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THE NEW ISO-, GENO-, AND HYPER-MATHEMATICS OF HADRONIC MECHANICS

Original content uploaded February 15, 1997. Revisions uploaded on February 22 and March 29, April 4, and June 15, 1997 thanks to numerous critical comments by various visitors, which are acknowledged with gratitude. Additional critical comments should be sent to ibr@gte.net and will be appreciated.

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This section lists open research problems in pure and applied mathematics. All interested mathematicians in all countries, including graduate students, are welcome to participate in the research.

Following the introductory section, individual open problems are presented via:

Papers resulting from the proposed research will be listed at the end of each section. We assume the visitor of this site is aware of the inability at this time to have technical symbols and formulae in the www. Therefore, the symbols used in the presentation below have been rendered as simple as possible and they do not correspond to the symbols generally used in the technical literature.


OPEN RESEARCH PROBLEMS IN MATHEMATICS

CONTENTS

I. INTRODUCTION

II. OPEN RESEARCH PROBLEMS IN NUMBER THEORY

III. OPEN RESEARCH PROBLEMS IN GEOMETRIES

IV. OPEN RESEARCH PROBLEMS IN FUNCTIONAL ANALYSIS

V. OPEN RESEARCH PROBLEMS IN LIE-SANTILLI THEORY

VI. OPEN RESEARCH PROBLEMS IN TOPOLOGY

VII. MISCELLANEOUS OPEN RESEARCH PROBLEMS




I. INTRODUCTION
Studies initiated in the late 1970's under the support of the Department of Energy at Department of Mathematics of Harvard University by the theoretical physicists Ruggero Maria Santilli have indicated that current mathematical knowledge is generally dependent on the assumption of the simplest conceivable unit e which assumes either the numerical value e = +1, or the n-dimensional unit form e = diag. (1, 1, . . . , 1).

Systematic studies were then initiated for the reformulation of contemporary mathematical structures with respect to a generalized unit E of the same dimension of the original unit e (e.g., an nxn matrix) but with an arbitrary functional dependence on a local chart r, its derivatives with respect to an independent variable (e.g., time t) v = dr/dt, a = dv/dt, and and any other needed variable,

under the conditions of being everywhere invertible and admitting e as a particular case.

Jointly, the conventional associative product axb among generic quantities a, b (e.g., numbers, vector fields, operators, etc.) is lifted into the form

where T is fixed, and axT , Txb are the original associative products, in which case the quantity E = 1/T is indeed the correct left and right unit of the new theory, E*a = a*E = a for all elements a of the original set. To achieve consistency, the dual liftings must be applied to the totality of the original mathematical structure.

By conception and construction, the new formulations are locally isomorphic to the original ones for all positive-definite generalized units E > 0. As a result, maps (3) do not yield "new mathematical axioms", but only "new realizations" of existing mathematical axioms and, for this reason, they were called "isotopic" in the Greek meaning of being "axiom-preserving".

When E is no longer Hermitean (e.g., it is nowhere singular and real-valued but non-symmetric), then we have the general loss of the original axioms in favor of more general axioms (see below for examples) and, for this reason maps (3) were called "genotopic" from the Greek meaning of being "axiom inducing". In this case we have two different units <E = 1/R and E> = 1/S, generally interconnected by the conjugation <E = (E>)› with corresponding ordered products to the left and to the right,

Additional classes of mathematical structures occur when the generalized units are multivalued, or subjected to anti-isomorphic conjugation (see below).

The new lines of mathematical inquiries emerged from these studies imply novel formulations of: number theory, functional analysis, differential geometries, Lieís theory, topology, etc. For example, ordinary numbers and angles, conventional and special functions and transforms, differential calculus, metric spaces, enveloping algebras, Lie algebras, Lie groups, representation theory, etc., must be all reformulated under isotopies for the generalized product a*b = axTxb in such a way to admit E(t, r,v, a, ... ) = 1/T as the new left and right unit, and a more general setting occurs under genotopies.

To illustrate the nontriviality of these .liftings it is sufficient here to recall that Lie's theory with familiar product [A, B] = AxB -- BxA (where A, B are vector fields on a cotangent bundle or Hermitean operators on a Hilbert space, and AxB, BxA are conventional associative products), is linear, local-differential and potential-Hamiltonian, thus possessing clear limitations in its applications.

The isotopies and genotopies of Lie's theory , called Lie-Santilli isotheory and genotheory, respectively, include the corresponding liftings of universal enveloping algebras, Lie algebras, Lie groups, transformation and representation theories, etc. and are based on the following corresponding generalized products first proposed by Santilli in 1978

where product (5a) is "axiom preserving" in the sense of preserving the original Lie axioms, while product (5b) is "axiom-inducing" in the sense of violating Lie's axioms in favor of the more general axioms of Albert's Lie-admissible algebras (a generally nonassociative algebra U with abstract elements a, b, c, and product ab is said to be Lie-admissible when the attached antisymmetric algebra U_, which is the same vector space as U equipped with the product [a, b]U = ab - ba, is Lie).

As expected, the theories with products (5a) and (5b) have been proved to provide an effective characterization of nonlinear, nonlocal and nonhamiltonian systems of increasing complexity (the former applying for stable-reversible condiions, and the latter for open-irreversible conditions, see the next Web Page 19). Their consistent treatment requires corresponding new mathematics, called iso- and genlo-mathematics, respectively. For instance, it would be evidently inconsistent to define an algebra with generalized unit E(t, r, ...) = E› over a conventional field of numbers with trivial unit e = +1, and the same happens for functional analysis, differential calculus, geometries, etc.

The studies initiated by Santilli were continued by numerous scholars including Gr. Tsagas, D. S. Sourlas, J. V. Kadeisvili, H. C. Myung, S. Okubo, S. I. Vacaru, B. Lin, D. Rapoport-Campodonico, R. Ohemke, A. K. Aringazin, M. Nishioka, G. M. Benkart, A. Kirukhin, J. Lohmus, J. M. Osborn, E. Paal, L. Sorgsepp, N. Kamiya, P. Nowosad, D. Juriev, C. Morosi, L. Pizzocchero,R.Aslander, S. Keles,and other scholars. A comprehensive list of contributions in related fields up to 1984 can be find in Tomber's Bibliography in Nonassociative Algebras, C. Baltzer et al., Editors, Hadronic Press, 1984. A bibliography on more recent contributions can be found in the monograph by J. Lohmus, A. Paal and L. Sorgsepp, Nonassociative Algebras in Physics, 1994, Hadronic Press (see Advanced Titles in Mathematics in this Web Site).

The mathematical nontriviality of the above studies is also illustrated by the fact that, at a deeper analysis, isotopies and genotopies imply the existence of SEVEN DIFFERENT LIFTINGS of current mathematical structures with a unit, each of which possess significant subclasses, as per the following outline:

1) ISODUAL MATHEMATICS. It is characterized by the so-called isodual map, first introduced by Santilli in 1985 (see [I-1] for a recent account), given the lifting of a generic quantity a (a number, vector-field, operator, etc.) into its anti-Hermitean form

which must also be applied, for consistency, to the totality of the original structure. This implies that the isodualities of conventional mathematics, called isodual mathematics, have a "negative-definite unit" and related new product, according to the liftings The above maps permitted the identification of new numbers with negative unit -1 (see Problem 1 below). In turn, the identification of new numbers permitted the identification of new spaces, algebras, geometries, etc. Note that in this first lifting the unit remains the number 1 and only changes its sign.

Since the norm of isodual numbers is negative-definite, isodual mathematics has resulted to be useful for a novel representation of antimatter (see Page 19).

The visitor should be aware that contemporary mathematics appears to be inapplicable for a physically consistent representation of antimatter at the CLASSICAL level, with corresponding predictable shortcomings at the particle level. In fact,we only have today one type of quantization, e.g., the symplectic quantization. As a result, the operator image of contemporary mathematical treatments of antimatter does not yield the needed charge conjugate state. At any rate, the map from matter to antimatter must be anti-automorphic (or, more generally,anti-isomorphic), as it is the case for charge conjugation in second quantization.

The only known map verifying these conditions at all levels of treatment is Santilli's isodual map (6). This yields a novel classical representation of antimatter with a corresponding novel isodual quantization which does indeed yield the correct charge conjugate state of particles (see Web Page 19). Thus, the isodual mathematics resolves the historical lack of equivalence in the treatment between matter and antimatter according to which the former is treated at all levels, from classical mechanics to quantum, field theories, while the latter was treated only at the level of second quantization.

To understand the implications, the visitor should keep in mind that contemporary mathematics does not appear to be applicable for an effective treatment of antimatter, thus requiring its reconstruction in an anti-isomorphic form.

2, 3) ISOTOPIC MATHEMATICS AND ITS ISODUAL. The isotopic mathematics or isomathematics for short [I-1] is today referred to formulations for which the generalized unit E, called isounit, has a nontrivial functional dependence and it is Hermitian, E = E(t, r, v, ...) = E›. An important case is the Lie-Santilli isotheory with basic isoproduct (5a).

The isodual isotopic mathematics is the image of the isomathematics under maps (6) and therefore has the unit isod(E) = -E› = -E.

These structures have been classified by the theoretical physicist J. V. Kadeisvili in 1991 into:

At this writing only Classes I, II and III have been preliminarily studied, while the remaining Classes IV and V are vastly unknown.

4, 5) GENOTOPIC MATHEMATICS AND ITS ISODUAL. The isotopies were proposed in 1978 as particular cases of the broader genotopies [I-1], which are characterized by two different generalized units <E = 1/R and E> = 1/S for the genomultiplication to the left and to the right according to Eqs. (4). The resulting genotopic mathematics, or genomathematics for short, is given by a duplication of the isomathematics, one for ordered products to the left and the other to the right.

The isodual genomathematics is the isodual image of the preceding one, and it is characterized by the systematic application of map (6) to each of the left and right genomathematics.

In an evening seminar delivered at ICM94 Santilli proved that the genotopies can also be axiom-preserving and can therefore provide a still broader realization of known axioms. The proof was presented for product (5b) which, when considered on ordinary spaces and fields with the conventional unit e, is known to verify Albertís axiom of Lie-admissibility. The same product was proved to verify the Lie axiom when each of the two terms A<B and B>A is computed in the appropriate genoenvelope and genofield with the corresponding genounit.

6, 7) HYPERSTRUCTURAL MATHEMATICS AND ITS ISODUAL. At the IBR meeting on multivalued hyperstructures held at the Castle Prince Pignatelli in August 1995, the mathematician Thomas Vougiouklis and R. M. Santilli presented a new class of hyperstructures, those with well defined hyperunits characterized by hyperoperations. A subclass of the latter hyperstructures important for applications is that with hyperunits characterized by ordered sets of non-Hermitean elements,

with corresponding multivalued hypermultiplications a<b = axRxb and a>b = axSxb. The latter structures evidently permit a third layer of generalized formulations which are also axiom-preserving when treated with the appropriate hypermathematics. The isodual hypermathematics is the isodual image of the above hypermathematics and is therefore itself multivalued.

Needless to say, the above studies are in their first infancy and so much remains to be done.

The material of this Web Page is organized following the guidelines of memoir [I-1] according to which there cannot be really new applications without really new mathematics, and there cannot be really new mathematics without new numbers. We shall therefore give utmost priority to the lifting of numbers according to the above indicated seven different classes. All remaining generalized formulations can be constructed from the novel base fields via mere compatibility arguments.

We shall then study the novel spaces and geometries which can be constructed over the new fields because geometries have the remarkable capability of reducing the ultimate meaning of both mathematical and physical structures to primitive, abstract, geometric axioms.

We shall then study: the generalized functional analysis which can be constructed on the preceding structures, beginning from Kadeisviliís new notions of continuity; the all fundamental Lie-Santilli theory; the underlying novel Tsagas-Sourlas integro-differential topology; and other aspects.

Only primary references with large bibliography are provided per each section. Subsequent calls to references of preceding sections are indicated with [I-1], [II-1], etc.

REFERENCE OF SECT. 1: We recommend to study the following memoir and some of the large literature quoted therein

[I-1] R. M. Santilli, Nonlocal-integral isotopies of differential calculus, mechanics, and geometries, Rendiconti Circolo Matematico Palermo, Supplemento No. 42, pages 7-83, 1996.

See also

[I-2] J. V. Kadeisvili, An Introduction to the Lie-Santilli Isotopic theory, Mathematical Methods in Applied Sciences, Vol. 19, pages 1349-1395, 1996.

[I-3] J. V. Kadeisvili, "Santilli's Isotopies of Contemporary Algebras, Geometries and Relativities, Second Edition, Ukraine Academy of Sciences, Kiev, 1997 (First Edition 1992).

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VERA VERITAS

II: OPEN RESEARCH PROBLEMS IN NUMBER THEORY

PROBLEM II.1: STUDIES ON THE ISODUAL NUMBER THEORY PROBLEMS II-2/II-3: STUDIED IN ISOFIELD THEORY AND ITS ISODUAL PROBLEM II-4/II-5: STUDIES ON GENONUMBERS THEORY AND ITS ISODUAL PROBLEM II-6/II-7: STUDIES IN HYPERNUMBER THEORY AND ITS ISODUAL

RECENT PAPERS APPEARED IN THE RESEARCH PROPOSED IN SECT. II

ON SANTILLI'S ISOTOPIES OF THE THEORY OF REAL NUMBERS,
COMPLEX NUMBERS, QUATERNIONS AND OCTONIONS
N. Kamiya
Department of Mathematics
Shimane University
Matsue 690, Japan
"New Frontiers in Algebras, Groups and Geometries",
Gr. Tsagas, Editor, Hadronic Press, Palm Harbor (1996), pp. 523-551.

A CHARACTERIZATION OF PSEUDOISOFIELDS
N. Kamiya
Department of Mathematics
Shimane University
Matsue 690, Japan
and
R. M. Santilli
Institute for Basic Research
P., O. Box 1577
Palm harbor, FL 34682, U.S.A.
ibr@gte.net
"New Frontiers in Algebras, Groups and Geometries",
Gr. Tsagas, Editor, Hadronic Press, Palm Harbor (1996), pp. 559-570.

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III: OPEN PROBLEM IN GEOMETRIES

PROBLEM III.1: STUDIES IN ISODUAL GEOMETRIES

PROBLEMS III-2/III-3: STUDIES ON ISOGEOMETRIES AND THEIR ISODUALS

PROBLEM III-2/III-3: PROBLEMS III-4/III-5: STUDIES ON GENOGEOMETRIES AND THEIR ISODUALS PROBLEMS III-6/III-7: STUDIES ON HYPERGEOMETRIES AND THEIR ISODUALS

RECENT PAPERS APPEARED IN THE RESEARCH PROPOSED OF SECT. III

ISOAFFINE CONNECTION AND SANTILLI'S ISORIEMANNIAN METRIC ON AN ISOMANIFOLD
Gr. Tsagas
Department of Mathematics
Aristotle University
Thessaloniki 54006, Greece
Algebras, Groups and Geometries, Vol. 13, pages 149-169, 1996

STUDIES ON SANTILLI'S LOCALLY ANISOTROPIC AND INHOMOGENEOUS
ISOGEOMETRIES, I: ISOBUNDLES AND GENERALIZED ISOFINSLER GRAVITY
Sergiu I. Vacaru
Institute of Applied Physics
Academy of Sciencves of Moldova
5,. Academy Street
CHISINAU 2028, REPOUBLIC OF MOPLDOVA
Fax +3732-738149, E-address lises@cc.acad.md
In press at Algebras Groupos and Geometries, Vol. 14, 1997

CARTAN'S STRUCTURE EQUATIONS ON SANTILLI-TASGAS-SOURLAS ISOMANIFOLDS
Recept Aslander
Inonu Universitesi
Egitim Fakultesi
Matematik Egitimi Bolumu
44100 Malatya, Turkey
and
Sadik Keles
Inonu Universitesi
Fen-Edebiyat Fakultesi
Matematik Bolumi
44100 Malatya, Turkey
In press at Algebras, Groups and Geometries, Vol. 14, 1997

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IV: OPEN RESEARCH PROBLEMS IN FUNCTIONAL ANALYSIS

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V. STUDIES ON THE LIE-SANTILLI ISO-, GENO- AND HYPER-THEORIES AND THEIR ISODUALS

PROBLEM V-1: STUDIES ON THE ISODUAL LIE THEORY PROBLEM V-2/V-3: STUDIES ON LIE-SANTILLI ISOTHEORY AND ITS ISODUAL PROBLEM V-4/V-5: STUDIES ON LIE-SANTILLI GENOTHEORY AND ITS ISODUAL PROBLEMS V-6/V-7: STUDIES ON THE LIE-SANTILLI HYPERTHEORY AND ITS ISODUAL

RECENT PAPERS APPEARED IN THE RESEARCH PROPOSED OF SECT. V

STUDIES ON THE CLASSIFICATION OF LIE-SANTILLI ALGEBRAS
Gr. Tsagas
Department of Mathematics
Aristotle University
Thessaloniki 54006, Greece
Algebras, Groups and Geometries Vol., 13, pp. 129-148, 1996

AN INTRODUCTION TO THE LIE-SANTILLI NONLINEAR, NONLOCAL AND NONCANONICAL ISOTOPIC THEORY
J. V. kadeisvili
Institute for Basic Research
P. O. Box 1577
Palm Harbor, FL 34682 U.S.A. (br> ibr@gte.net
Mathematical Methods in applied sciences Vol. 19, pp.1349-1395, 1996

REMARKS ON THE LIE-SANTILLI BRACKETS
M. Nishioka Yamaguchi University
Yamaguchi 753, Japan
"New Frontiers in Algebras, Groups and Geometries",
Gr. Tsagas, Editor, Hadronic Press, Palm Harbor (1996), pp.553-558.

COMMENTS ON A RECENT NOTE BY MOROSI AND PIZZOCCHERO ON ON SANTILLI'S ISOTOPIES OPF LIE'S THEORY
J. V. kadeisvili
Institute for Basic Research
P. O. Box 1577
Palm Harbor, FL 34682 U.S.A. (br> ibr@gte.net
Submitted for ppublication

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