Table Of ContentATLANTIS STUDIES IN MATHEMATICS FORENGINEERING AND SCIENCE
VOLUME3
SERIES EDITOR: C.K.CHUI
Atlantis Studies in
Mathematics for Engineering and Science
SeriesEditor:
C.K.Chui,StanfordUniversity,USA
(ISSN:1875-7642)
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AMSTERDAM –PARIS
(cid:2)c ATLANTISPRESS/WORLDSCIENTIFIC
The Hybrid Grand Unified
Theory
V. LAKSHMIKANTHAM
FloridaInstituteofTechnology
DepartmentofMathematicalSciences
Melbourne,USA
E. ESCULTURA
LakshmikanthamInstituteforAdvancedStudies
G.V.P.CollegeofEngineering
Madhuravada
Visakhapatnam,(A.P.)India
S. LEELA
FloridaInstituteofTechnology
DepartmentofMathematicalSciences
Melbourne,USA
AMSTERDAM –PARIS
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Copyright
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ISBN:978-90-78677-21-5 e-ISBN:978-94-91216-23-7
ISSN:1875-7642
(cid:2)c 2009ATLANTISPRESS/WORLDSCIENTIFIC
Preface
Thephysicistisinterestedindiscoveringthelawsofinanimatenatureandthemathemati-
cian uses the depth of his thought into exploring the mathematical concepts. But, the
symbiotic connection between physics and mathematics and the enormoususefulness of
mathematicsin the natural sciences is somethingquite mysterious. No rationaljustifica-
tion appearsto be satisfactory to understandthe uncannysuccess of mathematicsand its
roleinphysicaltheories.Apossibleexplanationisthatthelawsofnaturearewritteninthe
languageofmathematics.
A complete mechanism, natural or man-made, is understood fully by taking it apart,
studyingitspartsorcomponentsandtheirpropertiesandlearnhowthesepartsfittomake
the whole. It is really a natural way to understand any object or concept. The path of
progressinknowledgeissimilarlyacombinationofbothexpositionofpartsandtheirsyn-
thesis. Naturalsciencesandothersciencessuchasbiology,psychology,etc. arereminders
of these strides, taken in order to comprehend the world around us and there have been
attempts to catch a glimpse of the higher reality and understand it. The human mind is
limited and cannot go beyondcertain limitations of time, space and causation. Still, the
universeisexperiencedas a dynamicinseparablewhole, includingthe observerin an es-
sentialway.Inthisexperience,thetraditionalconceptsoftimeandspaceofisolatedobjects
and of cause and effectlose their meaning. Such an experienceis very similar to that of
the ancientscientists (Rishis). However,they repeatedlyinsisted thatthe ultimate reality
canneverbeanobjectofreasoning,deductionordemonstrableknowledge.Itcanneverbe
adequatelydescribedbywordsandMaxPlanckofthemodernscientificeraobservedthat
sciencecannotsolvetheultimatemysteryofnatureandthatisbecauseinthefinalanalysis,
weourselvesarepartofnatureandthereforepartofthemysterywearetryingtosolve.
The scientists in physics are happy with physical concepts that can be visualized or
observeddirectly. But most of the time they are forced to utilize mathematical concepts
to describe the physicalconcepts. Take for example,the case of the black hole. It is not
entirelyobservablebutitisaphysicalconceptbecausethereisanoverwhelmingevidence
ofitsexistence. Althoughtheyhaveverifieditsexistenceinthecoreofgalaxiesincluding
ourmilkyway,itisnotknownexactlywhatitisandtherefore,theyusethemathematical
concept of a singularity to describe it. However that is not amenable to computation or
studying its structure. Another such importantphysical conceptis that of the basic con-
v
vi TheHybridGrandUnifiedTheory
stituent of matter. It is very important to know what constitutes matter and it cannot be
ignoredbecauseofourdeepconvictionintheorderoftheuniverse.
Thisisourattempttoprovideahybridgrandunifiedtheorytounderstandtheuniverse,
bothinitsmicro/quantumaspectsaswellasmacro/galacticaspects.Itistrulyahybridthe-
oryasittriestoencompassboththemodernandancienttheoriesoftheuniverse,together
withitsfunctioningatalllevelsofhumancomprehension. Duringthisattemptitbecomes
necessary to acknowledge the ambiguity and limitations of mathematics concerning the
fundamentalconceptsofverylargeandverysmallnumbers,infinityandthelimitingpro-
cessin general. Althoughthemathematicalmodelingis themostadvancedmethodology
ofphysicsanditowesallitstremendousachievementstotheformer,theexistenceofun-
solvedproblemsandunansweredquestionssuggesttheneedforimprovement. Itisafact
that the origin of the concepts such as natural numbers, rational and irrational numbers,
zero,infinityandtheplacevaluesystem, amongothers, originatedinVedaSamhitas, an-
cientscientifictextsofIndia.However,becauseofthequirksofournarratedhumanhistory,
oursecondhandreceptionoftheseconceptsfromtheArabsandthebeliefthateverything
originated in Greece, the true original source appears to have been obscured and lost. It
is thereforenecessary to return to the sourceto clarify and understandthe basics to shed
somelightontheexistingrealnumbersystem.
InChapter1,werevisitthefundamentalsofmathematicstobringoutsourcesofambi-
guityincertainbasicmathematicalconcepts. Wealsoraisemanyquestionsthatarechal-
lengingthephysicistsanddiscusstherelevanceofmathematicalmethodologyinphysics.
Certainrelatedconceptssuchascosmicwavesarealsoconsidered.
Chapter2dealswiththemathematicsthatisessentialinaidingthedescriptionofphys-
icalconceptsthatarediscussedinChapter3.Inparticular,theexistingrealnumbersystem
isrevisedwiththepurposeofbringingouttheusefulnatureofdecimalnumerationsystem
in the current digital era of high accuracy computation with the aid of technology. The
ambiguityinvolvedin dealingwith infinity,limitingprocessandthe computationof very
largeandverysmallnumbersisminimized. Themathematicsofgeneralizedcurves,gen-
eralizedfractalsandchaosprovidesthe mathematicalmodelingofphysicalconceptsthat
ariseinthisgrandunifiedtheory.TheintegratedPontrjagin’smaximumprincipleisbriefly
discussedasithasbeeninstrumentalinsolvingthefamousn-bodyproblem.Theintroduc-
tionofdarknumbersaspartoftherefinedrealnumbersystempavesthewaytoquantize
thebasicconstituentofmatterandingeneral,darkmatter.
In chapter 3, the qualitative modelingis employed to study the search for basic con-
stituent of matter and its ramificationsin explaining quantumgravity and macro gravity.
Thetennaturallawsofnatureareenumeratedandthemathematicalmodelofasuperstring
is introduced to representthe structure of the basic constituentof matter. Verification of
this grand unified theory (GUT) is carried out by explaining various natural phenomena
suchasultra-energeticindexcosmicwaves,supernovaandtherarehitonearthbyasteroids
inspiteofbeingsoclosetotheastreoidbelt,etc.
In Chapter 4, the theoreticalconsiderationof n-bodyproblem and turbulence are de-
scribed. Also,physicsofthemindisinvestigatedsinceinthisgrandunifiedtheory,mind
Preface vii
isalsoanimportantpartinthestudyoftheuniversearoundus.
Finally, in Chapter 5, the hybrid nature of this book is really brought to the fore as
it deals with matters of mind and consciousness, evolution and involution, creation and
dissolution,etc.accordingtotheviewofancientscientists(Rishis). Itisreallylikecoming
full circle to consider how much alike are the theories of ancient and modern scientists,
relativetotheuniversearoundus.
Someoftheimportantfeaturesofthebookareasfollows:
(i) Itputstherealnumbersystemonsolidfoundationswithoutinconsistencyandempha-
sizestheappropriatenessofthedecimalsystemasacomputationtool;
(ii) Itimprovestheexistingrealnumbersystembyincorporatingthenotionofdarknum-
bersandtheirduals,personalandimpersonalinfinity;
(iii) Itintroducesthesuperstringasthebasicconstituentofmatterandthefractalnatureof
thesuperstringismodeledbytheCauchyrepresentationofdarknumber;
(iv) Itrelatesthedarkmatterofphysicsanddarknumbersofmathematics;
(v) Itemploysthetrulyhybridapproachofcombiningqualitativemathematicsandcom-
putationtodiscoverthenaturallawsinordertoexplaininaunifiedwayseveralnatural
phenomena,atmicro/quantumandmacro/galacticlevels;
(vi) Itattemptstoshowthecommonhybridinteractionbetweenmodernandancientscien-
tifictheoriesofnatureandsearchforhigherreality.
We wish to express our deepest appreciationto ProfessorsG.S. Osipenko, V.V. Gud-
kov,C.G.JesdudasonandT.GnanaBhaskarforthevariousscientificdiscussions. Weare
immenselythankfultoMs. SallyEllingsonfortheexcellenttypingofthemanuscriptinall
itsstages.
Contents
Preface v
1. BasicProblemsofMathematicsandPhysics 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 RetrospectionofFundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 TheProblemsofPhysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 RelatedConcepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4.1 WavesandVibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4.2 CosmicWaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4.3 TheSineandRelatedCurves . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4.4 GeneralizedFractal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2. MathematicsofGrandUnifiedTheory 13
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 HybridRealNumberSystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 TheConstructionofHybridRealNumberSystem . . . . . . . . . . . . . 14
2.2.2 TheNonterminatingDecimals . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.3 TheStructureofR∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.4 MorePropertiesofNonstandardNumbers . . . . . . . . . . . . . . . . . 23
2.2.5 g-LimitandtheLimit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 SpecialFunctionsandtheGeneralizedCurves . . . . . . . . . . . . . . . . . . . . 25
2.3.1 SomeMischievousFunctions . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.2 TheInfinitesimalZigzag. . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.3 SignificanceoftheInfinitesimalZigzag. . . . . . . . . . . . . . . . . . . 26
2.3.4 TheWildOscillationsin1 . . . . . . . . . . . . . . . . . . . . . . . . . . 27
x
2.3.5 RapidSpiralandOscillation. . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.6 Rectification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.7 ApplicationoftheInfinitesimalZigzag . . . . . . . . . . . . . . . . . . . 29
2.3.8 TheGeneralizedCurves . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4 GeneralizedFractalandChaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4.1 DefinitionsandExamples . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4.2 ThePeanoSpace-FillingCurve . . . . . . . . . . . . . . . . . . . . . . . 33
2.4.3 FillinguptheUnitCube . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4.4 TheInfinitesimalZigzagasLimitSetofFractal . . . . . . . . . . . . . . 35
2.4.5 AMoreGeneralFractal . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.5 TheIntegratedPontrjaginMaximumPrinciple . . . . . . . . . . . . . . . . . . . . 38
ix
