Ruggero Maria Santilli

P. O. Box 1577, Palm Harbor, FL 34682

Curriculum Vitae

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Last revision March 11, 2003

RUDIMENTS OF HADRONIC MECHANICS

**1. THE LARGE SCIENTIFIC IMBALANCE OF THE 20-TH CENTURY CAUSED BY NONLOCAL INTERACTIONS.**

One of the largest
scientific imbalances of the 20-th century has been the adaptation of
nonlocal-integral systems to pre-existing mathematics that
is notoriously local-differential. This approach has created serious
limitations and controversies that have remained unresolved despite attempts conducted
over a century, such as: 1) lack of numerically exact representations of
chemical characteristics; 2) historical
inability to achieve an exact representation of nuclear magnetic moments and other nuclear data; 3) absence of an exact representation of the Bose-Einstein correlation without *
ad hoc* free parameters to fit data; and other unresolved problems throughout all branches of science (see the remaining parts of this web site).

In all
these cases we have the mutual overlapping/penetration of particles and/or their
wavepackets at distances of the order of 10^{-13} cm, which conditions are strictly
nonlocal-integral because the volume of wave overlapping cannot be reduced to a finite set of isolated points. As such, these nonlocal conditions are beyond the exact applicability of the
mathematical structure, let alone physical laws of quantum mechanics.

It is appropriate here to recall that quantum mechanics and its underlying mathematics
permitted a * numerically exact* representation of * all* experimental data of
the **hydrogen atom**. By contrast, the same mathematics and quantum laws have
not permitted an equally exact representation of the experimental data of the
**hydrogen molecule,** since a historical 2 % of molecular binding energy has been
missed for about one century (we exclude here the use of
screened Coulomb laws since they can only be achieved via nonunitary transforms of the Coulomb law,
thus exiting the class of equivalence of quantum chemistry).

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One, among too many illustrations, is given by quantum treatments of the Bose-Einstein correlation, whose experimental data soon resulted in being structurally beyond the basic axioms of quantum mechanics because of the need of off-diagonal terms in the vacuum expectation values for the very formulation of the correlation, which terms are impossible under the axiom of quantum expectation values (see Refs. (112,113) for detail). Despite that, supporters of quantum mechanics introduce new parameters deprived of any physical meaning (the so-called "chaoticity parameters"), achieve in so doing a fit of the experimental data, and claim the exact validity of quantum mechanics for the Bose-Einstein correlation. As we shall see in these pages, numerous similar examples exist throughout all of quantitative sciences, including particle physics, nuclear physics, superconductivity, solid state physics, chemistry, biology, astrophysics and cosmology.

Besides prohibiting basic scientific advances in front of clear limitations of quantum mechanics, and causing major departures of physics from reality (as it has been the case for hadron physics), in this web site we shall show that the above debatable support for quantum mechanics has been directly responsible for our current alarming environmental problems. In fact, as we shall see, the resolutions of current environmental problems (such as recycling of nuclear waste, new clean energies, etc.) mandate the surpassing of quantum theories.

In reality, quantum mechanics has an axiomatic structure that can only be qualified as "majestic" (see next section) and an "exact" validity beyond doubt for the arena of its original conception (the hydrogen atom). Nevertheless, any claim of the universal validity of quantum mechanics is sheer scientific corruption, particularly when proffered by qualified physicists, because it is and it will be the fate of all present and future theories to have clear limitations.

In view of the above condition of physics, one of the biggest scientific challenges of the 20-th century has been the identification of at least ONE condition in which the limitations of quantum mechanics are so evident to discredit opposite views. Such a case is depicted in this figure and it is given by the following dichotomy:

-> The EXACT validity of quantum mechanics for ONE hydrogen atom (top view), compared to

-> the APPROXIMATE validity of the same theory for TWO hydrogen atoms bonded into the H-molecule (bottom view).

In fact, quantum mechanics has been able to represent ALL experimental data of the **H-atom** to any desired accuracy. By comparison, the rigid application of the same quantum axioms has prohibited in over seventy five years of research the achievement of an equally exact representation of the binding energy of the **H-molecule,** due to the historical error of about 2%. Such an error is often dismissed as "small." However, 2% of the binding energy of the H-molecule corresponds to about 1,000 Kcal/mole, while the energy released in the formation of the water molecule is of about 50 Kcal/mole. Therefore, said 2% error implies an error in thermochemical calculations that is 20 TIMES the quantities to be computed (see the recent monograph (59) for details).

In view of the above, supporters of quantum mechanics and chemistry have attempted the repetition of the same routine used for other limitations of the theories, namely, ad hod manipulations for the purpose of continuing to claim the exact validity of quantum theories. In fact, studies on molecular structures have been recently based on the so-called "screened Coulomb potentials," that is, the Coulomb potential V = e/r at the foundation of quantum mechanics adulterated into potentials of the type V' = f(r)e/r, where f(r) is an exponential or other function of the distance. In so doing, the representation of molecular data has been grossly improved, and the exact validity of quantum axioms for molecular structure has been claimed, a belief still widespread as of today.

However, this time supporters of the universal validity of quantum mechanics have lost any credibility for numerous reasons. The transition from the Coulomb to the screened Coulomb law can only be achieved via nonunitary transforms, thus exiting from the class of equivalence of quantum mechanics; the ad hoc functions and parameters in the screened potential have no visible physical or chemical meaning; and any screening of the Coulomb law loses the very notion of "quantum of energy," since the latter can only exist, as well known but not admitted, for stable orbits characterized by the real Coulomb law.

Irrespective of all the above problems, ** quantum mechanics and chemistry cannot represent the "strongly attractive force" between the identical valence electrons in molecular structures** (where the word "strong" has no connection with "strong interactions"). In fact, the two H-atoms are neutral; the two valence electrons have strongly repulsive Coulomb forces (because they have identical charges); and the molecular forces currently used in chemistry (exchange, van der Waals, and other forces) are known to be very weak and absolutely unable to represent the strength of valence bonds (see monograph (59) for technical details).

A similar situation exists for the **Cooper pair in superconductivity** that consists of two identical electrons in such a strong singlet bond that they have been detected to pass through potential barrier together. There is no doubt that quantum mechanics provides a representation of the **statistical ensemble** of Cooper pairs. However, any view that currently available descriptions provide a quantitative representation of the **structure of ONE Cooper pair,** including the ATTRACTIVE force between the two identical electrons, is outside the boundary of science. Numerous other cases of unequivocal insufficiencies of quantum mechanics will be reported later on.

*
This web site is intended to show that the ONLY position that can be considered "scientific" at this writing is that quantum mechanics and chemistry are not exactly valid for electron bonds, thus mandating the study of a suitable generalization with the opening of a new horizon of exciting basic research.*

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Since the sole difference between one isolated hydrogen atom and two
atoms coupled into the hydrogen molecule is given by the electron valence bonds, the above
occurrence illustrates the exact validity of quantum mechanics and related mathematics when the
systems can be effectively appropriate as being composed of point particles at sufficiently large mutual distances
(as it is the case for the structure of the hydrogen atom), while the same theory and related
mathematics have a merely *approximate* character when the systems contain interactions at short
distances (as it is the case for the mutual overlapping of the wavepackets of valence
electrons for the case of the hydrogen molecule).

The insufficiency is due to the fact that * the
mathematics of quantum mechanics is strictly local-differential in its structure, thus solely permitting the
representation of valence bonds as occurring between point particles interacting at large mutual distances.*
This representation is evidently valid in first approximation because electrons have indeed a
**point-like charge.** Nevertheless, electrons do not have a **point-like wave packet.** As a consequence, the local-differential representation is
insufficient because of the lack of treatment of the mutual penetration of the electron
wavepackets that has a strictly nonlocal-integral structure.

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When faced with the physical reality of this figure, supporters of the universal validity of quantum mechanics claim that the indicated new effects "do not exist" because **the electron has a point-like structure,** from which quantum mechanics is claimed to be exact for valence bonds and other similar conditions. The statement is an adulteration of the experimental reality that **the electron has a "point-like charge,"** because, if taken seriously, it implies that **the electron has a "point-like wavepacket,** a view that is outside science. When the extended character of the wavepackets of particles is admitted, contact, zero-range and nonpotential interactions due to deep wave-overlapping become unavoidable, with the consequential unavoidability of the limitations of quantum mechanics.

Unfortunately, the abstraction of **point-like wavepackets** has been widespread throughout most of the 20-th century physics with far reaching equivocal consequences for human knowledge. A vivid illustration is the assumption that **the hadronic constituents are point-like,** again, as a necessary condition to apply quantum mechanics. This basic belief has implied the current conception of hadrons as empty spheres with isolated point-particles in its interior, while, in the reality of nature, hadrons are the densest physical media measured by mankind until now. In fact, hadrons are composed of particles with extended wavepackets in condition of total mutual penetration (because the size of any wavepacket is of the same order of magnitude as the size of all hadrons).

When the extended character of the wavepackets of the hadronic constituents is admitted jointly with the extreme density of the medium in its interior, claims on the exact validity of quantum mechanics inside hadrons lose any scientific credibility, because they de fact imply that hadronic constituents orbit in the interior of the hyperdense hadrons in exactly the same way as the atomic constituent orbit in the empty vacuum around nuclei. It is evident that the dramatic differences in the structures of atoms and hadrons require corresponding differences in the mathematical and physical methods used for their treatment.

As we shall see in these pages, the assumption of the point-like abstraction of hadronic constituents, and the consequential exact validity of quantum mechanics in the interior of hadrons necessarily lead to the conjecture that the un-observable quarks are the physical constituents of hadrons. In turn, the inability to produce the constituents free, as it is the case for the nuclear and atomic constituents, has prohibited research on new **hadronic energies,** that is, originating from mechanisms in the interior of hadrons, as it occurs for **nuclear and atomic energies.**

On the contrary, the abandonment of point-like abstraction of hadronic constituents, the consequential inapplicability of quantum mechanics, and the validity within hadrons of covering theories, permit fundamentally new models of the structure of hadrons whose constituents can indeed be produced free, thus permitting basically** new clean hadronic energies** with mechanisms much similar to the energies obtainable by extracting the constituents of nuclear and atomic structures.

*
What does not appear to be sufficiently known is that the studies beyond quantum mechanics have long passed the level of "scientific" interest and are now under full "industrial" development with the investment of large corporate funds. The hope of these pages is therefore that of stimulating the participation of an increasing number of colleagues in the exciting and rewarding new scientific horizons.*

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It is then evident that a more adequate treatment of valence bonds in chemistry, as well as all
nonlocal-integral interactions in general, requires ** a new mathematics** that is partly
local-differential (to represent conventional Coulomb interactions) and partly
nonlocal-integral (to represent the overlapping of the wavepackets).

Additional physical
requirements establish that the needed mathematics must be ** nonlinear in
the wavefunction,** thus preventing the use of conventional quantum mechanics because
nonlinear Schroedinger's equations violate the superposition principle with
consequential inapplicability to composite systems [see references (45,46)].
It should be indicated that numerous "nonlocal interactions" exist in the physical
literature, but they have generally been adapted to be representable with a local
potential in a Hamiltonian (for theoretical and experimental studies on this type of
nonlocality, see, e.g., C. A. C. Dreismann (220-223) and references quoted
therein). However, the interactions occurring in valence bonds are dramatically more
general than the above inasmuch as they occur in the finite volume of wave overlappings
that, as indicated earlier, is not reducible to a finite number of isolated points.

Moreover, the interactions here considered are of **
nonpotential type** (that is, not admitting of a potential energy) because they are of contact, zero-range nature, thus prohibiting a
consistent representation with a Hamiltonian, let alone a potential. More
specifically, the granting of a potential energy to the deep wave-overlappings of
valence bonds would be the same as granting potential energy to the resistive force
experienced by a spaceship during re-entry in our atmosphere.

All in all, one century of failed attempt to achieve an exact representation of
molecular data, and unresolved problematic aspects and controversies in all branches of science, establish beyond credible doubt that the interactions occurring in valence bonds cannot be exactly treated via quantum mechanics and chemistry, thus mandating the construction of new disciplines.

In this web, site we shall report the research on the construction of a generalization of quantum mechanics under the name of **hadronic mechanics** and its underlying new mathematical structure, for the axiomatically consistent and invariant representation of the indicated nonlocal, nonlinear and nonpotential systems. The name of "hadronic" mechanics was selected to illustrate the intended primary applicability of the new discipline, with the understanding that the same discipline is applicable to all other systems with nonlocal, nonlinear and nonpotential internal effects, such as nuclei, molecules and stars.

These web pages are devoted to the outline of **new structure models of hadrons, nuclei and molecules permitted by hadronic mechanics, their experimental verifications and their new clean energies and fuels.** Also, we shall not be in a position to review the new mechanics and its eight different branches to avoid a prohibitive length. A necessary pre-requisite for the technical understanding of these pages is a knowledge of the mathematical structure of hadronic mechanics as available in the recent memoir (226), and of the physical profiles as available in memoir (31) paper (44).

Due to their numbers, references have been divided into primary groups identified by square brackets [1], [2], etc., while individual references are identified by curl brackets (1), (2), etc. References on the initiation of new theories are listed in their chronological orders of appearance. It should be indicated that the references listed in this presentation are only those directly relevant to hadronic mechanics and/or its classical foundations, that is, those directly dealing with **the invariant treatment of nonlocal, nonlinear and nonpotential effects in classical and operator mechanics realized via a generalization of the basic unit of the theory and related multiplication. ** Additional large lists of references, including connections with other theories indirectly related to hadronic mechanics, are available in the monographs [7] and conference proceedings [8].

The author solicits the indication of additional references that should be listed in these pages, with particular reference to references in the *origination* of the various new theories and their novel mathematical structure. Such references will be listed provided that they are directly related to hadronic mechanics and/or its classical foundation, namely, they are based on the *generalization of the basic unit,* or provide some alternative formulation of nonlocal, nonlinear and nonpotential interactions with a rigorously proved invariance.

Independent scientists are solicited to publish in Part VI of this web site their theories under the uncompromisable condition that they *are not* based on quantum mechanics or any of its versions within its class of unitary equivalence, and are based instead on *generalizations* of quantum mechanics. Qualified visitors are also solicited to present comments, irrespective of whether in favor or against this presentation, which comments will be published in Part VII of this web site provided that the authors release authorization for publishing their comments with their full identity and address. Authors should note that all presentations and/or comments intended for publication in Part VI and/or VII of this web site have to be sent already in htlm format to avoid costs and delays.

The ultimate aim of this presentation is the study of new clean energies and fuels much needed by our contemporary society, which clean energies and fuels can only be conceived and treated via a generalization of quantum mechanics, since all possible energies predicted by the latter have long been explored in the past century. Such a task has clear societal relevance, thus expecting the participation of the scientific community at large. It is therefore hoped that independent scientists answered this call with professional comments and presentations proved via the necessary mathematical elaboration, because any individual theory can at best contain one grain of truth. After all, nature is so complex and polyhedric to prevent its realistic reduction to any one single theory.

**2. CATASTROPHIC PHYSICAL AND MATHEMATICAL INCONSISTENCIES OF NONUNITARY GENERALIZATIONS OF QUANTUM MECHANICS.**

As indicated earlier, quantum mechanics has an axiomatic structure that is simply majectic due to its consistency on mathematical and physical grounds, despite its considerable diversification.

A reason for this consistency is its ** invariance,** namely, the preservation of numerical
values, physical laws and mathematical axioms under the time evolution of the theory.

Since all quantum time evolutions are expressible via Hermitean Hamiltonians, the above property is essentially expressed by the invariance under unitary transforms. In fact, quantum theories are expressible via a Hamiltonian H defined on a Hilbert space over the field C of complex numbers, as a result of which quantum mechanics is characterized by the following one-parameter Lie group of **unitary time evolution**

(2.1) U(t) = e^{ixHxt}, UxU^{+} = U^{+}xU = I, |(t)> = U(t)x|(0)>, H =
p^{2}/2m + V = H^{+}, ,

where "x" represents hereon the conventional associative product of matrices or operators. We then have the trivial invariance of the basic unit

(2.2) I -> I' = UxIxU^{+} = I.

To illustrate the less trivial invariance of numerical predictions, consider a quantum bound state with 1MeV energy at the initial time t = 0,

(2.3) Hx|(0)> = 1MeVx|(0)>.

The eigenvalue 1 MeV is then preserved at all subsequent times,

(2.4) Ux(Hx|(0)>) = (UxHxU^{+})x(UxU^{+})^{-1}x(Ux|(0)>) =

H'x|(t)> = Ux(1MeVx|(0)>) = 1MeVx|(t)>.

Similarly, it is easy to see that the original Hermiticity of H,

(2.5) (<(0)|xH^{+})x|(0)> = <(0)|x(Hx|(0)>), H^{+} = H,

is preserved at all times because

(2.6a) (<(0)|xU^{+})x(UxU^{+})^{-1}x(UxH^{+
}xU^{+})x|(t)>) = (<(t)|xH'^{+})x|(t) =

= <(t)|x(UxH^{+}xU)x(UxU^{+})^{-1}x(Ux|(0)>) =
= <(t)|x(H'x|(t)>),

(2.6b) H'^{+} = H'.

As a result, quantum mechanical quantities that are observable at the initial time remain observable at all times, as well known.

The axiomatic consistency of quantum mechanics then follows, e.g., its consistent application to measurements originating from the invariance of basic units, the invariance of numerical predictions, and the invariance of Hermiticity-observability. As graduate students in physics are expected to know well, the verification of causality and probability laws follows from the above invariances.

However, as pointed out in the preceding section, a necessary condition to achieve a consistent representation of nonlocal, nonlinear and nonpotential interactions occurring in deep wave-overlappings of particles at short distances is to exit from the class of equivalence of quantum mechanics, thus requiring a nonunitary theory. The serious difficulties in the generalization of quantum mechanics are then illustrated by the following

*THEOREM 2.1 (46): All classical theories with noncanonical time evolutions and all operator theories with nonunitary time evolutions do not possess invariant units, numerical predictions and observables, thus suffering from catastrophic mathematical and physical inconsistencies.*

Recall that all theories possessing unitary structure (2.1) are particular cases of quantum mechanics irrespective of whatever explicit form is assumed for the the Hamiltonian. As such, these theories are of no interest for these pages, not even indirect.
A fundamental necessary condition to represent nonlocal, nonlionear and nonpotential effects to exit from the class of equivalence of quantum mechanics is that the considered theory has the **nonunitary time evolution** when formulated on the Hilbert space over the field of complex numbers

(2.7a) |(t)> = U(t)x|(0)>, UxU^{+} ≠ I,

(2.7b) U(t) ≠ e^{ixHxt}.

The lack of conservation of the basic units is then inherent in the very definition of nonunitary transforms, e.g., for the unit of the basic Euclidean space we have the following **noninvarian ce**

(2.8) I = Diag. (1, 1, 1) -> I' = UxIxU^{+} ≠ I,

and it becomes transparent if one recalls that I = Diag (1, 1, 1) provides a dimensionless characterization of the assumed units of measurements, e.g., I = Diag. ( 1 cm, 1 cm, 1 cm).

Therefore,** theories with nonunitary structure cannot be consistently applied to experiments, because a pillar of the measurement theory is the invariance of the basic units.**

For the case of eigenvalues, assume again that the state considered at the initial time t = 0 has the energy of 1 MeV as in Eq. (2.3) and that 15 seconds later we have

(2.9) U(15)xU^{+}(15) = 1/50.

It is then easy to see that the energy of the same state 15 seconds later is 50 Mev, rather than 1 MeV, because

(2.10a) Ux(Hx|(0)>) = (UxHxU^{+})x(UxU^{+})^{-1}x(Ux|(0)>) =

= H'x(UxU^{+})^{-1}x|(15)> = Ux(1MeVx|(0)>) = 1MeVx(Ux|(0)>) = 1 MeVx|(15)>;

(2.10b) H'x|(15)> = (UxU^+)x1MeVx|(15)> = 50 MeVx|(15)>.

Therefore, **nonunitary theories do not admit invariant numerical predictions and, as such, have no known physical value.**

The lack of conservation of Hermiticity in time for nonunitary theories is consequential (this is called **Lopez's Lemma** (72,173)). In fact, under the initial assumption of Hermiticity of the Hamiltonian at time t = 0,

(2.11) (<(0)|xH^{+})x|(0)> = <(0)|x(Hx|(0)>), H^{+} = H,

we have its **lack** of conservation in time

(2.12) (<(0)|xU^{+})x(UxU^{+})^{-1}x(UxH^{+
}xU^{+})x|(t)>) = <(t)|xH'^{+})x|(t) ≠

≠
<(t)|x(UxH^{+}xU)x(UxU^{+})^{-1}x(Ux|(0)>) = <(t)|x(H'x|(t)>),

because, in view of the general lack of commutativity of T and H,

(2.13)TxH'^{+} \not = H'xT,

by therefore implying that

(2.14) H^{+} = H, H'^{+} ≠ H',

where one should note that the above expressions are computed in the *conventional* Hilbert space, thus preventing the time evolution of the complementary ket.**theories with a nonunitary time evolution do not possess consistent observables, and, as such, have no known physical meaning.**

In addition, the author had some of his graduate students prove that **theories with a nonunitary time evolution violate the principle of causality as well as probability laws,** and this explains additionally why they have no known physical contents.

The **mathematical inconsistencies** of nonunitary theories are equally catastrophic. In fact, nonunitary theories can indeed be consistently formulated over a conventional space defined on a conventional field at the initial time. However, at all subsequent times they lose the unit of the basic field, with consequential inapplicability of the carrier space and collapse of the entire mathematical structure. For more details, see Ref. (46) and various contributions reviewed therein, including **Okubo (171) no-quantization theorem** for generalizations of quantum mechanics with a nonassociative enveloping algebra.

In conclusion, *the achievement of a consistent treatment of nonlocal, nonlinear and nonpotential effects due to deep wave-overlappings of particles at short distances constitutes one of the most difficult problems of contemporary science inasmuch as it requires the achievement of a suitable generalization of the mathematics underlying quantum mechanics, and then a generalization of the physical theory itself.*

It should be stressed that ** numerous, rather popular contemporary theories suffer the catastrophic inconsistencies of Theorem 2.1,** including:

* All known deformations/generalizations of Lie algebras and quantum mechanics;

* All known q-, k- and other deformations of Lie algebras;

* All known particle and nuclear models with "imaginary potentials" (or non-Hermitean Hamiltonians);

* All known time evolutions with external terms as familiar in statistical mechanics;

* All known supersymmetries; >br>

* All known models of quantum gravity with a nonunitary structure;

* All known studies on Kac-Moody algebras; etc.

Researchers interested in studies on structural generalizations of quantum mechanics are suggested to study the literature on the above catastrophic inconsistencies (46,171-176) so as to avoid that their works are obsolete at the time of their printing.

**3. RUDIMENTS OF HADRONIC MECHANICS.**

The construction of a covering of quantum mechanics under the name of **hadronic mechanics** was proposed by R. M. Santilli (23,38) in 1978 for the specific objective of achieving a consistent quantitative treatment of nonlocal, nonlinear and nonpotential effects in deep wave-overlappings of particles at short distances.

Studies on hadronic mechanics were then conducted in over two decades by a considerable number of mathematicians, theoreticians and experimentalists, including:** A. O. E.
Animalu, A. K. Aringazin, R. Ashlander, C. Borghi, F. Cardone, J. Dunning-Davies, F.
Eder, J. Ellis, J. Fronteau, M. Gasperini, T. L. Gill, J. V. Kadeisvili, A. Kalnay, N.
Kamiya, S. Keles, C. N. Ktorides, M. G. Kucherenko, D. B. Lin, C.-X. Jiang, A. Jannussis,
R. Mignani, M. R. Molaei, N. E. Mavromatos, H. C. Myung, M. O. Nishioka, D. V. Nanopoulos,
S. Okubo, D. L. Rapoport, R. M. Santilli, D. L. Schuch, D. S. Sourlas, A. Tellez-Arenas, Gr. Tsagasm N.
F. Tsagas, E. Trell, R. Trostel, S. Vacaru, H. E. Wilhelm, W. Zachary,** and others (see Refs. [1-11]).

Since *there cannot be really new physical theories without really new mathematics,* Santilli's primary effort was the search of new mathematics specifically tailored for the problems considered. More particularly, since *there cannot be really new mathematics without new numbers,* Santilli devoted primary efforts in the identification of **new numbers** today known as **Santilli isonumbers, genonumbers, hypernumbers and their isoduals** [8-11], from which compatible mathematics and corresponding mechanics could be uniquely and unambiguously built.

In this way, Santilli's proposal was centered on the prior construction of **new mathematics** under the names of **isotopic and genotopic mathematics** (loc. cit.) today called **Santilli iso- and geno-mathematics** [loc. cit.], where the prefix "iso" was proposed and shall be used hereon in its Greek meaning of "axiom preserving," while the prefix "geno" is used in the Greek meaning of "axiom inducing."
Despite their rather broad character these new mathematics resulted to be excessively restrictive because they are single-valued. Therefore, Santilli (14) proposed in 1996 a particular form of multi-valued mathematics today known as **Santilli hypermathematics.**

Finally, hadronic mechanics needed the treatment of antimatter beginning at the purely *classical* level. This required the construction (11) in 1983 of yet new ,mathematics, today known as **santilli isodual mathematics,** that are anti-isomorphic images of the preceding mathematics from which to built corresponding mechanics suitable for the classical treatment of antimatter in such a way as to be equivalent to charge conjugation at the operator level.

The preceding new mathematics [2] then permitted the generalization (also called broadening and technically referred to as "lifting") of classical and quantum mechanics into **isomechanics [4], genomechanics [5], and hypermechanics [6]** for the treatment of matter in conditions of increasing complexity, and their **isodual mechanics [3]** for the treatment of antimatter, according to a structure outline in Figure 3.

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**Isomathematics** (23,38) is based on the lifting of the basic unit I of (classical and) quantum mechanics into a generalized unit I^ that is a nowhere singular but possesses un unrestricted, integro-differential, functional dependence on local coordinates q, momenta p, wavefunctions y, its derivatives, and any other needed variable,

(3.1) I -> I^(q, p, y,...) = 1/T > 0,

while jointly lifting the associative product AxB into the form

(3.2) AxB -> = A*B = AxTxA,

under which I^ is the correct right and left generalized unit of the new theory,

(3.3) IxA = AxI = A -> I^*A = (1/T)xTxA = A*I^ = A.

Since the new unit I^ and product A*B preserve the original axioms, they are called **isounit** and **isoproduct** respectively, while T is called the **isotopic element.** The lifting of the basic unit and product then imply the corresponding lifting of the totality of conventional mathematics into the covering isomathematics, including isonumbers, isospaces, isogeometries, isoalgebras, isogroups, isosymmetries, etc. The corresponding lifting of classical and quantum mechanics into the classical and operator isotopic mechanics is a mere consequence.

**Genomathematics (23,28) (hypermathematics )14))** are characterized by basic axioms (3.1)-(3.3) where I^ is invertible, non-Hermitean and single-valued (multi-valued) with corresponding compatible further lifting of conventional and iso-mathematics. The **isodual mathematics** (11) are based on the anti-isomorphic map, called **isoduality**,

(3.4) I^ -> I^^{d} = -I^^{+}(-q^{+}, -p^{+}, -y^{+},...),

which map characterizes the isodual images of conventional, iso-, geno-, and hyper-mathematics in a form ideally suited to represent antimatter.

The large time elapsed between the original proposal of 1978 to build hadronic mechanics and its applications was due to the lack of achievement of invariance. In fact, after having lifted all aspects of quantum into hadronic mechanics, invariance continued to elude for over a decade. The origin eventually resulted to be where expected the least, in the differential calculus.

Memoir (14) pointed out that, contrary to popular beliefs over centuries, the differential calculus is also crucially dependent on the assumed unit. When the generalized unit I^ is a constant, then the lifting of the differential calculus is trivial, because in this case d^q^ = dq (where d^ = Txd is the isodifferential and q^ = qxI^ is the isovariable). However, such a lifting is not trivial when the generalized unit has an explicit dependence on the differentiation variable, I^ = I^(q, ...) because in this case d^q^ ≠ dq. Full invariance of hadronic mechanics was achieved only in memoir (14) of 1996 following the construction of the iso-, geno-, hyper- and isodual differential calculus.

A recent general presentation of the above new mathematics and related generalized mechanics is available in memoir (226). Its study is recommended to visitors interested in acquiring a technical knowledge of the content of this site.

As of today, hadronic mechanics has clear experimental verifications in particle physics, nuclear physics, superconductivity, chemistry, biology, astrophysics and cosmology. Among them we recall:

* The first known optimization of the shape of extended
objects moving within resistive media via the optimal control theory (14);

* The exact representation from first principles of the anomalous behavior of the
meanlives of unstable particles with speed by Cardone et al (110,11);

* The exact representation from first principles of the
experimental data on the Bose-Einstein correlation by Santilli (112) and Cardone and
Mignani (113);

* The achievement of an exact confinement of quarks
by Kalnay (216) and Kalnay and Santilli (2.17) thanks to incoherence between the external
and internal Hilbert spaces;

* The proof by Jannussis and Mignani (186) of the convergence of
hadronic perturbative series when conventionally divergent;

* The initiation by Mignani (182) of a * nonpotential-nonunitary
scattering theory;* to represent scattering among extended particles;
* The first exact and invariant representation by
Santilli (114,115) of nuclear magnetic moments and other nuclear data;

* The first and only model by Animalu (170) and Animalu and
Santilli (116) of the Cooper pair in superconductivity with an * attractive force
between the two identical electrons* in excellent agreement with experimental data;

* The exact representation via isorelativity by Mignani (118) of the
large difference in cosmological redshifts between quasars and galaxies when physically
connected;

* The exact representation by Santilli (117) of the internal blueshift and
redshift of quasar's cosmological redshift;

* The elimination of the need for a missing mass in the universe by
Santilli (34); and other applications.

*BASIC METHODOLOGICAL ASSUMPTION:* in these web pages we shall assume that hadrons, nuclei and molecules are isolated from the rest of the universe, their structure is invariant under time reversal and their elementary constituents can be treated via single-valued mathematics. As a result, from here on we shall consider the applicable branches of hadronic mechanics given by isomathematics and isomechanics (Figure 3).

As well known, there are still limitations for the symbols available in the current htlm format of web sites. Therefore, we have to refer to memoir (226) for the symbols now generally used in studies of hadronic mechanics, while in these pages we are forced to a number of modifications.

**Notations:** all conventional products will be indicated with the symbol "x," e.g., AxB, while all isoproducts will be indicated with the symbol "*," e.g. A*B. All quantities belonging to quantum mechanics will be denoted with ordinary symbols, such as n, H, J, etc., while the corresponding quantities belonging to hadronic mechanics will be denoted with a "hat," e.g., n^, H^, J^, etc. To avoid confusion with the symbol used for the product "x", spacetime coordinates will be denoted with the letter q = (r, t). Total derivative will be denoted with the usual symbol d/dq, but partial derivative will be denoted with the best available symbol D/Dq. Equations will be quoted with the related part when needed, e.g., (II.3.5), denoting Part II of these web pages, section 3 and Equation 5. Roman numerals will be omitted when referring to equations of the part at hand.

The simplest possible method for the construction of isomathematics and isomechanics is given by the identification of the isounit with the basic nonunitary transform

(3.5) UxU^{+} = I^ = 1/T ≠ I,

and then applying the same transform to the *totality* of quantities and all their operations of conventional mechanics, including: the basic unit I; numbers n, m, etc.; their product nxm; operators A, B, etc.; their associative product AxB; exponentiations expX and other functions and transforms; differentials dq and derivatives d/dq; Hilbert inner product (|x|), eigenvalue equation Hx|> = Ex|> and expectation values (|xAx|)/(|x|); Heisenberg's equation idA/dt = [A, H] = AxH - HxA; etc.

(3.6a) I-> I^ = UxIxU^{+}

(3.6b) n -> n^ = UxnxU^{+} = nxI^,

(3.6c) nxm -> Ux(nxm)xU^{+} = n^*m^ = (nxm)xI^,

(3.6d) e^{X} -> Ux(e^{X})xU^{+} = I^x(e^{TxX}) = (e^{XxT})xI^,

(3.6e) dq -> = d^q^ = Txdq^,

(3.6f) d/dq -> d^/d^q^ = I^xd/dq^,

(3.6g) A -> A^ = UxAxU^{+}.

(3.6h) AxB -> A^*B^ = Ux(AxB)xU^{+} = (UxAxU^{+}x(UxU^{+})^{-1}x(UxBxU^{+}) = A^xTxB^,

(3.6i) (s|x|s) -> Ux(s|x|s)xU^{+} = (s^|xTx|s^)xI^,

(3.6j) (A) = (s|xAx|s)/(s|x|s) = (s^|xTxAxTx|s^)/(s^|xTx|s^) = (A^),

(3.6k) Hx|s> = Ex|s> -> Ux(Hx|s>) = H^xTx|s^> = H^*|s^> = Ux(Ex|s>) = E'*|s^>

etc.

where D/Dq represents partial derivative, and one should note the crucial change of the eigenvalue E -> E' (because the eigenvalue of H'xT is generally different than that of H).

Despite the simplicity of the above method for the construction of isomathematics and isomechanics, the visitor should be aware that ** specific applications generally require the study of realizations admitted by the isotopic axioms, yet not reachable via the above simple nonunitary map even though isoequations (3.6) will remain valid,** as we shall see for the isotopies of SU(2)-spin, the Dirac equation, etc.

The isomechanics characterized by the above new mathematics can be outlined as follows: recall that * there cannot be a really new mechanics without really new Newton's equations and, in turn, there cannot be a really new formulation of Newton's equations without a generalization of Newton-Leibnitz differential calculus.* The fundamental equations of the isotopic branch of hadronic mechanics is therefore given by the **iso-Newton equations**

(3.7a) m^*d^v^/d^t^ = - D^V^(r^)/D^r^,

(3.7b) v^ = vxI^_{v}(t, r, ...), t^ = txI^_{t},

(3.7c) I^_{v} = Diag. (n_{1}^{2}, n_{2}^{2}, n_{3}^{2}) x exp[F(t, r, ...)], I^_{t} = n_{4}^{2},

where: all fractions are tacitly assumed to be isotopic hereon (/^ = (/)xI^) unless otherwise specified; n_{1}^{2}, n_{2}^{2}, n_{3}^{2} represent the extended, nonspherical and deformable **shape** of the particle considered; n_{4}^{2} represents the **density** of the particle considered; V^(r^) characterizes all conventional potential forces; and F(t, r, ...) is a sufficiently smooth and dimensionless function characterizing all nonpotential forces.

As one can see, the iso-Newton equations have been conceived and constructed to coincide at the abstract level with the conventional equations. Yet they are *directly universal,* that is capable of representing all infinitely possible generally nonconservative Newtonian systems ("universality") directly in the frame of the experimenter, thus without transformation of local coordinates ("direct universality"). as the visitor is encouraged to verify (226).

In fact, Eqs. (3.7) can be written when projected in our conventional Euclidean space and time

(3.8) mxd(vxI^_{v>})/dt = - I^_{r}xDV(r)/Dr.

It is then easy to see that any given non-Hamiltonian Newtonian systems can always be represented via Eqs. (3.7) via a suitable selection of the function F. The construction of an endless number of illustrative examples is left to the interested visitor.

**Iso-Hamiltonian mechanics** is then characterized by the isoaction principle (see (226)) with the basic equations

(3.9a) d^r^/d^t^ = D^H^/D^/D^p^

(3.9b) d^p^/d^t^ = - D^H^/D^r^,

(3.9c) H^ = p^^{2}^/2^*m^ + V^(r^)

and corresponding isotopies of the Hamilton-Jacobi equations (loc. cit.). It is evident that Eqs. (3.9) are directly universal for all possible nonconservative Newtonian systems. For explicit realizations of the isounits see (226).

A simple isotopy of the naive or symplectic quantization here ignored for brevity, then yields the basic equations of the isotopic branch of hadronic mechanics (see then latest monograph (59) and historical notes quoted therein), that is characterized by the **iso-Schroedinger equations**

(3.11a)i^*D^|s^>/D^t^ = H^*|s^> = E^*|s^>

(3.11b) p^*|s^> = -ixD^|s^>/D^r^) = -ixI^_{r}xD|s^>/Dr,

and the **iso-Heisenberg equations**

(3.12a) idA/dt = [A, H] = AxH - HxA -> id^A^/d^t^ = [A^, H^]* = A^*H^ - H^*A^ =

= A^xT(q, p, y,...)xH^ - HxT(q, p, y,...)xA,

(3.12b) [r^, p^]* = ixI^, [r^, r^]* = [p^,p^]* = 0.

For additional aspects we are forced to refer the interested visitor to the technical literature, with particular reference to the latest presentations (59,226) to avoid a prohibitive length of this presentation. A knowledge of the isotopic branch of hadronic mechanics is however necessary for a true understanding of this web site. For instance, it is particularly instructive to see the proof that **all quantities that are observable for quantum mechanics remain observable for hadronic mechanics** (because the condition of iso-Hermiticity coincides with conventional Hermiticity).

We now remain with an illustration of the invariance of isomechanics without which theories have no known physical meaning (Theorem 2.1). Note first that, if isotopies (3.6) are subjected to another nonunitary transform, they **are not** generally invariant, e.g.,

(3.13a) WxW^+ ≠I,

(3.13b) I^ -> I^' = WxI^xW^{+} ≠ I^,

(3.13c)A*B = AxTxB -> Wx(A*B)xW^{+} = A'xT'xB' ≠ A'*B'.

However, such a treatment implies a mixture of the mathematics of quantum and hadronic mechanics, e.g., because the nonunitarity of W is formulated in the quantum Hilbert space. Full invariance is indeed achieved if such mixtures are avoided, and th a nonunitary transform is formulated with the mathematics of hadronic mechanics, i.,e., it is turned into the **isounitary transform**

(3.14) W = W'xT^{1/2}, WxW^{+} = W'*W^{+} = W'^{+}*W' = I^,

in which case the totality of the formalism of hadronic mechanics is indeed invariant, e.g.,

(3.15a) I^ -> I^' = W^*I^*W^^{+} = I^,

(3.15b) A*B -> A'*B' = W^*(A*B)*W^^{+} = A'xTxB', etc.,

where one should note the **numerical invariance of the isounit I^ and of the isotopic element T.**

The above invariance resolves all inconsistencies of Theorem 2.1 and illustrates that said inconsistencies are due to the treatment of a non-quantum theory via the old mathematics of quantum mechanics, while the construction of a new more appropriate mathematics resolves all indicated inconsistencies.

In closing this rudimentary review, it should be noted that hadronic mechanics is a **completion of quantum mechanics** much along the celebrated argument by Einstein, Podolsky and Rosen; the operator T is an explicit and concrete *operator* realization of the theory of "hidden variables" (because Hx|> and H*|> coincide at the abstract level); and Bell's inequalities admit a consistent nonunitary-isotopic image with a well-defined *classical image*, under which available studies on "local realism" require some substantial revisions (for these aspects, one may consult, e.g., Ref. (33)).

[1]

(1) I. Newton, * Philosophiae Naturalis Principia Mathematica* (1687),
translated and reprinted by Cambridge Univ. Press. (1934).

(2) J. L. Lagrange, * Mechanique Analytique* (1788), reprinted by
Gauthier-Villars, Paris (1888).

(3) W. R. Hamilton, * On a General Method in Dynamics* (1834), reprinted
in {\it Hamilton's Collected Works,} Cambridge Univ. Press (1940).

(4) S. Lie, * Over en Classe Geometriske Transformationer,* English translation by E.
Trell, Algebras Groups and Geometries {\bf 15}, 395 (1998).

(5) A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. {\bf 47}, 777 (1935).

(6) P. A. M. Dirac, * The Principles of Quantum
Mechanics,* Clarendon Press, Oxford, fourth edition (1958).

(7) A. A. Albert, Trans. Amer. Math. Soc. {\bf 64}, 552 (1948).

[2] ** BASIC MATHEMATICAL PAPERS:**

(8) R. M. Santilli, Nuovo Cimento {\bf 51}, 570 (1967).

(9) R. M. Santilli, Suppl. Nuovo Cimento {\bf 6}, 1225 (l968).

(10) R. M. Santilli, Hadronic J. {\bf 3}, 440 (l979).

(11) R. M. Santilli, Hadronic J. {\bf 8}, 25 and 36 (1985).

(12) R. M. Santilli Algebras, Groups and Geometries {\bf 10}, 273 (1993).

(13) R. M. Santilli and T. Vougiouklis, contributed paper in {\it New Frontiers in
Hyperstructures,} T., Vougiouklis, Editor, Hadronic Press, p. 1 (1996).

(14) R. M. Santilli, Rendiconti Circolo Matematico di
Palermo, Supplemento {\bf 42}, 7 (1996).

(15) R. M. Santilli, Intern. J. Modern Phys. D {\bf 7}, 351 (1998).

[3] ** ISODUAL FORMULATIONS:**

(16) R. M. Santilli, Comm. Theor. Phys. {\bf 3}, 153 (1993).

(17) R. M. Santilli, Hadronic J. {\bf 17}, 257 (1994).

(18) R. M. Santilli, Hadronic J. {\bf 17}, 285 (1994).

(19) R. M. Santilli, Communication of the JINR, Dubna, Russia,. No. E2-96-259 (1996).

(20) R. M. Santilli, contributed paper in {\it New Frontiers of Hadronic Mechanics,}
T.L.Gill, ed., Hadronic Press (1996).

(21) R. M. Santilli, Hyperfine Interactions, {\bf 109}, 63 (1997).

(22) R. M. Santilli, Intern. J. Modern Phys. A {\bf 14}, 2205 (1999).

[4] ** ISOTOPIC FORMULATIONS:**

(23) R.M.Santilli: Hadronic J. {\bf 1}, 224 (1978).

(24) R. M. Santilli, Phys. Rev. D {\bf 20}, 555 (1979).

(25) H. C. Myung and R. M. Santilli, Hadronic J. {\bf 5}, 1277 (1982).

(26) R. M. Santilli, Novo Cimento Lett. {\bf 37}, 545 (1983).

(27) R. M. Santilli, Hadronic J. {\bf 8}, 25 and 36 (1985).

(28) R. M. Santilli, JINR Rapid. Comm. {\bf 6}, 24 (1993).

(29) R. M. Santilli, J.Moscow Phys.Soc. {\bf 3}, 255 (1993).

(30) R. M. Santilli, Communication of the JINR, Dubna, Russia, # E4-93-352, 1993, published in Chinese J.Syst.Ing. \& Electr.{\bf 6}, 177 (1996).

(31) R. M. Santilli, Found. Phys. {\bf 27}, 635 (1997).

(32) R. M. Santilli, Found. Phys. Letters {\bf 10}, 307 (1997).

(33) R. M. Santilli, Acta Appl. Math. {\bf 50}, 177 (1998).

(34) R. M. Santilli, contributed paper to the {\it Proceedings of the International Workshop on
Modern Modified Theories of Gravitation and Cosmology,} E. I. Guendelman, Editor, Hadronic Press, p. 113 (1998).

(35) R. M. Santilli, contributed paper to the {\it Proceedings of the VIII M. Grossmann Meeting
on General Relativity,} Jerusalem, June 1998, World Scientific, p. 473 (1999).

(36) R. M. Santilli, contributed paper in {\it Photons: Old Problems in Light of New Ideas,} V. V.
Dvoeglazov, Editor, Nova Science Publishers, pages 421-442 (2000).

(37) R. M. Santilli, Found. Phys. Letters {\32}, 1111 (2002).

[5] ** GENOTOPIC FORMULATIONS:**

(38) R. M. Santilli: Hadronic J. {\bf 1},574 and 1267 (1978).

(39) R. M. Santilli, Hadronic J. {\bf 2}, 1460 (l979) and {\bf 3}, 914 (l980).

(40) R. M. Santilli, Hadronic J. {\bf 4}, 1166 (l981).

(41) R. M. Santilli, Hadronic J. {\bf 5}, 264 (l982).

(42) H. C. Myung and R. M. Santilli, Hadronic J. {\bf 5}, 1367 (l982).

(43) R. M. Santilli, Hadronic J. Suppl. {\bf 1}, 662 (l985).

(44) R. M. Santilli, Found. Phys. {\bf 27}, 1159 (1997).

(45) R. M. Santilli, Modern Phys. Letters {\bf 13}, 327 (1998).

(46) R. M. Santilli, Intern. J. Modern Phys. A {\bf 14}, 3157 (1999).

[6] ** HYPERSTRUCTURAL FORMULATIONS:**

(47) R. M. Santilli, Algebras, Groups and Geometries {\bf 15}, 473 (1998).

[7] ** MONOGRAPHS:**

(48) R. M. Santilli, {\it Foundations of Theoretical Mechanics}, Vol. I,
Springer--Verlag, Heidelberg--New York (1978).

(49) R. M. Santilli, {\it Lie-admissible Approach to the Hadronic Structure,}
Vol.I, Hadronic Press, Palm Harbor, Florida (1978).

(50) R. M. Santilli, {\it Lie-admissible Approach to the Hadronic Structure,} Vol.
II, Hadronic Press, Palm Harbor, Florida (1981).

(51) R. M. Santilli, {\it Foundations of Theoretical Mechanics}, Vol. II,
Springer--Verlag, Heidelberg--New York (1983).

(52) R. M. Santilli, {\it Isotopic Generalizations of Galilei and Einstein Relativities,}
Vol. I, Hadronic Press, Palm Harbor, Florida (1991).

(53) R. M. Santilli, {\it Isotopic Generalizations of Galilei and Einstein Relativities,}
Vol. II, Hadronic Press, Palm Harbor, Florida (1991).

(54) R. M. Santilli, {\it Elements of Hadronic Mechanics},
Vol I, Ukraine Academy of Sciences,
Kiev, Second Edition (1995).

(55) R. M. Santilli, {\it Elements of Hadronic Mechanics},
Vol II, Ukraine Academy of Sciences,
Kiev, Second Edition (1995).

(56) C. R. Illert and R. M. Santilli, {\it Foundations of Theoretical Conchology,}
Hadronic Press, Palm Harbor, Florida (1995).

(57) R. M. Santilli {\it Isotopic, Genotopic and Hyperstructural Methods
in Theoretical Biology}, Ukraine Academy of Sciences, Kiev (1996).

(58) R. M. Santilli, {\it The Physics of New Clean Energies and Fuels According to
Hadronic Mechanics,} Special issue of the Journal of New Energy, 318 pages (1998).

(59) R. M. Santilli, {\it Foundations of Hadronic Chemistry with Applications to New
Clean Energies and Fuels,} Kluwer Academic Publishers, Boston-Dordrecht-London
(2001).

(60) R. M. Santilli, {\it Ethical Probe of Einstein's Followers in the USA: An insider's view,} Alpha Publishing, Newtonville, MA (1984).

(61) R. M. Santilli, {\it Documentation of the Ethical Probe,} Volumes I, II and III, Alpha Publishing, Newtonville, MA (1985).

(62) H. C. Myung, {\it Lie Algebras and Flexible
Lie-Admissible Algebras,} Hadronic Press (1982).

(63) A. K. Aringazin, A. Jannussis, D.F. Lopez, M. Nishioka, and B. Veljanoski,
{\it Santilli's Lie-isotopic Generalization of Galilei's and Einstein's
Relativities,} Kostarakis Publishers, Athens (1991).

(64) D. S. Sourlas and G. T. Tsagas, {\it Mathematical
Foundations of the Lie-Santilli Theory,} Ukraine Academy of Sciences,
Kiev (1993).

(65) J. Lohmus, E. Paal and L. Sorgsepp, {\it
Nonassociative Algebras in Physics}, Hadronic Press, Palm Harbor, FL,
USA (1994).

(66) J. V. Kadeisvili, {\it Santilli's Isotopies of
Contemporary Algebras, Geometries and Relativities},
Second Edition, Ukraine Academy of Sciences, Kiev , Second Edition (1997).

(67) R. M. Falcon Ganfornina and J. Nunez Valdes, {\it Fondamentos de la Isoteoria de
Lie-Santilli,} (in Spanish) International Academic Press, America-Europe-Asia,
(2001), also available in the pdf file
http://www.i-b-r.org/docs/spanish.pdf

(68) Chun-Xuan Jiang, {\it Foundations of Santilli's Isonumber Theory,}
with Applications to New Cryptograms, Fermat's Theorem and Goldbach's Conjecture,
International Academic Press, America-Europe-Asia
(2002) also available in the pdf file
http://www.i-b-r.org/docs/jiang.pdf

[8] ** CONFERENCE PROCEEDINGS AND REPRINT VOLUMES:**

(69) H. C. Myung and S. Okubo, Editors, {\it Applications of Lie-Admissible Algebras
in Physics,} Volume I, Hadronic Press (1978).

(70) H. C. Myung and S. Okubo, Editors, {\it Applications of Lie-Admissible Algebras
in Physics,} Vol. II, Hadronic Press (1978).

(71) H. C. Myung and R. M. Santilli, Editor, {\it Proceedings of the Second Workshop on
Lie-Admissible Formulations,} Part I, Hadronic J. Vol. 2, no. 6, pp. 1252-2033 (l979).

(72) H. C. Myung and R. M. Santilli, Editor, {\it Proceedings of the Second Workshop on
Lie-Admissible Formulations,}Part II, Hadronic J. Vol. 3, no. 1, pp. 1-725 (l980.

(73) H. C. Myung and R. M. Santilli, Editor, {\it Proceedings of the Third Workshop on
Lie-Admissible Formulations,}Part A, Hadronic J. Vol. 4, issue no. 2, pp. 183-607 (l9881).

(74) H. C. Myung and R. M. Santilli, Editor, {\it Proceedings of the Third Workshop on
Lie-Admissible Formulations,} Part B, Hadronic J. Vo. 4, issue no. 3, pp. 608-1165 (l981).

(75) H. C. Myung and R. M. Santilli, Editor, {\it Proceedings of the Third Workshop on
Lie-Admissible Formulations,} Part C, Hadronic J. Vol. 4, issue no. 4, pp. 1166-1625
(l981).

(76) J. Fronteau, A. Tellez-Arenas and R. M. Santilli, Editor, {\it Proceedings of the First
International Conference on Nonpotential Interactions and their Lie-Admissible
Treatment,} Part A, Hadronic J., Vol. 5, issue no. 2, pp. 245-678 (l982).

(77) J. Fronteau, A. Tellez-Arenas and R. M. Santilli, Editor, {\it Proceedings of the First
International Conference on Nonpotential Interactions and their Lie-Admissible
Treatment,} Part B, Hadronic J. Vol. 5, issue no. 3, pp. 679-1193 (l982).

(78) J. Fronteau, A. Tellez-Arenas and R. M. Santilli, Editor, {\it Proceedings of the First
International Conference on Nonpotential Interactions and their Lie-Admissible
Treatment,} Part C, Hadronic J. Vol. 5, issue no. 4, pp. 1194-1626 (l982).

(79) J. Fronteau, A. Tellez-Arenas and R. M. Santilli, Editor, {\it Proceedings of the First
International Conference on Nonpotential Interactions and their Lie-Admissible
Treatment,} Part D, Hadronic J. Vol. 5, issue no. 5, pp. 1627-1948 (l982).

(80) J.Fronteau, R.Mignani, H.C.Myung and R. M. Santilli, Editors, {\it Proceedings of the
First Workshop on Hadronic Mechanics,} Hadronic J. Vol. 6, issue no. 6, pp. 1400-1989
(l983).

(81) A. Shoeber, Editor, {\it Irreversibility and Nonpotentiality in Statistical Mechanics,}
Hadronic Press (1984).

(82) H. C. Myung, Editor, {\it Mathematical Studies in Lie-Admissible Algebras,} Volume I,
Hadronic Press (1984).

(83) H. C. Myung, Editor, {\it Mathematical Studies in Lie-Admissible Algebras,} Volume II,
Hadronic Press (1984).

(84) H. C. Myung and R. M. Santilli, Editor, {\it Applications of Lie-Admissible Algebras
in Physics,} Vol. III, Hadronic Press (1984).

(85) J. Fronteau, R. Mignani and H. C. Myung, Editors, {\it Proceedings of the Second Workshop on
Hadronic Mechanics,} Volume I Hadronic J. Vol. 7, issue no. 5, pp. 911-1258 (l984).

(86) J.Fronteau, R.Mignani and H.C.Myung, Editors, {\it Proceedings of the Second Workshop on
Hadronic Mechanics,} Volume II, Hadronic J. Vol. 7, issue no. 6, pp. 1259-1759 (l984).

(87) D. M. Norris et al, {\it Tomber's Bibliography and Index in Nonassociative
Algebras,} Hadronic Press, Palm Harbor, FL (1984).

(88) H. C. Myung, Editor, {\it Mathematical Studies in Lie-Admissible Algebras,} Volume III,
Hadronic Press (1986).

(89) A. D. Jannussis, R. Mignani, M. Mijatovic, H. C. Myung B. Popov and A. Tellez Arenas,
Editors, {\it Fourth Workshop on Hadronic Mechanics and Nonpotential Interactions,}
Nova Science, New York (l990).

(90) H. M. Srivastava and Th. M. Rassias, Editors, {\it Analysis Geometry and Groups:
A Riemann Legacy Volume,} Hadronic Press (1993).

(91) F. Selleri, Editor, {\it Fundamental Questions in Quantum Physics and Relativity,}
Hadronic Press (1993).

(92) J. V. Kadeisvili, Editor, {\it The Mathematical Legacy of Hanno Rund}, Hadronic Press
(1994).

(93) M. Barone and F. Selleri Editors, {\it Frontiers of Fundamental Physics,} Plenum, New
York, (1994).

(94) M. Barone and F. Selleri, Editors, {\it Advances in Fundamental Physics,} Hadronic Press
(1995).

(95) Gr. Tsagas, Editor, {\it New Frontiers in Algebras, Groups and Geometries
,} Hadronic Press (1996).

(96) T. Vougiouklis, Editor, {\it New Frontiers in Hyperstructures,}
Hadronic Press, (1996).

(97) T. L. Gill, Editor, {\it New Frontiers in Hadronic Mechanics,} Hadronic Press (1996).

(98) T. L. Gill, Editor, {\it New Frontiers in Relativities,} Hadronic Press (1996).

(99) T. L. Gill, Editor, {\it New Frontiers in Physics,}, Volume I, Hadronic Press (1996).

(100) T. L. Gill, Editor, {\it New Frontiers in Physics,}, Volume II, Hadronic Press (1996).

(101) C. A. Dreismann, Editor, {\it New Frontiers in Theoretical Biology,} Hadronic Press
(1996).

(102) G. A. Sardanashvily, Editor,{\it New Frontiers in Gravitation,} Hadronic Press (1996).

(103) M. Holzscheiter, Editor, {\it Proceedings of the
International Workshop on Antimatter Gravity,} Sepino, Molise, Italy,
May 1996, Hyperfine Interactions, Vol. {\bf 109} (1997).

(104) T. Gill, K. Liu and E. Trell, Editors, {\it Fundamental Open Problems in Science at the
end of the Millennium,}} Volume I, Hadronic Press (1999).

(105) T. Gill, K. Liu and E. Trell, Editors, {\it Fundamental Open Problems in Science at the
end of the Millennium,}} Volume II, Hadronic Press (1999).

(106) T. Gill, K. Liu and E. Trell, Editors, {\it Fundamental Open Problems in Science at the
end of the Millennium,}} Volume III, Hadronic Press (1999).

(107) V. V. Dvoeglazov, Editor {\it Photon: Old Problems in Light of New Ideas,} Nova Science
(2000).

(108) M. C. Duffy and M. Wegener, Editors, {\it Recent Advances in Relativity Theory} Vol. I,
Hadronic Press (2000).

(109) M. C. Duffy and M. Wegener, Editors, {\it Recent Advances in Relativity Theory} Vol. I,
Hadronic Press (2000).

[9] ** EXPERIMENTAL VERIFICATIONS:**

(110) F. Cardone, R. Mignani and R. M. Santilli, J. Phys. G: Nucl. Part. Phys.
{\bf 18}, L61 (1992).

(111) F. Cardone, R. Mignani and R. M. Santilli, J. Phys. G: Nucl. Part. Phys. {\bf 18}, L141
(1992).

(112) R. M. Santilli, Hadronic J. {\bf 15}, Part I: 1-50 and Part II: 77�134 (l992).

(113) Cardone and R. Mignani, JETP {\bf 88}, 435 (1995).

(114) R. M. Santilli, Intern. J. of Phys. {\bf 4}, 1 (1998).

(115) R. M. Santilli Communications in Math. and Theor. Phys. {\bf 2}, 1 (1999).

(116) A. O. E. Animalu and R. M. Santilli, Intern. J. Quantum Chem. {\bf 26},175 (1995).

(117) R. M. Santilli, contributed paper to {\it Frontiers of Fundamental Physics,} M. Barone and
F. Selleri, Editors Plenum, New York, pp 41�58 (1994).

(118) R. Mignani, Physics Essays {\bf 5}, 531 (1992).

(119) R. M. Santilli, Comm. Theor. Phys. {\bf 4}, 123 (1995).

(120) Yu. Arestov, V. Solovianov and R. M. Santilli, Found. Phys. Letters {\bf 11}, 483 (1998).

(121) R. M. Santilli, contributed paper in the {\it Proceedings of the International Symposium on
Large Scale Collective Motion of Atomic Nuclei,} G. Giardina, G. Fazio and M. Lattuada,
Editors, World Scientific, Singapore, p. 549 (1997).

(122) J. Ellis, N. E. Mavromatos and D. V. Napoulos in {\sl Proceedings of the
Erice Summer School, 31st Course: From Superstrings to the Origin of Space--Time},
World Sientific (1996).

(123) C. Borghi, C. Giori and A. Dall'OIlio Russian J. Nucl. Phys. {\bf 56}, 147 (1993).

(124) N. F. Tsagas, A. Mystakidis, G. Bakos, and L. Seftelis, Hadronic J. {\bf 19}, 87 (1996).

(125) R. M. Santilli and D. D. Shillady, Intern. J. Hydrogen Energy {\bf 24}, 943 (1999).

(126) R. M. Santilli and D., D. Shillady, Intern. J. Hydrogen Energy {\bf 25}, 173 (2000).

(127) R. M. Santilli, Hadronic J. {\bf 21}, pages 789-894 (1998).

(128) M.G. Kucherenko and A.K. Aringazin, Hadronic J. {\bf 21}, 895 (1998).

(129) M.G. Kucherenko and A.K. Aringazin, Hadronic Journal {\bf 23}, 59 (2000).

(130) R. M. Santilli and A. K. Aringazin, "Structure and Combustion of
Magnegases", e-print http://arxiv.org/abs/physics/0112066, to be published.

,
[10] ** MATHEMATICS PAPERS:**

(131) S. Okubo, Hadronic J. {\bf 5}, 1564 (1982).

(132) J. V. Kadeisvili, Algebras, Groups and Geometries {\bf 9}, 283 and 319 (1992).

(133) J. V. Kadeisvili, N. Kamiya, and R. M. Santilli, Hadronic J. {\bf 16}, 168 (1993).

(134) J. V. Kadeisvili, Algebras, Groups and Geometries {\bf 9}, 283 (1992).

(135) J. V. Kadeisvili, Algebras, Groups and Geometries {\bf 9}, 319 (1992).

(136) J. V. Kadeisvili, contributed paper in the {\it Proceedings of the
International Workshop on Symmetry Methods in Physics,} G. Pogosyan et al., Editors,
JINR, Dubna, Russia (1994).

(137) J. V. Kadeisvili, Math. Methods in Appl. Sci. {\bf 19} 1349 (1996).

(138) J. V. Kadeisvili, Algebras, Groups and Geometries, {\bf 15}, 497 (1998).

(139) G. T. Tsagas and D. S. Sourlas, Algebras, Groups and Geometries {\bf 12}, 1 (1995).

(140) G. T. Tsagas and D. S. Sourlas, Algebras, Groups and Geometries {\bf 12}, 67 (1995).

(141) G. T. Tsagas, Algebras, Groups and geometries {\bf 13}, 129 (1996).

(142) G. T. Tsagas, Algebras, Groups and geometries {\bf 13}, 149 (1996).

(143) E. Trell, Isotopic Proof and Reproof of Fermat�s Last Theorem Verifying Beal�s
Conjecture. Algebras Groups and Geometries {\bf 15}, 299-318 (1998).

(144) A. K. Aringazin and D. A. Kirukhin,, Algebras, Groups and
Geometries {\bf 12}, 255 (1995).

(145) A.K. Aringazin, A. Jannussis, D.F. Lopez, M. Nishioka, and B. Veljanoski, Algebras, Groups and Geometries {\bf 7}, 211 (1990).

(146) A. K. Aringazin, A. Jannussis, D. F. Lopez, M.
Nishioka, and B. Veljanoski, Algebras, Groups and
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(147) D. L. Rapoport, Algebras, Groups and Geometries, {\bf 8}, 1 (1991).

(148) D. L. Rapoport, contributed paper in the{\it
Proceedings of the Fifth International Workshop on
Hadronic Mechanics,} H.C. Myung, Edfitor, Nova Science Publisher (1990).

(149) D. L. Rapoport, Algebras, Groups and Geometries {\bf 8}, 1 (1991).

(150) C.-X. Jiang, Algebras, Groups and Geometries {\bf 15}, 509 (1998).

(151) D. B. Lin, Hadronic J. {\bf 11}, 81 (1988).

(152) R. Aslaner and S. Keles, Algebras, Groups and Geometries {\bf 14}, 211 (1997).

(153) R. Aslander and S. Keles,. Algebras, Groups and Geometries {\bf 15}, 545 (1998).

(154) M. R. Molaei, Algebras, Groups and Geometries {\bf 115}, 563 (1998) (154).

(155) S. Vacaru, Algebras, Groups and Geometries {\bf 14}, 225 (1997) (155).

(156) N. Kamiya and R. M. Santilli, Algebras, Groups and Geometries {\bf 13}, 283 (1996).

(157) S. Vacaru, Algebras, Groups and Geometries {\bf 14}, 211 (1997).

(158) Y. Ylamed, Hadronic J. {\bf 5}, 1734 (1982).

(159) R. Trostel, Hadronic J. {\bf 5}, 1893 (1982).

[11] **PHYSICS PAPERS:**

(160) J. P. Mills, jr, Hadronic J. {\bf 19}, 1 (1996).

(161) J. Dunning-Davies, Foundations of Physics Letters, {\bf 12}, 593 (1999).

(162) E. Trell, Hadronic Journal Supplement {\bf 12}, 217 (1998).

(163) E. Trell, Algebras Groups and Geometries {\bf 15}, 447-471 (1998).

(164) E. Trell, "Tessellation of
Diophantine Equation Block Universe," contributed paper to {\it Physical
Interpretations of Relativity Theory,} 6-9 September 2002, Imperial College, London.
British Society for the Philosophy of Science, in print, (2002).

(165) J. Fronteau, R. M. Santilli and A. Tellez-Arenas, Hadronic J. {\bf 3}, 130 (l979).

(166) A. O. E. Animalu, Hadronic J.{\bf 7}, 19664 (1982).

(167) A. O. E. Animalu, Hadronic J. {\bf 9}, 61 (1986).

(168) A. O. E. Animalu, Hadronic J. {\bf 10}, 321 (1988).

(169) A. O. E. Animalu, Hadronic J. {\bf 16}, 411 (1993).

(170) A. O. E. Animalu, Hadronic J. {\bf 17}, 349 (1994).

(171) S. Okubo, Hadronic J. {\bf 5}, 1667 (1982).

(172) D.F.Lopez, in {\it
Symmetry Methods in Physics}, A.N.Sissakian, G.S.Pogosyan and X.I.Vinitsky, Editors (
JINR, Dubna, Russia (1994).

(173) D. F. Lopez, {\it Hadronic J.} {\bf 16}, 429 (1993).

(174) A.Jannussis and D.Skaltsas,{\it Ann. Fond. L.de Broglie} {\bf
18},137 (1993).

(175) A. Jannussis, R. Mignani and R. M. Santilli,
{\it Ann.Fonnd. L.de Broglie} {\bf 18}, 371 (1993).

(176) A. O. Animalu and R. M. Santilli, contributed paper in {\it Hadronic
Mechanics and Nonpotential Interactions} M. Mijatovic, Editor,
Nova Science, New York, pp. 19-26 (l990).

(177) M. Gasperini, Hadronic J. {\bf 6}, 935 (1983).

(178) M. Gasperini, Hadronic J. {\bf 6}, 1462 (1983).

(179) R. Mignani, Hadronic J. {\bf 5}, 1120 (1982).

(180) R. Mignani, Lett. Nuovo Cimento {\bf 39}, 413 (1984).

(181) A. Jannussis, Hadronic J. Suppl. {\bf 1}, 576 (1985).

(182) A. Jannussis and R. Mignani, Physica A {\bf 152}, 469 (1988).

(183) A. Jannussis and I.Tsohantis, Hadronic J. {\bf 11}, 1 (1988).

(184) A. Jannussis, M. Miatovic and B. Veljanosky, Physics Essays {\bf 4}, (1991).

(185) A. Jannussis, D. Brodimas and R. Mignani, J. Phys. A: Math. Gen. {\bf 24}, L775 (1991).

(186) A. Jannussis and R. Mignani, Physica A {\bf 187}, 575 (1992).

(187) A. Jannussis, R. Mignani and D. Skaltsas Physics A {\bf 187}, 575 (1992).

(188) A.Jannussis et al, Nuovo Cimento B{\bf 103}, 17 and 537 (1989).

(189) A. Jannussis et al., Nuovo Cimento B{\bf 104}, 33 and 53 (1989).

(190) A. Jannussis et al. Nuovo Cimento B{\bf 108} 57 (l993).

(191) A. Jannussis et al., Phys. Lett. A {\bf 132}, 324 (1988).

(192) A. K. Aringazin, Hadronic J. {\bf 12}, 71 (1989).

(193) A. K. Aringazin, Hadronic J. {\bf13}, 183 (1990).

(194) A. K. Aringazin, Hadronic J. {\bf 13}, 263 (1990).

(195) A. K. Aringazin, Hadronic J. {\bf 14}, 531 (1991).

(196) A. K. Aringazin, Hadronic J. {\bf 16}, 195 (1993).

(197) A. K. Aringazin and K. M. Aringazin,Invited paper, in the {\it Proceedings of the Intern.
Conference 'Frontiers of Fundamental Physics',} Plenum Press, (1993).

(198) A. K. Aringazin, K. M. Aringazin, S. Baskoutas, G. Brodimas, A. Jannussis, and K. Vlachos,
contributed paper in the {\it Proceedings of the Intern. Conference 'Frontiers of Fundamental
Physics',} Plenum Press, (1993).

(199) A. K. Aringazin, D.A. Kirukhin, and R. M. Santilli, Hadronic J. {\bf 18}, 245 (1995).

(200) A. K. Aringazin, D.A.Kirukhin, and R. M. Santilli, Had-ronic J. {\bf 18}, 257 (1995).

(201) T. L. Gill, Hadronic J. {\bf 9}, 77 (1986).

(202) T. L. Gill and J. Lindesay, Int. J. Theor,
Phys {\bf} 32}, 2087 (1993).

(203) T. L. Gill, W.W. Zachary, M.F. Mahmood and J. Lindesay,
Hadronic J. {\bf 16}, 28 (1994).

(204) T. L. Gill, W.W. Zachary and J. Lindesay,
Foundations of Physics Letters {\bf 10}, 547 (1997).

(205) T. L. Gill, W.W. Zachary and J.
Lindesay, Int. J. Theo Phys {\bf 37}, 22637 (1998).

(206) T. L. Gill, W.W. Zachary and J.
Lindesay, Foundations of Physics, {\bf 31}, 1299 (2001).

(207) D. L. Schuch, K.-M. Chung and H. Hartmann,
J. Math. Phys. {\bf 24}, 1652 (1983).

(208) D. L. Schuch and K.-M. Chung, Intern. J. Quantum
Chem. {\bf 19}, 1561 (1986).

(209) D. Schich 23, Phys. Rev. A {\bf 23}, 59, (1989).

(210) D. Schuch, contributed paper in {\it New
Frontiers in Theoretical Physics,} Vol. I, p 113, T. Gill Editor, Hadronic Press, Palm Harbor
(1996); D. L. Schuch, Hadronic J. {\bf 19}, 505 (1996).

(211) S.L. Adler, Phys.Rev. {\bf 17}, 3212 (1978).

(212) Cl. George, F. Henin, F. Mayene and I. Prigogine, Hadronic J. {\bf 1},
520 (1978).

(213) C. N. Ktorides, H. C. Myung and R. M. Santilli, Phys. Rev. D {\bf 22}, 892
(1980).

(214) R. M. Santilli, Hadronic J. {\bf 13}, 513 (1990).

(215) R. M. Santilli, Hadronic J. {\bf 17}, 311 (1994).

(216) A. J. Kalnay, Hadronic J. {\bf 6}, 1 (1983).

(217) A. J. Kalnay and R. M. Santilli, Hadronic J. {\bf 6}, 1873 (1983).

(218) M. 0. Nishioka, Nuovo Cimento A {\bf 82}, 351 (1984).

(219) G. Eder, Hadronic J. \underline{\bf 4,} (1981) and {\bf 5}, 750 (1982).

(220) H. E. Wilhelm, contributed paper in {\it
Space, Time and Motion: Theory and Experiments}, H., E. Wilhelm and K. Liu, editors,Chinese J.
Syst Eng. Electr., {\bf 6}, issue 4 (1995).

(221) C. A. Chatzidimitriou-Dreismann, T. Abdul-Redah, R.M.F. Streffer, J.Mayers
Phys. Rev. Lett. {\bf 79}, 2839 (1997).

(222) C. A. Chatzidimitriou-Dreismann, T. Abdul-Redah, B. Kolaric
J. Am. Chem. Soc. {\bf 123}, 11945 (2001).

(223) C. A. Chatzidimitriou-Dreismann, T. Abdul-Redah,
R. M. F. Streffer, J.Mayers J. Chem.
Phys. {\bf 116}, 1511 (2002) (see also
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(224) R. M. Santilli, Hadronic J. Suppl. {\bf 1}, 662 (l985).

(225) R. M. Santilli, Found. Phys. {\bf 11}, 383 (l981).

(226) R. M. Santilli, "Iso-, geno-, hyper-mechanifor matter and their isoduals for antimatter," in press at *Journal of Dynamical Systems and Geometric Theories,* also available in pdf format.

(227) R. M. Santiulli, Comm. Thgeor. Phys. {\bf 4} 123 (1995).

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(229) H. Rauch et al., Phys. Lett. A {bf 54}, 425 (1975). G. Badurek et al., Phys. Rev. D {bf 14}, 1177 (1976). H. Rauch et al., Z. Physik B {\bf 29}, 281 (1978). H. Kaiser et al., Z. Physik A 291, 231 (1979).

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(231) L. B. Redei, Phys. Rev. {\bf 145}, 999 (1966)

(232) D. Y. Kim, Hadronic J. {\bf 1}, 1343 (1978)

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(236) B. Lorstad, Int. J. Mod. Phys. A {\bf 4}, 2861 (1989).

(237) V. V. Burov et al., JINR Communication E4-95-440 (1995), Dubna, Russia.

(238) R. M. Santilli, in *Proceedings of "Deuteron 1993",* JINR, Dubna, Russia (1994).

(239) R. M. Santilli, Intern. J. Hydrogen Energy {\bf 28}, 177 (2003).

(240) A. Enders and G.
Nimtz, J. Phys. France {\bf 2}, 1693 (1992).

(241) G. Nimtz and W. Heitmann,
Progr. Quantum Electr. {\bf 21}, 81 (1997).

(242) F. Mirabel and F. Rodriguez, Nature {\bf 371}, 464 (1994).

(243) J. Tingay et al., Nature {\bf {374}}, 141 (1995).

(244) D. Baylin et al., IAU Comm. 6173 (1995).

(246) P. Saari and K. Reivelt, Phys. Rev. Letters {\bf 79} (1997), in press.

(247) E. Recami and R. M. Santilli, Hadronic Journal {\bf 23}, 279 (2000).

(248) H. Arp, * Quasars Redshifts and Controversies,* Interstellar media, Berkeley, 1987.

(249) H. Arp, contributed paper in * Frontiers of Fundamental Physics, F. Selleri and M. Barone, Editors, Editors, Plenum, New York (1994).
*

*
(250) J. W. Sulentic, Astrophys. J. {\bf 343}, 54 (1989).
*

*
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*

*
(252) R. M. Santilli, Hadronic J. Suppl. 4A, issues 1 (252a), 2 (244b), 3 (252c) and 4 (252d), 1988.
*

*
(253) R. M. Santilli, contributed paper in Frontiers of Fundamental Physics, F. Selleri and M. Barone, Editors, Editors, Plenum, New York (1994).*

*
(254) P. Marmet, IEEE Trans. Plasma Sci. {\bf 17}, 238 (1988)
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