Studies on Santilli’s Isonumber Theory
American Journal of Modern Physics
Volume 5, Issue 2-1, March 2016, Pages: 17-36
Received: Aug. 10, 2015; Accepted: Aug. 11, 2015; Published: May 18, 2016
Views 3514      Downloads 108
Author
Arun S. Muktibodh, Mohota College of Science, Nagpur, India
Article Tools
Follow on us
Abstract
Beginning with studies in the 1980s at the Department of Mathematics of Harvard University, the Italian-American scientist R. M. Santilli discovered new realizations of the abstract axioms of numeric fields with characteristic zero, based on an axiom-preserving generalization of conventional associative product and consequential positive-definite generalization of the multiplicative unit, today known as Santilli isonumbers [1], and the resulting novel numeric fields are known as Santilli isofields. By remembering that 20th century mathematics was formulated on numeric fields, their generalization into isofields stimulated a corresponding generalization of all of 20th century mathematics and its application to mechanics, today known as Santilli isomatheatics and isomechanics, respectively, which is used for the representation of extended-deformable particles moving within physical media under Hamiltonian as well as contact non-Hamiltoian interactions. Additionally, Santilli discovered a second realization of the abstract axioms of a numeric field, this time with arbitrary (non-singular) negative definite generalized unit and related multiplication, today known as Santilli isodual isonumber [1] that have stimulated a second covering of 20th century mathematics and mechanics known as Santilli isodual isomathematics and isodual isomechanics. The latter methods are used for the classical as well as operator form of antimatter in full democracy with the study of matter. In this paper, we present a comprehensive study of Santilli's epoch making discoveries of isonumbers and their isoduals along with their application to isomechanics and its isodual for matter and antimatter, respectively.
Keywords
Isonumber, Isodual Number, Isodual-Isonumber, Genonumber
To cite this article
Arun S. Muktibodh, Studies on Santilli’s Isonumber Theory, American Journal of Modern Physics. Special Issue: Issue II: Foundations of Hadronic Mechanics . Vol. 5, No. 2-1, 2016, pp. 17-36. doi: 10.11648/j.ajmp.2016050201.12
References
[1]
R. M. Santilli, isonumbers and genonumbers of dimension 1,2,4,8, their isoduals and pseudoduals, and “hidden numbers" of dimension 3,5,6,7, Algebras, Groups and Geometries 10, 273-322 (1993).
[2]
R. H. Bruck, A Survey of Binary Systems, Springer Verlag, (1958).
[3]
E.Fermi, Nuclear Physics, Univ. of Chicago Press (1950).
[4]
W. R. Hamilton, Lectures on quaternions, Dublin, Ireland (1853).
[5]
R. M. Santilli, Hadronic J.1, 223(1978), Addendum 1, 1343 (1978).
[6]
R. M. Santilli, Lie-Admissible Approach to Hadronic Structure, Vol. I: NonApplicability of the Galilei and Einstein Relativities ?, Hadronic Press, Tarpon Springs, FL (1978)
[7]
R. M. Santilli, Lie-Admissible Approach to Hadronic Structure, Vol. II: Coverings of the Galilei and Einstein Relativities ?,(1982) and Vol. III: Identification of the Hadronic constituents with Physical Partcles ?, Hadronic Press, Tarpon Springs, FL.
[8]
R. M. Santilli, On a possible Lie-Admissible covering of the Galilei relativity in Newtonian mechanics fo conservative and Galilei form-noninvariant systems, Hadronic J. 1. 223-423 (1978) and Addendum, Hadronic J. I. 1279-1342 (1978).
[9]
R. M. Santilli, Foundations of Theoretical Mechanics, Vol.II: Birkhoffian Generalization of Hamiltonian Mechanics, Springer-Verlag, Heidelberg, New York (1982).
[10]
C. F. Gauss, Die 4 Beweise der Zeriengung ganzer algebraischet Funktionen (1799), Edited by E.Netto, Ostwalds Klassiker der exkten Wissenscheften n. 144.
[11]
N. H. Abel, Overes Completes, Christiania Publisher, (1881).
[12]
A. Cayley, Phyl. Magaz. and J. of Science 3, 210 (1845).
[13]
E. Galois, Overers Mathematiques, Gautier-Villars, Paris, (1897).
[14]
O. Haupt and Sengenhorst, Algebraic extension of a field, in Fundamenatals of Mathematics, Vol.I, Edited by H. Benhnke et. al., MIT Press, Cambridge MA (1974).
[15]
A. Hurwitz, Uber die Composition der quadratischen Formen vonbeliebigvielen Variabeln, Nach.kon.Gesell.der Wiss. Gottingen, 309-316 (1898).
[16]
A. A. Albert, Ann.Math. 43, 161, (1942).
[17]
N. Jacobson, Composition Algebras and their automorphism, Rendiconti Circolo Matematico Palermo 7, series II, 55 (1958, Modern Algebra, edited by A. A. Albert, Prentice Hall, New York (1963).
[18]
R. M. Santilli, Elements of Hadronic Mechanics, Vol. I and II, Ukraine Academy of Sciences, Kiev, second edition 1995 http://www.santilli-foundation.org/docs/Santilli-300.pdf http://www.santilli-foundation.org/docs/Santilli-301.pdf.
[19]
A. S. Muktibodh, Foundations of Isomathematics, American Institute of Physics, AIP Conference Proceedings, Vol. 1558, 707, (2013).
[20]
G. Pickertand H. Steiner, Complex numbers and quaternions, in Fundamentals of Mathematics, edited by Behnek et. al., MIT Press (1974).
[21]
C. H. Curtis, The Four and Eight Square Problem and Division Algebras, in Studies in Modern Algebras, Edited by A. A. Albert, Prentice Hall (1963).
[22]
E. Kleinfeld, A characerization of the Cayley numbers , in Studies in Mathematics, Edited by A. A. Albert, Prentice Hall, New York (1963).
[23]
R. M. Santilli, Isotopies of contemporary mathematical structures, I; Isotopies of fields, vector spaces, transformation theory, Lie Algebras, analytic mechanics and space-time symmetries, Algebras, groups and Geometries 8, 169-266 (1991).
[24]
R. M. Santilli, Isotopies of contemporary mathematical structures, II; Isotopies of symplecticgeometry, affine geometry, Riemanian geometry and Einstein gravitation, Algebras, Groups and Geometries, 8, 275-390 (1991).
[25]
R. M. Santilli, Hadronic Mathematics, Mechanics and Chemistry. Vol.III: Iso-, Geno-, Hyper-,Formulations for Matter and their isoduals for Antimatter. Palm Harbor, FL34682, U.S.A.; International Academic Press, (2008).
[26]
R. M. Santilli, Isodual Theory of Antimatter with Application to Antigravity, Grand Unification and Cosmology, Vol.151 of Fundamental Theories of Physics. 3300 A A Dordrecht, The Netherlands: Springer, ( 2006).
[27]
R. M. Santilli, Lie-Isotopic liftings of Lie symmetries. I. General Considerations, Hadronic J., Vol. 8, pp. 25-35, January, 1985.
[28]
R. M. Santilli, Lie-Isotopic liftings of Lie symmetries. II., Liftin of Rotations, Hadronic J. Vol.8, pp. 36-51, January,(1985).
[29]
R. M. Santilli, Foundations of Hadronic Mechanics, Hadronic J. Suppl. 4B,1-25(1989).
[30]
R. M. Santilli, Isodual spaces and antiparticles, Comm. Theor. Phys. Vol. 3 (1994).
[31]
R. M. Santilli, Isotopic Generalization of Galilei’s and Einstein’s Relativities, Vol. II, Classical isotopies, 1-st edition Hadronic Press, Palm Harbor, FL(1991): 2-nd edition Ukrain Academy of Sciences, Kiev.
[32]
J. V. Kadeisvili, Santilli’s Lie-isotopic generalizations of Contemporary Algebras, Geometries and Relativities, 1-st edition (1992), Hadronic Press, Palm Harbor, FL,2-nd Edition, Ukrain Academy of Sciences, Kiev.
[33]
R. M. Santilli, Non-local Formulation of the Bose-Einstein correlation within the context of hadronic mechanics, Hadronic J. 15, 1-99, (1992).
[34]
A. A. Bhalekar, Santilli’s New Mathematics for Chemists and Biologists. An Introductory Account Hadronic J. (2013).
[35]
J. V. Kadeisvili, Elements of Functional isoanalysis, Algebras Groups and Geometries, 9, 283-318,(1992).
[36]
A. K. Aringazin, a.Jannussis, D.F. Lopez, M. Nishioka and B. Veljanoski, Santilli’s Lie-isotopic Generalization of Galliei’s and Einstein’s Relativities, Kostarakis Publisher, Athens, Greece (1991).
[37]
R. M. Santilli, Nonlinear, nonlocal, noncanonical axiom-preserving isotopies, Q-operator deformations of Lie symmetries, Proceedings of third international Wigner Symposium, Oxford University, Sept. (1993).
[38]
R. M. Santilli, Lie-isotopic lifting of the special relativity for extended-deformable particles, Lettere Nuovo Cimento 37, 545-555 (1983).
[39]
R. M. Santilli, Isotopic Generalization of Galilei’s and Einstein’s Relativities, Vol. I, Mathematical Foundations, 1-st edition Hadronic Press, Palm Harbor, FL (1991): 2-nd edition Ukrain Academy of Sciences, Kiev.
[40]
J. V. Kadeisvilli, M. Kamiya and R. M. Santilli, A characterization of isofields and their isoduals, Algebras Groups and Geometries, Vol 10, 168-185(1993).
[41]
R. M. Santilli, Contributed paper to the proceedings, Third International Wigner Symposium, Oxford University(1993), and JINR Rapid Comm. 6, 24 (1993).
[42]
Klimyk, R. M. Santilli, Algebras, Groups and Geometries 10 , 323 (1993).
[43]
R. M. Santilli, Algebras, Groups and Geometries, 10, 273 (1993).
[44]
A. S. Muktibodh, Iso-Galois fields, Hadronic J., Vol. 36 (2), 225 (2013).
[45]
D. S. Sourlas and G. T. Tsagas, Mathematical foundations of Lie-Santilli theory, Ukraine Academy of Sciences, Kiev.
[46]
R. M. Santilli, Hadronic Mathematics, Physics and Chemistry, Vols. I [4a], II[4b], III[4c], IV[4d] and V[4c], New York: International Academic Press.
[47]
H. Davenport, Aritmetica superiore, Un’introduzione alla teoria dei numeri, Bologna: Zanichelli, (1994).
[48]
R. M. Santilli, Circolo matematico di Palermo,(Special issue on Santilli isotopies) Supp., Vol. 42 (1996).
[49]
C. X. Jiang, Fundamentals of the Theory of Santillian numbers, International Academic Press, America-Europe-Asia (2002).
[50]
D. E. R. Denning, Cryptography and Data Security, Addison Wesly, Reading M A (1983).
[51]
T. Vougiouklis, Hyperstructures and their Representations, Hadronic Press, Palm Harbor, FL (1994).
[52]
S. Geoeriev, J. Kadeisville, Foundations of Iso-Differential Calculus, Vol.I and Vol.2 (2014).
[53]
A. S. Muktibodh, Iso-Galois Field and Isopermutation Group, Hadronic J. 13, Vol. 38, I (2015).
[54]
A. S. Muktibodh, Isopermutation Group, Clifford Analysis Cliford Algebra and Applications, Vol 3, No. 3 (2014).
ADDRESS
Science Publishing Group
1 Rockefeller Plaza,
10th and 11th Floors,
New York, NY 10020
U.S.A.
Tel: (001)347-983-5186