No representation is required for the 3 and 5 ends. 5.2.4. Enumeration of RNA Graphs To estimate the size of RNA’s structural space and to aid in finding new RNA folds, we now consider the enumeration of tree and dual graphs. Specifically, we seek to describe all possible RNA topologies (NV) for a fixed number of vertices (V); the number of vertices is a measure of RNA chain length. Enumeration of RNA tree graphs The number of possible RNA graphs (NV ) for a given number of vertices (V ) can be counted using tree enumeration theorems of Cayley for labeled trees and Harary-Prins for unlabeled trees (trees with equivalent vertices). These theorems are cornerstones in the subarea of graph theory that deals with graphical enumeration. Labeled trees refer to graphs with labeled vertices, as illustrated in Figure 3; graph vertices are not labeled in an unlabeled graph. Enumeration of unlabeled trees considers the number of non-isomorphic graphs, i.e., topologically distinct trees irrespective of the vertex identity. The labeled trees, on the other hand, allow distinction of specific bulges, loops, junctions and ends in RNA graphs. Both Cayley and Harary-Prins approaches are relevant to the counting of RNA’s structural repertoire. The number of labeled trees for any V is given by the Cayley formula: NV = V V 2

(7)

For unlabeled trees, Harary and Prins obtained the counting polynomial t(x) whose coefficient NV is the number of distinct graphs with V vertices: t(x) =

∞ X

NV xV = T (x) −

V =1

where T (x) = x · exp

 1 2 T (x) − t(x2 ) 2

∞ hX 1 r=1

r

i

T (xr )

(8)

(9)

The counting polynomial (equations 8 and 9) up to the first 12 terms is t(x) = x+x2 +x3 +2x4 +3x5 +6x6 +11x7 +23x8 +47x9 +106x10 +235x11 +551x12 +. . . (10) In this polynomial, the coefficients of the first, second and third terms, for example, indicate that there is only one distinct graph each for V = 1, 2 and 3; the Harary-Prins enumeration polynomial is derived based on the Pölya method. Clearly, the number of distinct graphs (NV ) as a function of vertex number according to Cayley’s formula ( 7 ) for labeled trees grows faster than Harary-Prins’s formula ( 8 ) for unlabeled trees. Based on the counting polynomial (equation 10 ), we can estimate the number of distinct secondary motifs for a given RNA size. Since a tree edge roughly corresponds to 20 nt, a

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tree with six vertices (five edges or 100 nt) has six possible motifs, whereas an 11-vertex (10 edges or 200 nt) tree has 235 possible motifs. As the RNA size increases from 100 to 200 nt, the number of possible motifs increases by a factor of 39, indicating the potential of large RNAs to form many more novel secondary motifs. Enumeration of RNA dual graphs The enumeration of dual graphs, unlike trees, simultaneously yields tree, pseudoknot and other possible topological motifs as defined by the dual graph rules (D1-D3). We have heuristically enumerated all such graphs for the cases of V = 2, 3 and 4, which correspond to 3, 8 and 30 possible dual graphs, respectively. In addition to RNA trees (T ) and pseudoknots (P ), enumerated motifs in Figure 5.2.4. reveal graphs involving single-edge connectors; we call such motifs bridge graphs ( B) or simply bridges. Bridges are biologically important since they suggest existence of independent RNA submotifs and thereby help in the modular design of RNAs. Examples of RNA bridges are box H/ACA snoRNA, hepatitis C virus (HCV) RNA and group I intron. The distribution of tree, pseudoknot and bridge types in V = 2, 3 and 4 motif sets is as follows. For the V = 2 motif set, three graphs correspond to one tree, one pseudoknot and one bridge; for V = 3 set, eight graphs correspond to two trees, three pseudoknots and three bridges; and for V = 4 set, 30 graphs include four trees, 20 pseudoknots and 13 bridges (thus seven graphs are both pseudoknots and bridges). These enumeration results imply that the number of bridges (N V bridge ), trees (N V tree ), and pseudoknots (N V pseudo ) within a given topological set follow: N V tree ≤ N V bridge ≤ N V pseudo The complexity and number of the dual graphs increase quickly with vertex number, making it non-trivial to determine the number of topological possibilities for a given V . General enumeration theorems for RNA dual graphs are not available. 5.2.5. Conclusion We estimate the number of distinct RNA tree motifs based on the Cayley and Harary-Prins enumeration theorems. These theorems imply that the RNA topology space is much smaller than the sequence space, which renders our topological approach potentially effective for finding novel RNAs. Our surveys of existing RNAs identified a number of motifs in nature but showed that many hypothetical motifs do not exist. Since not all enumerated motifs are probable RNAs, energetic, functional and evolutionary aspects of RNA folds must be taken into consideration to provide better future estimates of RNA’s repertoire.

5.3. Octonions and Transylvania Lottery Developed by John Baez’s Stuff from Department of Mathematics. University of California (USA). The quaternions, H, are a 4-dimensional algebra with basis 1, i, j, k. To describe the product we could give a multiplication table, but it is easier to remember that:

Algebraic Topics on Discrete Mathematics

1. 1 is the multiplicative identity, 2. i, j and k are square roots of −1,

171

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3. we have , ij = k, ji = −k, and all identities obtained from these by cyclic permutations of (i, j, k). We can summarize the last rule in a picture:

i

k

j

When we multiply two elements going clockwise around the circle we get the next one: for example, ij = k. But when we multiply two going around counterclockwise, we get minus the next one: for example, ji = −k. We can use the same sort of picture to remember how to multiply octonions:

This is the Fano plane, a little gadget with 7 points and 7 lines. The ‘lines’ are the sides of the triangle, its altitudes, and the circle containing all the midpoints of the sides. Each pair of distinct points lies on a unique line. Each line contains three points, and each of these triples has has a cyclic ordering shown by the arrows. If ei , ej and ek are cyclically ordered in this way then ej ei = −ek ei ej = ek Together with these rules: 1. 1 is the multiplicative identity, 2. e1 , e2, . . ., e7 are square roots of −1

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173

the Fano plane completely describes the algebra structure of the octonions. Indexdoubling corresponds to rotating the picture a third of a turn. This is certainly a neat mnemonic, but is there anything deeper lurking behind it? Yes! The Fano plane is the projective plane over the 2-element field Z2 . In other words, it consists of lines through the origin in the vector space Z32 . Since every such line contains a single nonzero element, we can also think of the Fano plane as consisting of the seven nonzero elements of Z32 . If we think of the origin in Z32 as corresponding to 1 ∈ O, we get the following picture of the octonions:

Note that planes through the origin of this 3-dimensional vector space give subalgebras of O isomorphic to the quaternions, lines through the origin give subalgebras isomorphic to the complex numbers, and the origin itself gives a subalgebra isomorphic to the real numbers. What we really have here is a description of the octonions as a ‘twisted group algebra’. Given any group G, the group algebra R[G] consists of all finite formal linear combinations of elements of with real coefficients. This is an associative algebra with the product coming from that of G . We can use any function α : G2 → ±1 to ‘twist’ this product, defining a new product ? : R[G]xR[G] → R[G] by: g ? h = α(g, h)gh where g, h ∈ G ⊂ R[G]. One can figure out an equation involving α that guarantees this new product will be associative. The category of Z32 -graded vector spaces provides a context in which the octonions are commutative and associative. So far this idea has just begun to be exploited. Example 5.2. A curious application of finite geometry is how to win in Transylvanian lottery. The Transylvanian lottery is a lottery where three numbers between 1 and 14 are picked by the player, and three numbers are chosed randomly. The player wins if two of his

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numbers are among the random ones. The problem of how many tickets the player must buy in order to be certain of winning can be solved by the use of the Fano plane. The solution is to buy a total of 14 tickets, comprised of two sets of seven. One set of seven is every line of a Fano plane with the numbers 1-7, the other with 8-14, i.e.: 1 − 2 − 5, 1 − 3 − 6, 1 − 4 − 7, 2 − 3 − 7, 2 − 4 − 6, 3 − 4 − 5, 5 − 6 − 7, 8 − 9 − 12, 8 − 10 − 13, 8 − 11 − 14, 9 − 10 − 14, 9 − 11 − 13, 10 − 11 − 12, 12 − 13 − 14

Because at least two of the winning numbers must be either high (8-14) or low (1-7), and every high and low pair is represented by exactly one ticket, you would be guaranteed at least two correct numbers on one ticket with these 14 purchases. 21 each 26 of the time you will have one ticket with two numbers matched. If all three winning numbers are either high or low you would either have one ticket with all three numbers (1 each 26 chance of this occurring), or three different tickets that each matched two (4 each 26 chance).

5.4. Discrete Mathematics with Mathematica Based in "Computational Discrete Mathematics" by Sriram Pemmaraju and Steven Skiena from Department of Computer Science. Universities of Iowa and New York (USA). We recommend Mathematica programming language to solve discrete mathematic problems with computational tools. Firstly we need to load Combinatorica package: 0, α 6= 1), as the following objective function: (Model 6)

max

X i∈I

s. t.

wi Ki

x1−α i 1−α

(2) − (13),

where wi is a fixed parameter. Due to the strict concavity of the function to be maximized, this defines a unique allocation referred to as α-bandwidth allocation. This allocation corresponds to the maximum throughput criterion when α → 0, to proportional fairness when α → 1, to the potential delay criterion when α → 2, and to max-min fairness when α → ∞ [32].

7.7 ε-Proportionally Fair Allocation Wang and Luh [43], [45] transformed the different QoS measurements onto a normalized scale by using achievement functions. Depending on the specified aspiration and reservation levels, ai and ri , respectively, Wang and Luh [43], [45] proposed the proportional fair

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bandwidth allocation by using the achievement function ofxi constructed as follows: (Model 7)

max

X

wi log αi

i∈I

s. t.

xi ri

(2) − (13),

where wi is a fixed parameter and αi = ai /ri . It is a strictly increasing function of xi , having value 1 if xi = ai , and value 0 if xi = ri . The use of the logarithmic function prevents the possibility of assigning zero flow to any user, and on the other hand makes it non-profitable to assign too much flow to the users. Note that this allocation is equivalent to proportionally fair allocation. In equilibrium, connections that share the same links do not necessarily equally share the available bandwidth. Rather, their shares reflect how they value the bandwidth as expressed by their utility functions and how their use of the bandwidth implies a cost on others. This could be a basis to provide differentiated services in terms of different bandwidth allocations. An equilibrium bandwidth allocation is usually characterized in terms of its fairness to users. Thus given a fixed number of users and fixed network capacities, one can typically arrange through an appropriate control mechanism to achieve an equilibrium which represents, according to some criterion, an equitable bandwidth allocation among users (see [20], [22], [26], [31], etc.)

8

Implication

We introduce the concept of majorization1 to provide the fairness. For any m-dimensional vector x=( x1 , . . . , xm ) of reals, let x(1) ≤ . . . ≤ x(m) denote the components of x in increasing order. P Pm Pk Definition 9 For x and t in Rm , x ≤M t if m i=1 s(i) = i=1 t(i) and i=1 x(i) ≥ Pk i=1 t(i) , for k = 1, . . . , m − 1. When x ≤M t then x is said to be majorized by t. If x ≤M t, then the allocation x is more fair than t. Next, we have the following definition. Definition 10 A function g : Rm → R is called Schur-concave, if x ≤M t implies g(x) ≥ g(t). Thus, we have the following theorem taken from [33]. Theorem 11 Let h be an arbitrary real function and defineg(x)= then g is Schur-concave if and only if h is concave.

Pm

i=1

h(xi ) for x ∈ Rm ,

1 Multiple criteria optimization defines the dominance relation by the standard vector inequality. The theory of majorization includes the results which allow us to express the relation of fair (equitable) dominance as a vector inequality on the cumulative ordered outcomes (see [29] and [33]).

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Recall that the achievement function P fi is a concave function. According to this definition, we know that the function fi is Schur-concave. Next, we consider a generic resource allocation problem defined as an optimization problem withm objective functions fi (x): max{ f (x) : x ∈ Q }, (16) where f (x) is a vector-function that maps the decision space Rm into the criterion space Rm , Q denotes the feasible set, and x ∈ Rm denotes the vector of decision variables. In the following, we will introduce the concept of fairness by using the fair aggregation function (see [19], [21], [33], [34]). Typical solution concepts for multiple criteria problems are defined by aggregation functions g : Rm → R to be maximized. Thus, (16) implies max{g(f (x)) : x ∈ Q}

(17)

The simplest aggregation functions commonly used for the multiple criteria problem (16) are defined as the sum of outcomes X fi (x), (18) g(f (x)) = i∈I

or the worst outcome g(f (x)) = min fi (x) i∈I

(19)

An aggregation (17) is fair if it is defined by a strictly increasing and strictly Schur-concave function g. Definition 12 An aggregation function g satisfying all the following requirements (20), (21) and (22), we call the corresponding problem (17) a fair aggregation of problem (16). For all i ∈ I = {1, 2, . . . , m} g(f1 (x), . . . , fi−1 (x), fi0 (x), fi+1 (x), . . . , fm (x)) < g(f1 (x), . . . , fm (x)),

(20)

whenever fi0 (x) < fi (x). For any permutation π of I, g(fπ(1) (x), fπ(2) (x), . . . , fπ(m) (x)) = g(f1 (x), f2 (x), . . . , fm (x))

(21)

For any 0 <  < zi0 − zi00 , we have g(f1 (x), . . . , fi0 (x) − , . . . , fi00 (x) + , . . . , fm (x)) > g(f1 (x), f2 (x), . . . , fm (x)) (22) P We know that fi is a fair aggregation function according to this definition. Every optimal solution to the fair aggregation (17) of a resource allocation problem (16) defines some fair allocation scheme. In order to guarantee the consistency of the aggregated problem (17) with the maximization all individual objective functions in the original multiple criteria problem, the aggregation function must be strictly increasing with respect to every coordinate, i.e. (20). In order to guarantee the fairness of the solution concept, the aggregation function must be additionally symmetric (impartial), i.e. (21). Symmetric functions satisfying the requirement (22), are called strictly Schur-concave functions. Next, we have the following two theorems taken from [33].

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195

Theorem 13 For a strictly concave, increasing function fi : R → R, the function g(f (x)) =

X

fi (x)

(23)

i∈I

is a strictly monotonic and strictly Schur-concave function. Theorem 14 For a strictly concave, increasing function fi : R → R, the optimal solution of the problem X fi (x) : x ∈ Q} (24) max{ i∈I

is a fair solution for resource allocation problem (16).

9

Blocking Probability with Predetermined Optimal Solutions

The blocking probability is an important performance measurement of network system. In our situation here, the blocking is due to the failure of setting up the number of end-toend paths, Ki , for each class i. In this section, we study the blocking probability of an end-to-end transmission system with predetermined optimal solutions, including optimal bandwidth allocation, x∗i , and Ki optimal end-to-end paths, pi,j . At the source node o, connections arrive at random times to enter the core network. The predetermined number, Ki is used to denote the limit on the number of connections in class i. A new connection in class i can not enter the source node o and is lost when all Ki end-to-end path are busy. That is, for each class i, a connection gets dropped on its arrival when the number of connections occupying the end-to-end paths equalsKi . Otherwise, it will be routed through an end-to-end path pi,j with allocated bandwidth ∗ xi predetermined by MILP. The principal quantity of interest is the blocking probability of different QoS classes, that is, the steady-state probability that allKi end-to-end paths in class i are busy. Our objective is to estimate these blocking probabilities. Assume that connections in class i arrive to the source node o in accordance with independent Poisson processes at rate λi , but the packet sizes have a general distribution G with mean σi . For each class i, we define µi = x∗i /σi , where x∗i is the optimal bandwidth allocation for each connection of class i. The average service time corresponds to the packet transmission time and is equal to average pack size divided by bandwidth. That is, σi 1 = ∗. µi xi

(25)

Hence, for each class i, the service times of connections occupying the end-to-end paths have a general distribution G with mean 1/µi = σi /x∗i . Suppose that connections occupy the end-to-end paths in the order they arrive and that packet sizes, which need to be transmitted from o to d, are identically distributed, mutually independent, and independent of the inter-arrival times. Under these assumptions, we analyze this end-to-end transmission system as M/G/Ki /Ki loss systems [7], that is, Poisson arrivals, general service, Ki end-to-end paths

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with identical bandwidth allocation x∗i , and no waiting space. For each class i, we can derive the steady-state occupancy probabilities from Erlang loss system [39]. We have    σi ni λi ni λi ∗ xi µi = Pi (0) Pi (ni ) = Pi (0) ni ! ni ! 

where ni = 1, 2, . . . , Ki , i = 1, 2, . . . , m. P i Solving for Pi (0) in the equation K ni =0 Pi (ni ) = 1, we obtain that      j −1  j −1 Ki Ki X X λ σ 1 1 λ i i i  =  Pi (0) =  j! µi j! x∗i j=0

(26)

j=0

and then

Pi (ni ) =

=

   j −1 ni X Ki λ 1 i   j! µi j=0     ni X j −1 Ki 1 σi λi 1 σi λi   ni ! x∗i j! x∗i 1 ni !



λi µi

j=0

for ni = 1, 2, . . . , Ki . For each class i, the blocking probability is given as follows: 1 Pi (Ki ) = Ki !



σi λi x∗i

   Ki X j −1 Ki σ 1 λ i i   j! x∗i

(27)

j=0

under conditions of Poisson arrival, general service time, and onlyKi end-to-end paths. Eq. (27) is referred to as Erlang’s loss formula [39]. If we denote the traffic load ρi =

σi λi , Ki x∗i

(28)

then Eq. (27) can be rewritten

Pi (Ki ) =

 −1 Ki (Ki ρi )Ki X (Ki ρi )j  Ki ! j! j=0

=

(Ki ρi )Ki [exp(Ki ρi ) − Ri (Ki )]−1 Ki !

(29)

where Ri (Ki ) is the Ki th-degree Taylor remainder term of exp(Ki ρi ) [3]. It is valid for all service distributions and only depends on the traffic load, ρi . From Taylor’s formula with remainder, we have the following results.

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Proposition 15 There exists a real number ξi ∈ (0, Ki ρi ), such that exp(Ki ρi ) = Ki X (Ki ρi )j + Ri (Ki ) as j! j=0

Ri (Ki ) =

exp(ξi )(Ki ρi )Ki +1 . (Ki + 1)!

Moreover, lim Ri (Ki ) = 0.

Ki →∞

Harel [16] proved that the fraction of customers lost in the M/G/K/K system is convex in the arrival rate, if the traffic intensity is below someρ∗ and concave if the traffic intensity is greater than ρ∗ . Some convexity properties of the blocking probability (29) are listed below. These results are consistent with convexity properties showed by Harel [16]. Proposition 16 For each Ki , there exists a ρ∗i such that for all ρi < (>)ρ∗i , the blocking probability (29) is strictly convex (concave) inρi . Proposition 17 The blocking probability (29) is strictly decreasing and strictly convex in x∗i /σi , provided λi and Ki fixed.

10

Conclusions and Future Research

A backbone link cannot be sized in arbitrarily small portions, but are built up from fixedsized modules. Furthermore, even demand volume is usually restricted to allocation in discrete portions. As a consequence, the resulting mathematical programmes have to be solved over a partly integral-valued domain, rendering so-called mixed-integer programming problems (MIP problems). Large MIP problems are known to be very difficult to handle and often require heuristics, in order to be solvable in a reasonable amount of time. However, accounting for the requirement of modularity is a most challenging problem and the research community of network design as well as the industry are addressing research in this direction. To summarize, it is our conviction that future large-scale networks, particularly the next generation Internet, will require a large portion of sophisticated network engineering, in order to meet the demands of service guarantees, connection reliability, cost efficiency and structured sharing of transmission media. It is reasonable to assume that the sharing of media will be based on fairness principles, possibly integrated in some sort of prioritization scheme, and that the network robustness will become the network design fundament. Since the next generation Internet will be a large and indeed a very complex network of networks (internetwork), a spirit of generality has to characterize the derivation of the new engineering methodologies. A future aim will also be to take the common requirement of QoS guarantees into account. In the network optimization context such approaches are referred to as demandoriented. Our target is to make efforts in this direction, by combination of the novel techniques of network calculus along with the classical theory of optimization.

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Acknowledgments. This research was supported in part by the National Science Council, Taiwan, R.O.C., under NSC 95-2221-E-004-007.

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[46] Wang, C. H., Yue, W., Luh, H., Performance Evaluation of Predetermined Bandwidth Allocation for Heterogeneous Networks, Technical Report of IEICE, 107 (6), 37–42 (2007). [47] Wang, C. H., Luh, H., Two-Phase Modeling of QoS Routing in Communication Networks. International Workshop on Performance Modeling and Evaluation in Computer and Telecommunication Networks (PMECT07) in conjunction with IEEE ICCCN2007, Honolulu, Hawaii, USA, August 16, 2007. [48] Wang, C. H., Luh, H., Blocking Probabilities of QoS Routing in IP Networks with Multiple Classes, The Second Asia-Pacific Symposium on Queueing Theory and Network Applications (QTNA2007), International Conference Center, Kobe, Japan, August 1-4, 2007. [49] Wang, C. H., Luh, H., A Fair QoS Scheme for Bandwidth Allocation by Precomputation-based Approach, International Journal of Information and Management Sciences, 19 (3), accepted for publication (2008). [50] Wu, H., Jia, X., He, Y., Huang, C., Bandwidth-guaranteed QoS routing of multiple parallel paths in CDMA/TDMA ad hoc wireless networks, International Journal of Communication Systems, 18, 803–816 (2005). [51] Xiao, X., Ni, L. M., Internet QoS: A big picture, IEEE Network, 13 (2), 8–18 (1999). [52] Ye, H. Q., Qu, J., Stability of data networks: Stationary and bursty models. Operations Research, 53 (1), 107–125 (2005).

* A version of this chapter was also published in Leading-Edge Applied Mathematical Modeling Research, edited by Matas P. lvarez published by Nova Science Publishers, Inc. It was submitted for appropriate modifications in an effort to encourage wider dissemination of research.

In: Mathematics and Mathematical Logic: New Research ISBN 978-1-60692-862-2 c 2010 Nova Science Publishers, Inc. Editors: Peter Milosav and Irene Ercegovaca

Chapter 8

R EVERSIBLE L OGIC Alexis De Vos Department of Electronics and Information Systems, University of Ghent, Ghent, Belgium

Abstract Reversible logic circuits are beneficial to both classical and quantum computer design. Present-day logic building-blocks (like OR gates and NAND gates) are logically irreversible and therefore cannot be used for designing reversible computers. Thus reversible computation needs an appropriate design methodology. In contrast to conventional digital logic circuits, reversible logic circuits (of a same logic width w) form a mathematical group. The reversible circuits of width w form a group isomorphic to the symmetric group S2w . Its Young subgroups allow systematic and efficient synthesis of an arbitrary reversible circuit. We can choose either a left coset, a right coset, or a double coset approach. The optimal design is reminiscent of the so-called banyan networks of telecommunication. As an illustration, three experimental prototypes (in c-MOS chip technology) of reversible computing devices are presented. Special care has been taken to avoid as much as possible the appearance of garbage bits. The examples illustrate how, in a near future, reversible computers will outperform conventional computers, in terms of power dissipation and heat generation.

1.

Introduction

Reversible computing [1] [2] is useful both in low-power classical computing [3] [4] and in quantum computing [5]. According to the Landauer theorem, the only way to make classical digital computing lossless, is by taking care that, at each stage of the computation, no information is lost. Indeed, each bit of information that is thrown away, causes the generation of a quantum of heat (with magnitude kT log(2), where k is the Boltzmann constant and T is the temperature of the computer hardware). This can be easily seen, using thermodynamics. For instance, resetting a bit to 0 makes one forget about its original contents. So, an unknown bit has become a fully specified one and thus the (macroscopic) entropy

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3

1

adding

?

?

4

computer

adding computer

4

Figure 1. A logically irreversible pocket calculator. of the machine has diminished by k log(2). Because of the second law of thermodynamics, at least an amount k log(2) of microscopic entropy has to be created simultaneously. Therefore an amount kT log(2) of heat is released into the environment [6] [7] [8] [9]. As stated above, reversible computing also has its uses in quantum computing, as in the latter technology only reversible operations are possible. In the present review paper however, we will only deal with classical circuits. Thus, if we want to avoid any heat generation in these circuits, we have to avoid any loss of information during the computational process. This means we have to construct a logically reversible computer. What is a reversible computer? Figure 1 gives a counterexample. It is a small calculator, designed for one particular task: the computation of the sum of two numbers. This computation is logically irreversible, because, if we have forgotten the input values (i.e. 3 and 1), knowledge of the output value (i.e. 4) is not sufficient to recover what the inputs have been. Indeed, 4 could have been equally well 2 + 2 or 4 + 0 or 0 + 4 or ... Figure 2 gives an actual example of a reversible computer. Again we have a pocket calculator, designed for one particular task: the computation of the sum and the difference of two numbers. This computation is reversible, because, if we have forgotten the input values (3 and 1), knowledge of the output values (4 and 2) is sufficient to recover what the inputs have been. Indeed, if A and B designate the two input numbers and P and Q the two output numbers, then P

= A+B

Q = A−B suffices to calculate backwards: A =

1 (P + Q) 2

B =

1 (P − Q) . 2

Reversible Logic

3

1

3

1

205

4

adding & subtracting computer

adding & subtracting computer

2

4

2

Figure 2. A logically reversible pocket calculator. Thus, the outputs contain enough information to reconstruct the inputs. In other words: the outputs contain the same information as the inputs. In the present paper, we will demonstrate the application of group theory to the detailed design of reversible classical hardware. Reversible logic circuits distinguish themselves from arbitrary logic circuits by two properties: • the number of output bits always equals the number of input bits and • for each pair of different input words, the two corresponding output words are different. For instance, it is clear that an AND gate (thruth table in Table 1a) is not reversible, as • it has only one output bit, but two input bits and • for three different input words (i.e. for 00, 01, and 10), the three corresponding output words are equal. Analogously, neither the OR gate (Table 1b), nor the NAND gate, nor the NOR gate are reversible. Table 2a, on the other hand, gives an example of a reversible circuit. Here, the number of inputs equals the number of outputs, i.e. two. This number is called the width w of the reversible circuit r. The table gives all possible input words (A, B). We see how all the corresponding output words (P, Q) are different. Thus the four (P, Q) words are merely a permutation of the four (A, B) words. This particular truth table may be replaced by a set of w Boolean equations: P (A, B) = A ⊕ B Q(A, B) = A . We will use the following short-hand notations for the basic Boolean functions: X = NOT X

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Table 1. Truth table of two irreversible logic circuits: (a) the AND gate and (b) the OR gate.

AB

P

AB

P

00 01 10 11

0 0 0 1

00 01 10 11

0 1 1 1

(a) XY

(b) = X AND Y

X+Y

= X OR Y

X⊕Y

= X XOR Y ,

where XOR is an abbreviation of EXCLUSIVE OR. In contrast to arbitrary logic circuits, reversible logic circuits form a group. For a group, we need a set (S) as well as an operation under which each pair (x, y) of elements of the set corresponds to a third element of the set (written xy). We are allowed to speak of a group, • if the operation is associative, i.e. if x(yz) equals (xy)z (therefore denoted simply xyz), • if the set S contains an identity element i, i.e. if xi = ix = x for each x ∈ S, and • if for each element x of S there exists an inverse element x−1 (such that xx−1 = x−1 x = i). In our case, the operation working on two circuits is the cascading of the two circuits. Table 2b gives the truth table of the identity gate i and Table 2c gives r−1 , i.e. the inverse of circuit r. The reader will easily verify that not only the cascade rr−1 , but also the cascade r−1 r equals i. In the present paper, we will apply different techniques from group theory to the synthesis and the analysis of reversible circuits. In Sections 2 and 3, we discuss an important group theoretical tool, i.e. the subgroup concept. Sections 4, 5, and 6 discuss cosets and double cosets and how to use them for synthesis of a reversible logic circuit. In Section 7 we demonstrate how to change a logically irreversible gate into a reversible one. Finally, in Section 8, we discuss the relationship with some experimental prototypes.

2.

Group Theory

All reversible circuits of the same width form a group. If we denote by w the width, then the truth table of an arbitrary reversible circuit has 2w rows. As all output words have to be

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Table 2. Truth table of three reversible logic circuits of width 2: (a) an arbitrary reversible circuit r, (b) the identity gate i, and (c) the inverse r−1 of r.

AB

PQ

AB

PQ

AB

PQ

00 01 10 11

00 10 11 01

00 01 10 11

00 01 10 11

00 01 10 11

00 11 01 10

(a)

(b)

(c)

different, they can merely be a repetition of the input words in a different order. In other words: the 2w output words are a permutation of the 2w input words. There exist (2w )! ways to permute 2w objects. Therefore there exist exactly (2w )! different reversible logic circuits of width w. The number 2w is called the degree of the group; the number (2w )! is called the order of the group. The group is isomorphic to a group well-known by mathematicians: the symmetric group S2w . For further reading in the area of symmetric groups in particular, and groups, subgroups, cosets, and double cosets in general, the reader is refered to appropriate textbooks [10] [11]. The symmetric group has a wealth of properties. For example, it has a lot of subgroups, of which most have been studied in detail. Some of these subgroups naturally make their appearance in the study of reversible computing. An example is the subgroup of conservative logic circuits, studied in detail by Fredkin and Toffoli [12]. Table 3a gives an example. In each of its rows, the output (P, Q, R, ...) contains a number of 1s equal to the number of 1s in the corresponding input (A, B, C, ...). Table 4 gives C(w), the number of conservative logic circuits of width w. An even more important subgroup is the subgroup of linear reversible circuits. Linear reversible circuits have been studied in detail by Patel et al. [13]. A logic circuit is linear iff each of its outputs P , Q, ... is a linear function of the inputs A, B, ... In its turn, a linear function is defined as follows. A Boolean function f (A, B, ...) is linear iff its Reed–Muller expansion [14] only contains terms of degree 0 and terms of degree 1. The reversible circuit of Table 3a is not linear. Indeed it can be written as a set of three Boolean equations: P

= B ⊕ AB ⊕ AC

Q = A R = C ⊕ AB ⊕ AC . Whereas the function Q(A, B, C) is linear, the function P (A, B, C) is clearly not (its Reed–Muller expansion containing two terms of second degree). Table 3b is an example of

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Alexis De Vos

Table 3. Truth table of three reversible logic circuits of width 3: (a) a conservative circuit, (b) a linear circuit, and (c) an exchanging circuit. ABC

P QR

ABC

P QR

ABC

P QR

000 001 010 011 100 101 110 111

000 001 100 101 010 110 011 111

000 001 010 011 100 101 110 111

100 000 001 101 111 011 010 110

000 001 010 011 100 101 110 111

000 001 100 101 010 011 110 111

(a)

(b)

(c)

a linear circuit: P

= 1⊕B ⊕C

Q = A R = A⊕B . Based on pioneering work by Kerntopf [15], De Vos and Storme [16] have proved that an arbitrary Boolean function can be synthesized by a (loop-free and fanout-free) wiring of a finite number of identical reversible gates, provided the gate is not linear. In other words: all non-linear reversible circuits can be used as a universal building block. Thus the linear reversible circuits constitute the ‘weak’ ones. Indeed, any wiring of linear circuits (be they reversible or not, be they identical or not) can yield only linear Boolean functions at its outputs. The linear reversible circuits form a group isomorphic to what is called in mathematics the affine general linear group AGL(w, 2). Its order equals 2(w+1)w/2 w!2, where w!2 is the bifactorial of w, the q-factorial being a generalization of the ordinary factorial w! = w!1: w!q = 1(1 + q)(1 + q + q 2 )...(1 + q + ... + q w−1 ) . Table 4 gives the number of different linear reversible circuits. We see that (at least for w > 2) a vast majority of the reversible circuits are non-linear and thus can act as universal gates. Now we descend the hierarchy of subgroups one step, by imposing that ‘each of the outputs equals one of the inputs’. Table 3c is such a circuit: P

= B

Q = A R = C.

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Table 4. The number R of different reversible circuits, the number C of different conservative reversible circuits, the number L of different linear reversible circuits, and the number E of different exchanging reversible circuits, as a function of the circuit width w. w 1 2 3 4

R 2 24 40,320 20,922,789,888,000

C 1 2 36 414,720

L

E

2 24 1,344 322,560

1 2 6 24

Such circuits are called exchangers (a.k.a. SWAP gates). They form a subgroup isomorphic to Sw of order w!. Also this number is given in Table 4. Finally, we can impose that ‘each of the outputs equals the corresponding input’: P

= A

Q = B R = C. This results in the trivial subgroup I of order 1, merely consisting of one circuit, i.e. the identity gate i. We have thus constructed a chain of subgroups: S2w ⊃ AGL(w, 2) ⊃ Sw ⊃ I , with subsequent orders (2w )! > 2(w+1)w/2 w!2 > w! > 1 , where we have assumed w > 1. Here, the symbol ⊃ reads ‘is proper supergroup of’. For the example w = 3, this becomes: S8 ⊃ AGL(3, 2) ⊃ S3 ⊃ I , with subsequent orders 40, 320 > 1, 344 > 6 > 1 .

3.

Control Gates

Besides using the letters A, B, C, ... for the input bits and P , Q, R, ... for the output bits, we will also use A1 , A2, ..., Aw for the inputs and P1 , P2, ..., Pw for the outputs, because sometimes this is more convenient.

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Alexis De Vos

Table 5. Truth table of three reversible logic circuits of width 3: (a) an arbitrary circuit, (b) a controlled NOT gate, and (c) a twin circuit. A1 A2 A3

P1 P2 P3

A1 A2 A3

P1 P2 P3

A1 A2A3

P1 P2 P3

000 001 010 011 100 101 110 111

100 101 110 000 111 011 010 001

000 001 010 011 100 101 110 111

001 000 011 010 101 100 110 111

000 001 010 011 100 101 110 111

001 011 010 000 100 101 111 110

(a)

(b)

(c)

We now define a special class of reversible logic circuits, called control gates, by their relationship between the outputs P1 , P2, ..., Pw and the inputs A1 , A2, ..., Aw. In a control gate, we always have P1 = A1, P2 = A2 , ..., Pu = Au , where u is an integer obeying 0 < u < w. The other outputs, i.e. Pu+1 , Pu+2 , ..., and Pw , are controlled by means of some Boolean function f of the u inputs A1, A2 , ..., Au : • If f (A1 , A2, ..., Au) = 0, then we additionally have Pu+1 = Au+1 , Pu+2 = Au+2 , ..., Pw = Aw . • If however f (A1 , A2, ..., Au) = 1, then the values of Pu+1 , Pu+2 , ..., Pw follow from the values of Au+1 , Au+2 , ..., Aw by the application of a reversible circuit g of width v = w − u. Thus: if f = 0, then we apply the v-bit follower to Au+1 , Au+2 , ..., Aw, else we apply the vbit circuit g to them. We call A1 , A2, ..., Au the controlling bits and Au+1 , Au+2 , ..., Au+v the controlled bits. Whereas w is the width, u is the controlling width and v is the controlled width. We call f the control function and g the controlled circuit. u There exist 22 Boolean functions f of u binary variables. Together with the XOR operation, they form a group isomorphic to the direct product group S2 × S2 × ...× S2 = u S22 . Therefore, the control gates with a same controlled gate g form a group isomorphic to u u S22 . Its order is 22 . We also assign an icon to the control gate: Figure 3b (where u = 4 and v = 2, such that w = 6). Note that each of the controlling bits is labelled by a small square. However, in the special case where the controlling function f (A1 , A2, ..., Au) is an AND of some controlling bits whether inverted or not, then circular tags are used: a filled circle if the variable is not inverted, an open circle if the variable is inverted. Figure 3c shows an example: f (A1 , A2, A3, A4) = A1 A3 A4. We consider two special cases in detail:

Reversible Logic

211

f

a

g

g

b

c

Figure 3. Icons of reversible circuits of width w = 6: (a) arbitrary circuit, (b) control gate with arbitrary control function f , and (c) control gate with AND control function.

• If v = 1, then only two possibilities exist for the controlled circuit g: either it is the trivial follower or it is the inverter. We only have to consider the latter choice. Then the controlled gate is represented by a cross: see Figure 4a. We call such gates controlled NOTs. An example is given in Table 5b. Its controlling function is f (A1, A2) = A1 + A2 . Note that, in the truth table of a controlled NOT, the first two rows are either permuted or not, the second two rows are either permuted or not, etcetera. Therefore the controlled NOTs form a subgroup isomorphic to S2 × S2 ×...× w−1 [17]. If the gate’s control function f is an AND function, we call it S2 of order 22 a TOFFOLI gate (e.g. Figure 4b with control function f = A1 A3 A4 ).

• If v = 2, then 4! = 24 possibilities exist for the controlled circuit g. The most interesting case is where g is a SWAP gate. Then the controlled gate is represented by a cross-over: see Figure 4c. We call such gates controlled SWAPs. They form a subw−2 group of order 22 . If additionally the gate’s control function is an AND function, we call it a FREDKIN gate (e.g. Figure 4d with control function f = A2 A3 ).

Note that the functionallity of the controlled NOT gate can be written as a set of w Boolean equations:

P1 = A1 P2 = A2 ... Pw−1 = Aw−1 Pw = f (A1 , A2, ..., Aw−1) ⊕ Aw

(1)

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Alexis De Vos

a

b

c

d

Figure 4. Icons of control gates: (a) arbitrary controlled NOT gate, (b) specific TOFFOLI gate, (c) arbitrary controlled SWAP gate, and (d) specific FREDKIN gate. and so can the functionallity of the controlled SWAP gate: P1 = A1 P2 = A2 ... Pw−2 = Aw−2 Pw−1 = f (A1, A2, ..., Aw−2)(Aw−1 ⊕ Aw ) ⊕ Aw−1 Pw = f (A1, A2, ..., Aw−2)(Aw−1 ⊕ Aw ) ⊕ Aw .

4.

(2)

Cosets

Subgroups are at the origin of a second powerful tool in group theory: cosets. For convenience, we will, as much as possible, adopt the following style: a group is represented by a bold-faced capital letter; its order is denoted by an upper-case letter, whereas an element of the group is represented by a lower-case letter. E.g. a and b might be two elements among the C elements of some group C. Already in Table 4, we used upper-case letters (i.e. R, C, L, and E) for group orders. If H (with order H) is a subgroup of the group G (with order G), then H partitions G G classes, all of the same size H. These equipartition classes are called cosets. We into H distinguish left cosets and right cosets. The left coset of the element a of G is defined as all elements of G which can be written as a cascade ba, where b is an arbitrary element of H. Such left coset forms an equipartition class because of the following property: if c is member of the left coset of a, then a is member of the left coset of c. Right cosets are defined in an analogous way. Note that H itself is one of the left cosets of G, as well as one of its right cosets. What is the reason of defining cosets? They are very handy in synthesis. Assume we want to make an arbitrary element of the group G in hardware. Instead of solving this problem for each of the G cases, we only synthesize the H circuits b of H and a single G − 1). If we can make each of these representative ri of each other left coset (1 ≤ i ≤ H

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213

G H+H − 1 circuits, we can make all the others by merely making a short cascade bri. If we G − 1 is much smaller than G. cleverly choose the subgroup H, we can guarantee that H + H G We call the set of H + H − 1 building-blocks the library for synthesizing the G circuits of G. The clever choice of the subgroup H of G is the challenge of the designer. He/she can G − 1)/dH = 0, what leads to e.g. aim for minimizing the size of the library: d(H + H √ √ √ H = G. Of course, seldom G will have a subgroup of order G. In most cases, G is not even an integer number, but an irrational number instead. Then, the designer has to √ look for a subgroup√with order ‘in the neighbourhood’ of G. In passing, we remark that the condition H = G can be rewritten as

log(G) =2. log(H) Ratios of logarithms of sizes will play more important parts in our story. See e.g. Appendix B. Maslov and Dueck [18] present a method for synthesizing an arbitrary reversible circuit of width three. As a subgroup H of the group G = S8 , they propose all circuits with output (P, Q, R) equal (0,0,0) in case of the input (A, B, C) = (0, 0, 0). This subgroup is isomorphic to S7 . Thus the supergroup has order G = 8! = 40, 320, whereas the subgroup has order H = 7! = 5, 040. The subgroup partitions the supergroup into 8 cosets. Interesting is the fact, that the procedure can be repeated: for designing each of the 5,040 members of S7 , Maslov and Dueck choose a subgroup of S7 . They choose all reversible circuits where (P, Q, R) equals (0,0,0) in case (A, B, C) = (0, 0, 0) and equals (0,0,1) in case (A, B, C) = (0, 0, 1). This is a subgroup isomorphic to S6 of order 6! = 720, which partitions S7 into seven cosets. Etcetera. Figure 5a illustrates one step of the procedure: the 24 elements of S4 are fabricated by means of the 6 elements of its subgroup S3 plus the representatives of the 3 other cosets in which S4 is partitioned by S3 . Thus Maslov and Dueck apply the following chain of subgroups: S8 ⊃ S7 ⊃ S6 ⊃ S5 ⊃ S4 ⊃ S3 ⊃ S2 ⊃ S1 = I ,

(3)

with subsequent orders 40, 320 > 5, 040 > 720 > 120 > 24 > 6 > 2 > 1 .

(4)

They need, for synthesizing all 40,320 members of S8 , a library of only (7+6+...+1)+1 = 29 elements (the identity gate included). For an arbitrary circuit width w, synthesis of all (2w )! members of S2w needs a library of 22w−1 − 2w−1 + 1 elements. Van Rentergem et al. [19] [20] also present a coset method for synthesis, however based the following subgroup H: all circuits from G = S2w possessing the property P1 = A1. Such circuits consist of the cascade of two control gates with u = 1 and v = w − 1. See Figure 6. If A1 = 0, then g is applied to A2 , A3 , ..., and Aw , else (thus if A1 = 1) h is applied. We will call such circuit a twin circuit. An example is given in Table 5c. Note that, in the truth table of a twin circuit, the upper 2w−1 rows are permuted among themselves and so are the lower 2w−1 rows. Therefore the twin circuits form a subgroup isomorphic to S2w−1 × S2w−1 . The twin circuits of width 3 form a group isomorphic to S4 × S4 = S24 of order

214

Alexis De Vos S4

S3

S4

S 2 x S2

a

b

Figure 5. The symmetric group S4 partitioned (a) as the four left cosets of S3 and (b) as the three double cosets of S2 ×S2 . Note: the dots depict the elements of S4 ; the bold-faced dots depict the elements of the subgroup and the representatives of the (double) cosets.

g

h

Figure 6. A twin circuit of width 4: if A1 = 0 then apply g else apply h. (4!)2 = 576. The subgroup S24 partitions its supergroup S8 into 70 cosets. Subsequently, the members of S4 are partitioned into six cosets by use of its subgroup S22 , etcetera. Thus, finally, Van Rentergem et al. apply the following chain of subgroups: S8 ⊃ S24 ⊃ S42 ⊃ S81 = I ,

(5)

40, 320 > 576 > 16 > 1 .

(6)

with subsequent orders If we denote 2w by n, then Dueck et al. apply the subgroup Sn−1 of the group Sn , whereas Van Rentergem et al. apply the subgroup S n2 × S n2 . We note that both the group Sn−1 (which can also be written Sn−1 × S1 ) and the group S n2 × S n2 are special cases of Young subgroups of Sn . In general, a Young subgroup [21] [22] [23] of the symmetric group Sn (with n an arbitrary integer number) is any subgroup isomorphic to Sn1 ×Sn2 ×...×Snk , with (n1 , n2, ..., nk) a partition of the number n, i.e. with n1 + n2 + ... + nk = n. Also the conservative gates [12], mentioned in Section 2, form a Young subgroup of w S2 , based on the binomial partition 2w = 1 + (1w ) + (2w ) + ... + (w−1 w ) + 1.

Reversible Logic

5.

215

Double Cosets

Even more powerful than cosets are double cosets. The double coset of a, element of G, is defined as the set of all elements that can be written as b1ab2 , where both b1 and b2 are members of the subgroup H. A surprising fact is that, in general, the double cosets, in which G is partitioned by H, are of different sizes (ranging from H to H 2). The number of double cosets, in which G is partitioned by H, therefore is not easy to predict. It is some number G G between 1 + G−H H 2 and H . Usually, the number is much smaller than H , leading to the (appreciated) fact that there are far fewer double cosets than there are cosets. This results in smaller libraries for synthesis. However, there is a price to pay for such small library. Indeed, if the chain of subgroups considered has length n, then the length of the synthesized cascade is 2n − 1 (instead of n as in the single coset synthesis). The subgroup Sa−1 partitions its supergroup Sa into only two double cosets, a small one of size (a − 1)! and a large one of size (a − 1)!(a − 1). Therefore, a double coset approach using the Maslov–Dueck subgroup chain (3) needs only 2w library elements. However, a w synthesized cascade can be 22 − 1 gates long. Therefore this subgroup chain is not a good choice in combination with double coset synthesis. For the problem of synthesizing all members of S8 , Van Rentergem, De Vos and Storme [24] have chosen the double cosets of the above mentioned subgroup obeying P1 = A1 . They conclude that, for synthesizing all 40,320 members of S8 , they need a library of only (4 + 2 + 1) + 1 = 8 elements. These suffice to synthesize an arbitrary member of S8 by a cascade with length of seven or less. For an arbitrary circuit width w, synthesis of all (2w )! members of S2w needs a library of 2w − 1 elements. Figure 5b illustrates one step of the procedure: the 24 elements of S4 are fabricated by means of the 4 elements of its subgroup S2 ×S2 plus the representatives of the two other double cosets in which S4 is partitioned by S2 ×S2 . Figure 7a shows how an arbitrary member g of S16 is decomposed with the help of two members (b1 and b2) of S8 × S8 and one representative a of the double coset of g. Van Rentergem et al. have demonstrated that it is always possible to construct a representative that is a controlled NOT gate: P1 = f (A2, A3, ..., Aw) ⊕ A1 P2 = A2 P3 = A3 ...

...

Pw = Aw , where (in contrast to Section 3) A1 is the controlled bit (instead of Aw ). A proof is given in Appendix A (case p = 2w−1 and q = 2). We conclude that the present synthesis for an arbitrary circuit of width w consists of the cascade of • a first twin circuit, • a controlled NOT gate, and • a second twin circuit.

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Alexis De Vos

a

b

Figure 7. An arbitrary circuit g (member of the group S16 ) decomposed as b1ab2 with the help of double cosets generated (a) by its subgroup S28 and (b) by its subgroup S82 . We illustrate the Van Rentergem procedure with an example where G = S8 and H isomorphic to S24 and thus G = 40, 320 and H = 576. We choose the truth table of Table 5a. Figure 8a gives the result of repeated application of the procedure, until all subcircuits are member of S2 , i.e. are equal to either the 1-bit identity gate or the 1-bit inverter. The nested schematic can easily be translated into a chain of controlled NOTs, i.e. the conventional way of writing down a reversible circuit: Figure 8b. This particular circuit consists of eigth controlled NOT gates, of which seven are simply TOFFOLI gates. We now introduce a cost function, called ‘gate cost’: we assign to each controlled NOT a unitary cost (whatever the number of variables in the control function). The gate cost of Circuit 8b thus is 8. Note that the present double coset approach ends up with a chain of cost of the order 4w .

6.

Double Cosets Again

Instead of applying a subgroup H isomorphic to the Young subgroup S2w−1 × S2w−1 = S22w−1 and taking in each double coset a representative which is member of S2 × S2 × ...× w−1 S2 = S22 , we can also work the other way around: choose a subgroup H isomorphic to the w−1 and look whether in each double coset there exists a representative Young subgroup S22 2 which is member of S2w−1 . See Figure 7b. This is indeed always possible [25] [26] [27]. Again, a proof is provided in Appendix A (case p = 2 and q = 2w−1 ). Thus, we may conclude that a synthesis of an arbitrary circuit of width w may consist of the cascade of A

P

B C

Q R

A B C

P Q R

a

b

Figure 8. Decomposition of the example circuit of width 3 (Table 5a): (a) written as nested controlled gates and (b) written as a chain of controlled NOTs.

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217

• a first controlled NOT gate, • a twin circuit, and • a second controlled NOT gate. Note that Figure 7b is like Figure 7a inside out. We say that the two circuits are each other’s dual. They are indeed based on two different Young subgroups. These subgroups are based on two dual partitions of the number 2w (i.e. the degree of the supergroup S2w ) : 2w = 2w−1 + 2w−1 = 2 + 2 + ... + 2

(2w−1 terms) .

Because of its importance, we give here in detail [25] [26] the synthesis algorithm using the latter partition:

Algorithm A In the algorithm, we use the notations like Ai (j), where the subscript i refers to the column in the truth table, whereas the number j refers to the row. Thus these counters obey 1 ≤ i ≤ w and 1 ≤ j ≤ 2w . For finding the three parts of the decomposition, we proceed as follows. We add to the given truth table (consisting of w input columns A and w output columns P ) two extra sets of columns F and J. These are filled in, in three steps: • First, we fill in the w − 1 columns F2 , F3 , ..., Fw by merely copying columns A2 , A3 , ..., Aw and, analogously, fill in the w − 1 columns J2, J3 , ..., Jw by merely copying columns P2 , P3 , ..., Pw . • Then we construct a ‘coil’ of 0s and 1s, starting from F1 (1) = 0. • Then we construct a second coil, starting from the non-filled-in F1 (j) with lowest j, etcetera, until all F1 (and thus also all J1 ) are filled in. Above, a coil consists of a finite number of ‘windings’. Here, a winding is a fourbit sequence F1 (k) = X, F1 (l) = X, J1 (l) = X, and J1 (m) = X, where l results from the condition that the string F2 (l), F3(l), ..., Fw(l) has to be equal to the string F2 (k), F3(k), ..., Fw(k) and where m results from the condition that the string J2 (m), J3(m), ..., Jw(m) has to be equal to the string J2 (l), J3(l), ..., Jw(l). Although the above text might suggest that the algorithm is complicated, it is in fact very straightforward. Figure 9 provides an illustration of this fact, by giving in detail the synthesis of a reversible circuit of width w = 2, i.e. the circuit with two inputs (A1 and A2 ) and two outputs (P1 = A2 and P2 = A1 , in this particular case). First, between the columns (A1, A2) and (P1, P2) of the truth table, we insert the four empty columns (F1 , F2) and (J1 , J2). Subsequently, columns F2 and J2 are filled in by simply copying A2 and P2 , respectively. This step is displayed in boldface. Next comes the tricky part: filling in the columns F1 and J1 . For this purpose, we start at F1 (1). We may set this bit arbitrarily, but we choose to set it to 0. This starting choice is marked by a small square in Figure 9. As a consequence, we automatically can fill in a lot of other bits in columns

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Alexis De Vos

A1 A2

F1 F2

0 0 1 1

0 1 1 0

0 1 0 1

0 1 0 1

J1 J 2

0 1 1 0

1 1 0 0

P1 P2

0 1 0 1

1 1 0 0

Figure 9. Example of a synthesis according to the basic algorithm ( w = 2): expanded truth table. F1 and J1 . Indeed, as all computations need to be reversible, F1 (1) = 0 automatically leads to F1 (3) = 1. Then we impose J1 (3) = F1 (3), i.e. J1 (3) = 1. Again, reversibility requires that J1 (3) = 1 infers J1 (4) = 0. Etcetera, until we come back at the starting point F1 (1). The arrows in Figure 9 show the order of filling in. Here, everything is filled in when the travelling around is closed. So, this synthesis is finished after a single coil. This example illustrates that during the application of the algorithm, we ‘walk in circles’, while repeatedly assigning the bit sequence 0, 1, 1, 0, 0, 1, 1, ..., 1, 1, 0, 0, 1, 1, 0 . In case the first coil is closed before columns J1 and F1 are completely filled in, the designer just has to start a second coil, etcetera. The fact that the above algorithm always comes to an end with the extended truth table being completely filled, constitutes an additional proof that the theorem of Appendix A is true for the special case p = 2 (and thus q = 2w−1 ). As a result, Algorithm A yields a decomposition of an arbitrary reversible circuit a (Figure 10a) into the desired cascade (Figure 10b) of a first controlled NOT gate with controlled bit on the first wire, a twin circuit a1 leaving the first bit unaffected (P1 = A1 ), and a second controlled NOT gate with controlled bit on the first wire. Note that circuit a1 in Figure 10b is simpler than circuit a in Figure 10a, as a1 obeys P1 = A1. Algorithm A can now be ‘deepened’ as follows. By applying the decomposition of a1 into three circuits, we obtain Figure 10c, where the circuit a2 is again simpler than the circuit a1 , because it fulfils both P1 = A1 and P2 = A2 . Etcetera, until we obtain Figure 10d, where the circuit aw−1 obeys P1 = A1, P2 = A2 , ..., and Pw−1 = Aw−1 . These properties reveal that aw−1 is nothing but a control gate with controlled bit Aw . Therefore, Figure 10d is equivalent to Figure 10e, such that we have decomposed a into 2w − 1 controlled NOT gates. This procedure automatically leads us to Algorithm B:

Algorithm B We add to the given truth table (consisting of w input columns A and w output columns P ) not two extra sets of columns, but 2(w − 1) sets of columns. We call them A1, A2, ...,

Reversible Logic

a

a

b

a1

c

a2

d

a3

219

e

Figure 10. Decomposition of a reversible logic circuit of width w = 4 : (a) original logic circuit, (b), (c), and (d) intermediate steps, (e) final decomposition. Aw−2 , Aw−1 , P w−1 , P w−2 , ..., P 2 , and P 1 . Together they make 2(w − 1)w new columns. These are filled in, in the following steps: • First, we fill in all A1 columns except column A11 , by copying the w−1 corresponding A columns, and analogously, we fill in all P 1 columns except column P11 , by copying the w − 1 corresponding P columns. • Then we fill in the two columns A11 and P11 , by constructing a coil, starting from bit A11(1), then constructing a new coil, starting at the non-filled-in A11(j) with lowest j, etcetera, until all A11 (and thus also all P11 ) are filled in. • Then, we fill in all A2 columns except column A22 , by copying the w − 1 correspond-

220

Alexis De Vos Table 6. Expanded truth table according to Algorithm B.

A1A2 A3 0 0 0 0 1 1 1 1

0 0 1 1 0 0 1 1

0 1 0 1 0 1 0 1

A11 A12 A13 0 1 0 1 1 0 1 0

0 0 1 1 0 0 1 1

0 1 0 1 0 1 0 1

A21 A22 A23 0 1 0 1 1 0 1 0

0 0 1 1 1 0 0 1

0 1 0 1 0 1 0 1

P12 P22 P32 0 1 0 1 1 0 1 0

0 0 1 1 1 0 0 1

0 1 0 0 1 1 0 1

P11 P21 P31 0 1 0 1 1 0 1 0

0 0 1 0 1 1 1 0

0 1 0 0 1 1 0 1

P1 P2 P3 1 1 1 0 1 0 0 0

0 0 1 0 1 1 1 0

0 1 0 0 1 1 0 1

ing A1 columns, and analogously, we fill in all P 2 columns except column P22 , by copying the w − 1 corresponding P 1 columns. • Then we fill in the two columns A22 and P22 , by constructing the appropriate number of coils, starting from bit A22 (1), until all A22 (and thus also all P22 ) are filled in. w−1 • Etcetera, until finally all Aw−1 w−1 (and thus also all Pw−1 ) are filled in. At that moment, we have all 2w2 2w entries of the extended table.

We illustrate our ‘deepened’ procedure for the example circuit in Table 5a. By applying the above procedure, we obtain Table 6. The first step of the procedure is displayed in bold face, whereas the second step is emphasized in italic. (The reader may verify that here this step requires two coils, the former having three windings, the latter having only one winding.) The third step of the algorithm is underlined. The above procedure thus yields a decomposition of the logic circuit into five logic circuits (one computing A1 from A, one computing A2 from A1 , ..., and one computing P from P 1 ). All five subcircuits are automatically controlled NOT gates. By merely inspecting Table 6, we find their subsequent control functions: f (A2 , A3) = A3 , f (A11 , A13) = A11 A13 , f (A21 , A22) = A21 A22 , f (P12 , P32 ) = P12 ⊕ P32 , and f (P21 , P31) = P31 . Figure 11a shows the final synthesis of Table 5a with its five controlled NOT gates. Note that this gate cost of 5 is lower than the gate cost of 8 in Figure 8b. Noteworthy is the automatic V-shape of the positions of the five crosses (i.e. controlled NOTs) in the figure. When we apply the same procedure to each of the 8! = 40,320 circuits of the group S8 , sometimes one or more of the five control functions equals 0. This means that one or more of the five controlled NOTs is the identity gate and thus in fact is abscent, such that there is a total of less than five gates. We thus yield a statistical distribution of gate cost ranging from 0 to L = 2w − 1 = 5. The average gate cost turns out to be about 4.2. We stress that a gate cost 2w − 1 is very close to optimal. No synthesis method can do better than 2w−3. This is proven in Appendix B. This constitutes a significant improvement over the method described in Section 5, as the latter yields cascade lenghts rising as 4w . As

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221

A1

P1

A2

P2

A3

P3

A1

P1

A2

P2

A3

P3

a

b

Figure 11. Decomposition of the example circuit of width 3 (Table 5a), according to Algorithm B: (a) into controlled NOT gates and (b) into TOFFOLI gates. w−1

lead to (almost) optimal decompositions, we conclude that subgroups isomorphic to S22 the controlled NOTs form a natural library for synthesis. We stress that such library is larger than libraries with merely TOFFOLI gates. The latter are of the type of Figure 4b, controlled by AND functions. In the synthesis approach presented here, we fully make use of building blocks of the type of Figure 4a, controlled by arbitrary control functions. Whereas the synthesis method of Section 5 is based on a subgroup chain like (5), i.e. of a set of subgroups of ever diminishing size, the Algorithm B is based on a set of w subgroups w−1 all of the same order S22 . These w subgroups are conjugate to each other. This means that one subgroup Hi can be obtained from another one Hj as follows: Hi = c Hj c−1 , where c is some member of the supergroup G. In our case we have that c = c−1 is the SWAP gate exchanging the wires # i and # j. Figure 12 illustrates the two approaches: the former with ever smaller subgroups and the latter with equal-size subgroups. In order to be efficient, the conjugate subgroups should overlap as little as possible. In our synthesis method, they ovelap very little, as two subgroups have only one element in common, i.e. the identity element (or w-bit follower): Hi ∩ Hj = I . Sometimes, for practical purposes, the library of controlled NOT gates is considered as too large. Then a library consisting of only TOFFOLI gates is a possibility. In that case, we may proceed as follows: each controlled NOT is decomposed into TOFFOLI gates, by merely replacing the control function by its Reed–Muller expansion. Such expansion may contain up to 2w−1 terms. From Appendix B, we see that such expansion is not optimal. Better results can be obtained by one of the so-called ESOP expansion 1 algorithms [28] [29]. Figure 13 gives an example: the controlled NOT gate with control function f = A1 + A2 , i.e. an OR function. The Reed–Muller expansion (Figure 13b) reads f = A1 ⊕ A2 ⊕ A1 A2 , 1

The name ESOP is derived from the expression ‘exclusive-or sum of products’. It is a variant of the SOP or ‘sum of producs’, where ‘sum’ stands for the Boolean OR function and ‘product’ for AND. Thus an ESOP is a Boolean expression which is written as a XOR of ANDs. In contrast to the Reed–Muller expansion, an ESOP expansion allows NOT functions (besides the XOR and the AND functions).

222

Alexis De Vos S

S

8

8

4

S2 S

4

2 4

S2

I I

4 S 2

4

S2

b

a

Figure 12. Two sets of efficient subgroups: (a) a subgroup chain and (b) a subgroup flower. whereas one of the minimum-ESOP expansions (Figure 13c) is: f = A1 ⊕ A1 A2 . Applying either the Reed–Muller decomposition or a minimum- ESOP expansion to the circuit of Figure 11a yields Figure 11b, with six TOFFOLI gates.

7.

Garbage Bits

In the above sections, synthesis means: finding a hardware implementation for a given reversible truth table. However, often a synthesis job is defined by an irreversible truth table. A famous example is the design of a full adder: see Table 7a. It has three input bits: the augend bit A, the addend bit B, and the carry-in bit Ci ; it has two output bits: the sum

a

b

c

Figure 13. Decomposition of (a) a controlled NOT gate into (b) its Reed–Muller expansion and into (c) one of its minimal ESOP expansions.

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223

Table 7. Full adders: (a) irreversible and (b) reversible.

ABCi

Co S

000 001 010 011 100 101 110 111

00 01 01 10 01 10 10 11 (a)

ABCi P

Co SG1G2

0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111

0000 1000 0100 1100 0101 1101 1001 0001 0111 1111 1011 0011 1010 0010 1110 0110 (b)

bit S and the carry-out bit Co . Basicly, the table gives eight additions of three numbers: 0+0+0 = 0 0+0+1 = 1 ... 1+1+0 = 2 1+1+1 = 3. The truth table surely is not reversible: • the number of output bits is not equal to the number of input bits and • various output words appear more than once in the table, the output Co S = 10 e.g. appearing three times in Table 7a. In order to implement this calculation in a revesible computer, we have to expand the table, such that the original table is embedded in a large reversible one: see e.g. Table 7b. All bits from Table 7a are repeated (in boldface) in Table 7b. The new table has two extra output columns: G1 and G2. Those bits are called garbage bits. They are not asked for in the

224

Alexis De Vos

A B Ci

Co

0

S

Figure 14. Full adder with three TOFFOLI gates and one FREDKIN gate. first place, but are added in order to guarantee that all output words are different. Because there are now four output bits, there have to be four input bits as well. Therefore, we have added one additional input column: bit P , called the preset. For the desired application, P will always be put equal to 0. We note that there is no unique way to embed an irreversible table in a larger reversible table. The embedding should thus be done carefully, in order to minimize the resulting reversible hardware cost. Figure 14 gives the cheapest reversible implementation of Table 7a [30]. It consists of three TOFFOLI gates, one FREDKIN gate, and two SWAP gates. Its gate cost is 4 (as SWAPS are considered free of cost). We note that a controlled NOT can be interpreted as a reversible embedding of the calculation of an irreversible Boolean function. Indeed, assume that we like to calculate the Boolean function f (A, B, C) of three Boolean variables, as defined by its truth table in Table 8a. The table is, of course, irreversible. Indeed, it has less output columns than input colums. Besides, the output row 0 appears not less than five times and the output row 1 appears three times. Table 8b shows the truth table of the controlled NOT gate with control function f (A, B, C), with controlling bits A, B, and C, and with controlled bit the extra input D. We have the following relations between outputs and inputs: P = A, Q = B, R = C, and S = f (A, B, C) ⊕ D, in accordance with (1). The output S equals the desired function f , if the input D is preset to zero. The extra outputs P , Q, and R are garbage outputs. We note that the reversible embedding (Table 8b) unfortunately has a total number of columns which is double the number of columns in the original problem (Table 8a). Thus, reversible digital circuits have the disadvantage to generate a lot of garbage output, not desired for the application. Because of Landauer’s principle, we are not allowed to throw them away. We thus have to take them all the way through the following computational steps. Only two garbage bits in a full adder is already a matter of great concern. Indeed, such full adder is just a building block in, say, a 32-bit adder. The latter circuit will itself be a building block for a 32-bit multiplier. Such multiplier is used many times in e.g. a digital filter. Many filters make up a filter bank; many filter banks make up e.g. a speech processor. Each time we step from one architectural level to the next, the number of garbage bits explodes. This proliferation of garbage will cause huge costs because of extra gates and extra interconnections. Therefore, the design challenge consists of designing, at each level of abstraction, circuits that generate as little garbage as possible. The clever Cuccaro adder [31] succeeds in avoiding one of the two garbage bits G1 and G2 in Table 7b and thus is highly recommendable [32]. The trick consists in avoiding to calculate Co and S simultaneously, in a single circuit. Cuccaro et al. first calculate Co and afterwards compute S from Co , when the information of Co is not necessary anymore.

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225

Table 8. Computing the Boolean function f (A, B, C): (a) irreversibly and (b) reversibly.

ABC

f

000 001 010 011 100 101 110 111

0 0 0 1 1 0 0 1

(a)

ABCD

P QRS

0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111

0000 0001 0010 0011 0100 0101 0111 0110 1001 1000 1010 1011 1100 1101 1111 1110 (b)

8.

Experimental Prototypes

We can use either left cosets, or right cosets, or double cosets in the synthesis procedure; we can choose one subgroup or another. Whatever choices we make, we obtain a procedure for synthesizing an arbitrary circuit by cascading a small number of standard cells from a limited library. By appropriate choice of the representatives of the (double) cosets, we can see to it that all building blocks in the library are member of either of the two following special subgroups: • the subgroup of controlled NOT gates and • the subgroup of controlled SWAP gates. Among the controlled NOT gates, we note three special elements: • If f is identically zero, then Pw is always equal to Aw . Then the gate is the identity gate i. • If f is identically one, then Pw always equals 1 ⊕ Aw . Then the gate is an inverter or NOT gate: Pw = Aw .

226

Alexis De Vos

• If f (A1, A2, ..., Aw−1) equals the (w − 1)-bit AND function A1 A2 ...Aw−1, then the gate is a TOFFOLI gate: whenever A1 A2...Aw−1 equals 0, then Pw simply equals Aw ; but whenever A1 A2...Aw−1 equals 1, then Pw equals NOT Aw . For physical implementation, dual logic is very convenient. It means that, within the hardware, any logic variable X is represented by two physical quantities, the first representing X itself, the other representing NOT X. Thus, e.g. the physical gate realizing logic gate of Table 2a has four physical inputs: A, NOT A, B, and NOT B, or, in short-hand notation: A, A, B, and B. It also has four physical outputs: P , P , Q, and Q. Such approach is common in electronics, where it is called dual-line or dual-rail electronics. Also some quantum computers make use of dual-rail qubits [33]. As a result, half of the input pins are at logic 0 and the other half at logic 1, and analogous for the output pins. In this way, dual electronics is physically conservative: the number of 1s at the output equals the number of 1s at the input (i.e. equals w), even if the truth table of the reversible logic gate is not conservative. As a result, we get the advantages of conservative logic, without having to restrict ourselves to conservative logic. Dual-line hardware allows very simple implementation of the inverter. It suffices to interchange its two physical lines in order to invert a variable. In other words: in order to hardwire the NOT gate: • output P is simply connected to input A and • output P is simply connected to input A. Controlled NOTs are implemented as NOT gates which are controlled by switches. A first example is the controlled NOT gate with a single controlling bit (i.e. the bit A): P

= A

Q = A⊕B . These logic relationships are implemented into the physical world as follows: • output P is simply connected to input A, • output P is simply connected to input A, • output Q is connected to input B if A = 0, but connected to B if A = 1, and • output Q is connected to input B if A = 0, but connected to B if A = 1. The last two implementations are shown in Figure 15a. In the figure, the arrow heads show the position of the switches if the accompanying label is 1. A second example is a TOFFOLI gate with two controlling bits (i.e. the bit A and the bit B): P

= A

Q = B R = AB ⊕ C . Its logic relationships are implemented into physical world as follows:

Reversible Logic

227

• output P is simply connected to input A, • output P is simply connected to input A, • output Q is simply connected to input B, • output Q is simply connected to input B, • output R is connected to input C if either A = 0 or B = 0, but connected to C if both A = 1 and B = 1, and • output R is connected to input C if either A = 0 or B = 0, but connected to C if both A = 1 and B = 1. The last two implementations are shown in Figure 15b. Note that in both Figures 15a and 15b, switches always appear in pairs, of which one is closed whenever the other is open and vice versa. It is clear that the above design philosophy can be extrapolated to a controlled NOT gate with arbitrary control function f . Suffice it to wire a square circuit like in Figures 15a and 15b, with the appropriate series and parallel connections of switches. If we apply such hardware wiring to the controlled NOT with control function f = A + B, we realise that direct implementation of Figure 13a needs eight switches. In contrast, Figure 13b needs sixteen switches and Figure 13c needs twelve switches. This fact illustrates that decomposing a controlled NOT into TOFFOLIs is not necessarily advantageous. Now that we have an implementation approach, we can realize any reversible circuit in hardware. We will demonstrate here some examples of implementation into electronic chip. In electronic circuits, a switch is realized by the use of a so-called transmission gate, i.e. two MOS-transistors in parallel (one n-MOS transistor and one p-MOS transistor). Here, MOS stands for metal-oxide-semiconductor, whereas n refers to ‘negative’ type and p to ‘positive’ type. Circuits containing both n-type transistors and p-type transistors are called c-MOS circuits, where c stands for ‘complementary’. Readers, familiar with electronics, will notice that circuits like in Figure 15 are not of the conventional style called ‘restoring logic’, but of a less well-known style called ‘pass-transistor logic’. The pass-transistor logic families allow good control of leakage currents and therefore yield lower energy consumption [34]. As an example, Figure 16 shows a 4-bit reversible ripple adder [35], implemented in full-custom 2.4 µm standard c-MOS technology, consisting of sixteen TOFFOLI gates with a total of 192 transistors. This prototype chip was fabricated in 1998. The circuit functions equally well from left to right as it works from right to left. Conventional digital chips do not have this property. A second example [36] (Figure 17) was fabricated in 2000, in submicron technology: a 4-bit carry-look-ahead adder, implemented in 0.8 µm standard cMOS technology, containing four TOFFFOLIs with w = 2, four controlled NOTs of width w = 3, and one complex controlled NOT of width w = 13. It contains a total of 320 transistors. Switches not only can decide whether an input variable is inverted or not. We can apply switches also in order to decide whether two input variables are swapped or not. This

228

Alexis De Vos B

_ Q

A

A

a

A

A

_ B

Q C

A

A

_ R

B

B

A

B

b

A

R

B

A

c

R

A

B

_ C

B

A

A

A

A

R

A

A

Q

C

Q

C

Figure 15. Schematic for (a) a controlled NOT gate (u = 1, v = 1), (b) a controlled NOT gate (u = 2, v = 1), and (c) a controlled SWAP gate (u = 1, v = 2). concept leads to the implementation of the controlled SWAP gate, e.g. the FREDKIN gate: P

= A

Q = AB ⊕ AC ⊕ B R = AB ⊕ AC ⊕ C , where A is the controlling bit and B and C are the two controlled bits. The set of equations corresponds to (2), with control function f (A) = A. Figure 15c shows the physical implementation. The reader will easily extrapolate the design philosophy to reversible logic gates of width w = u+v, where u controlling bits decide, by means of a control function f , whether the v controlled bits are either subjected or not to some particular swapping and/or inverting. One application [32] (Figure 18) is an 8-bit Cuccaro adder, implemented in 0.35 µm standard c-MOS technology, containing 17 TOFFOLI and 16 FREDKIN gates. It contains

Reversible Logic

229

Figure 16. Microscope photograph (140 µm × 120 µm) of a 2.4-µm 4-bit reversible ripple adder.

Figure 17. Microscope photograph (610 µm × 290 µm) of a 0.8-µm 4-bit reversible carrylook-ahead adder. a total of 392 transistors. The prototype chip was fabricated in 2004. The reader will observe that full-custom prototyping at a university lab follows Moore’s law, with a couple of years delay with respect to industry. Indeed, many commercial chips nowadays use 0.18 or 0.13 µm transistors. Some companies have entered the nanoscale era, by introducing 90 nm, 65 nm and even 45 nm products [37] to the market. Moore’s law, i.e. the continuing shrinking of the transistor sizes, leads to a continuing decrease of the energy dissipation per computational step. This heat generation Q is of the order of magnitude of CVt2 , where Vt is the threshold voltage of the transistors and C is the total capacitance of the capacitors in the logic gate [38]. We have C of the order of magnitude of 0  LW t , where L, W , and t are the length, the width, and the oxide thickness of the transistors, whereas 0 is the permittivity of vacuum (8.85 ×10−12 F/m) and  is the

230

Alexis De Vos

Figure 18. Microscope photograph (140 µm × 230 µm) of a 0.35-µm 8-bit reversible Cuccaro adder. dielectric constant of the oxide (either 3.9 for silicon oxide SiO 2 or 22 for hafnium oxide HfO2). Table 9 gives some typical numbers. Note that a technology is named after its value for the transistor length L. We see how Q becomes smaller and smaller, as L shrinks. However, this dissipation in electronic circuits still is about four orders of magnitude in excess of the Landauer quantum kT log(2), which amounts (for T = 300 K) to about 3 ×

Reversible Logic

231

Table 9. Moore’s law for dimensions L, W , and t, and for threshold voltage Vt , as well as for resulting capacitance C and heat dissipation Q. technology (µm)

L (µm)

2.4 0.8 0.35

2.4 0.8 0.35

W (µm)

t (nm)

Vt (V)

2.4 2.0 0.5

42.5 15.5 7.4

0.9 0.75 0.6

C (fF)

Q (fJ)

46.8 3.6 0.82

38 2.0 0.30

1 femtojoule

1000

100

Q (attojoule)

10

1

C V_t ^2

1 attojoule

0.1

0.01 Landauer quantum 0.001

1 zeptojoule 2000

2010

2020

2030

2040

Figure 19. Adiabatic heat generation Q in future technologies. 10−21 J or 3 zeptojoule. Further shrinking of L and W and further reduction of Vt ultimately will lead to a Q value in the neighbourhood of kT log(2). According to the International Technology Roadmap of Semiconductors [39], Moore’s law will continue in the near future. Figure 19 shows how CVt2 will continue to decrease, ultimately approaching Landauer’s quantum around 2034... That day, digital electronics will have good reason to be reversible... This, however, does not mean that the reversible MOS circuits are useless today. Indeed, as they are a reversible form of pass-transistor topology, they are particularly suited for adiabatic addressing [40], leading to substantial power saving. In subsequent gates, switches are opening and closing, one after the other, like dominoes, transferring information from the inputs to the outputs of the overall circuit [41]. Figure 20 shows an example of a quasi-adiabatic experiment [42]. We see two transient signals: one of the input variables and one of the

232

Alexis De Vos

Figure 20. Oscilloscope view of a 0.35-µm full adder.

resulting output bits. In practice, such procedure leads to a factor of about 10 in power reduction [38]. The reduction of the power dissipation is even more impressive if standard c-MOS technology is replaced by SOI (silicon-on-insulator) technology. Indeed, in the latter process, the threshold voltage Vt can be controlled better, such that low-Vt technologies are possible [43] [44].

9.

Conclusion

We have demonstrated how the symmetric group S2w has various interesting subgroups. Many (but not all) of them turn out to be Young subgroups. We have shown how cosets and double cosets are particularly helpful for synthesizing arbitrary reversible circuits. E.g., we have demonstrated the power of subgroup chains of the form S2w ⊃ S2w −1 ⊃ S2w −2 ⊃ ... w−2 w−1 w ⊃ S22 ⊃ S21 for ⊃ S3 ⊃ S2 ⊃ S1 and the form S2w ⊃ S22w−1 ⊃ S42w−2 ⊃ ... ⊃ S24 single coset synthesis. A particularly efficient synthesis is obtained by the double cosets of w−1 w different but conjugate subgroups of the type S22 . w−1

Controlled NOTs belong to a subgroup isomorphic to S22 , whereas controlled SWAPs w−2 belong to a subgroup isomorphic to S22 . They form the building blocks for hardware implementation. Silicon prototypes of adder circuits, in three different standard technologies (2.4 µm, 0.8 µm, and 0.35 µm), have been fabricated and tested. They illustrate how energy dissipation and heat generation in chips keep on shrinking according to Moore’s law, eventually leading to computer hardware in which Landauer’s principle will be more than academic...

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233

Acknowledgements The author wishes to thank his Ph.D. students Bart Desoete, Filip Beunis, and Yvan Van Rentergem, as well as prof. Leo Storme, for valuable collaborations. He further thanks the Invomec division of Imec v.z.w. (Leuven, Belgium) and the Eurochip and Europractice organisations, for processing the chips at Alcatel Microelectronics (Oudenaarde, Belgium), Austria Mikro Systeme (Unterpremst¨atten, Austria), and AMI Semiconductor (Oudenaarde, Belgium).

Appendix A: A Remarkable Theorem from Combinatorics Let n be an integer, i.e. a number of objects {1,2,...,n}. Let p be a divisor of n, i.e. we have n = pq, with both p and q integers. We arrange the n objects into q rows, each of p objects, the arrangement being called a Young tableau [45]. We consider an arbitrary permutation of the n objects. During such a permutation, the q subsets exchange q 2 flows Fij . These Fij form a matrix F with all matrix elements equal to 0, 1, 2,..., or p. The matrix element Fij denotes the number of objects which move from row # i to row # j. There are 2q constraints X

Fij

= p

Fij

= p,

j

X i

of which 2q − 1 are independent. As an example, we consider an arbitrary permutation of the 35 objects {1,2,...,35}. Figure 21a shows the permutation a in the Young tableau for n = 35, p = 7, and q = 5. The tableau consists of the five sets {1,2,...,7}, {8,9,...,14}, ..., and {29,30,...,35}. These sets exchange a number of objects according to the flow matrix 

   F =  

6 1 0 0 0

0 5 1 0 1

0 0 5 0 2

1 0 0 6 0

0 1 1 1 4



    ,  

where e.g. F53 = 2 expresses that in Figure 21a two objects are mapped from the fifth row {29,30,...,35} to the third row {15,16,...,21}. Note that the q×q flow matrix F contains incomplete information about the permutation a. The complete information on a is given by the corresponding n × n permutation matrix. Theorem : each permutation a can be decomposed as a = h1 vh2, where both h1 and h2 only permute objects within rows and where v only permutes objects within columns. Figure 21a shows the permutation a, as a mapping. Figures 21b, 21c, and 21d show the corresponding permutations h1 , v, and h2 to be performed subsequently: • The vertical permutation v (Figure 21c) is found as follows: the cycles of a are projected vertically, yielding one or more vertical cycles.

234

Alexis De Vos

a

c

b

d

Figure 21. Decomposition of (a) an arbitrary permutation of 35 objects into (b) a first ‘horizontal’ permutation, (c) a ‘vertical’ permutation, and (d) a second ‘horizontal’ permutation.

• The horizontal permutation h1 (Figure 21b) merely consists of horizontal arrows which map the arrow tails of vertical and oblique arrows in Figure 21a to the corresponding arrow tails of Figure 21c. Subsequently additional horizontal arrows are added, in order to form closed horizontal cycles. • Finally, the horizontal permutation h2 (Figure 21d) simply equals v −1h−1 1 a. This theorem forms the basis of the existence of Clos networks [46], more precisely rearrangeable (non-blocking) Clos networks [47] [48] [49]. In the past, the approach has been applied succesfully in telephone switching systems. Nowadays, it is fruitful in internet routing [50]. These conventional applications of the theorem are concerned with permutations of wires, i.e. with the decomposition of the members of the subgroup of exchangers (isomorphic to Sw ). Here we apply the theorem not to the w wires, but to the 2w messages, i.e. to the full group S2w . Whereas Figure 10b is reminiscent of Clos networks (but here with n = 2w instead of n = w), Figure 10e is reminiscent of so-called banyan networks (but here again with n = 2w instead of n = w). Finally, the coils/windings procedure of Section 6 is reminiscent of the ‘looping algorithm’, presented of Hui [48]. The fact that it is always possible to construct an appropriate vertical permutation v is a direct consequence of either Birkhoff’s theorem [51] (a.k.a. the Birkhoff–von Neumann theorem) or Hall’s marriage theorem or K¨onig’s theorem or the maximum-flow theorem, all these notorious theorems in combinatorics being equivalent [52]. As an example, one of the formulations of the Birkhoff theorem says: any matrix of size q × q and line sum p can be decomposed as the sum of p matrices of the same size and unit line sum [53]. In other words: any flow matrix with line sum p can be decomposed as the sum of p permutation matrices. E.g. the above matrix F of size 5 × 5 with line sum 7 can be decomposed as the

Reversible Logic

235

following sum of seven permutation matrices: 

F =4

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

  +

1 0 0 0 0

0 1 0 0 0

0 0 0 0 1

0 0 0 1 0

0 0 1 0 0

  +

1 0 0 0 0

0 0 0 0 1

0 0 1 0 0

0 0 0 1 0

0 1 0 0 0

  +

0 1 0 0 0

0 0 1 0 0

0 0 0 0 1

1 0 0 0 0

0 0 0 1 0



.

It suffices to apply in each of the p columns of the Young tableau one of these p permutations, in order to obtain the desired vertical permutation v. In Figure 21c, we indeed recognize four empty columns (corresponding to the four identity matrices in the matrix sum) and three non-empty columns (corresponding to the remaining three matrices in the matrix sum). Thus, the theorem says that one permutation of n objects can be decomposed into • a product of q subpermutations, each of p objects, followed by • a product of p subpermutations, each of q objects, and finally • a second product of q subpermutations, each of p objects. This is depicted in Figure 22. In brief: one big spaghetti is decomposed into 2q + p small permutations. The theorem can be interpreted in terms of group theory: if we define • N as the group of all permutations of the n objects, • H as the group of all ‘horizontal permutations’ of these objects, and • V as the group of all ‘vertical permutations’, then we have that • N is isomorphic to the symmetric group Sn , • H is isomorphic to the Young subgroup Sp × Sp × ...×Sp = Sqp , and • V is isomorphic to the Young subgroup Sq × Sq × ...×Sq = Spq . Note that the Young subgroups Sqp and Spq are based on two so-called dual partitions of the number n : n = p + p + ... + p

(q terms) and

n = q + q + ... + q

(p terms).

Therefore these two subgroups are referred to as dual Young subgroups. The combinatorial theorem says the following: if the group N is partitioned into double cosets by means of the subgroup H, then we can choose in each double coset a representative which is member of V. In how many double cosets the supergroup N is partitioned by the subgroup H is a very difficult problem. We will call this number X(p, q). Fortunately, knowledge of the value of X is not important; essential is the fact that (thanks to Birkhoff’s theorem) each of the X double cosets contains (at least) one member of V. We now consider the special case where n is even. We distinguish two special subcases [26]:

236

Alexis De Vos p planes

q planes

q planes

Figure 22. The 3-dimensional decomposition of an arbitrary permutation of 35 objects into five ‘horizontal’ subpermutations, seven ‘vertical’ subpermutations, and a second five ‘horizontal’ subpermutations. • p = 2 and thus q = n/2 • p = n/2 and thus q = 2. In the latter subcase, the group Sn is partitioned by the subgroup isomorphic to S2n/2 into exactly n n X( , 2) = + 1 2 2 double cosets [24]. This partitioning is applied in Section 5. In the former subcase, the n/2 number of double cosets in which Sn is partitioned by the subgroup S2 is surprisingly complicated [54]: n/2 [( n2 )!]2 X 2k (n − 2k)! n . X(2, ) = 2 2n k=0 [( n2 − k)!]2 k!

This partitioning is applied in Section 6.

Appendix B: Optimal Syntheses In the present appendix, we follow the reasoning method introduced by Even et al. [55], by Maslov and Dueck [18], and by Patel et al. [13]. Let us assume a set of N circuits, to be synthesized with the help of a library of B different building blocks (a subset of the former set, including the trivial identity gate). We build all possible cascades of length l. We call m(l) the number of different circuits which can be synthesized by these cascades. Note that these m circuits automatically include all circuits synthesized by a shorter cascade, because we have included the identity building-block in the library. The question arises: which cascade length L is necessary, in order to guarantee that these cascades contain the synthesis of all N given circuits, i.e. to guarantee that m(L) ≥ N ? A hypothetical ‘most efficient’ library (where all different cascades yield different circuits) would realize m(l) = B l .

Reversible Logic Therefore B L ≥ N and thus L=



log(N ) log(B)



237

.

(7)

Here dxe stands for the ceiling of x, i.e. the smallest integer larger than or equal to x. We now apply this general result to three different cases: (1) In a first application, N is the total number of reversible circuits of width w and B is the number of controlled NOT gates of width w. Thus: = (2w )!

N

B = w (22

w−1

− 1) + 1 .

Using the Stirling inequalities [56] √

2π nn+1/2 e−n < n! <

√ 1 ), 2π nn+1/2 e−n (1 + 4n

we obtain that w

N > 2[w−log(2)]2 and finally conclude that L > 2w − 4 .

Therefore our decomposition in Section 6 (with l = 2w − 1) is close to optimal. (2) If we keep N = (2w )!, but we choose for B is the number of controlled SWAP gates of width w, then B is somewhat smaller: B=

w(w − 1) 2w−2 (2 − 1) + 1 , 2

leading to L > 4w − 10 . This proofs that the synthesis method of Section 6 cannot work with 2w−1 controlled SWAP gates. The reader is kindly invited to verify that this (negative) result is not in conflict with the theorem of Appendix A for n equal to a multiple of 4 and p = 4 and q = n4 . (3) If we apply the general result (7) to the decomposition of controlled NOT gates into TOFFOLI gates, we have N

= 22

w−1

B = 3w−1 and thus obtain L>

2w . 2 log2(3)w

Therefore, the Reed–Muller decomposition, needing a cascade of up to is not optimal.

2w 2

building blocks,

238

Alexis De Vos

References [1] Markov, I. An introduction to reversible circuits. Proceedings of the 12 th International Workshop on Logic and Synthesis , Laguna Beach (May 2003), pp. 318 - 319. [2] Frank, M. Introduction to reversible computing: motivation, progress, and challenges. Proceedings of the 2005 Computing Frontiers Conference , Ischia (May 2005), pp. 385 - 390. [3] De Vos, A. Lossless computing. Proceedings of the I.E.E.E. Workshop on Signal Processing, Pozna´n (October 2003), pp. 7 - 14. [4] Hayes, B. Reverse engineering. American Scientist 94, 107 - 111 (March–April 2006). [5] Feynman, R. Quantum mechanical computers. Optics News 11, 11 - 20 (1985). [6] Landauer, R. Irreversibility and heat generation in the computational process. I.B.M. Journal of Research and Development 5, 183 - 191 (1961). [7] Keyes, R. and Landauer, R. Minimal energy dissipation in logic. I.B.M. Journal of Research and Development 14, 153 - 157 (1970). [8] Bennett, C. Logical reversibility of computation. I.B.M. Journal of Research and Development 17, 525 - 532 (1973). [9] Bennett, C. and Landauer, R. The fundamental physical limits of computation. Scientific American 253, 38 - 46 (July 1985). [10] Scott, W. Group theory. New York: Dover Publications (1964). [11] Hall, P. The theory of groups. Providence: AMS Chelsea Publishing (1968). [12] Fredkin, E. and Toffoli, T. Conservative logic. International Journal of Theoretical Physics 21, 219 - 253 (1982). [13] Patel, K., Markov, I. and Hayes, J. Optimal synthesis of linear reversible circuits. Proceedings of the 13 th International Workshop on Logic and Synthesis , Temecula (June 2004), pp. 470 - 477. [14] Wang, L. and Almaini, A. Optimisation of Reed–Muller PLA implementations. I.E.E. Proceedings – Circuits, Devices and Systems 149, 119 - 128 (2002). [15] Kerntopf, P. On universality of binary reversible logic gates. Proceedings of the 5 th Workshop on Boolean Problems , Freiberg (September 2002), pp. 47 - 52. [16] De Vos, A. and Storme, L. r-Universal reversible logic gates. Journal of Physics A: Mathematical and General 37, 5815 - 5824 (2004). [17] De Vos, A., Raa, B. and Storme, L. Generating the group of reversible logic gates. Journal of Physics A: Mathematical and General 35, 7063 - 7078 (2002).

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[18] Maslov, D. and Dueck, G. Reversible cascades with minimal garbage. I.E.E.E. Transactions on Computer-Aided Design of Integrated Circuits and Systems 23, 1497 - 1509 (2004). [19] Van Rentergem, Y., De Vos, A. and De Keyser, K. Using group theory in reversible computing. Proceedings of the I.E.E.E. World Congress on Computational Intelligence, Vancouver (July 2006), pp. 8566 - 8573. [20] Van Rentergem, Y., De Vos, A. and De Keyser, K. Six synthesis methods for reversible logic. Open Systems & Information Dynamics 14, 91 - 116 (2007). [21] Kerber, A. Representations of permutation groups I. Lecture Notes in Mathematics 240, Berlin: Springer Verlag (1970), pp. 17 - 23. [22] James, G. and Kerber, A. The representation theory of the symmetric group. Encyclopedia of Mathematics and its Applications 16, 15 - 33 (1981). [23] Jones, A. A combinatorial approach to the double cosets of the symmetric group with respect to Young subgroups. European Journal of Combinatorics 17, 647 - 655 (1996). [24] Van Rentergem, Y., De Vos, A. and Storme, L. Implementing an arbitrary reversible logic gate. Journal of Physics A: Mathematical and General 38, 3555 - 3577 (2005). [25] Van Rentergem, Y. and De Vos, A. Synthesis and optimization of reversible circuits. Proceedings of the Reed–Muller 2007 Workshop , Oslo (May 2007), pp. 67 - 75. [26] De Vos, A. and Van Rentergem, Y. Young subgroups for reversible computers. Advances in Mathematics of Communications . 2, 183 - 200 (2008). [27] De Vos, A. and Van Rentergem, Y. Networks for reversible logic. Proceedings of the 8 th International Workshop on Boolean Problems , Freiberg (September 2008). [28] Gaidukov, A. Algorithm to derive minimum ESOP for 6-variable function. Proceedings of the 5 th International Workshop on Boolean Problems , Freiberg (September 2002), pp. 141 - 148. [29] Pogosyan, G., Rosenberg, I. and Takada, S. Building minimum ESOPs through redundancy elimination. Proceedings of the 6 th International Workshop on Boolean Problems, Freiberg (September 2004), pp. 201 - 206. [30] Van Rentergem, Y. and De Vos, A. Optimal design of a reversible full adder. International Journal of Unconventional Computing 1, 339 - 355 (2005). [31] Cuccaro, S., Draper, T., Kutin, S. and Moulton, D. A new quantum ripple-carry addition circuit. Proceedings of the 8 th Workshop on Quantum Information Processing , Cambridge (June 2005), arXiv:quant-ph/0410184.v1. [32] Skoneczny, M., Van Rentergem, Y. and De Vos, A. Reversible Fourier transform chip. Proceedings of the 15 th International Conference on Mixed Design of Integrated Circuits and Systems, Pozna´n (June 2008).

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[33] Chuang, I. and Yamamoto, Y. The dual-rail quantum bit and quantum error correction. Proceedings of the 4 th Workshop on Physics and Computation , Boston (November 1996), pp. 82 - 91. [34] Singh M., Giacomotto C., Zeydel B. and Oklobdzija V. Logic style comparison for ultra low power operation in 65 nm technology. Proceedings of the 17 th International PATMOS Workshop, G¨oteborg (September 2007), pp. 181 - 190. [35] Desoete, B., De Vos, A., Sibi´nski, M. and Widerski, T. Feynman’s reversible logic gates, implemented in silicon. Proceedings of the 6 th International Conference on Mixed Design of Integrated Circuits and Systems , Krak´ow (June 1999), pp. 497 - 502. [36] Desoete, B. and De Vos, A. A reversible carry-look-ahead adder using control gates. Integration, the V.L.S.I. Journal 33, 89 - 104 (2002). [37] Bohr, M., Chau, R., Ghani, T. and Mistry, K. The high-k solution. I.E.E.E. Spectrum 44, 23 - 29 (October 2007). [38] De Vos, A. and Van Rentergem, Y. Energy dissipation in reversible logic addressed by a ramp voltage. Proceedings of the 15 th International PATMOS Workshop , Leuven (September 2005), pp. 207 - 216. [39] Zeitzoff, P. and Chung, J. A perspective from the 2003 ITRS. I.E.E.E. Circuits & Systems Magazine 21, 4 - 15 (2005). [40] Patra, P. and Fussell, D. On efficient adiabatic design of MOS circuits. Proceedings of the 4 th Workshop on Physics and Computation , Boston (November 1996), pp. 260 269. [41] Alioto, M. and Palumbo, G. Analysis and comparison on full adder block in submicron technology. I.E.E.E. Transactions on Very Large Scale Integration Systems 10, 806 823 (2002). [42] Van Rentergem, Y. and De Vos, A. Reversible full adders applying Fredkin gates. Proceedings of the 12 th International Conference on Mixed Design of Integrated Circuits and Systems, Krak´ow (June 2005), pp. 179 - 184. [43] Belleville, M. and Faynot, O. Low-power SOI design. Proceedings of the 11 th International PATMOS Workshop , Yverdon (September 2001), pp. 8.1.1 - 8.1.10. [44] Nagaya, M. Fully-depleted type SOI device enabling an ultra low-power solar radio wristwatch. O.K.I. Technical Review 70, 48 - 51 (2003). [45] Yong, A. What is ... a Young tableau? Notices of the A.M.S. 54, 240 - 241 (2007). [46] Clos, C. A study of non-blocking switching networks. Bell Systems Technical Journal 32, 406 - 424 (1953). [47] Hwang, F. Control algorithms for rearrangeable Clos networks. I.E.E.E. Transactions on Communications 31, 952 - 954 (1983).

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[48] Hui, J. Switching and traffic theory for integrated broadband networks . Boston: Kluwer Academic Publishers (1990), pp. 53 - 138. [49] Chao, J., Jing, Z. and Liew, S. Matching algorithms for three-stage bufferless Clos network switches. I.E.E.E. Communications Magazine 41, 46 - 54 (2003). [50] Jajszczyk, A. Nonblocking, repackable, and rearrangeable Clos networks: fifty years of the theory evolution. I.E.E.E. Communications Magazine 41, 28 - 33 (2003). [51] Bhatia, R. Matrix analysis. New York: Springer (1997), pp. 28 - 56. [52] Borgersen, R. Equivalence of seven major theorems in combinatorics. http:// home.cc.umanitoba.ca/˜umborger/Presentations/GS-05R-1.pdf (2004). [53] de Werra, D. Path coloring in bipartite graphs. European Journal of Operational Research 164, 575 - 584 (2005). [54] Comtet, L. Advanced combinatorics. Dordrecht: Reidel Publishing Company (1974), pp. 124 - 125. [55] Even, S., Kohavi, I. and Paz, A. On minimal modulo 2 sums of products for switching functions. I.E.E.E. Transactions on Electronic Computers 16, 671 - 674 (1967). [56] Courant, R. Differential and integral calculus . London: Blackie & Son Limited (1970), p. 361.

In: Mathematics and Mathematical Logic: New Research ISBN 978-1-60692-862-2 Editors: Peter Milosav and Irene Ercegovaca © 2010 Nova Science Publishers, Inc.

Chapter 9

IMAGINARY CUBIC OSCILLATOR AND ITS SQUARE-WELL APPROXIMATIONS IN X –AND P– REPRESENTATION Miloslav Znojil∗ Theory Group, Nuclear Physics Institute ASCR,, Czech Republic

ABSTRACT Schrödinger equation with imaginary PT symmetric potential V (x) = i x3 is studied using the numerical discretization methods in both the coordinate and momentum representations. In the former case our results con¯rm that the model generates an infinite number of bound states with real energies. In the latter case the differential equation is of the third order and a square-well, solvable approximation of kinetic energy is recommended and discussed. One finds that in the strong-coupling limit, the exact PT symmetric solutions converge to their Hermitian predecessors.

PACS 03.65.Ge

ACKNOWLEDGEMENTS Work supported by the GA·CR grant Nr. 202/07/1307, by the M·SMT \Doppler Institute" project Nr. LC06002 and by the Institutional Research Plan AV0Z10480505.

∗

e-mail: [email protected]

244

1

Miloslav Znojil

Introduction

An interest in imaginary cubic anharmonic oscillators dates back to their perturbation analysis by Caliceti et al [1]. The simplified homework example with the mere two-term non-Hermitian Hamiltonian HBZ = p2 + i x3 has been proposed by D. Bessis and J. Zinn-Justin who had in mind its possible applicability in the context of statistical physics [2]. The example has been revitalized by C. Bender et al due to its possible methodical relevance in relativistic quantum field theory [3]. They emphasized that the apparent reality of the spectrum of energies EBZ is quite puzzling. The conjecture of a full compatibility of similar Hamiltonians with the postulates of quantum mechanics [4] opened many new and interesting questions. The current Hermiticity of Hamiltonians was replaced by a weaker condition of their commutativity with a product PT of the spatial parity P and of the complex conjugation T . The latter factor is to be understood as a one-dimensional version of the operator of time reversal. The discussion between C. Bender and A. Mezincescu [5] pointed out that one of the key problems of the new studies lies in the ambiguity of the spectrum which depends quite crucially on our choice of the boundary conditions which can be, in general, complexified [6]. The fragile character of the reality of the energies has been confirmed by the WKB and perturbative studies [7, 8] and by the quasi-exact and exact models [9] where the admissible unavoided level crossings [10] prove sometimes followed by the spontaneous breakdown of PT symmetry [11]. In such a context we propose here an extremely elementary approach to similar models replacing interactions which admit just a numerical treatment (typically, V (x) = ix3 ) by their exactly solvable square-well analogues.

2

Models in coordinate representation

In a search for analogies between the solvable and unsolvable models in one dimension, all of the possible forms of a confining well are often being approximated by the ordinary real and symmetric square well (

V

(SQW )

(x) =

S

S 2 , x ∈ (−∞, −π) (π, ∞), 0, x ∈ (−π, π).

(1)

In this spirit one can also replace the antisymmetric and imaginary homework potential VBZ (x) = i x3 by its elementary square-well analogue  2   −i T , x ∈ (−∞, −π),

x ∈ (−π, π), V (ISQW ) (x) =  0,  +i T 2 , x ∈ (π, ∞).

2

Imaginary Cubic Oscillator and Its...

245

Schr¨odinger equation which appears in such a setting, "

#

h ¯ d − + V (ISQW ) (x) ψ(x) = Eψ(x) 2m dx2

(2)

will be complemented by the standard L2 (lR) boundary conditions ψ(±∞) = 0.

(3)

The well known PT symmetric normalization convention will be employed, with a free real parameter G in the unbroken PT -symmetry requirements [12] ψ(0) = 1,

∂x ψ(0) = i G.

(4)

Putting h ¯ = 2m = 1 and using the ansatz (

ψ(x) =

cos k x + B sin k x, x ∈ (0, π), k 2 = E, (L + i N ) exp(−σ x), x ∈ (π, ∞), σ 2 = i T 2 − k 2 ,

(5)

we shall guarantee the full compatibility of such a convention with the symmetry requirements (4) by the choice of the purely imaginary constant B = i G/k in wave functions (5).

3

Matching conditions at x = π

Let us split σ = p + i q in its real and imaginary part with a fixed sign, p, q ≥ 0. This gives p2 + k 2 = q 2 and 2pq = T 2 . These rules are easily re-parameterized in terms of a single variable α, p = q cos α,

k = q sin α,

q=√

T , 2 cos α

α ∈ (0, π/2).

(6)

The standard matching at the point of discontinuity is immediate, cos kπ + B sin kπ = (L + i N ) exp(−σ π), σ − sin kπ + B cos kπ = − (L + i N ) exp(−σ π). k After we abbreviate σ/k = − tan Ωπ, we get an elementary complex condition of matching of logarithmic derivatives at x = π, G = −i k tan(k + Ω)π.

(7)

The real part defines our first unknown parameter, G = G(α). Due to our normalization conventions, the imaginary part of the right-hand-side expression must vanish, Re[tan(k +Ω)π] = 0. An elementary re-arrangement of such an equation acquires the form of an elementary quadratic algebraic equation for X = tan kπ. Its two explicit solutions read p−q p+q X2 = X1 = , (8) k k 3

246

Miloslav Znojil

or, after all the insertions, "

#

#

"

π − α(+) πT sin α(+) , tan √ = tan 2 2 cos α(+) "

#

(9)

#

"

πT sin α(−) α(−) √ tan . (10) = tan − 2 2 cos α(−) In implicit manner these equations specify the two respective infinite series of appropriately bounded real roots α = αn(±) ∈ (0, π/2).

4

Energies

For α ∈ (0, π/2) the left-hand-side arguments [. . .] in eqs. (9) and (10) run from zero to infinity. Their tangens functions oscillate infinitely many times from minus infinity to plus infinity. Within the same interval, the limited variation of the argument α makes both the eligible right-hand side functions monotonic, very smooth and bounded, tan[(π − α(+) )/2] ∈ (1, ∞) and tan[α(−) /2] ∈ (0, 1). A priori this indicates that our roots k = k(αn(±) ) will all lie within well determined intervals, ¶

µ

kn(+)

1 1 , ∈ n + ,n + 4 2 µ

n = 0, 1, . . . ,

¶

3 ∈ m + ,m + 1 m = 0, 1, . . . . 4 After such an approximate localization of the roots, an unexpected additional merit of our parametrization (6) manifests itself in an unambiguous removal of the tangens operators from both eqs. (9) and (10). This gives the following two relations, (−) km

kn(+) = n +

1 ωn(+) − , 2 4

(−) km =m+1−

(−) ωm , 4

ωn(±) =

2αn(±) ∈ (0, 1). π

After an elementary change of notation with ωn(+) = ω2n and ωn(−) = ω2n+1 , we may finally combine the latter two rules in the single secular equation µ

π sin ωN 2

¶

2N + 2 − ωN = · 4T

s

µ

2 cos

π ωN 2

¶

N = 0, 1, . . . ,

(11)

In a graphical interpretation this equation represents an intersection of a tangens-like curve with the infinite family of parallel lines. This is illustrated in Figure 1. The equation generates, therefore, an infinite number of the real roots ωN ∈ (0, 1) at all the non-negative integers N = 0, 1, . . ..

4

Imaginary Cubic Oscillator and Its...

247

8

2k 6

4

2

0

0.1

0.2

0.3

0.4

0.5

y

Figure 1: Graphical solution of eq. (11) (y = ωN /2, T = 1).

5

Wave functions in the weak coupling regime

Equation (7) in combination with eqs. (9) and (10) determines the real parameter G = G(±) = −

k2 q±p

(12)

responsible for the behavior of the wave functions near the origin [remember that B = iG/k in eq. (5)]. For its analysis let us introduce an auxiliary linear function of ω and N , q

2N + 2 − ωN R(ωN , N ) = ∈ 4T

Ã

N + 1/2 N + 1 , 2T 2T

!

.

Our secular eq. (11) can be then read as an algebraic quadratic equation with the unique positive solution, µ

cos

π ωN 2

¶

=

1

R(ωN , N ) +

q

R2 (ωN , N ) + 1

.

(13)

This is an amended implicit definition of sequence ωN . As long as the right hand side expression is very smooth and never exceeds one, the latter formula re-verifies that the root ωN is always real and bounded as required. In the domain of the large and almost constant R À 1 (i.e., for small square-well heights T or at the higher excitations), our new secular equation (13) gives a better picture of our bound-state parameters ωN = 1 − ηN which all lie very close to one. The estimate 1 1 π 5 √ ≈ ηN = arcsin − + ... 2 2 2R 48 R3 R+ R +1 5

248

Miloslav Znojil

represents a quickly convergent iterative algorithm for the efficient numerical evaluation of the roots ωN . One can conclude that in a way compatible with our a priori expectations, the value of p = pN = Reσ ≈ q/2R lies very close to zero. As a consequence, the asymptotic decrease of our wave functions remains slow. We have q = qN = Imσ ≈ k so that, asymptotically, our wave functions very much resemble free waves exp(−ikx). In the light of eq. (12) we have ψ(x) ≈ exp(−ikx) near the origin.

6

Wave functions in the strong coupling regime

For the models with a very small R (i.e., for the low-lying excitations in a deep well with T À 1) we get an alternative estimate π ωN = arcsin 4

s

³√ ´i 1h 1 1 π R− 1 + R2 − 1 ≈ R − R2 + . . . ¿ . 2 2 4 4

In the limit R → 0 the present spectrum of energies moves towards (and precisely coincides with) the well known levels of the infinitely deep Hermitian square well of the same width I = (−π, π) (cf. eq. (1) with S → ∞). The complex-rotation transition from the Hermitian well V (SQW ) (x) of eq. (1) (with S À 1) to its present non-Hermitian PT symmetric alternative V (ISQW ) (x) of eq. (2) (with T À 1) proves amazingly smooth. Wave functions exhibit a similar tendency. In outer region, they are proportional to exp(−px) and decay very quickly since p = O(R−1/2 ). Parameter G(±) becomes strongly superscript-dependent, G(+) = −

k2 = O(R3/2 ), q+p

G(−) = −(q + p) = O(R−1/2 ).

In the interior domain of x ∈ (−π, π) the wave functions with superscripts (+) and (−) become dominated by their spatially even and odd components cos kx and sin kx, respectively. The superscript mimics (or at least keeps the trace of) the quantum number of the slightly broken spatial parity P. We can summarize that our present PT symmetric model is quite robust. Independently of the coupling T the spectrum is unbounded from above and remains constrained by inequalities (N + 1)2 (N + 1/2)2 ≤ EN ≤ . (14) 4 4 The analogy between our exactly solvable square-well model and the standard or “paradigmatic” PT symmetric Hamiltonian HBZ appears closer than expected.

7

Transition to the momentum representation

Let us turn our attention to one-dimensional harmonic oscillator H (HO) = p2 + x2 which is exactly solvable and which appears in virtually any textbook on quantum

6

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249

mechanics. In PT symmetric quantum mechanics a similar guiding role can be and has been attributed to the non-Hermitian cubic Hamiltonian H (CO) = p2 + i x3 of Bender et al [4]. We have seen that a straightforward numerical and semi-classical analysis of the related Schr¨odinger equation H (CO) |ψn i = En(CO) |ψn i,

n = 0, 1, . . .

(15)

supports a highly plausible conjecture that the spectrum of energies is real, discrete and bounded below. The conjectured absence of its imaginary components is indicated by the Hilbert-Schmidt analysis [5] and by the perturbation calculations in both the weak-coupling regime [13] or in its strong-coupling, purely numerically generated re-arrangement [12]. Certainly, the problem deserves a change of the traditional perspective. Let us, therefore, move now to its momentum representation. This would give the momentum as a mere number, p ∈ (−∞, ∞) while the coordinate x becomes represented by the differential operator xˆ = i ∂p . Equation (15) then acquires a new form containing the purely real differential Schr¨odinger operator H (CO) − E of third order, "

#

d3 + p2 ψ(p) = E ψ(p) , dp3

ψ(p) ∈ L2 (IR).

This gives an unusual formulation of our bound-state problem where the quadratic p−dependence of the kinetic term T (p) = p2 does not seem to make the equation any easier to solve. For this reason we shall drastically simplify the kinetic term and deduce some consequences.

8

Piecewise constant approximate kinetic energy

In a way proposed by Pr¨ ufer [14] many wave functions can be visualized as certain deformations of solutions which correspond to a locally constant potential, ψ(x) ≈ c1 sin[%(x)] + c2 cos[%(x)]. In the standard quantum mechanics such a trick found immediate applications in numerical computations [15] while it still admits an easy interpretation via some traditional Sturm Liouvillean oscillation theorems [16, 17]. Using this idea as a methodical guide let us now replace the kinetic energy operator T (p) = p2 by the most elementary square well of a finite depth Z > 0, T (p) =

   Z,  

p ∈ (−∞, −1), p ∈ (−1, 1), p ∈ (1, ∞).

0, Z,

In the bounded range of energies E < Z this splits our toy model in the two separate differential equations, "

#

d3 − 8 α3 ψ(p) = 0, 3 dp

7

p ∈ (−1, 1),

250

Miloslav Znojil "

#

d3 + 8 β 3 ψ(p) = 0, dp3

p ∈ (−∞, −1)

[

(1, ∞).

The two auxiliary parameters α = α(E) > 0 and β = β(E) > 0 are defined in such a way that Z = E + 8 β 3 > E = 8 α3 . They appear in the three independent (exponential) solutions of our equation. Their general superpositions are complex but they may be given the real, trigonometric form. Near the origin we have √ ψ0 (p) = d e2α p + f e−α p cos(˜ α p + θ), p ∈ (−1, 1), α ˜ = 3α where the symbols d, f and θ stand for the three undetermined constant parameters. In the right and left asymptotic regions we obtain the similar formulae. After we omit their exponentially growing and normalization-violating unphysical components we get the one-parametric family ψ+ (p) = g e−2β p ,

p ∈ (1, ∞).

The two-parametric left-barrier counterpart of this formula reads ψ− (p) = c eβ p cos(β˜ p + η),

p ∈ (−∞, −1),

β˜ =

√

3β .

At the right discontinuity p = 1 we have to guarantee the continuity of ψ(p), ∂p ψ(p) and ∂p2 ψ(p). This is equivalent to the three matching conditions d e2α + f e−α cos(˜ α√+ θ) = g e−2β , α + θ)] = −2β g e−2β , 2αd e2α − α f e−α [cos(˜ α + θ) + 3√sin(˜ 4α2 d e2α − 2α2 f e−α [cos(˜ α + θ) − 3 sin(˜ α + θ)] = 4β 2 g e−2β .

(16)

The weighted sum of these equations re-scales and interrelates the unknown coefficients d and g in terms of a new energy parameter t = t(E) = β(E)/α(E) > 0 or rather R = R(E) = (1 − t + t2 )−1/2 > 0, d e2α ≡ D(E) =

G(E) , 3 R2 (E)

G(E) ≡ g e−2β .

After we eliminate G from the last two equations (16) which are linear in d and f we obtain the elementary formula which defines the shift θ = θ(E), √ 3 . tan(˜ α + θ) = 2/t − 1 The trigonometric factors become fixed up to their common sign ε = ±1, √ 3 ε t(E) R(E). cos(˜ α + θ) = ε [1 − t(E)/2] R(E), sin(˜ α + θ) = 2 The same sign enters our last decoupled definition f e−α ≡ F (E) =

2[t(E) + 1] G(E) . 3 ε R(E) 8

Imaginary Cubic Oscillator and Its...

251

Up to an overall normalization (say, g = 1) in our wave functions matched at p = 1, all the free parameters become specified as functions of the energy. At the second, p = −1 discontinuity we have to satisfy three matching conditions as well. In terms of abbreviations L1 = d e−2α + f eα cos(−˜ α+ √ θ), −2α α 3 sin(−˜ L2 = 2αd e − α f e [cos(−˜ α + θ) + √ α + θ)], L3 = 4α2 d e−2α − 2α2 f eα [cos(−˜ α + θ) − 3 sin(−˜ α + θ)] they read L1 (α, d, f, θ) = c e−β cos(−β˜√+ η), L2 (α, d, f, θ) = β c e−β [cos(−β˜ + η) − √ 3 sin(−β˜ + η), L3 (α, d, f, θ) = −2β 2 c e−β [cos(−β˜ + η) + 3 sin(−β˜ + η). They determine the values of c = c(E) and η = η(E). Their properly weighted sum gives √

1 − t + t2 3 F (E) sin(−˜ α + θ) + (2t − 1) F (E) cos(−˜ α + θ) + 2 D(E) e−6α = 0 . t+1

After its slight re-arrangement we arrive at the amazingly transparent relation 2

√

(1 − 4 t + t ) cos(2 3α) +

√

Ã

√

2

3 (1 − t ) sin(2 3α) =

1 − t + t2 −3α e 1+t

It should be read as an implicit definition of physical energies E.

10

y 5

0

0.01

0.02

0.03

0.04

0.05

-5

-10

Figure 2: Graphical solution of equation (17) at Z = 1/1000.

9

!2

.

(17)

252

9

Miloslav Znojil

The Z−dependence of the spectrum

It is quite instructive to search for the physical energies numerically. Starting from the very-shallow-well extreme in eq. (17) we find the two clearly distinguished energy roots. The qualitative features of the graph of secular determinant remain unchanged in a broad interval of the inverse strengths 1/Z. Its shape is sampled in Figure 2 at Z = 10−3 . We have tested that even the approximate height ≈ −5 of its left plateau stays virtually unchanged between Z = 10−5 and Z = 10−3 . Within the same interval of the shallowest wells the left zero grows from the value 0.00280 till 0.0241. Beyond the broad, downwards-oriented peak one finds the second, right zero moving from the value 0.01047 (found very close to the instantaneous threshold 0.01077) up the value 0.1056 (not very far from its threshold 0.1077, either) within the same interval of 1/Z.

0.0004

y 0.0002

0 0.6235

0.624

0.6245

0.625

0.6255

-0.0002

-0.0004

Figure 3: Local maximum giving the new doublet of roots at Z = 5.3005. A new feature emerges around Z = 10−1 (with the left zero at 0.0445 and with the right zero 0.222 still quite close to the threshold 0.232) and Z = 1 (with the left zero at 0.072 and with the right zero 0.446, not that close already to the threshold 1/2) in the left half of the picture. The plateau develops a local, safely negative maximum. In the subsequent domain of Z > 1 we have to switch our attention back to the right half of our graph. Immediately before the coupling reaches the integer value of Z = 5, the end of the curve returns to the negative half-plane near the maximal (i.e., threshold) energy. This means that there emerges the third energy level there. The total number of bound states grows to N = 3 (cf. the leftmost items in our Table 1).

10

Imaginary Cubic Oscillator and Its...

253

Table 1. Number of levels N and its changes ∆ with growing Z. N ∆

2

3 1

5

3

2

−2

4 1

5 1

7

8

2

1

6 −2

7 1

9 2

10 1

Beyond Z = 5, our attention has to return quickly to the left half of the picture where the very slow growth of the local maximum creates a new quality at last. The top of the local bump touches and crosses the horizontal axis at Z ≈ 5.3003 and E ≈ 0.6244. At Z = 5.3005 a new doublet of energies is formed in a way illustrated in Figure 3. The number of levels jumps to N = 5.

1

1.2

1.3

1.4

1.5

1.6

0

-1

-2 y -3

-4

-5

Figure 4: The local maximum not giving the doublet of roots at Z = 35. A smooth deformation of the graph takes place when the value of Z grows on. During this evolution we discover that our (originally broad), downward-oriented peak shrinks quite quickly and moves comparatively slowly to the right. It gets close to the rightmost and, to its bad luck, slightly more slowly moving zero number five. The magnified picture of the resulting “collision” is displayed here in Figure 4. At Z = 35 it shows that in the threshold region of the energies, • the wavy motion of the threshold end of our graph still did not manage to reach the zero axis; • the downwards-oriented peak has already left the positive part (and moved to the negative part) of the curve in question. As a consequence, the number of levels drops, quite unexpectedly, down to 3 again (cf. Table 1).

11

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Miloslav Znojil

10

y 5

0

1

0.5

1.5

2

2.5

-5

-10

Figure 5: Typical Z À 1 graph of eq. (17) (Z = 190). In the vicinity of Z = 40 the new, rightmost energy root emerges at last. Up to Z = 100 and beyond, the number of levels stays equal to 4. Then it increases to 5, due to the emergence of the next threshold zero. Only after that, the slowly moving downward peak reaches the domain of the fourth zero. At almost exactly Z = 190 its left (and temporarily negative) local maximum reaches the zero value again (cf. Figure 5). At this moment the number of states jumps up by two to seven. The magnified graphical proof is offered by Figure 5 at Z = 200.

1

y0.5

0

2.26

2.28

2.3

2.32

2.34

-0.5

-1

Figure 6: Quasi-degeneracy of the doublet of roots at Z = 200. The latter Figure illustrates nicely the rapid shrinking of our peak with Z. The numerical detection of its position becomes more and more difficult. Although this position plays a crucial role in the practical determination of the number of levels at

12

Imaginary Cubic Oscillator and Its...

255

a given Z, we must be very careful in distinguishing the subgraph of Figure 6 (with three zeros) from a simple straight line with the single zero. The pattern is deceitful and the standard software which searches for roots has to be used with due care. Vice versa, the above analysis enables us to take into account all the specific features of the Z−dependence of the graph in eq. (17). We get a regular pattern summarized in Table 1 and exhibiting a certain regularity of the Z−dependence of the number of levels N = N (Z).

10

Energies in the square-well approximation

The main consequence of the presence of the above-mentioned narrow peak is an unusual irregularity observed in the emergence of the new levels in the deeper wells. We may conclude that this irregularity is not an artifact of the computation method. The energy formula (17) for our square-well toy model is exact and the seemingly unpredictable emergence of its roots just reflects the fact that our Schr¨odinger equation is of the third order. In particular, there exists no symmetry/antisymmetry with respect to the parity p → −p etc.

10

y 5

0

1

2

3

4

5

-5

-10

Figure 7: Numerical invisibility of the narrow peak and of the new threshold root at Z = 1200. Methodical consequences of our analysis are a bit discouraging. Firstly, the very symbolic-manipulation derivation of our present formulae proved unexpectedly complicated even in comparison with the multiple standard square wells in textbooks. That’s why we did not move to any further piece-wise constant approximations of T (p). Secondly, even our use of the most elementary solvable example revealed quite clearly a very real danger of the possible loss of certain levels. For an illustration let us imagine that our numerical study would have been started in the deep-well domain, i.e., at the large Z. It is quite easy to generate the graphs of eq. (17) there.

13

256

Miloslav Znojil

In the standard and routine finite-precision computer arithmetics one discovers that the results are very smooth and look virtually the same, say, in the interval of Z ∈ (1000, 1200). Let us pick up, for definiteness, the larger sample Z = 1200. We get a picture (cf. our last Figure 7) which is regular and, deceptively, indicates that N (1200) = 7. Unfortunately, the correct answer (appearing at the right end of our Table 1) is N (1200) = 10. Its derivation requires the use of a significantly enhanced precision. Otherwise, whenever we use just the standard 14 digits and Figure 7, we would have missed as much as three (i.e., cca 30 % of all) energy levels. In the light of our preceding considerations, an easy explanation of the latter numerical paradox lies in the presence of the narrow peak. A priori, it is hardly predictable of course. It is necessary to spot it by brute force. One finds that at Z = 1000, this anomalous peak still lives safely below the sixth energy level. The related number of levels is reliably confirmed as equal to seven, indeed. In between Z = 1000 and Z = 1200, it is necessary to work in an enhanced precision arithmetics. One finds that the upper, threshold end of the curve crosses the horizontal axis only slightly above Z = 1100. Due to the very steep slope of the curve in this region, this crossing is not visible even at z = 1200 in Figure 7. One has to trace the narrow peak carefully. It overtakes the sixth energy level at Z ≈ 1190 (and E = 4.217), in an arrangement resembling our Figure 5 above. Thus, one concludes, finally, that the new, almost degenerate pair of the energy levels emerges immediately beyond this point.

11

Wave functions and their zeros

A marginal merit of our use of the square-well-shaped T (p) lies in the availability of the explicit wave functions. For the lack of space we have to omit illustrative pictures, mentioning just a few of their most characteristic features. In the first step we notice that in the rightmost interval of p the absence of any nodal zero in the wave function is in fact very similar to the usual Sturm Liouville behavior. Less expectedly, at the exact energy value one encounters an infinity of the nodal zeros in the leftmost subinterval of p ∈ (−∞, −1). In this domain we are fortunate in studying the exactly solvable case. The very presence of this infinite “left” set of nodal zeros is extremely sensitive to the numerical level of precision we use. Indeed, the errors are proportional to the unphysical ψ (unphys.) (p) ∼ exp(−2β p) which is growing rapidly at p ¿ −1. After the smallest deviation of the energy E from its absolutely precise boundstate value even the non-numerical and absolutely precise wave functions will be dominated by the growing asymptotics ψ (unphys.) (p) ∼ exp(−2β p) near the left infinity. The change of sign of the asymptotics is a reliable source of information about the fact that the energy crossed it physical value. This observation survives in the shooting numerical algorithms [18] as well as in the rigorously proved versions of the method of Hill determinants [19]). In this context our present numerical experiments could be perceived as opening a number of new questions. Some of them emerge in

14

Imaginary Cubic Oscillator and Its...

257

ufer-type althe purely numerical context of an appropriate generalization of the Pr¨ gorithms. Especially in the vicinity of the correct physical energies they could lead to reliable and robust right-to-left shooting numerical recipes.

12

Outlook

In the x−representation of our problem our main emphasis has been put on the exact solvability of its replacement by the purely imaginary square well model. New light has been thrown on some properties of wave functions. One can expect that the further detailed study of the PT symmetric square wells will give new answers to the puzzles concerning the irregular behavior of the nodal zeros in the complex plane as formulated in ref. [20]. Our present study indicates that some complexified versions of the Sturm Liouville oscillation theorems should be developed for the study of zeros of the separate real and imaginary parts of PT symmetric wave functions. After the standard Fourier-transformation transition to the p−representation of our imaginary cubic oscillator the underlying eigenvalue problem can be seen from a different perspective. Its Hamiltonian is being replaced by a real differential expression. On a suitable Hilbert space this specifies the Hamiltonian operator with the numerical range (and, hence, spectrum) which is, obviously, real. This complements the extensive discussion of this topic in [5]. Among several immediate constructive consequences of the latter observation we underlined the consistency of the approximations imposed directly upon the kinetic term T (p). As long as the behavior of wave-function asymptotics at large |p| À 1 differs in the left and right infinity, several new qualitative aspects of the problem emerge and became clarified by our schematic piece-wise constant approximation of T (p). At small p ≈ 0 the emergence and motion of the nodal zeros can be interpreted in a graphical manner explaining some features of the N (Z) dependence. In particular, the puzzling loss of its monotonicity seems confirmed by our solvable model. The use of the momentum representation proved able to throw a new light on the counterintuitive bound states in PT symmetric quantum mechanics. The emergence/disappearance of our quasi-degenerate doublets should be emphasized as, perhaps, analogous to the unavoided level crossings in harmonic oscillators [10] and/or to the anomalous doubling of levels in the models of Natanzon type [21]. Similar irregularities in the spectra could be, perhaps, attributed to a peculiar combination of the analyticity and non-Hermiticity in PT symmetric systems. In a brief summary of our numerical observations let us point out the regularity of the Z−dependence of the number N (Z) of the bound states. This indicates that one should search for an improved application of Sturm-Liouville theory in complex domain [22]. The possibility of deduction of new oscillation-type theorems exists, first of all, in the middle interval of p ∈ (−1, 1) where, for a continuously growing energy parameter E, a steady right-ward movement of the nodal zeros competes with the exponential terms which are varying slowly.

15

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Miloslav Znojil

References [1] Caliceti E, Graffi S and Maioli M 1980 Commun. Math. Phys. 75 51 [2] D. Bessis 1992 private communication [3] Bender C M and Milton K A 1997 Phys. Rev. D 55 R3255 and 1998 Phys. Rev. D 57 3595 and 1999 J. Phys. A: Math. Gen. 32 L87 [4] Bender C M and Boettcher S 1998 Phys. Rev. Lett. 80 5243; Bender C M, Boettcher S and Meisinger P N 1999 J. Math. Phys. 40 2201; Znojil M and Tater M 2001 J. Phys. A: Math. Gen. 34 1793 [5] Mezincescu G A 2000 J. Phys. A: Math. Gen. 33 4911; Bender C M and Wang Q 2001 J. Phys. A: Math. Gen. 34 3325; Mezincescu G A 2001 J. Phys. A: Math. Gen. 34 3329 [6] Bender C M and Turbiner A 1993 Phys. Lett. A 173 442; Buslaev V and Grecchi V 1993 J. Phys. A: Math. Gen. 26 5541; Fernandez F, Guardiola R, Ros J and Znojil M 1999 J. Phys. A: Math. Gen. 32 3105 [7] Alvarez G 1995 J. Phys. A: Math. Gen. 27 4589 [8] Delabaere F and Pham F 1998 Phys. Lett. A 250 25; Delabaere F and Trinh D T 2000 J. Phys. A: Math. Gen. 33 8771 [9] Bender C M and Boettcher S 1998 J. Phys. A: Math. Gen. 31 L273; Cannata F, Junker G and Trost J 1998 Phys. Lett. A 246 219; Bagchi B, Cannata F and Quesne C 2000 Phys. Lett. A 269 79 Znojil M 2000 J. Phys. A: Math. Gen. 33 L61 and 4203 and 4561 and 6825; L´evai G and Znojil M 2000 J. Phys. A: Math. Gen. 33 7165 [10] Znojil M 1999 Phys. Lett. A. 259 220 and 264 108 [11] Khare A and Mandel B P 2000 Phys. Lett. A 272 53; Bender C M, Berry M, Meisinger P N, Savage V M and Simsek M 2001 J. Phys. A: Math. Gen. 34 L31 [12] Fernandez F, Guardiola R, Ros J and Znojil M 1998 J. Phys. A: Math. Gen. 31 10105 [13] Bender C M and Weniger E J 2001 J. Math. Phys. 42 2167 [14] Pr¨ ufer H 1926 Math. Ann. 95 499

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´ I, Havl´ıˇcek M. and Hoˇrejˇs´ı J 1981 Phys. Lett. A 82 64 [15] Ulehla [16] Fl¨ ugge S 1971 Practical quantum mechanics I (Berlin: Springer), p. 153 [17] Ince E L 1956 Ordinary differential equations (New York: Dover), p. 223 [18] Killingbeck J P, Gordon N A and Witwit M R M 1995 Phys. Lett. A 206 279; Znojil M 1997 Phys. Lett. A 230 283 [19] Hautot A 1986 Phys. Rev. D 33 437; Znojil M 1992 J. Math. Phys. 33 213 [20] Bender C M, Boettcher S and Savage V M 2000 J. Math. Phys. 41 6381 [21] Znojil M, L´evai G, Roy P and Roychoudhury R 2001 Phys. Lett. A 290 249 [22] Hille E 1969 Lectures on Ordinary Differential Equations (Reading: AddisonWesley)

17

INDEX A academic, 232 accounting, 197 accuracy, 13 achievement, 181, 192, 193, 194, 200 ACM, 198, 199, 200 adaptation, 13, 17 adiabatic, 231, 240 adult, 11, 13, 17 adulthood, 13 adults, 12, 13, 53 Ag, 97, 98, 100, 102, 105 agent, 162 aggregation, 68, 70, 71, 72, 76, 82, 85, 194 agricultural, 161, 162 aid, 11, 18, 52, 169 air, 256 algorithm, 168, 191, 199, 200, 217, 218, 220, 234, 248 alienation, 15 alternative, 6, 36, 69, 74, 75, 78, 79, 82, 83, 183, 248 alternatives, viii, 67, 68, 69, 74, 75, 76, 77, 82, 83 ambiguity, 21, 131, 244 amplitude, 23, 138, 143, 144 AMS, 52, 55, 238 angular momentum, 93, 94, 110 animal cognition, 11 animals, 11, 15, 17, 18, 34 anomalous, 256, 257 anthropic principle, 14 antithesis, 23, 40 appendix, 236 application, 10, 23, 28, 38, 58, 82, 84, 89, 96, 115, 136, 138, 139, 148, 164, 173, 185, 205, 210, 216, 218, 224, 228, 237, 257

archetype, 3, 5, 24, 25, 26, 27, 31, 34, 35, 37, 38, 39, 41, 42, 46, 47, 48, 49, 50, 52 argument, 9, 10, 33, 38, 176, 246 Aristotelian, 53 Aristotle, 3, 4 arithmetic, 11, 12, 17, 20, 26, 97 Asia, 201 aspiration, 192 assumptions, 5, 6, 21, 28, 29, 69, 195 asymptotic, 248, 250 asymptotically, 248 asymptotics, 256, 257 Athens, 1, 4 Atiyah, 49, 52 ATM, 199 atom, 93 atoms, 18, 30, 38, 47, 88, 95, 110, 167 Austria, 233 autonomy, 27 availability, viii, 67, 180, 256

B Balanced Scorecard, viii, 67, 68, 77, 83, 84 bandwidth, ix, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 195, 196, 198, 199 bandwidth allocation, 180, 185, 187, 190, 191, 192, 193, 195, 196, 199 banks, 224 barrier, 250 base pair, 167, 168 behavior, 24, 26, 184, 247, 256, 257 Belgium, 203, 233 bell, 152 benzene, 94, 95 binomial distribution, 97 bipolar, 25 birth, 11, 40

262

Index

blocks, ix, 37, 161, 162, 163, 203, 213 Bohr, 240 Boltzmann constant, 203 bonds, 167 borderline, 26 Borel, 49, 52 Bose, 155, 177 boson, 90, 91 bosons, ix, 87, 89, 90, 100 Boston, 54, 84, 85, 177, 240, 241 bottleneck, 188 bottlenecks, 181 boundary conditions, 244, 245 bounds, 185 brain, 11, 12, 13, 15, 16, 17, 18, 46 Brazil, 67, 83 breakdown, 244 broadband, 179, 185, 200, 241 Broadband, 185 Bromide, 162 browsing, 182 building blocks, 34, 37, 221, 225, 232, 236, 237

C calculus, 33, 68, 83, 197, 241 capacitance, 229, 231 carrier, 46 category a, 41, 43, 44, 47 cation, 181, 187, 190, 195 cerebral hemisphere, 16 chaos, 27, 34, 40, 41 chemical structures, 167 children, 12, 13, 16, 17, 53 chiral, 97 chirality, 97 classes, ix, 6, 89, 94, 95, 107, 115, 130, 137, 146, 147, 151, 153, 154, 159, 165, 180, 182, 186, 187, 188, 190, 195, 212 classical, vii, ix, 7, 10, 12, 22, 23, 39, 45, 51, 58, 88, 179, 182, 183, 190, 191, 197, 203, 204, 205, 249 classification, viii, 12, 87, 89, 129 closure, 121, 122, 126 codes, 33 coding, 33 cognition, 11, 13 cognitive development, 11, 13 cognitive domains, 12 cognitive science, 11 coil, 217, 218, 219 coke, 61, 64 coke formation, 61, 64 collective unconscious, 3

colors, 97, 137, 141, 142, 144, 145, 176 combinatorics, viii, 87, 89, 97, 178, 234, 241 commodity, 183, 186 communication, ix, 40, 179, 180, 181, 183, 185, 186, 189, 199, 258 communities, 183 community, 2, 181, 182, 197 commutativity, 146 comparison task, 12 compatibility, 244, 245 competence, 11 complementarity, 52 complex numbers, 5, 7, 8, 48, 146, 173 complexity, 5, 170 compliance, 93 components, 181, 183, 193, 248, 249, 250 composites, 42 composition, 10, 31, 41, 42, 43, 45, 46, 47, 50, 64, 130, 134 computation, ix, 2, 180, 185, 203, 204, 238, 255 computing, ix, 178, 203, 204, 207, 220, 238, 239 conception, 4, 10, 15, 40 concordance, 39, 71, 72 confession, 30 configuration, 89, 92, 94, 95 conflict, 237 confusion, 14, 58, 131, 149 Congress, 84, 239 congruence, 148, 150 conjecture, 10, 25, 36, 154, 155, 177, 244, 249 conjugation, 244 connectivity, 181 consciousness, 4, 14, 23, 27, 28, 46, 49, 51, 52 consensus, viii, 67, 68, 69, 71, 72, 73, 74, 77, 78, 79, 83, 84, 85 conservation, 189 constraints, 57, 58, 62, 65, 189, 190, 192, 233 construction, 5, 7, 8, 9, 10, 16, 17, 25, 27, 28, 32, 76, 121, 123 constructivist, 18, 27 consumption, 227 continuity, 13, 250 control, 12, 28, 161, 162, 180, 184, 191, 192, 193, 199, 200, 210, 211, 212, 213, 216, 218, 220, 221, 224, 227, 228, 240 convection, 61 convergence, 17, 199 conversion, 63 convex, 197 conviction, 197 Copenhagen, 162 correlation, 15, 39, 40, 70, 89, 105, 107, 110, 111, 112

Index correlations, 63, 105, 110 costs, 62, 63, 183, 186, 224 counterbalance, vii, 1 coupling, x, 88, 107, 110, 243, 247, 248, 249, 252 cross-fertilization, 36 culture, 13, 14, 16, 18, 53, 54 customers, 191, 197 Cybernetics, 18, 84, 85 cycles, 73, 74, 96, 135, 140, 142, 143, 144, 233, 234 Czech Republic, 243

D danger, 10, 255 data communication, ix, 179, 181, 183, 184 database, 182, 188, 190 death, 29, 37 decay, 248 decision making, viii, 58, 59, 67, 68, 69, 74, 84, 85 decisions, 72, 185 decomposition, 64, 140, 142, 143, 144, 217, 218, 219, 220, 222, 234, 236, 237 deduction, 29, 257 definition, 8, 9, 19, 20, 45, 72, 126, 130, 131, 132, 133, 148, 150, 153, 159, 187, 193, 194, 247, 250, 251 deformation, 253 degenerate, 22, 94, 95, 108, 256, 257 degradation, 44 delivery, 184, 185 Department of Energy, 112 derivatives, 245 designers, 185 detection, 254 developed countries, 162 deviation, 256 dichotomy, 14 dielectric constant, 230 differentiation, 34 digit magnitude, 12 digitization, ix, 179 discipline, 25, 51, 185 discontinuity, 245, 250, 251 discordance, 68, 71, 73, 74 discourse, 41, 53, 68 discreteness, 40 discretization, x, 243 distribution, viii, 15, 57, 58, 59, 90, 91, 92, 94, 97, 161, 170, 183, 195, 220 distribution function, viii, 57 division, 130, 150, 151, 233 dominance, 193

263

Doppler, 243 dream, 38 duality, 199 dust, 38 dynamical system, 46, 50 dyscalculia, 12

E earth, 40 ecological, 161 education, 54 educational psychology, 11 effluent, 62, 64 ego, 40 elasticity, 57 electron, 88, 89, 90, 91, 92, 93, 94 electronic circuits, 227, 230 electronic systems, 89, 93 electrons, ix, 18, 30, 87, 88, 90, 92, 93, 94, 95, 107, 110 email, 182, 184 emotion, 27, 31 emotional, 25, 27, 31 emotions, 27 Empiricism, 9, 49 empowerment, 15 energy, viii, x, 24, 46, 87, 88, 92, 96, 227, 229, 232, 238, 243, 249, 250, 251, 252, 254, 255, 256, 257 energy consumption, 227 enterprise, viii, 16, 67, 68, 77, 83 entropy, 203, 204 environment, 39, 52, 65, 68, 84, 85, 130, 204 epistemological, 39 epistemology, 25 equality, 33, 191 equilibrium, 64, 193 equity, 184 ESR, 96 estimating, 166 estimator, 187 ethane, 61, 63, 64 ethylene, 61 Euclidean space, 35 Europeans, 152 evolution, 4, 12, 16, 17, 18, 31, 37, 40, 241, 253 evolutionary process, 17 expansions, 222

F factorial, 208 failure, 2, 38, 195

264

Index

fairness, ix, 179, 180, 181, 186, 187, 190, 191, 192, 193, 194, 197, 199 family, 7, 130, 246, 250 fear, 27 feedback, 25 Fermat, 16 fermions, ix, 87, 89, 90, 93 fertilizer, 161 fertilizers, 161 Feynman, 238, 240 fiber, 184 field theory, vii filters, 224 Finland, 53, 113 flexibility, 57, 68, 69 flow, 9, 64, 181, 182, 183, 186, 189, 193, 199, 233, 234 flue gas, 61 fluid, 17 Fourier, 239, 257 free will, 51 frequency, 98, 100, 102 Freud, 34 fullerene, 96 function values, 58 functional magnetic resonance imaging (fMRI), 11, 13 future, 112 fuzzy logic, 57 Fuzzy set theory, 68 fuzzy sets, viii, 22, 57, 58, 65, 75, 77

G garbage, ix, 203, 223, 224, 239 gas, 64 Gaussian, 64, 146 generalization, 8, 10, 34, 97, 208, 257 generation, vii, x, 89, 91, 94, 197, 203, 204, 229, 231, 232, 238 generators, 91, 94, 138, 139, 182 genomics, 166 geography, 183 Germany, 53 goals, 57, 65, 68, 180, 181, 186 God, 6, 40 gold, 88 Goodman, 54 government, iv GPRS, 182 graph, 158, 167, 169, 170, 177, 178, 186, 252, 253, 254, 255 Greece, 1 grouping, 161

groups, 15, 43, 89, 94, 95, 96, 107, 110, 112, 131, 133, 134, 150, 162, 177, 178, 207, 238, 239 growth, 77, 179, 180, 253 GSM, 182 Guaranteed Service, 185 guidance, 73 gyrus, 11, 12 Gyrus, 54

H H2, 64 hafnium, 230 Hamiltonian, 88, 107, 244, 248, 249, 257 happiness, 16 harm, 3, 4, 178 harmony, 3, 4 Harvard, 54, 84 Hawaii, 201 heat, vii, x, 64, 203, 204, 229, 231, 232, 238 height, 23, 252 Heisenberg, 29 hemisphere, 11 hepatitis, 170 hepatitis C, 170 herbs, 162 heterogeneous, 180 Hilbert, 2, 10, 28, 29, 54, 249, 257 Hilbert space, 257 holistic, vii, 1, 52 homework, 244 homology, 11 homomorphism, 133, 136, 148, 149 host, 12, 185 human, viii, 4, 11, 12, 13, 14, 15, 16, 17, 18, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 39, 40, 46, 52, 65, 67, 68, 71, 72, 73 human animal, 13 human brain, 12, 13, 15, 17, 18 human experience, 23 human nature, 52, 68 humanism, 14 humanity, 15, 34, 35 humans, 5, 11, 13, 18, 39, 46 hybrid, 200 hydro, 62 hydrocarbon, 63 hydrocarbons, 62 hydrogen, 62 hypothesis, 11, 26, 27, 122

I icosahedral, 96, 107

Index identity, 9, 19, 26, 27, 41, 42, 44, 45, 46, 47, 88, 116, 130, 131, 135, 136, 146, 153, 169, 171, 172, 206, 207, 209, 213, 216, 220, 221, 225, 235, 236 illusion, 16 images, 16, 24, 26, 27, 38 imaging, 11, 53 implementation, 12, 192, 222, 224, 226, 227, 228, 232, 238 inattention, 12 incidence, 162 inclusion, 9 income, 190 independence, 32, 51 Indian, 17 indices, 59, 97 individual differences, 50 individuality, 68 induction, 20 industrial, 161 industry, ix, 129, 197, 229 ineffectiveness, 191 inequality, 58, 193 infants, 11, 13, 15, 35 inferior parietal region, 12, 16 infinite, ix, 8, 13, 21, 28, 31, 32, 35, 36, 38, 50, 88, 115, 116, 119, 121, 122, 123, 124, 125, 132, 177, 178, 243, 246, 256 Information System, 203 infrastructure, 180 injections, 122 inorganic, 25 insects, 34, 35 insight, 4, 10 inspiration, 10 instability, 10 instruction, 16 integration, 2 intellect, 25, 27, 32 intelligence, 22 interaction, 58, 96 interactions, 175, 244 interdisciplinary, vii, 1 Internet, ix, 179, 180, 181, 182, 183, 184, 185, 191, 197, 199, 201, 234 Internet Protocol, 181 interval, 23, 71, 246, 252, 256, 257 intervention, 12 intrinsic, viii, 67, 68, 71 intron, 170 intuition, 15, 35 invariants, 142 inventiveness, 15

265

inventories, 142 inversion, 97 Investigations, 56 IP, 182, 183, 184, 186, 198, 199, 200, 201 IPv6, 182 irrationality, 29 isomorphism, 7, 19, 133, 148 Israel, 177

J Jaffe, 49, 52, 55 Japan, 201 Japanese, 154 joining, 143, 144 judgment, 31 Jung, vii, viii, 1, 2, 23, 24, 25, 27, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 49, 50, 52, 53, 55 Jungian, 3, 28, 55 justice, 3 justification, 10

K Kant, 30, 54 kernel, 133, 148 kinetic energy, x, 249 King, 87 Kobe, 201

L labeling, 6 labor, 58 labor force, 58 land, 162 language, 11, 12, 16, 17, 24, 26, 27, 33, 35, 115, 116, 133, 174 latency, 184 Latin squares, ix, 152, 153, 154, 155, 156, 178 law, 10, 12, 14, 41, 42, 149, 204, 229, 231, 232 laws, 23 leakage, 227 learners, 15 learning, 11, 17, 40, 77 Leibniz, 34 lesions, 12 life forms, 18 light, 88 likelihood, 32 limitation, 184 limitations, 26, 35, 39, 51 linear, 2, 8, 12, 24, 58, 59, 62, 173, 207, 208, 209, 238, 247, 250 linear function, 59, 207, 247

266

Index

linear programming, 58 linear systems, 25 linguistic, vii, viii, 24, 57, 67, 68, 69, 70, 71, 74, 77, 78, 83, 84, 85 linguistic information, viii, 67, 68, 70, 84 links, 180, 181, 186, 187, 188, 193 loading, 183 localization, 246 location, 190 locus, 13 logical deduction, 29 London, 53, 54, 55, 56, 84, 241 Los Angeles, 53 losses, viii, 67, 184 low power, 240 lying, 248

M magma, 130, 131 magnetic, 11 magnetic resonance imaging, 11, 13 Maine, 55 management, 69, 84, 180, 182, 185, 186, 199 manipulation, 16, 255 manufacturing, 58 mapping, 33, 233 market, 229 Markov, 238 Markovian, 198 marriage, 234 mathematical disabilities, 11 mathematical knowledge, 15 mathematical programming, 57, 190 mathematicians, 2, 5, 7, 10, 14, 15, 16, 17, 18, 28, 29, 30, 35, 36, 52, 207 mathematics, ix, 12, 16, 18, 33, 47, 129, 174, 177, 178, 208 matrices, 113 matrix, 30, 74, 147, 162, 163, 233, 234, 235 Max-Min Fairness, ix meanings, 58 measurement, 2, 161, 195 measures, 59, 60, 61, 62, 64, 71, 73 media, 181, 197 medicine, ix, 129 membership, viii, 6, 22, 23, 57, 58, 59, 70, 71, 74, 75, 76, 77 memory, vii, 1 mental image, 16 mental power, 4, 5 mental representation, 2, 12, 16, 17 messages, 23, 26, 234 metaphor, 13

metaphors, 13 methane, 62 metric spaces, vii Middle Ages, 40 Mind-Body, 56 MIP, 197 MIT, 53, 84 modeling, viii, 67, 84 models, v, vii, 3, 6, 8, 10, 18, 20, 25, 65, 74, 77, 82, 83, 126, 179, 183, 186, 201, 244, 248, 257 modules, 197 molecules, viii, 87, 88, 89, 95, 96, 110 momentum, vii, x, 93, 94, 110, 243, 248, 249, 257 MOS, ix, 203, 227, 228, 231, 232, 240 Moscow, 56 motion, 23, 253, 257 motivation, 238 movement, 2, 16, 29, 34, 140, 257 multicultural, 15 multimedia, 182 multiplication, 43, 44, 45, 47, 50, 89, 125, 130, 146, 147, 149, 170 multiplicity, 6, 99 multiplier, 224 music, 29 mutations, 140

N National Science Foundation, 112 natural, vii, viii, 1, 4, 5, 6, 7, 8, 9, 18, 19, 20, 21, 23, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 41, 42, 45, 46, 47, 48, 49, 51, 52, 57, 65, 69, 74, 88, 125, 129, 130, 137, 149, 153, 160, 191, 221 network, ix, 179, 180, 181, 182, 183, 184, 185, 186, 188, 189, 190, 191, 192, 193, 195, 197, 199, 241 neurobiological, 16 neuroimaging, 12, 13 neuronal circuits, 16 neurons, 11, 12 neurophysiology, 12, 13 neuropsychology, 11, 12 neuroscience, 11 New Jersey, 198 New Science, 55 New York, 55, 65, 113, 114, 166, 174, 177, 178, 199, 200, 238, 241, 259 Newton, 34 next generation, 197 Ni, 201 NMR, 96, 102

Index nodes, 180, 181, 185, 186, 188 non-human, 13 non-uniform, 8 nonverbal, 17 normal, 88, 89, 107, 118, 133, 136, 138, 157 normalization, 77, 245, 250, 251 norms, 26 NSC, 198 nuclear, viii, 87, 89, 90, 91, 93, 96, 97, 99, 100, 102, 105, 106 nuclei, 88, 96, 100, 102 nucleus, 90, 97 number-words, 9 numerical computations, 249

O objective reality, 51 observations, 21, 257 obsolete, 183 operator, ix, 14, 88, 179, 190, 191, 244, 249, 257 opposition, 26, 34, 40, 152 optical, ix, 179, 183 optical transmission, 183 optimism, 63, 64 optimization, vii, ix, 57, 58, 59, 84, 154, 179, 181, 190, 193, 194, 197, 239 orbit, 88, 96, 97, 107, 110, 137, 139, 145, 165, 177 organic, 25 orientation, 90 orthogonal Latin squares, 155, 156 orthogonality, 155 oscillation, 249, 257 oscillator, 243, 248, 257 oxide, 227, 229, 230 oxide thickness, 229 ozone, 162

P Pacific, 201 packets, 180 pairing, ix, 115, 122, 123, 124, 125, 126 paradox, 10, 46, 256 parameter, 15, 64, 68, 71, 72, 73, 77, 78, 83, 192, 193, 245, 247, 250, 257 Pareto, 76, 188, 190, 199 Pareto optimal, 188, 190 Parkinson, 65 particles, ix, 30, 38, 87, 89, 90, 91, 93, 102 partition, 90, 91, 92, 97, 119, 120, 130, 137, 139, 159, 198, 214, 217 pathogens, 162

267

patients, 34 pattern recognition, 35 pedagogical, 4 perception, 17, 21, 40, 68, 69, 72 periodic, 89, 93, 94, 96, 97, 102, 105, 108, 110 Periodic Table, viii, 87, 89 periodicity, viii, 87, 88, 89, 91, 94, 95, 96, 97, 102, 105, 106, 107, 108, 110, 112 permit, viii, 67 permittivity, 229 personality, 40 perturbation, 249 Philadelphia, 65 philosophers, 3, 7, 15 philosophical, vii, 1, 7, 10, 14, 33, 48, 49 philosophy, 3, 4, 14, 29, 227, 228 physical environment, 15 physical world, 13, 17, 38, 226 physicists, 16, 18, 38 physics, 30, 244 physiological, vii, 1 planar, 167, 168 planets, 3, 14 planning, viii, 67, 68, 83, 161, 187, 189 Plato, 4, 25, 27, 34, 39, 50 play, 13, 47, 65, 77, 89, 130, 153, 213 pleasure, 28 plurality, 2 Poisson, 183, 195, 196 polarity, 14, 34 polynomial, ix, 89, 95, 96, 129, 140, 145, 149, 150, 151, 169, 176 polynomials, 91, 97, 149, 151 poor, 180, 192 population, 12, 15, 161 power, vii, x, 4, 15, 27, 31, 32, 36, 69, 84, 93, 97, 116, 132, 149, 155, 160, 161, 163, 177, 203, 231, 232, 240 powers, 32, 89, 97, 132, 160 predators, 17 predicate, 20, 21, 33, 117, 125, 126 pre-existing, 50 preference, viii, 67, 74, 75, 76, 77, 79, 82, 83, 84, 85, 186 prefrontal cortex, 12 press, 113 pressure, 63 prices, 199 primary products, 62 primate, 12, 17 private, 184, 258 probability, 29, 65, 195, 196, 197 probe, 96

268

Index

production, 26, 61 productivity, 162 profit, 190 prognosis, 11 program, 4, 5, 197 programming, 57, 58, 65, 174, 200 proliferation, 224 propagation, 187 properties, 88 property, 15, 20, 27, 31, 34, 130, 146, 181, 212, 213, 227 Proportional Fairness, ix, 179 proposition, 14, 34, 36, 37, 38, 150 protocols, 184, 186, 198 protons, 88, 102 prototype, 3, 227, 229 prototyping, 229 pseudo, 170 psyche, 2, 24, 31, 34, 37, 38, 50 psychic process, 24 psychological functions, 2, 39 Psychological Perspective, 53 psychologist, 28 psychology, 3, 37 psychophysics, 12 PT, x, 243, 244, 245, 248, 249, 257 public, 191 Public Switched Telephony Network PSTN), ix, 179 pyrolysis, 62

Q QoS, v, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 195, 197, 198, 199, 200, 201 quality of service, 185, 198 quantum, vii, ix, 2, 88, 89, 92, 96, 99, 102, 203, 204, 226, 230, 231, 239, 240, 244, 248, 249, 257, 259 quantum computing, 203, 204 quantum field theory, 244 quantum mechanics, vii, 88, 244, 249, 257, 259 quarks, 18, 38 qubits, 226 Quinn, 49, 52, 55

R radiation, 61 radio, 240 rail, 226, 240 random, 23, 27, 161, 174, 195 range, 8, 180, 249, 257

reading, 207 real numbers, 5, 13, 22, 23, 32, 46, 48, 59 real time, 180 realism, 10, 33 reality, 3, 11, 13, 14, 16, 18, 24, 26, 30, 34, 36, 37, 38, 39, 49, 51, 52, 244 reasoning, 8, 13, 22, 118, 119, 120, 121, 122, 123, 124, 125, 236 recalling, 118 recognition, 18, 35 reconciliation, 26 recovery, 62 recursion, 20 reflection, 145 regional, 182 regular, 96, 142, 143, 144, 147, 255, 256 rehabilitation, 11 relationship, viii, 4, 12, 15, 38, 40, 87, 206, 210 relationships, viii, 3, 23, 35, 36, 87, 226 relativity, 52, 88 relevance, 244 reliability, 184, 191, 197 religion, 14, 16, 17, 25 Renaissance, 34, 40 repetitions, 133 research, 112 Research and Development, 238 resolution, 34, 165 resource allocation, 180, 181, 194, 195, 200 resources, ix, 58, 179, 180, 181, 183, 184, 199 retail, 47 returns, 252 revenue, 181, 191 rhythm, 3 right hemisphere, 12 rings, ix, 43, 129, 146, 148, 150 risk, 183 RNA, ix, 129, 166, 167, 168, 169, 170 robustness, 197 rotations, 89, 137, 138, 140, 142, 143, 144, 165 routing, 180, 181, 182, 185, 186, 188, 190, 198, 199, 200, 201, 234

S sacred, 23, 27, 31 sample, 161, 256 satisfaction, 58, 63, 77, 181 scalar, 11, 70, 71 scheduling, 185, 198 schema, 8, 24 school, 13, 14, 15, 28 Schrödinger equation, x, 243 scientific community, 2

Index search, 11, 12, 24, 49, 154, 177, 186, 241, 244, 252, 257 searches, 255 searching, 35 secular, 246, 247, 252 security, 5 Self, 25, 40, 41, 56 semantic, 68, 77 semantics, 3, 58, 59, 115 semiconductor, 227 semigroup, 131, 146 sentences, 32, 118, 126 series, ix, 20, 30, 31, 35, 179, 227, 246 service quality, 180 services, ix, 179, 180, 181, 182, 183, 184, 185, 191, 193 set theory, 58, 68 shape, 13, 43, 64, 90, 91, 92, 133, 220, 252 shaping, 10 Shapiro, 56 shares, 68, 193 sharing, ix, 179, 184, 185, 197, 199, 200 Shell, 93 sign, 2, 6, 12, 88, 107, 116, 203, 245, 250, 256 signals, 231 signs, 6, 18, 146 silicon, 230, 232, 240 similarity, 84 Singapore, 53 singular, 147 skeleton, 50 skin, 162 SMS, 182 social construct, 15, 17, 18 social context, 15 social responsibility, 15 Socrates, 25 software, 255 SOI, 232, 240 soil, 161 solar, 240 sounds, 14 South America, 35 space-time, 26, 39 Spain, 129, 164 spare capacity, 185 spatial, 12, 244, 248 special theory of relativity, 88 species, 17, 89, 96, 97, 101, 102, 154 spectroscopy, viii, 87, 88, 89, 93, 96 spectrum, 22, 36, 98, 244, 248, 249, 252, 257 speculation, 35 speech, 180, 224

269

speed, 88, 187, 198 speed of light, 88 spin, viii, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 105, 106, 107, 110, 111 spin-1, 90, 91 spiritual, 3, 4, 25, 27, 52 stability, 34, 184, 199 stabilizers, 138 stages, 25, 34 standard model, 8, 20, 125 standards, 21 starvation, 186 statistics, 161 stochastic, 183 Stochastic, 200 stochastic processes, 183 strain, 35 strategies, 9 streams, 181, 185 stress, 27, 162, 220, 221 structuralism, 10, 20 structuring, 50 students, vii, 1, 15, 233 subgroups, ix, 131, 133, 138, 161, 203, 207, 208, 209, 213, 214, 215, 217, 221, 222, 225, 232, 235, 239 subjective, 15, 72 subjectivity, 68 subrings, 147, 148 subscribers, 182, 191 substitution, 97 subtraction, 16, 130 superstitious, 27 supervision, 68 suppliers, 58 surprise, 22 survival, 17 switching, 95, 182, 234, 240, 241 symbolic, 3, 13, 17, 24, 26, 35, 70, 255 symbols, 31, 33, 34, 35, 38, 89, 93, 94, 97, 116, 131, 151, 152, 153, 250 symmetry, viii, 24, 45, 87, 88, 94, 95, 96, 105, 107, 112, 145, 244, 245, 255 syntactic, 49, 68 synthesis, ix, 26, 40, 50, 179, 206, 212, 213, 215, 216, 217, 218, 220, 221, 222, 225, 232, 236, 237, 238, 239 systems, 5, 6, 7, 8, 27, 35, 46, 58, 65, 89, 93, 96, 107, 110, 112, 195, 198, 234, 257

T Taiwan, 179, 198

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

Index 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

Index World Wide Web, 175

271

Z Y

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

zeitgeist, 51 Zen, 50