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Santilli called the new fields F* isofields,, the new numbers n* real, complex or quaternionic isonumbers, the new product n*∗m* isoproduct, and the new unit I* isounit where the prefix "iso" is used in the Greek meaning of being "axiom-preserving," In the same paper [1], Santilli discovered that the isounit I* does not need to be an element of the original field, and can be any desired non-singular quantity, such as a number, function, matrix, operator, etc. This led to the classification of isonumbers of the first (second) kind depending on when the isounit is (is not) an element of the original field.

The non-triviality of Santilli isonumbers should be indicated to prevent predictable misjudgments. As an illustration, Santilli isofields imply that prime numbers do not have an absolute meaning since their numerical value depends on the assumed unit. Consider the real isofields of the second kind F*(n, ∗, I*) where n*≈n since I* ∈ F. Then, for I* = 3, the number 4 is a prime number. Among various independent studies on Santilli isonumber theory, we indicate monograph [2] by the Chinese mathematician C-X. Jiang and studies by the Italian physicist C. Corda [3] and Indian Chemist A. A. Bhalekar [4].

In memoir [5] of 1996, Santilli identified the foundation of isomathematics which is characterized by an isotopic lifting of the totality of 20th century mathematics defined over a field of characteristics zero, thus including the isotopies of functional analysis, metric spaces, differential calculus, Lie's theory, geometries, symmetries, topologies, etc. Particularly significant are Santilli's step-by-step isotopies of the various branches of Lie's theory with isoproduct [A, B]* = A∗B - B∗A which have been constructed (when Santilli was at Harvard University in 1978 [6]) to characterize, apparently for the first time in a consistent way, non-linear, non-local and non-Hamiltonian systems. Particularly important is also Santilli's isodifferential calculus that permitted the isotopies of Newton's, Hamilton's, Schroedinger, Heisenberg's, Dirac's and other basic equations of physics. These broader dynamical equations and their underlying isomathematics characterize the isotopic branch of hadronic mechanics, or isomechanics as a shot range covering of quantum mechanics to avoid excessive linear, local and Hamiltonian approximations of complex physical or chemical systems.

An important application and verification of isomathematics and related isomechanics, for which the new mathematics was proposed for, has been the first representation at the non-relativistic and relativistic levels of all characteristics of the neutron (and not only its mass) in its synthesis from a proton and an electron in the core of a star, p⁺ + e^- → n + ν. In this case, the conventional Schroedinger equation of quantum mechanics does not yield physically meaningful solutions because the rest energy of the neutron is higher by 0.782 MeV than the sum of the rest energy of the proton and the electron, thus requiring a positive binding energy which is anathema for quantum mechanics. Thanks to their no-unitary structure, the Shcroedinger-Santilli isoequation for the non-relativistic treatment and the Dirac-Santilli is isoequation for the relativistic treatment have permitted the removal of these insufficiencies for the most fundamental nuclear synthesis in nature (see the review by J. V. Kadeisvili [7]). Comprehensive treatments of isomathematics, isomechanics and their applications in numerous fields are available in Santilli's monographs [5-11]. For independent studies we indicate monographs [12-15] and quoted references by the mathematicians Gr. Tsagas, D. S. Sourlas, J. V. Kadeisvili, R. M. Falcon Ganfornina, J. Nunez Valdes, S. Georgiev, and others.

OUTLINE OF GENOMATHEMATICS

Santilli has dedicated his research life to the representation of irreversible systems due to their evident significant for energy-releasing processes. The research initiated during his graduate studies at the University of Turin, Italy, in the mid 1960s. Santilli recognized that 20th century theories are reversible over time because based on Lie algebras whose product is invariant under anti-Hermiticity [A, B] = AB - BA = -[A, B]⁺. As a condition to assure irreversibility over time, Santilli therefore searched for a covering of Lie algebras whose product is neither antisymmetric nor symmetric, selected Lie-admissible algebras according to the American mathematician A. A. Albert [16], and published in 1967 paper [17] the embedding of Lie algebras into covering Lie-admissible algebras illustrated with the first formulation of (p,q)-deformations (A, B) = pAB - qBA that was later on followed by a large number of papers in the simpler deformations AB - qBA.

Also during his Ph. D. thesis, Santilli proved a No Reduction Theorem according to which macroscopic irreversible systems cannot be consistently reduced to a finite number of quantum mechanical particles all in reversible conditions.Alternatively, Santilli No reduction Theorem implies that known thermodynamical laws cannot be consistently eliminated via the reduction of systems to quantum mechanical events. The theorem established that macroscopic irreversibility, rather than "disappearing" at the elementary particle level, originate instead at the ultimate structure of physical systems, thus establishing the need for their quantitative studies preferably as a covering of reversible quantum descriptions.

Mathematical maturity in the formulation of Lie-admissible treatments was initiated with paper [1] of 1993 in which Santilli discovered that, besides the unit of a field being an arbitrary non-singular quantity, the multiplication can be ordered either to the right n>m or to the left n<m while preserving all axioms of a field. This allowed Santilli to introduce the forward genofields F^>(n^>, >, I^>) with forward real, complex or quaternionic genonumbers n^> = n⋅I^>, forward genoproduct n^>>m^> = n^>⋅S⋅m^>, and forward genounit I^> = 1/ S, with conjugate backward genofields^<F(^<n, <, ^<I), ^<I = 1 / R. The property R≠S then assures the lack of violation of causality and other physical laws in the representation of energy releasing processes.

Genomathematics is characterized by the dual, forward and backward genotopies of 20th century mathematics over a field of characteristic zero, thus including the forward and backward genotopies of numbers, functional analysis, differential calculus, metric spaces, algebras, geometries, topologies, etc. which genotopies, to be consistent, must be formulated over a basic genofield. The Lie-admissible covering of Lie algebras [6(b)] is characterized by two universal enveloping genoassociative algebras, one for the product ordered to the right and one to the left, resulting in a geno-bimodular structure namely, a bimodular structure with action to the right A>|) = HS|) and to the left (|<H = (|RA, R≠S. Such a geno-bimodule characterizes a Lie-admissible algebra with product (A, B) = ARB - BSA first identified by Santilli in 1978 [6], with genoequations i dA/dt = (A, H) = A<H - H<A = ARH - HSA. These equations are at the foundation of the genotopic branch of hadronic mechanics, or genomechanics, that has already permitted the scientific identification and industrial development of new fuels and energies by U. S. publicly traded companies and their foreign associates.

A presentation of Lie-admissible formulations as of 1995 is available in Refs. [6,8,9,10, 11]. The latest presentation is available in memoir [18] of 2006. Among a numbers of independent studies in Lie-admissible treatments we mention refs. [19-27] by the physicists S. Adler, P. Roman, H. C. Myung, J. Ellis, E. Mavromatos, D. V. Nanopoulos, J. Dunning Davies, A. A. Bhalekar, V. M. Tangde and others. Additional more recent references are available in the Proceedings of the Third International Conference on Lie-admissible Treatments of Irreversible Processes[28].

OUTLINE OF HYPERMATHEMATICS

Hypermathematics is the most general mathematics that can be currently conceived by the human mind. It is based on multi-valued, forward and backward hyperlifting of genomathematics via hyperoperations called "hopes" verifying the axioms of the Hv hyperstructures by the Greek mathematician T. Vougiouklis (see paper [29] of 2013 and references quoted therein). Hypermathematics admits as particular cases the preceding geno- and iso-mathematics and characterizes the hyperstructural branch of hadronic mechanics, also calledhypermechanics [9, 18]. It should be noted that hypermathematics achieves compatibility with our sensory perception because it is multi-valued rather than "multi-dimensional" as it is the case for various theories. A primary aim of hypermathematics is to stimulate a new era in the study of biological structures, such as the DNA, whose complexity is outside realistic capabilities of 20th century mathematics as well as isomathematics and genomathematics. Monograph [30] by I. Gandzha and J. V. Kadeisvili is suggested for a general independent review.

FINANCIAL SUPPORT FOR PARTICIPATION

Scientist interested in participating are suggested to apply for financial support by sending via email a one page motivation and the CV to
The R. M. Santilli Foundation
http://www.santilli-foundation.org/
http://www.world-lecture-series.org/
Email: board(at)santilli-foundation(dot)org

Deadline for applications: August 15, 2014

[1] R. M. Santilli, "Isonumbers and Genonumbers of Dimensions 1, 2, 4, 8, their Isoduals and Pseudoduals, and ;Hidden Numbers; of Dimension 3, 5, 6, 7," Algebras, Groups and Geometries Vol. 10, 273 (1993),
http://www.santilli-foundation.org/docs/Santilli-34.pdfbr>

[2] Chun-Xuan Jiang, Foundations of Santilli Isonumber Theory, International Academic Press (2001),
http://www.i-b-r.org/docs/jiang.pdf

[3] C. Corda, (a) "Introductionto Santilli iso-numbers American Institute of Physics Conf. Proc. 1479 , 1013 (2012); and (b) "Introduction to Santilli iso-mathematics", American Institute of Physics Conf. Proc. 1558, 685 (2013)
http://www.santilli-foundation.org/docs/Corda-iso-mathematics.pdf

[4] A. A. Bhalekar, "Santilli's New Mathematics for Chemists and Biologists. An Introductory Acount,", CACAA, 3(1), 15-86 (2014)
http://www.santilli-foundation.org/docs/Bhalekar-Math-2013.pdf

[5] R. M. Santilli, "Nonlocal-Integral Isotopies of Differential Calculus, Mechanics and Geometries," in Isotopies of Contemporary Mathematical Structures, P. Vetro Edi- tor, Rendiconti Circolo Matematico Palermo, Suppl. Vol. 42, 7-82 (1996), http://www.santilli-foundation.org/docs/Santilli-37.pdf

[6] R. M. Santilli, Foundation of Theoretical Mechanics, Volume (a) I (1978) and (b) II (1982) Springer-Verlag, Heidelberg, Germany,
http://www.santilli-foundation.org/docs/Santilli-209.pdf
http://www.santilli-foundation.org/docs/santilli-69.pdf

[7] J. V. Kadeisvili, "The Rutherford-Santilli Neutron," Hadronic Journal , 31, 1-114 (2008)
http://www.i-b-r.org/Rutherford-Santilli-II.pdf

[8] R. M. Santilli, Isotopic Generalizations of Galilei and Einstein Relativities, Volumes (a) I and (b) II, International Academic Press (1991) ,
http://www.santilli-foundation.org/docs/Santilli-01.pdf
http://www.santilli-foundation.org/docs/Santilli-61.pdf

[9] R. M. Santilli, Elements of Hadronic Mechanics, Volumes 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

[10] R. M. Santilli, Hadronic Mathematics, Mechanics and Chemistry,, Vol. I [a], II [b], III [c], IV [d] and [e], International Academioc Press, (2008), available as free downlaods from http://www.i-b-r.org/Hadronic-Mechanics.htm

[11] R. M. Santilli, Foundations of Hadronic Chemistry, with Applications to New Clean Energies and Fuels, Kluwer Academic Publishers (2001),
http://www.santilli-foundation.org/docs/Santilli-113.pdf .

[12] S. Sourlas and G. T. Tsagas, Mathematical Foundation of the Lie-Santilli Theory, Ukraine Academy of Sciences (1993),
http://www.santilli-foundation.org/docs/santilli-70.pdf

[13] J. V. Kadeisvili, Santilli's Isotopies of Contemporary Algebras, Geometries and Relativities, Ukraine Academy of Sciences, Second edition (1997),
http://www.santilli-foundation.org/docs/Santilli-60.pdf

[14] R. M. Falcon Ganfornina and J. Nunez Valdes, Fundamentos de la Isoteoria de Lie- Santilli, International Academic Press (2001),
http://www.i-b-r.org/docs/spanish.pdf

[15] S. Georgiev and J. V. Kadeisvili, Foundations of the IsoDifferential Calculus, Vol. I, to appear.

[16] A. A. Albert,"Onthepower-associativerings,"Trans.Amer.Math.Soc.,64, 552-593 (1948).

[17] R. M. Santilli, "Embedding of Lie-algebrasintoLie-admissiblealgebras, " Nuovo Ci- mento, 51, 570 (1967)
http://www.santilli-foundation.org/docs/Santilli-54.pdf

[18] R. M. Santilli, "Lie-admissible invariant representation of irreversibility for matter and antimatter at the classical and operator levels," Nuovo Cimento B 121, 443 (2006),
http://www.santilli-foundation.org/docs//Lie-admiss-NCB-I.pdf

[19] S. Adler, Phys. Rev. 17, 3212 (1978)

[20] P. Roman and R. M. Santilli, "A Lie-admissible model for dissipative plasma," Let- tere Nuovo Cimento 2, 449-455 (l969)

[21] H. C. Myung and R. M. Santilli, "Bimodular-genotopic Hilbert Space Formulation of the Interior Strong Problem," Hadronic J. 5, 1367-1404 (l982).

[22] J. Fronteau, R. M. Santilli andA. Tellez-Arenas, "Lie-admissible structure of statis- tical mechanics," Hadronic Journal 3, 130-145 (1979).

[23] J. Ellis, N. E. Mavromatos and D. V. Nanopoulos in Proceedings of the Erice Sum- mer School, 31st Course: From Superstrings to the Origin of Space-Time, World Scientific (1996).

[24] J. Dunning-Davies, "Lie-admissible formulation of thermodynamoical laws," ZZZZZZZZZZ to be completed.

[25] J. Dunning-Davies, (a) "Thermodynamics of Antimatter via Santilli's Isodualities", Found. Phys. Lett., 12(6), 593-599 (1999); (b) "The Thermodynamics Associated with Santilli's Hadronic Mechanics",, Prog. Phys., 4, 24-26 (2006).

[26] A. A. Bhalekar, (a) "Santilli's Lie-admissible mechanics. The only option commensurate with irreversibility and nonequilibrium thermodynamics", AIP Conf. Proc., 1558, 702-706 (2013); (b) "Geno-nonequilibrium thermodynamics.I. Background and preparative aspects", CACAA, 2(4), 313-366 (2013); (c) "On the Geno-GPITT framework", AIP Conf. Proc., 1479, 1002-1005 (2012); (c) "Studies of Santilli's isotopic, genotopic and isodual four directions of time", AIP Conf. Proc., 1558, 697-701 (2013).
http://www.santilli-foundation.org/docs/RhodesGreece-AAB.pdf
http://www.santilli-foundation.org/docs/TimeArrows-AAB-2.pdf
http://www.santilli-foundation.org/docs/ANIL%20BHALEKAR-Rhodes-I.pdf
http://www.santilli-foundation.org/docs/ANIL%20BHALEKAR-Rhodes-II.pdf

[27] V. M. Tangde, (a) "Advances in hadronic chemistry and its applications", CACAA, 2(4), 367-392 (2013); (b) "Elementary and brief introduction of hadronic chemistry", AIP Conf. Proc., 1558, 652-656 (2013).
http://www.santilli-foundation.org/docs/paper244.pdf
http://www.santilli-foundation.org/docs/Tangde.pdf
http://www.santilli-foundation.org/docs/hadronic-chemistry-2013.pdf
http://www.santilli-foundation.org/docs/Vijay%20Tangde-Hadronic-Chemistry.pdf

[28] C. Corda, editor, {\it Proceedings of the 2011 International Conference on Lie-admissible Formulations for Irreversible Processes,} Kathmandu University, Nepal (2011)
http://www.santilli-foundation.org/docs/Nepal-2011.pdf

[29] R. M. Santilli and T. Vougiouklis, (a) "Isotopies, Genotopies, Hyperstructures and their Applications,"Prooc. Int. Workshop in Monderoduni: New Frontiers in Hyperstructures and Related Algebras, Hadronic Press (1996), 177-188; and (b) "Lie-admissible hyper algebras," Italian Journal of Pure and Applied Mathematics, Vol. 31, pages 239-254 (2013)
http://www.santilli-foundation.org/Lie-adm-hyperstr.pdf

[30] I. Gandzha and J Kadeisvili, New Sciences for a New Era: Mathematical, Physical and Chemical Discoveries of Ruggero Maria Santilli, Sankata Printing Press, Nepal (2011),
http://www.santilli-foundation.org/docs/RMS.pdf

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