# User:Vuara/hypermathematics-hypermechanics

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Elements of Iso-, Geno-, Hyper-Mathematics for Matter, Their Isoduals for Antimatter, and Their Applications in Physics, Chemistry, and Biology

```Foundations of Physics    September 2003, vol. 33, no. 9,   pp. 1373-1416(44)

```

Santilli R.M.[1]

[1] Institute for Basic Research, P.O. Box 1577, Palm Harbor, Florida 34682. ibr@gte.net, http://www.i-b-r.org

Abstract:

Pre-existing mathematical formulations are generally used for the treatment of new scientific problems. In this note we show that the construction of mathematical structures from open physical, chemical, and biological problems leads to new intriguing mathematics of increasing complexity called iso-, geno-, and hyper-mathematics for the treatment of matter in reversible, irreversible, and multi-valued conditions, respectively, plus anti-isomorphic images called isodual mathematics for the treatment of antimatter. These novel mathematics are based on the lifting of the multiplicative unit of ordinary fields (with characteristic zero) from its traditional value +1 into: (1) invertible, Hermitean, and single-valued units for isomathematics; (2) invertible, non-Hermitean, and single-valued units for genomathematics; and (3) invertible, non-Hermitean, and multi-valued units for hypermathematics; with corresponding liftings of the conventional associative product and consequential lifting of all branches of mathematics admitting a (left and right) multiplicative unit. An anti-Hermitean conjugation applied to the totality of quantities and their operation of the preceding mathematics characterizes the isodual mathematics. Intriguingly, the emerging formulations preserve the abstract axioms of conventional mathematics (that based on the unit +1). As such, the new formulations result to be new realizations of existing abstract mathematical axioms. We then show that the above mathematical advances permit corresponding liftings of conventional classical and quantum theories with a resolution of basic open problems in physics, chemistry, and biology, numerous experimental verifications, as well as new industrial applications.

Keywords: antimatter; nonlocality; irreversibility; multivaluedness

+++

http://www.i-b-r.org/docs/Iso-Geno-Hyper-paper.pdf

# (x2)d = (¡xt)2d = (xd) £d (gd) £d (xtd) £d Id

n22
n23

ˆ A( ˆ w) = (ˆeˆi

ˆ£ ˆX ˆ£ ˆ w) ˆ£ ˆ A(ˆ0) ˆ£(ˆe¡ˆi ˆ£ ˆ wˆ£ ˆX ) = (ei£ ˆX £ˆ T£w) £ A(0) £ (e¡i£w£ˆ T£ ˆX ); (3:39) where ˆ w 2 ˆR are the isoparameters; the isorepresentation theory; etc. The non-triviality of the above liftings is expressed by the appearance of the isotopic element ˆ T(x; :::) at all levels (I), (II) and (III) of the isotheory. The arbitrary functional dependence of ˆ T(x; :::) then implies the achievement of the desired main features of the isotheory which can be expressed by the following: 24 LEMMA 3.2 (14): Lie-Santilli isoalgebras on conventional spaces over conventional fields are generally nonlocal, nonlinear and noncanonical, but they verify locality, linearity and canonicity when formulated on isospaces over isofields. To illustrate the Lie-Santilli isotheory in the operator case, consider the eigenvalue equation on H over C, H(x; p; Ã; :::) £ jÃ >= E £ jÃ >. This equation is nonlinear in the wavefunction, thus violating the superposition principle and preventing the study of composite nonlinear systems, as indicated earlier. However, under the factorization H(x; p; Ã; :::) = H0(x; p) £ ˆ T(x; p; Ã; :::); (3:40) the above equation can be reformulated identically in the isotopic form H(x; p; Ã; :::) £ jÃ >= H0(x; p) £ ˆ T(x; p; Ã; :::) £ jÃ >= H0 ˆ £jÃ >= E £ jÃ >= ˆE ˆ £jÃ >; (3:41) whose reconstruction of linearity on isospaces over isofields (called isolinearity (14)) is evident and so is the verification of the isosuperposition principle with resulting applicability of isolinear theories for the study of composite nonlinear systems. Similar results occur for the reconstruction on isospace over isofields of locality (called isolocality) and canonicity (called isocanonicity). A main role of the isotheory is then expressed by the following property: LEMMA 3.3 (29): Under the condition that ˆI is positive-definite, isotopic algebras and groups are locally isomorphic to the conventional algebras and groups, respectively. Stated in different terms, the Lie-Santilli isotheory was not constructed to characterize new Lie algebras, because all Lie algebras over a field of characteristic zero are known. On the contrary, the Lie-Santilli isotheory has been built to characterize new realizations of known Lie algebras generally of nonlinear, nonlocal and noncanonical character as needed for a deeper representation of valence bonds or, more generally, systems with nonlinear, nonlocal and noncanonical interactions. The mathematical implications of the Lie-Santilli isotheory are significant. For instance, Gr. Tsagas (142) has shown that all simple non-exceptional Lie algebras of dimension N can be unified into one single Lie-Santilli isotope of the same dimension, while studies for the inclusion of exceptional algebras in this grand unification of Lie theory are under way In fact, the characterization of different simple Lie algebras, including the transition from compact to noncompact Lie algebras, can be characterized by different realizations of the isounit while using a unique form of generators and of structure constants (see the first examples for the SO(3) algebra in Ref. (23) of 1978 and numerous others in the quoted literature). The physical implications of the Lie-Santilli isotheory are equally significant. We here mention the reconstruction as exact at the isotopic level of Lie symmetries when believed to be broken under conventional treatment. In fact, R. M. Santilli has proved: the exact reconstruction of the rotational symmetry for all ellipsoidical deformations of 25 the sphere (12); the exact SU(2)-isospin symmetry under electromagnetic interactions (28,33); the exact Lorentz symmetry under all (sufficiently smooth) signature-preserving deformations of the Minkowski metric (26); and the exact reconstruction of parity under weak interactions (55). R. Mignani (180) has studied the exact reconstruction of the SU(3) symmetry under various symmetry-breaking terms. In all these cases the reconstruction of the exact symmetry has been achieved by merely embedding all symmetry breaking terms in the isounit. The construction of the isodual Lie-Santilli isotheory for antimatter is an instructive exercise for interested readers. The main physical theories characterized by isomathematics are given by: 3.6: Iso-Newtonian Mechanics and its isodual. As it is well known, Newton (1) had to construct the differential calculus as a necessary pre-requisite for the formulation of his celebrated equations. Today we know that Newton’s equations can only represent point-particles due to the strictly local-differential character of the underlying Euclidean topology. The fundamental character of Newtonian Mechanics for all scientific inquiries is due to the preservation at all subsequent levels of study (such as Hamiltonian mechanics, quantum mechanics, quantum chemistry, quantum field theory, etc.) of: 1) The underlying Euclidean topology; 2) The differential calculus; and 3) The notion of point particle. By keeping in mind Newton’s teaching, the author has dedicated primary efforts to the isotopic lifting of the conventional differential calculus, topology and geometries (14) as a necessary pre-requisite for a structural generalization of Newton’s equations into a form representing extended, nonspherical and deformable particles under action-at-adistance/ potential as well as contact/nonpotential forces. The need for such a lifting is due to the fact that point particles cannot experience contact-resistive forces. This feature has lead to subsequent theories, such as Hamiltonian and quantum mechanics, which solely admit action-at-a-distance/potential forces among point particles. Such a restriction is indeed valid for a number of systems, such as planetary systems at the classical level and atomic systems at the operator level, because the large distances among the constituents permit an effective point–like approximation of particles. However, when interactions occur at short distances, as in the case of electron valence bonds (Figure 2) or the mutual penetration of the wavepackets of particles in general, the point-like approximation is no longer sufficient and a representation of the actual, extended, generally nonspherical and deformable shape of particles is a necessary prerequisite to admit contact nonpotential interactions. By recalling the fundamental character of Newtonian mechanics for all of sciences, the achievement of a consistent representation of the contact interactions of valence electron bonds at the operator level requires the prior achievement of a consistent Newtonian representation. To outline the needed isotopies, let us recall that Newtonian mechanics is formulated 26 on the Kronecker product Stot = St£Sx£Sv of the one dimensional space St representing time t, the tree dimensional Euclidean space Sx of the coordinates x = (xk ®) (where k = 1; 2; 3 are the Euclidean axes and ® = 1; 2; :::; n represents the number of particles), and the velocity space Sv; v = dx=dt. It is generally assumed that all variables t; x; and v are defined on the same field of real numbers R. However, the unit of time is the scalar I = 1, while the unit of the Euclidean space is the matrix I = Diag:(1; 1; 1). Therefore, on rigorous grounds, the representation space of Newtonian mechanics Stot = S1 £ Sx £ Sv must be defined on the Kronecker product of the corresponding fields Rtot = Rt £ Rx £ Rv with total unit ITot = 1 £ Diag:(1; 1; 1)x £ Diag:(1; 1; 1)v. Newtonian systems requested for the isotopies are given by the so-called closed-isolated non-Hamiltonian systems (51), namely, systems which are closed-isolated from the rest of the universe, thus verifying all ten Galilean total conservation laws, yet they admit internal non-Hamiltonian forces due to contact interactions. A typical illustration is given by the structure of Jupiter which, when considered as isolated from the rest of the universe, does indeed verify all Galilean conservation laws, yet its internal structure is clearly non-Hamiltonian due to vortices with varying angular momentum and similar internal dissipative effects. In essence, contact nonpotential forces produce internal exchanges of energy, linear and angular momentum but always in such a manner to verify total conservation laws. A Newtonian representation of closed-isolated non-Hamiltonian systems of extended particles is given by(Ref. (51), page 236) m® £ ak® = m® £ dvk® dt = F®(t; x; v) = FSA ® (x) + FNSA ® (t; x; v); (3:42a) X ®=1;:::;n FNSA ® = 0; (3:42b) X ®=1;:::;n x®KFNSA ® = 0; (3:42c) X ®=1;:::;n x®^FNSA ® = 0; (3:42d) where: SA (NSA) stands for variational selfadjointness (variational nonselfadjointness), namely, the verification (violation) of the integrability conditions for the existence of a potential, and conditions (3.9b), (3.9c) and (3.9d) assure the verification of all ten Galilean conservation laws (for the total energy, linear momentum, angular momentum, and uniform motion of the center of mass). The restrictions to FNSA verifying the above conditions is tacitly assumed hereon. The isotopies of Newtonian mechanics, also called Newton-Santilli isomechanics (63- 68), requires the use of the isotime ˆt = t £ ˆIt with isounit ˆIt = 1= ˆ Tt and related isofield ˆR t, the isocoordinates ˆx = (ˆxk ®) = x £ ˆIx; with isounit ˆIx = 1= ˆ Tx and related isofield ˆRx, and the isospeeds ˆv = (vk®) = v £ ˆIv with isounit ˆIv = 1= ˆ Tv and related isofield ˆRv. 27 IsoNewtonian Mechanics is then formulated on the Kronecker product of isospaces ˆ STot = ˆ St £ ˆ Sx £ ˆ Sv over the Kronecker product of isofields ˆRt £ ˆRx £ ˆRv. The isospeed is the given by ˆv = ˆ dˆx ˆ dˆt = ˆIt £ d(x £ ˆIx) dt = v £ ˆIt £ ˆIx + x £ ˆIt £ dˆIx dt = v £ ˆIv; (3:43a) ˆIv = ˆIt £ ˆIx £ (1 + x £ ˆ Tx £ dˆIx dt

(3:43b)

The Newton-Santilli isoequation and its isodual, first proposed in memoir (14) of 1996 (where the isodifferential calculus was first achieved) can be written ˆm® ˆ£ ˆ dˆvk® ˆ dˆt = ¡ ˆ@ ˆ V (ˆx) ˆ@ˆxk ®

(3:44)

namely, the equations are conceived in such a way to formally coincide with the conventional equations for selfadjoint forces, FSA = ¡@V=@x; while all nonpotential forces are represented by the isounits or, equivalently, by the isodifferential calculus. Such a conception is the only one known which permits the representation of extended particles with contact interactions which is invariant (thus avoiding the catastrophic inconsistencies of Theorem 3.1) and achieves closure, namely, the verification of all ten Galilean conservation laws. An inspection of Eqs. (3.10) is sufficient to see that iso-Newtonian mechanics reconstructs canonicity on isospace over isofields, thus avoiding Theorem 3.1. Note that this would not be the case if nonselfadjoint forces appear in the right hand side of Eqs. (3.10) as in Eqs. (3.9a). The verification of all Galilean conservation laws is equally established by a visual inspection of Eqs. (3.10) since their symmetry, the iso-Galilean symmetry with structure (3.8), is the Galilean symmetry, only formulated on isospace over isofields (53). By recalling that conservation laws are represented by the generators of the underlying symmetry, conventional total conservation laws then follow from the fact that the generator of the conventional Galilean symmetry and its isotopic lifting coincide. When projected in the conventional Newtonian space STot, Eqs. (3.10) can be explicitly written ˆmˆ£ ˆ dˆv ˆ dˆt = m £ ˆIt £ d(v £ ˆIv) dt = = m £ a £ ˆIt £ ˆIv + m £ v £ ˆIt £ dˆIv dt = ¡ ˆ@ ˆ V (ˆx) ˆ@ˆx = ¡ˆIx £ @V @x

(3
46)

that is m £ a = ¡ˆ Tt £ ˆ Tv £ ˆIx £ @V @x ¡ m £ v £ ˆ Tv £ dˆIv dt

(3
47)

with necessary and sufficient conditions for the representation of all possible SA and NSA forces ˆIt £ ˆIv £ ˆIx = I; ˆIx = 1= ˆ Tt £ ˆ Tx; (3:48a) 28 m £ v £ ˆ Tv £ dˆIv dt = FNSA(t; x; v); (3:48b) which always admit a solution, since they constitute a system of 6n algebraic (rather than differential) equations in the 6n + 1 unknowns given by ˆIt, and the diagonal ˆIx and ˆIv. As an illustration, we have the following equations of motion of an extended particle with the ellipsoidal shape experiencing a resistive force FNSA = ¡° £ v because moving within a physical medium m £ a = ¡° £ v (3:49a) ˆIv = Diag:(n21

n22
n23

) £ e°£t=m: (3:49b) Interested readers can then construct the representation of any desired NSA forces (see also memoir (14) for other examples). Note the natural appearance of the velocity dependence, as typical of resistive forces. Note also that the representation of the extended character of particles occurs only in isospace because, when Eqs. (3.10) are projected in the conventional Newtonian space, all isounits cancel out and the point characterization of particles is recovered. Note finally the direct universality of the Newton-Santilli isoequations, namely, their capability of representing all infinitely possible Newton’s equations in the frame of the observer. As indicated earlier, Eqs. (3.42) can only describe a system of particles. The construction of the isodual Newton-Santilli isoequations for the treatment of a system of antiparticles is left to the interested reader. We finally indicate that the invariance of closed non-Hamiltonian systems (3.42) is given by the Galilei-Santilli isosymmetry ˆG(3:1) and their isoduals by ˆGd(3:1) (see Refs. (52,53) for brevity). 3.7: Iso-Hamiltonian Mechanics and its isodual. Eqs. (3.10) admit the analytic representation in terms of the following isoaction principle (14) ˆ± ˆ A(ˆt; ˆx) = ˆ± ˆ Z (ˆpk® ˆ£p ˆ dˆxk ®) ¡ ˆH ˆ£ t ˆ dˆt = = ± ˆ Z [pk® £ ˆ Tx(t; x; p; :::) £ d(xk ® £ ˆIx) ¡ H £ ˆ Tt(t; x; p; :::)d(t £ ˆIt) = 0: (3:50) Note the main result permitted by the isodifferential calculus, consisting in the reduction of an action functional of arbitrary power in the linear momentum (arbitrary order) to that of first power in p first order. Since the optimal control theory and the calculus of variation depend on the first order character of the action functional, the above reduction has important implications, such as the treatment of extended objects moving within resistive media apparently for the first time via the optimal control theory, since a first order conventional action is impossible for the systems considered.. Note that when the isounits are constant, isoaction the isoaction coincides with the conventional action. This illustrates the apparent reason why the isotopies of the action principle creeped in un-noticed for over one century. 29 It is easy to prove that the above isoaction principle characterizes the Hamilton-Santilli isoequations (14) ˆ dˆx ˆ dˆt = ˆ@ ˆH ˆ@ ˆp = ˆp ˆm = ˆp ˆm

# ( ˆW £ ˆ T £ A £ ˆ T £ ˆW y) £ ( ˆ T £ ˆW y)¡1 £ ˆ T £ ( ˆW £ ˆ T)¡1 £ ( ˆW £ ˆ T £ ˆB £ ˆ T £ ˆW y)

ˆ° =

1 (1 ¡ ˆ ¯2)1=2

(3:77e)

For the general expression interested readers can inspect Ref. (55). Ref. (26) proved that, contrary to another popular belief throughout the 20-th century, the Lorentz symmetry remains exact for all possible signature preserving (+,+,+,1) deformations of the Minkowski space, of course, when treated with the appropriate mathematics. The above exact reconstruction of the Lorentz symmetry can be geometrically visualized by noting that the light cone x23 ¡ c2 o £ t2 = 0 can only be formulated in vacuum while within physical media we have the generic hyperboloid r2 3 ¡ c2 o £ t2=n2(!; :::) = 0. However, it is an instructive exercise for interested readers to prove that the isolight cone (that is, the light cone on isospace over isofields) is the perfect cone ˆr2 3 ¡ c2 o £ ˆt = 0 with the exact symmetry ˆO(3:1) while its projection on conventional space is given by r2 3 ¡ c2 o £ t2=n2(!; :::) = 0 with broken Lorentz symmetry. (3) The isotranslations ˆ T (4) : ˆx0 = ˆ T (ˆa; :::)ˆ] £ x = ˆx + ˆ A(ˆa; x; :::); ˆa = a £ ˆIa 2 ˆRa which can be written x¹0 = x¹ + A¹(a; :::); (3:78a) A¹ = a¹(g¹¹ + a® £ [g;i¹ˆ;P®]=1! + :::); (3:78b) where there is no summation on the ¹ indices. Note that the isotranslations are highly nonlinear (thus non-inertial) in conventional spacetime although they are isolinear (thus inertial) in isospace. This illustrates the reason why conventional notion of relativity are solely applicable in spacetime, thus illustrating the reason of the name ”isorelativity.” (4) The novel isotopic invariance ˆI : ˆx0 = ˆ wˆ£ ˆx = w £ ˆx; ˆI0 = w £ ˆI, where w is a constant (29), ˆI ! ˆI0 = ˆ wˆ£ ˆI = w £ ˆI = 1= ˆ T0; (3:79a) ˆxˆ2 = (x¹ £ ˆ´¹º £ xº ) £ ˆI ´ ˆx0ˆ2 = [x¹ £ (w¡1 £ ˆ´¹º) £ xº ] £ (w £ ˆI); (3:79b) Therefore, the Poincar´e-Santilli isosymmetry can be written ˆ P(3:1) = ˆO (3:1) ˆ£ ˆ T (4) ˆ£ ˆI (3:80) thus having eleven (rather than ten) dimensions with parameters µk; vk; a¹;w; k = 1; 2; 3; ¹ = 1; 2; 3; 4; the 11-th dimension being characterized by invariant (3.78). Note that, contrary 39 to popular beliefs, the conventional Poincar´e symmetry is also eleven dimensional since invariance (3.78) also holds for conventional spacetime. The simplest possible realization of the above formalism for isorelativistic kinematics can be outlined as follows (see Section 3.13 for the isogravitational realization). The first application of isorelativity is that of providing an invariant description of locally varying speeds of light propagating within physical media. For this purpose a realization of isorelativity requires the knowledge of the density of the medium in which motion occurs. The simplest possible realization of the fourth component of the isometric is then given by the function g44 = n24 (x; !; :::) normalized to the value n4 = 1 for the vacuum (note that the density of the medium in which motion occur cannot be described by special relativity). Representation (3.68) then follows with invariance under ˆ P(3:1). In this case the quantities nk; k = 1; 2; 3; represent the inhomogeneity and anisotropy of the medium considered. For instance, if the medium is homogeneous and isotropic (such as water), all metric elements coincide, in which case ˆI = Diag:(g11; g22; g33; g44) = n24 £ Diag:(1; 1; 1; 1); (3:81a) ˆxˆ2 = x2 n24 £ n24 £ I ´ x2; (3:81b): thus confirming that isotopies are hidden in Minkowskian axioms, and this may be a reason why they have nog been discovered until recently. Next, isorelativity has been constricted for the invariant description of systems of extended, nonspherical and deformable particles under Hamiltonian and non-Hamiltonian interactions. Practical applications then require the knowledge of the actual shape of the particles considered, here assumed for simplicity as being spheroidal ellipsoids with semiaxes n21

n22
n23

. Note that them minimum number of constituents of a closed non- Hamiltonian system is two. In this case we have shapes represented with n®k; ® = 1; 2; ; :::; n. Applications finally require the identification of the nonlocal interactions, e.g., whether occurring on an extended surface or volume. As an illustration, two spinning particles denoted 1 and 2 in condition of deep mutual penetration and overlapping of their wavepackets (as it is the case for valence bonds), can be described by the following Hamiltonian and total; isounit total isounit H = p1 £ p1 2 £ m1 + p2 £ p2 2 £ m2 + V (r); (3:82a) ˆITot = Diag:(n2 11; n2 12; n2 13; n2 14) £ Diag:(n2 21; n2 22; n2 23; n2 24)£ £eN£( ˆ Ã1=Ã1+ ˆ Ã2=Ã2)£R ˆ Ã1"(r)y£ ˆ Ã2#(r)£dr3

(3
82b)

where N is a constant. Note the nonlinearity in the wavefunctions, the nonlocal-integral character and the lack of representation of all the above features via a Hamiltonian. From the above examples interested readers can then represent any other closed non- Hamiltonian systems. 40 The third important part of isorelativity is given by the following isotopies of conventional relativistic axioms which, for the case of motion along the third axis, can be written (29): ISOAXIOM I. The projection in our spacetime of the maximal causal invariant speed is given by: VMax = co £ g1=2 44 g1=2 33 = co n3 n4 = c n3

(3:83)

This isoaxioms resolves the inconsistencies of special relativity recalled earlier for particles and electromagnetic waves propagating in water. In fact, water is homogeneous and isotropic, thus requiring that g44 = g33 = 1=n2, where n is the index of refraction. In this case the maximal causal speed for a massive particle is co as experimentally established, e.g., for electrons, while the local speed of electromagnetic waves is c = co=n., as also experimentally established. Note that such a resolution requires the abandonment of the speed of light as the maximal causal speed for motion within physical media, and its replacement with the maximal causal speed of particles. It happens that in vacuum these two maximal causal speeds coincide. However, even in vacuum the correct maximal causal speed remains that of particles and not that of light, as generally believed. At any rate, physical media are generally opaque to light but generally not to particles. Therefore, the assumption as the maximal causal speed as that of light which cannot propagate within the medium considered would be evidently vacuous. It is an instructive exercise for the interested readers to prove that the maximal causal speed of particles on isominkowski space over an isofield remains co. ISOAXIOM II. The projection in our spacetime of the isorelativistic addition of speeds within physical media is given by: vTot = v1 + v2 1 + v1£g33£v2 co£g44£co = v1 + v2 1 + v1£n24 £v2 co£n23 £co (3:84) We have again the correct occurrence that the sum of two maximal causal speeds in water, Vmax = co£(n3=n4), yields the maximal causal speed in water, as the reader is encouraged to verify. Note that the such a result is impossible for special relativity. Note also that the isorelativistic sum of two speeds of lights in water, c = co=n, does not yield the speed of light in water, thus confirming that the speed of light within physical media, assuming that they are transparent to light, is not the fundamental maximal causal speed. ISOAXIOM III. The projection in our spacetime of the isorelativistic laws of dilation of time to and contraction of length `o and the variation of mass mo with speed are given by: t = ˆ° £ to; ` = ˆ°¡1 £ `o;m = ˆ° £ mo: (3:85) 41 Note that in water these values coincide with the relativistic one as it should be since particles such as the electrons have in water the maximal causal speed c0. Note again the necessity of avoiding the interpretation of the local speed of light as the maximal local causal speed. Note that the mass diverges at the maximal local causal speed, but not at the local speed of light.. ISOAXIOM IV. The projection in our spacetime of the iso-Doppler law is given by (for 90o angle of aberration): ! = ˆ° £ !o: (3:86) This isorelativistic axioms permits an exact, numerical and invariant representation of the large differences in cosmological redshifts between quasars and galaxies when physically connected. In this case light simply exit the huge quasar chromospheres already redshifted due to the decrease of the speed of light, rather than the speed of the quasars (118). Isoaxiom IV also permits a numerical interpretation of the internal blue- and red-shift of quasars due to the dependence of the local speed of light on its frequency. Finally, Isoaxiom IV predicts that a component of the predominance toward the red of sunlight at sunset is of iso-Doppler nature in view of the bigger decrease of the speed of light at sunset as compared to the same speed at the zenith (evidently because of the travel within a comparatively denser atmosphere). ISOAXIOM V. The projection in our spacetime of the isorelativistic law of equivalence of mass and energy is given by: E = m £ c2 o £ g44 = m £ c2 o n24

(3:87)

# (A;H; FNSA)

53 Figure 3: An illustration via sea shells growth of the third scientific unbalance of the 20-th century, the lack of a structurally irreversible mathematics (that is, a mathematics whose basic axioms are not invariant under time reversal) for quantitative representations of irreversible processes. The unbalance is due to the fact that all formulations used until now are of Hamiltonian type, while all known Hamiltonians and their background mathematics are reversible, thus implying the study of irreversible systems via fully reversible formulations. = ( @A @xk ® £ @H @pk® ¡ @A @pk® £ @H @xk ® ) + @A @pk® £ FNSA k® : (4:3) Since all known Lagrangians and Hamiltonians are reversible in time, according to the teaching of Lagrange and Hamilton, irreversibility is characterized, again, by the external terms representing contact zero-range interactions among extended particles. At the beginning of the 20-th century, Lagrange’s and Hamilton’s external terms were truncated, resulting in analytic equations d dt @L(x; v) @v ¡ @L(x; v) @x = 0; (4:4a) dx dt ¡ @H(x; p) @p = 0; dp dt + @H(x; p) @x = 0; (4:4b) with time evolution characterized by the familiar Lie brackets dA dt

# [A;H]

= @A @xk ® £ @H @pk® ¡ @A @pk® £ @H @xk ®

(4
5)

which are fully reversible. 54 The above occurrence was due to the successes of the truncated analytic equations for the representation of planetary and atomic structures, resulting in their use for virtually all scientific inquiries of the 20-th century. In turn, the assumption of the truncated analytic equations as the ultimate formulation of science implied the scientific unbalance under consideration here because planetary and atomic structures are fully reversible, thus lacking sufficient generalities for all of nature. 4.3: Catastrophic inconsistencies of formulations with external terms. More recent studies (23,38) have shown that the true Lagrange’s and Hamilton’s equations (those with external terms) cannot be used in applications due to a number of insufficiencies, such as: (1) the lack of invariant numerical predictions in accordance with Theorem 3.1 (due to their evident noncanonical character); (2) the lack of characterization of any algebra by the brackets of the time evolution, let alone the loss of all Lie algebras, because brackets (A;H; FNSA) of Eqs. (4.3) violate the right distributive and scalar laws as necessary to characterize an algebra commonly understood in contemporary mathematics since they are triple systems); (3) the lack of a topology suitable to represent contact nonpotential interactions among extended particles since the topology of conventional Hamiltonian formulation is strictly local-differential, thus solely characterizing point particles; and other limitations. The only resolution of these problematic aspects known to this author was the construction of the novel structurally irreversible mathematics indicated earlier. Stated in different terms, the manifestly inconsistent reduction of irreversible macroscopic systems to elementary particles in reversible conditions was due, again, to insufficiencies of the used mathematics. It should be noted that the isomathematics of the preceding section is also reversible in time because the isounit is Hermitean, thus lacking the mathematical characterization of time reversal, and confirming the need of constructing of a broader mathematics specifically suited to represent irreversibility. 4.4: Initial versions of irreversible mathematics. The achievement of a structurally irreversible mathematics resulted to be a long scientific journey due to the need of achieving invariance under irreversible conditions. The first studies can be traced back to Ref. (8) of 1967 which presented the first known parametric deformation of Lie algebras with product (A;B) = p £ A £ B ¡ q £ B £ A =

# v £ (A £ B ¡ B £ A) + w £ (A £ B + B £ A)

= v £ [A;B] + w £ fA;Bg; (4:6) where p, q, and p § q are non-null parameters, v = p + q;w = q ¡ p, and A, B are Hermitean matrices. 55 The studies continued with the first known presentation in Ref. (38) of the operator deformations of Lie algebra with product (Aˆ;B) = A £ P £ B ¡ B £ Q £ A =

# (A £ T £ B ¡ B £ T £ A) + (A £W £ B + B £W £ A)

etc
(4:20)

58 Particularly intriguing are the genogeometries (loc. cit.) because they admit nonsymmetric metrics, such as the genoriemannian metrics g>(x) = ˆ T>(x) £ ´; (4:21) where ´ is the Minkowski metric and ˆ T>(x) is a real-values, nowhere singular, 4 £ 4 nonsymmetric matrix, while bypassing known inconsistencies since they are referred to the nonsymmetric genounit ˆI> = 1= ˆ T>: (4:22) In this way, genogeometries are structurally irreversible and actually represent irreversibility with their most central geometric notion, the metric. 4.6: Lie-Santilli genotheory and its isodual. Particularly important for this note is the lifting of Lie’s theory permitted by genomathematics, first identified by R. M. Santilli in Ref. (23) of 1978, and today knows as the Lie-Santilli genotheory [7,8], which is characterized by: (1) The forward and backward universal enveloping genoassociative algebra ˆ»>;< ˆ», with infinite-dimensional basis characterizing thePoincar´e-Birkhoff-Witt-Santilli genotheorem ˆ»> : ˆI; ˆXi; ˆXi > ˆXj ; ˆXi > ˆXj > ˆXk; :::; i · j · k; (4:23a) < ˆ» : ˆI; ˆXi; ˆXi < ˆXj ; ˆXi < ˆXj < ˆXk; :::; i · j · k; (4:23b) where the ”hat” on the generators denotes their formulation on genospaces over genofields and their Hermiticity implies that ˆX > =< ˆX = ˆX ; (2) The Lie-Santilli genoalgebras characterized by the universal, jointly Lie-and Jordanadmissible brackets (4.7), < ˆL> : ( ˆXiˆ; ˆXj) = ˆXi < ˆXj ¡ ˆXj > ˆXi = ˆ Ck ij ˆ£ ˆX k; (4:24) although now formulated in an invariant form (see below); (3) The Lie-Santilli genotransformation groups < ˆG> : ˆ A( ˆ w) = (ˆeˆi ˆ£ ˆX ˆ£ ˆ w > ) > ˆ A(ˆ0) < (<ˆe¡ˆi ˆ£ ˆ wˆ£ ˆX ) = = (ei£ ˆX £ˆ T>£w) £ A(0) £ (e¡i£w£< ˆ T£ ˆX ); (4:25) where ˆ w> 2 ˆR> are the genoparameters; the genorepresentation theory, etc. The mathematical implications of the Lie-Santilli genotheory are significant because of the admission as particular cases of all possible algebras, as well as because, when computed on the genobimodule < ˆ»£ˆ»> Lie-admissible algebras verify all Lie axioms, while deviations from Lie algebras emerge only in their projection on the bimodule <» £ »> of the conventional Lie theory. This is due to the fact that the computation of the left action A < B = A £< ˆ T £ B on < ˆ» (that is, with respect to the genounit < ˆI = 1=< ˆ T) yields the save value as the computation of the conventional product A£B on <» (that is, with 59 respect to the trivial unit I), and the same occurs for the value of A > B on ˆ»>. In this way, thanks to genomathematics, Lie algebras acquire a towering significance in view of the possibility of reducing all known algebras to primitive Lie axioms. The physical implications of the Lie-Santilli genotheory ar equally significant. In fact, Noether’s theorem on the reduction of conservation laws to primitive Lie symmetries can be generalized to the reduction of, this time, nonconservation laws to primitive Lie- Santilli genosymmetries. As a matter of fact, this reduction was the very first motivation that suggested the construction of the genotheory in memoir (23) (see also monographs (49,50)). The reader can then foresee similar liftings of all remaining physical aspects treated via Lie algebras. The construction of the isodual Lie-Santilli;i genotheory is an instructive exercise for readers interested in learning the new methods. The physical theories characterized by genomathematics can be summarized as follows. 4.7: Geno-Newtonian Mechanics and its isodual. Recall that, for the case of isotopies, the basic Newtonian systems are given by those admitting nonconservative internal forces restricted by certain constraints which verify total conservation laws (closed non- Hamiltonian systems). For the case of the genotopies under consideration here, the basic Newtonian systems are the conventional nonconservative systems (4.1) without subsidiary constraints (open non-Hamiltonian systems). In this case irreversibility is characterized by nonselfadjoint forces, as indicated earlier. The forward geno-Newtonian mechanics and its isodual is a generalization of Newtonian mechanics for the description of motion forward in time of the latter systems via a structurally irreversible mathematics. The new mechanics is characterized by (14): the forward genotime ˆt> = t £ ˆI> t with (nowhere singular and non-Hermitean) forward time genounit ˆI> t = 1= ˆ T> t 6= ˆI>y t , related forward time genospace ˆ S> t over the forward time genofield ˆR> t ; the forward genocoordinates ˆx> = x £ ˆI> x with (nowhere singular non-Hermitean) forward coordinate genounit ˆI> x = 1= ˆ T> x 6= ˆI>y x with forward coordinate genospace ˆ S> x and related forward coordinate genofield ˆR> x ; and the forward genospeeds ˆv> = ˆ d>ˆx>= ˆ d>ˆt> with (nowhere singular and non-Hermitean) forward speed genounit ˆI> v = 1= ˆ T> v 6= ˆI>y v with related forward speed genospace ˆ S> x and forward speed genofield ˆR > v . Note that, to verify the condition of non-Hermiticity, the time genounits should be at least complex valued, and the same then occurs for the other genounits. The representation space is then given by the Kronecker product ˆ S> Tot = ˆ S> t £ ˆ S> x £ ˆ S> v ; (4:26) defined over the genofield ˆR > tot = ˆR> t £ ˆR> x £ ˆR> v ; (4:27) with total genounit ˆI> tot = ˆI> t £ hatI> x £ ˆI> v : (4:28) The basic equations are given by the forward geno-Newton equations, also known as Newton-Santilli genoequations, first proposed in memoir (14) via the genodifferential 60 calculus, also known as forward Newton-Santilli genoequations [8-11] ˆm> >® ˆ d>ˆv> k® ˆ d>ˆt> = ¡ ˆ@> ˆ V > ˆ@>ˆx>k ®

(4:29)

The backward geno-Newton equations is characterized by backward genounits can be obtained via transpose conjugation of the forward formulation. As one can see, the representation of Newton’s equations is done in a way similar to the isotopic case, the main difference being that the basic unit is now no longer symmetric. Note that in Newton’s equations the nonpotential forces are part of the applied force F, while in the geno-Newton equations nonpotential forces are represented by the forward genounits, or, equivalently, by the forward genodifferential calculus, in a way essentially similar to the case of isotopies. The main difference is that isounits are Hermitean, thus implying the equivalence of forward and backward motions, while genounits are non- Hermitean, thus implying irreversibility. Note also that the topology underlying Newton’s equations is the conventional, Euclidean, local-differential topology which, as such, can only represent point particles. By contrast, the topology underlying the geno-Newton equations is the Santilli-Sourlas- Tsagas genotopology (14,139) for the representation of extended, nonspherical and deformable particles via forward genounits, e.g., of the diagonal type ˆI> = Diag:(n21

n22
n23
n2

4) £ ¡>(t; x; v; :::); (4:30) where n2 k; k = 1; 2; 3 represents the semiaxes of an ellipsoid, n24 represents the density of the medium in which motion occurs (with more general nondiagonal realizations here omitted for simplicity), and ¡> represents contact interactions occurring for the motion forward in time. The construction of the isodual image of the above geno-Newtonian mechanics is instructive to understand the difference between isoduality and motion backward in time. 4.8: Geno-Hamiltonian mechanics and its isodual. The most effective setting to introduce real-valued and non-Hermitean (thus non-symmetric) genounits is in the 6ndimensional forward genocotangent bundle (geno-phase-space) with local genocoordinates and their conjugate ˆa>¹ = a½ £ ˆI>¹ 1½ ; (ˆa>¹) = Ã ˆx>k ® ˆp> k® !; ˆR> ¹ = R½ £ ˆI>½ 2¹ ; ( ˆR> ¹ ) = (ˆpk®; ˆ0); (4:31a); ˆI> 1 = 1= ˆ T> 1 = (ˆI> 2 )T = (1= ˆ T> 2 )T ; k = 1; 2; 3; ® = 1; 2; :::; n; ¹; ½ = 1; 2; :::6n; (4:31a) where the superscript T stands for transposed, with nowhere singular, real-valued and non-symmetric genometric and related invariant ˆ±> = ±6n£6n £ ˆ T> 1 6n£6n; (4:32a) ˆa>¹ > ˆR> ¹ = ˆa>½ £ ˆ T>¯ 1½ £ ˆR > ¯ = a½ £ ˆI>¯ 2½ £ R¯: (4:32b) 61 In this case we have the following genoactionprinciple (14) ˆ±> ˆ A> = ˆ±> ˆ Z > [ ˆR> ¹ >a ˆ d>ˆa> ¡ ˆH > >t ˆ d>ˆt>] = = ±Z [R¹ £ ˆ T>¹ 1º (t; x; p; :::) £ d(a¯ £ ˆI>º 1¯ ) ¡ H £ dt] = 0; (4:33) where the second expression is the projection on conventional spaces over conventional fields and we have assumed for simplicity that the time genounit is 1. It is easy to prove that the above genoprinciple characterizes the following forward geno-Hamilton equations, also called forward Hamilton-Santilli genoequations (originally proposed in Ref. (23) of 1978 with conventional mathematics and in ref. (14) of 1996 with genomathematics; see also Refs. (28,51,52,55)) ˆ!¹º ˆ£ ˆ dˆaº ˆ dˆt ¡ ˆ@ ˆH(ˆa) ˆ@ˆa¹ = = Ã 0 ¡1 1 0 !£ Ã dx=dt dp=dt !¡ Ã 1 K 0 1 !£ Ã @H=@x @H=@p ! = 0; (4:34a) ˆ! = ( ˆ@Rº ˆ@ˆa¹ ¡ ˆ@ ˆR¹ ˆ@ˆaº ) £ ˆI = Ã 0 ¡1 1 0 !£ ˆI; (4:34b) K = FNSA=(@H=@p): (4:34c) The time evolution of a quantity ˆ A>(ˆa>) on the forward geno-phase-space can be written in terms of the following brackets d ˆ A> dt

# ( ˆ A>; ˆH >)

ˆ@> ˆ A> ˆ@>ˆa>¹ ˆ£ ˆ!¹º ˆ£ ˆ@> ˆH > ˆ@ˆa>º = = @ ˆ A> @ˆa>¹ £ S ¹º £ artoa; ˆH > @ˆa>º = = ( @ ˆ A> @ˆx>k ® £ @ ˆH> @ ˆp> k® ¡ @ ˆ A> @ ˆp> k® £ @ ˆH > @ˆx>k ® ) + @ ˆ A> @ ˆp> k® £ Kk k £ @ ˆH > @ ˆp> k®

(4:35a)

S>¹º = !¹½ £ ˆI2º ½ ; !¹º = (jj!®¯jj¡1)¹º; (4:35b) where !¹º is the conventional Lie tensor and, consequently, S¹º is Lie-admissible in the sense of Albert (7). As one can see, the important consequence of genomathematics and its genodifferential calculus is that of turning the triple system (A;H; FNSA) of Eqs. (4.3) in the bilinear form (Aˆ;B) of brackets (4.35a),m thus regaining the existence of a consistent algebra in the brackets of the time evolution, for which central purpose genomathematics was built (since the multiplicative factors represented by K are fixed for each given system). The invariance of such a formulation will be proved shortly. 62 It is easy to verify that the above identical reformulation of Hamilton’s historical time evolution (4.3) correctly recovers the time rate of variations of physical quantities in general, and that of the energy in particular, dA dt = [ ˆ A>; ˆH >] + @ ˆ A> @ ˆp> k® £ FNSA k® : (4:36a) dH dt = [ ˆH >; ˆH >] + @ ˆH > @ ˆp> k® £ FNSA k® = vk ® £ FNSA k® : (4:36b) It is easy to show that genoaction principle (4.33) characterizes the following Hamilton- Jacobi-Santilli genoequations ˆ@>A> ˆ@>ˆt> + ˆH > = 0; (4:37a) ( ˆ@>A> ˆ@>ˆa>¹ ) = ( ˆ@>A> ˆ@>x>k ®

ˆ@>A> ˆ@>p> k ® ) = ( ˆR> ¹ ) = (ˆp> k®; ˆ0); (4:37b) which confirm the property (crucial for genoquantization as shown below) that the genoaction is indeed independent of the linear momentum. Note the direct universality of Eqs. (4.33) for the representation of all infinitely possible Newton equations (4.1) (universality) directly in the fixed frame of the experimenter (direct universality). Note also that, at the abstract, realization-free level, Geno-Hamilton equations (4.34) coincide with Hamilton’s equations without external terms, yet represent those with external terms. The latter are reformulated via genomathematics as the only known way to achieve invariance while admitting a consistent algebra in the brackets of the time evolution (38). Therefore, genohamilton equations (4.34) are indeed irreversible for all possible reversible Hamiltonians, as desired. The origin of irreversibility rests in the contact nonpotential forces according to Lagrange’s and Hamilton’s teaching. Note finally that the extension of Eqs. (4.9) to include nontrivial genotimes implies a major broadening of the theory we cannot review for brevity (14,55). The above geno-Hamiltonian mechanics requires, for completness, three additional formulations, the backward geno-Hamiltonian mechanics for the description of matter moving backward in time, and the isoduals of both the forward and backward mechanics for the description of antimatter. The construction of these additional mechanics is lefty to the interested reader. 4.9: Genotopic Branch of Hadronic Mechanics and its isodual. A simple genotopy of the naive or symplectic quantization applied to Eqs. (4.37) yields the genotopic branch of hadronic mechanics defined on the forward genotopic Hilbert space ˆH> with forward genoinner product < ˆ Ãj > j ˆ Ã > £ˆI> 2 ˆ C>. The resulting genotopy of quantum mechanics is characterized by the forward geno-Schroedinger equations (first formulated in Refs. (42,179) via conventional mathematics and in Ref. (14) via genomathematics) ˆi > > ˆ@> ˆ@>ˆt> j ˆ Ã> >= ˆH > > j ˆ Ã> >= 63 = ˆH (ˆx; ˆv) £ ˆ T>(ˆt>; ˆx>; ˆp>; ˆ Ã>; ˆ@> ˆ Ã>::::) £ j ˆ Ã> >= E> > jÃ> >; (4:38a) ˆp> k ˆ £j ˆ Ã> >= ¡ˆi> > ˆ@> k j ˆ Ã> >= ¡i £ ˆI>i k £ @ij ˆ Ã> >; ˆI> > j ˆ Ã> >= j ˆ Ã> >; (4:38b) with conjugate backward equations obtained via Hermitean conjugation. Note the crucial independence of isoaction ˆ A> in principle (4.33) from the linear momentum, as expressed by the Hamilton-Jacobi-Santilli genoequations (4.37). In fact, such independence assures that genoquantization yields a genowavefunction solely dependent on time and coordinates, ˆ Ã> = ˆ Ã>(t; x). Other geno-Hamiltonian mechanics do not verify such a condition, thus implying genowavefunctions with an explicit dependence also on linear momenta, ˆ Ã> = ˆ Ã>(t; x; p) which violate the abstract identity of quantum and hadronic mechanics andwhose treatment in any case is beyond our operator knowledge at this writing. The complementary geno-Heisenberg equations are given by in their finite and infinitesimal forms (first formulated in Ref. (38) via conventional mathematics and in Ref. (14) via genomathematics) ˆ A(ˆt) = (ˆeˆi ˆ£ ˆH ˆ£ ˆt > ) > ˆ A(ˆ0) < (<ˆe¡ˆi ˆ£ ˆt ˆ£ ˆH ) = = (ei£ ˆH£ˆ T>£t) £ A(0) £ (e¡i£t£< ˆ T£ ˆH ); (4:39a) ˆi ˆ£ ˆ d ˆ A ˆ dˆt

# ( ˆ Aˆ; ˆH ) = ˆ A < ˆH ¡ ˆH > ˆ A

+++

http://home1.gte.net/ibr/ir00018.htm

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.

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:

a) a brief summary of the topic. b) a statement of the open problem(s) suggested for study. c) the motivation for the proposed research. d) the suggested IBR member(s) and/or Editor(s) for technical assistance. e) representative references.

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 Prepared by J. V. KADEISVILI, IBR

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,

(1) e -> E = E(t, e, v, a, . . . ), 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

(2) axb -> a*b = axTxb, T = 1/E, 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 (3) e -> E(t, r, v, a, ... ), ab -> a*b = axTxb, E = 1/T, 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>)Ý with corresponding ordered products to the left and to the right,

(4a) <E = 1/R, a = 1/S, a>b = AxSxb, R = SÝ.

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, Lie’s 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

(5a) [A, B]* = A*B - B*A = AxT(t, r, ...)xB - BxT[t, r,...)xA, T = TÝ, (5b) (A, B) = AA = AxR(t, r, ...)xB - BxS(t, r, ...)xA, R = SÝ,

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, ...) = EÝ 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

(6) a -> isod(a) = - aÝ,

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 (7) e = 1 -> E = - 1, axb -> a*b = (-bÝ)x(-1)x(-aÝ) = - bÝxaÝ. 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, ...) = EÝ. 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) = -EÝ = -E.

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

CLASS I, when E is positive-definite (isotopies), CLASS II when E is negative definite (isodualities), CLASS III, given by the union of Classes I and II, CLASS IV, including the preceding classes plus null values of E (or singular values of T), and CLASS V, when E is arbitrary, e.g., a step-function or a distribution.

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 Albert’s axiom of Lie-admissibility. The same product was proved to verify the Lie axiom when each of the two terms AA 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,

(8a) <E = {<A, <B, ...} = 1/R = {1/R1, 1/R2, ...}, (8b) E> = {A>, B>, ... } = 1/S = {1/S1, 1/S2, ...},

with corresponding multivalued hypermultiplications ab = 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 Kadeisvili’s 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.

[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).

II: OPEN RESEARCH PROBLEMS IN NUMBER THEORY

Prepared by the IBR staff. PROBLEM II.1: STUDIES ON THE ISODUAL NUMBER THEORY DEFINITION II.1: Let F = F(a,+,x) be a conventional field of (real R, complex C or quaternionic Q) numbers a with additive unit 0, multiplicative unit e = 1, sum a+b and product axb. Santilli’s isodual field [II-1] isodF = isodF(isoda, isod+,isodx) is a ring of elements isoda = -aÝ, called isodual numbers, with isodual sum (isoda)isod+(isodb) = (-aÝ-bÝ) and isodual multiplication (isoda)isodx(isodb) = (-aÝ)(-1)(-bÝ) = -(aÝ)(bÝ) under which the additive unit is isisod0 = 0 and the multiplicative unit is isode = -e = -1. LEMMA II-1 [II-1]: The isodual field is a field (i.e., it verifies all axioms of a field).

PROPOSITION II-1 [loc. cit.]: The map F -> isodF is anti-isomorphic.

PROPOSED RESEARCH II-1: Study the isodual number theory, including theorems on prime, factorization, etc.

SIGNIFICANCE: Isodual numbers have a negative norm, thus being useful to represent antimatter.

FOR TECHNICAL ASSISTANCE: Contact the IBR staff at ibr@gte.net

PROBLEMS II-2/II-3: STUDIED IN ISOFIELD THEORY AND ITS ISODUAL DEFINITION II-2: Let F = F(a,+,x) be a conventional field of (real R, complex C or quaternionic Q) numbers a with additive unit 0, multiplicative unit e = 1, sum a+b and product axb. Santilli’s isofields [II-1] F* = F*(a*,+*,x*) are rings of elements a* =axE called isonumbers, where a is an element of F, axE is the multiplication in F, and E = 1/T is a well behaved, everywhere invertible, Hermitean and positive-definite quantity (e.g., a matrix, vector field, operator, etc. of Kadeisvili’s Class I) generally outside the original field F, equipped with the isotopic sum a*+*b* = (a+b)xE and product a*x*b* = a*xTxb* = (axb)xE, with additive unit 0* = 0 and multiplicative unit E, called isounit. Santilli’s isodual isofields isodF*(isoda*,isod+*,isodx*) are the anti-Hermitean images of F* under the isodualities of the elements of F* and all its operations. LEMMA II-2 [II-2]: Isofields verify all axioms of a field (including closure under the combined associative and distributive laws). The lifting F -> F* is therefore an isotopy.

PROPOSITION II-2 (ref.[II-1], p. 284): When E is an element of the original field F (e.g., an ordinary real number for F = R), the isofield F*(a,+*,x*) is also a field (i.e., closure occurs for conventional numbers a without need to use the isonumbers a* = axE).

PROPOSED RESEARCH II-2: Formulate the real isonumber theory: 1) with a basic unit given by an arbitrary, positive, real number E = n > 0; and 2) under isoduality to a negative-definite unit isodE = -n < 0. These problems can be studied via the simplest possible class of Santilli isofields R*(a,+*,x*) and their isoduals isodR*(isoda,isod+*,isodx*) in which the elements a are not lifted, as per Proposition II.2 above. The study implies the re-inspection of all conventional properties of number theory in order to ascertain which one is dependent on the selected unit. As an example, it is known that the notion of prime depends on the selected unit [II-1] because, e.g., the number 4 becomes prime for the isounit E = 3.

SIGNIFICANCE: An important advance of memoir [II-1] is that the axioms of a field need not to be restricted to the simplest possible unit +1 dating back to biblical times, because they equally hold for arbitrary units. This basic property has far reaching implications. In mathematics the property implies the lifting of all structures defined on numbers; in physics the broadening of the unit implies basically novel applications (See the subsequent Web Page 19 on Open Research Problems in Physics); and in biology it implies a structural revision of current theories (see the subsequent Web Page 20 on Open Research Problems in Biology).

FOR TECHNICAL ASSISTANCE: Contact the IBR staff at ibr@gte.net

PROBLEM II-4/II-5: STUDIES ON GENONUMBERS THEORY AND ITS ISODUAL DEFINITION II-3: Let F = F(a,+,x) be a conventional field of (real, complex C or quaternionic Q) numbers a with additive unit 0, multiplicative unit e = 1, sum a+b and product axb. Santilli’s genofields to the right [II-1] F> = F>(a>,+>,x>) are rings of elements a> = axE> called genonumbers, where a is an element of F, axE> is the multiplication in F, and E> = 1/S is a well behaved, everywhere invertible and non-Hermitean quantity generally outside F, equipped with all operations ordered to the right, i.e., the ordered genosum to the right (a>)+>(b>) = (a+b)xE>, the ordered genoproduct to the right (a>)>(b>) = (a>)xSx(b>) = (axb)xE>, etc., conventional additive unit to the right 0> = 0 and generalized (left and right) unit E> for the multiplication to the right called genounit. Santilli’s genofields to the left <F(<a,<+,<) are rings with genonumbers <a = <Exa, all operations ordered to the left, such as genosum (<a)<+(<b) = <Ex(a+b), genoproduct (<a)<(<b) = <Ex(axb), etc., with genoadditive unit to the left <0 = 0 and multiplicative genounit to the left <E = 1/R which is generally different than that the right. The isodual left and right genofields are the isodual images of the left and right genofields. REMARKS: In the definition of fields and isofields there is no ordering of the multiplication in the sense that in the products axb and a*b one can either select a multiplying b from the left , a>b or b multiplying a from the right a<b, because a>b = ab = b>a and a<b = b<a, but in general a>b ‚ a<b. Note that in each case the genounit is the left and right unit because (E>)>(a>) = (a>)>(E>) = a> for all possible a>.

LEMMA II-3 [II-1]: Each individual genofield to the right F> or to the left <F is a field. Thus each lifting F -> F> and F -> <F is an isotopy.

PROPOSITION II-3 [II-1]: When E> and <E are elements of an ordinary field F, each genofield F>(a,+>,x>) and <F(a,<+,<x) is a field.

PROPOSED RESEARCH II-4/II-5: Formulate the number theory with 1) a basic unit given by a positive real number E> = n in which all operations are ordered to the right; 2) formulate the same theory under an ordering to the left with a different positive-definite genounit <E = b; 3) construct the isoduals of both theories. These problems can be studied via the simplest possible class of Santilli genofields F>(a,+>,x>) and <F(a,<+,<x) in which the elements are not lifted, as per Proposition II-3 above. The study implies the re-inspection of all conventional properties of the isonumber theory.

SIGNIFICANCE: Another significant advance of memoir [II-1] is that the axioms of a field, not only do not need the restriction to the unit +1, but the operations can be all restricted to be EITHER to the right OR to the left. This simple property has additional far reaching mathematical, physical and biological implications. In mathematics, it implies a dual lifting of all isotopic structures. In physics it implies an axiomatic representation of the irreversibility of the physical world via the most fundamental mathematical notion, the unit. In fact, operations ordered to the right can represent motion forward in time, while operations ordered to the left can represent motion backward in time.

Irreversibility is then reduced to the differences between E> and <E or, equivalently, between a>b and a<b. The inclusion of the isodualities implies the capability to represent all possible four different motions in time: motion forward to future time E>, motion backward to past time <E, motion forward from past time isod(<E), and motion backward from future time isod(E>). In theoretical biology, Santilli’s genonumbers are the foundation of the first known consistent mathematical representation of the irreversibility of biological structures. The addition of isoduality then permits the mathematical representation of certain bifurcations in biology whose treatment is simply beyond conventional mathematics.

SUGGESTED TECHNICAL ASSISTANCE: Contact the IBR staff at ibr@gte.net

PROBLEM II-6/II-7: STUDIES IN HYPERNUMBER THEORY AND ITS ISODUAL Repeat the studies of Problem II-4/II-5 but for the case when the genounits to the right and to the left and given by an ordered set of nonhermitean invertible elements equipped, first, with conventional and, then, with hyperoperations. For a definition of hyperfields with conventional operations see ref. [I-1] of Sect. I. For their broader definition with hyperoperations see ref. [II-2] below. SUGGESTED TECHNICAL ASSISTANCE: Consult

Prof. T. Vougiouklis Department of Mathematics Democritus University of Thrace GR-67100 Xanthi, Greece, fax +30-551-39348, or

Prof. M. Stefanescu Department of Mathematics Ovidius University, Bd. Mamaia 124 Costanta 8700, Romania,

REFERENCES OF SECT. II:

[II-1] R. M. Santilli, Isonumbers and genonumbers of dimension 1, 2, 4, 8, their isoduals and pseudoduals, and “hidden numbers” of dimension 3,5,6,7, Algebras, Groups and Geometries Vol. 10, pages 273-322, 1993 [II-2] T. Vougiouklis, Editor, New Frontiers in Hyperstructures, Hadronic Press, 1996.

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.

III: OPEN PROBLEM IN GEOMETRIES

Prepared by the IBR staff. PROBLEM III.1: STUDIES IN ISODUAL GEOMETRIES DEFINITION III-1: Let S = S(r,g,R) be a conventional n-dimensional metric or pseudo-metric space with local chart r, nowhere singular, real-valued and symmetric metric g and invariant (rt)xgxr (where rt denotes transposed of r) over a conventional field R = R(a,+,x) of real numbers. Santilli’s isodual spaces isodS = isodS(isodr,isodg,isodR) is the vector space with local chart isodr = -r, isodual metric isodg = -g and isodual invariant (isodrt)x(isodg)x(isodr) = [(rt)xgxr](isode) on isodR. Isaodual geometries are the geometries on isodual spaces, thus based on negative-definite units. PROPOSITIONS III-1 [I-1]: Isodual spaces are anti-isomorphic to the original space.

PROPOSED RESEARCH III-1: Study the isodual Euclidean, isodual Minkowskian, isodual Riemannian, isodual symplectic and other isodual geometries, including the isodual calculus, the isodual sphere (i.e., the sphere with negative radius), the isodual light cone, etc. [I-1].

SIGNIFICANCE: Isodual geometries are fundamental for the recent isodual representation of antimatter, e.g., to characterize the shape of an antiparticle with negative units.

SUGGESTED TECHNICAL ASSISTANCE: Contact the IBR staff at ibr@gte.net

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

DEFINITION III-2: Let S = S(r,g,R) be a conventional n-dimensional metric or pseudo-metric space with local chart r, nowhere singular, real-valued and symmetric metric g and invariant (rt)xgxr (where rt denotes transposed) over a conventional real field R = R(a,+,x). Santilli’s isospaces [I-1] S*(r*,G*,R*) are vector spaces with local isocoordinates r* = rxE, isometric G* = g*xE = TxgxE and isoinvariant (r*t)*G*r* = [(rt)x(g*)xr]xE over an isofield R* = R*(a*,+*,x*) with a common isounit E = 1/T of Kadeisvili Class I. Santilli’s isogeometries [III-1] are the geometries of isospaces. The isodual isospaces and isodual isogeometries are the corresponding images under isoduality [I-1,III-1]. LEMMA III-2 [I-1,III-1]: Isospaces S*(x*,G*,F*) are locally isomorphic to the original space S(x,g,F). The lifting S -> S* is therefore an isotopy.

Proof. Each component of the metric g is lifted by the corresponding element of T, while the unit is lifted by the corresponding inverse amount E = 1/T, thus preserving the original geometric axioms.q.e.d.

PROPOSITION III-3 [III-2]: The axioms of the Euclidean geometry in n-dimension admit as particular cases all possible well behaved, real-valued, symmetric and positive-definite metrics of the same dimension.

REMARKS.To be consistently defined, Santilli’s isogeometries require the isotopies of the totality of the mathematical aspects of the original geometry, all formulated for a common isounit E with the same dimension of the isospace. This requires, not only the isotopies of fields and vector spaces, but also those of all other aspects.

PROBLEM III-2/III-3: Provide a mathematical formulation of Santilli’s isoeuclidean, isominkowskian, isoriemannian, isosymplectic and other isogeometries and their isoduals which have been only preliminarily studied for physicists in ref. [III-1]. SIGNIFICANCE: The mathematical and physical implications are significant indeed. Mathematically, the studies permit advances such as: the unification of all geometries of the same dimension into one single isotope; the admission under the Riemannian axioms of metric with arbitrary, nonlinear, integro-differential dependence in the velocities and other variables; the representation of nonhamiltonian vector-fields in the local chart of the observer (see the alternative to Darboux’s theorem, ref. [I-1], p. 63 motivated by the fact that, in view of their nonlinearity, Darboux’s transforms cannot be used in physics because the transformed frames cannot be realized in experiments and, in any case, they violate the axioms of Galilei’s and Einstein’s special relativity due to their highly noninertial character). Physically, the studies permit truly basic advances, such as the first quantitative research on the origin of the gravitational field, a geometric unification of the special and general relativity via the isominkowskian geometry in which the isometric is a conventional Riemannian metric, a novel operator formulation of gravity verifying conventional quantum axioms; and other advances (see Web Pages 19 and 20).

SUGGESTED TECHNICAL ASSISTANCE: Contact the IBR staff at ibr@gte.net, or

Prof. D. Sourlas Department of Mathematics University of Patras Gr-26100 patras, Greece Fax +30-61-991 980

Prof. Gr. Tsagas Department of Mathematics Aristotle University Thessaloniki 54006, Greece Fax +30-31-996 155

Prof. R. Miron Department of Mathematics “Al. I. Cuza” University 6600 Iasi, Romania rmiron@uaic.ro

PROBLEMS III-4/III-5: STUDIES ON GENOGEOMETRIES AND THEIR ISODUALS DEFINITION III-3: Let S = S(r,g,R) be a conventional n-dimensional metric or pseudo-metric space with local chart r, nowhere singular, real-valued and symmetric metric g and invariant (rt)xgxr over a conventional real field R = R(a,+,x). Santilli’s n-dimensional genospaces to the right [I-1] S>(r>,G>,R>) are vector spaces with local genocoordinates to the right r> = rxE>, genometric G> = (g>)x(E>) = SxgxE>, genoinvariant (r>t)>(G>)(r>) = [(rt)x(g>)xr]xE> over the genofield R> = R>(a>,+>,x>), common genounit to the right E> = 1/S given by an everywhere invertible, real-valued, non-symmetric nxn matrix, and all operations ordered to the right. Santilli’s genogeometries [III-1] are the geometries of genospaces. The isodual genospaces and isodual genogeometries are the corresponding images under isoduality [I-1,III-2]. Santilli’s n-dimensional genospaces to the left [I-1] <S(<r,<+,<R) are genospaces over genofields with all operations ordered to the left and a common nxn-dimensional genounit to the left <E = 1/R which is generally different than that to the right E> = 1/S. Genogeometries to the left and their isoduals are the geometries over the corresponding genospaces. LEMMA III-3: Genospaces to the right S> and, independently, those to the left <S, are locally isomorphic to the original spaces S. Proof. The original metric g is lifted in the form g -> Sxg, but the unit is lifted by the inverse amount I -> E = 1/S, thus preserving the original axioms. q.e.d.

PROPOSED RESEARCH: Provide a mathematical formulation of Santilli’s genoeuclidean, genominkowskian, genoriemannian, genosymplectic and other genogeometries to the left, their corresponding forms to the right and their isoduals which have been preliminarily studied in ref. [III-1] for physicists.

SIGNIFICANCE: Another important aspect of memoir [I-1] is that the Riemannian axioms do not necessarily need a symmetric metric because the metrics can also be nonsymmetric with structure g> = Sxg, S nonsymmetric, provided that the geometry is formulated on an isofield with isounit given by the INVERSE of the nonsymmetric part, E = 1/S, and the same occurs for the left case. This property has permitted the first quantitative studies on irreversibility of interior gravitational problems via the conventional Riemannian axioms, as it occurs in the physical reality, e.g., the irreversibility of the structure of Jupiter or of a collapsing star, for which purpose the genogeometries were constructed in the first place [III-1].

SUGGESTED TECHNICAL ASSISTANCE: Contact the IBR staff at ibr@gte.net

PROBLEMS III-6/III-7: STUDIES ON HYPERGEOMETRIES AND THEIR ISODUALS Extend the studies of the genogeometries to the right and to the left to the case when the corresponding genounits are given by an ordered set of invertible, real-valued and nonsymmetric nxn elements, first, with ordinary operations and then with hyperoperations. The existence of these geometries has been only indicated in ref. [I-1] without any detailed treatment.

SUGGESTED TECHNICAL ASSISTANCE: Consult the IBR staff at ibr@gte.net

REFERENCES OF SECT. III:

[III-1] R. M. Santilli, Elements of Hadronic Mechanics, Vol. I, Mathematical Foundations, Ukraine Academy of Sciences, Kiev, Second Edition, 1995.

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

IV: OPEN RESEARCH PROBLEMS IN FUNCTIONAL ANALYSIS

Conventional and special functions and transforms and functional analysis at large are dependent on the assumed basic unit. As an example, a change of the two-dimensional unit of the Gauss plane implies a change in the very definition of angles and trigonometric functions, and the same happens for hyperbolic functions, Fourier, Laplace and other transforms, Dirac and other distributions, etc. To be operational, the seven classes of novel mathematical methods of the preceding sections require seven corresponding generalized forms of functional analysis, which are here recommended for study in a progressive way, beginning with the simplest possible case of isoduality.

Mathematical work done to date in these new topics has been rather limited. We here mention: Santilli has preliminarily studied the structure of the seven forms of differential calculus [I-1], isotrigonometric and isohyperbolic functions, the isofourier transforms and few other aspects {III-1]; Kadeisvili [IV-1,2] has studied basic definitions of isocontinuity and its isodual and reinspected some of the studies in the field; H. C. Myung and R. M. Santilli [IV-3] studied the isotopies of the Hilbert space, Dirac delta distributions and few other notions; A. K. Aringazin, D. A. Kirukhin and R. M. Santilli [IV-4] have studied the isotopies of Legendre, Jacobi and Bessel functions and their isoduals; M. Nishioka [IV-5] studied the Dirac-Myung-Santilli isodelta distribution (see [III-1] for a review up to 1995).

SUGGESTED TECHNICAL ASSISTANCE: Contact Prof. J. V. Kadeisvili at ibr@gte.net, or A. K. Aringazin and D. A. Kirukhin at aringazin@kargu.krg.kz

REFERENCE FOR SECT. IV:

[IV-1] J. V. Kadeisvili, Elements of functional isoanalysis, Algebras, Groups and Geometries vol. 9, pages 283-318, 1992. [IV-2] J. V. Kadeisvili, Elements of the Fourier-Santilli isotransforms, Algebras, Groups and Geometries Vol. 9, pages 319-242, 1992 [IV-3] H. C. Myung and R. M. Santilli, Modular-isotopic Hilbert space formulation of the exterior strong problem, Hadronic J. Vol. 5, pages 1277-1366, 1982. [IV-4] A. K. Aringazin, D. A. Kirukhin and R. M. Santilli, Isotopic generalization of Legendre, Jacobi and Bessel functions, Algebras, Groups and Geometries Vol. 12, pages 255-359, 1995. [IV-5] M. Nishioka, Extension of the Dirac-Myung-Santilli delta functions to field theory, Lett. Nuovo Cimento Vol. 39, pages 369-372, 1984.

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

Prepared by the IBR staff PROBLEM V-1: STUDIES ON THE ISODUAL LIE THEORY DEFINITION V-1: Let L be an n-dimensional Lie algebra with ordered Hermitean basis X = {A,B, ...} = XÝ, conventional commutator [A, B] = AxB - BxA (where AxB is conventionally associative) over a field F (of characteristics zero). A Lie-Santilli isodual algebra [I-1] isodL is the image of L under the isodual map (5), thus including isodual generators isodX = -XÝ = -X, isodual commutator isod[A, B] = (isodA)isodx(isodB) - (isodB)isodx(isodA) = - [A, B], etc., all defined on an iusodualF with negative-definite unit E = -Diag. (1, 1, ..., 1), and norm. LEMMA V-1 [I-1]: IsodL is anti-isomorphic to L.

PROPOSED RESEARCH V-1: Reformulate Lie's theory (enveloping associative algebras, Lie algebras, Lie groups, transformation and representation theories, etc.) for the Lie-Santilli isodual theory with an n-dimensional negative-definite unit E = - Diag(1, 1, ... 1).

SIGNIFICANCE: Isodual symmetries characterize antiparticles.

SUGGESTED TECHNICAL ASSISTANCE: Contact the IBR staff at ibr@gte.net

PROBLEM V-2/V-3: STUDIES ON LIE-SANTILLI ISOTHEORY AND ITS ISODUAL DEFINITION V-2: Let U(L) be the enveloping associative algebra of a Lie algebra L with infinite-dimensional basis (Poincare’-Birkhoff-Witt theorem) 1, X, (Xi)x(Xj )(i ¾ j), .. and conventional exponentiation expA = 1 + A/1! + AxA/2! + ...over a field F. Santilli’s universal enveloping isoassociative algebra U*(L) of L [V-1] of Class I is characterized by the isotopies of the Poincare’-Birkhoff-Witt theorem) with infinite-dimensional isobasis (9) E, X, (Xi)*(Xj) (i ¾ j), ...,

where the isounit E = 1/T is of Kaidesvilki Class I (positive-definite) has the same dimension of the representation at hand and A*B = AxTxB, with base isofield F* characterized by the same isounit E of U*(L). The isodual isoassociative envelope is the image of U* under isoduality. REMARKS. The conventional exponentiation is no longer applicable for U*(L) and must be replaced by the isoexponentiation

(10) isoexpA = E + A/1! + A*A/2! + ... = {exp(AxT)}xE = Ex{exp(TxA)}.

The symbol U*(L) rather than U*(L*) is used to indicate that the basis X of L is unchanged under isotopy and merely redefined in isospace,thus X* = X. This is the most significant case on physical grounds because generators of Lie symmetries represent quantities such as energy, linear momentum, angular momentum, etc. which, as such, cannot be changed. Only the operations defined in them can be changed. Since Lie’s theory leaves invariant its basic unit, the functional dependence of the isounit E is left unrestricted and, therefore, can depend on an independent variable t (say time), local chart r of the carrier space, its derivatives v = dr/dt, a = dv/dt and any other needed variable, E = E(t, r, v, a, ...) = 1/T. The nontriviality of the Lie-santilli isotheory can therefore be seen up-front because it implies the appearance of a nonlinear, integro-differential element T in the EXPONENT of the group structure, Eq. (10).

DEFINITION V-3. Let L be an n-dimensional Lie algebra as per Definition V-1. A Lie-Santilli isoalgebra [V-1] L* of Class I is the algebra homeomorphic to the antisymmetric algebra [U*(L)]- of U*(L). It can be defined as an isovector space with the same ordered basic X* = X of L equipped with the isoproduct

(11) [A, B]* = A*B - B*A = AxT(t, r, v, a, ...)xB - BxT(t, r, v, a, ...)xA

REMARKS. Lie-Santilli isoalgebras verify the conventional Lie axioms (anti-commutativity of the product and Jacobi identify), only formulated in isospace (that is, with respect to the isoassociative produce A*B) over the isofield F*. LEMMA V-3 [III-1]: Lie-Santilli isoalgebras are left and right isolinear, i.e., they verify the left linearity conditions on L* as an isovector space over F*,

(12a) {[(a*)*A* + (b*)*B*, C*]* = (a*)*[A*,B*]* + (b*)*[B*, C*]*

(12b) [(A*)*B*, C*] = (A*)*[B*, C*]* + [A*, C*]*(B*)

and corresponding right conditions. Isoalgebras L* are isolocal (in the sense of being everywhere local-differential except at the isounit E) and isocanonical (in the sense of admitting a canonical structure in isospace, see [III-1] for brevity). DEFINITION V-4: Let G be an n-dimensional connected Lie transformation group r’ = K(w)xr on a space S(r,F), where w are the parameters in F, verifying the usual conditions (differentiability of the map GxS -> S, invariance of the basic unit e = I, and linearity), as well as the conditions to be derived from the Lie algebra L via exponentiation

(13a) Q(w) = {exp(ixXxw)} x Q(0) x {exp(-ixwxX)}

(13b) i [Q(dw) - Q(0) ] / dw = QxX - XxQ = [ Q, X].

A connected Lie-Santilli isotransformation group [V-1] G* of Class I is the set of isotransforms (14) r*’ = Q*(w*)] * (r*) = [Q*[w*)] x T(t, r, v, a, ...) x r

on a Class I isospace S*(r*,F*), where now the isoparameters w* = wxE belong to F*, which verifies the usual conditions in their isotopic form (Kadeisvili’s isodifferentiability of the isomap (G*)*S* -> S*, invariance of the isounit E, and isolinearity), as well as the conditions to be derivable from the Lie-Santilli isoalgebra L* [V-1] (15a) Q*(w*) = {isoexp[i(X*)*(w*)]} * [Q*(0)] * {isoexp[-i(w*)*(X*)]} =

= { exp (i X x T x w) } x Q*(0) x { exp(-iwTX) },

(15b) i [Q*(dw*) - Q*(0) ] / dw* = (Q*) *(X*) - (X*) * (Q*) = [ Q*, X*]*

with isogroup laws [V-1] (16) [Q*(w*)]*[Q*(w*’)] = Q*(w* + w*’] , [Q*(w*)]*[Q*(-w*)] = Q*(0*) = E

The isodual Lie-Santilli isogroups isodG* are the isodual image of G* under map (5). LEMMA V-3 [V-1]: Lie-Santilli isoenvelopes U*, isoalgebras L* and isogroup G* are locally isomorphic to the original structures U, L, and G, respectively for all possible positive-definite isounit E (not so otehrwise). The liftings

(17) U -> U*, L -> L* and G -> G*

are therefore isotopies. PROPOSED RESEARCH V-2: Conduct mathematical studies on the Lie-Santilli isotheory of Class I and its isodual with particular reference to: the isostructure theory; the isorepresentation theory; and related aspects.

REMARKS. At the abstract, realization-free level, isoenvelopes U*, isoalgebras L* and isogroups G* coincide with the conventional envelopes U, algebras L and groups G, respectively, by conception and construction for all positive-definite isounits E (not necessarily so otherwise). This illustrates the insistence by Santilli in indicating that the isotopies do not produce new mathematical structures, but only new realizations of existing abstract axioms.

As a result of the, the isorepresentation theory of U, L and G on isospaces over isofields is expected to coincide with the conventional representations of the original structures U, L and G on conventional spaces over conventional fields. The aspect of the isorepresentation theory which is important for applications is the PROJECTION of the isorepresentation on conventional spaces. Stated differently, Lie’s theory admits only one formulation, the conventional one. On the contrary, the covering Lie-Santilli isotheory admits two formulations, one in isospace over isofield and one given by its projection on conventional spaces over conventional fields.

The latter are important for applications, e.g., because the physical space-time is the conventional Minkowski space, while the isominkowski space is a mathematical construction. As a result, the isorepresentation theory of the Poincare’-Santilli isosymmetry [V-4] on isominkowski space over isofields is expected to coincide with that of the conventional symmetry on the conventional Minkowski space over the conventional field of real numbers. The mathematically and physically significant aspects are given by the PROJECTION of the isorepresentation on the conventional Minkowski space-time.

SIGNIFICANCE: The isotheory characterizes all infinitely possible, well behaved, arbitrarily nonlinear, nonlocal-integral and nonhamiltonian, classical and operator systems by reducing them to identical isolinear, isolocal and isocanonical forms in isospaces over isofields, thus permitting a significant simplification of notoriously complex structures.

PROPOSED RESEARCH V-3: Study the Lie-Santilli isotheories of Classes III (union of Class I with positive-definite and II with negative definite isounits E), Class IV (Class III plus null isounit E) and Class V arbitrary isounits E, including discontinuous realzioations). As a particular case unify all simple Lie algebras of the same dimension in Cartan's classification into one single isoalgebra of the same dimension of Class III, whose study has been initiated by Tsagas and Sourlas [V-4].

REMARK 1. In his original proposal on the isotopies of Lie's theory of 1978 (see the references inn [V-1]), Santilli proved the loss at the abstract level of all distinction between compact and noncompact Lie algebras of the same dimension provided that the isounit has an arbitrary positive- or negative-definite signature (Class III). This was illustrated via the algebra of the rotation group in three dimension O(3). When its conventional generators X1, X2, X3 (the components of the angular momentum) are equipped with the isounit E = Diag. (+1, +1., -1) and isoproduct (11) they characterize the noncompact O(2.1) algebra. The isoalgebra O*(3) with the fixed generators X1, X2, X3 equipped with isoproduct (11) and a isotopic element T of Class III therefore unifies all simple Lie algebras of dimension 3. This result has been proved to hold also for all orthogonal and unitary algebras, and it is expected to hold for all possible Lie algebras, including the exceptional ones.

REMARK 2. As indicated in the subsequent Web Page 19, the zeros of the isounit represent gravitational singularities. The study of the Lie-Santilli theory of Class IV is therefore important for applications. No study in on record at this writing in this field which requires the prior study of numbers, spaces, geometries, etc., whose units can be psoitive, negative as wel as null. No study is also on record on the isotoppies of Class V.

IMPORTANT NOTE. Visitors of this page should be aware that the treatment of the isoproduct [A, B]* = AxTxB - BxTxA on conventional spaces over conventional fields is not invariant under the group action and, as such, it has no known physical value. In fact, when realized on a Hilbert space over a conventional field, isogroups G* are characterized by nonunitary transforms WWÝ ‚ I. As a result, the base unit of a conventional treatment of the isoproduct [A, B]* is not left invariant by the isogroup, I -> I’ = WxIxWÝ ‚ I, and, consequently, the isoproduct itself is not invariant, [A, B]* -> Wx[A, B]*xWÝ = A’xT’xB’ - B’xT’xA’, where T’ = (WÝ to -1)xTx(W to -1) ‚ T. The loss of the traditional invariance of Lie’s theory then implies the lack of meaningful applications.

On the contrary, when treated via the isotopic mathematics, that is, formulated on isospaces over isofields, the isoproduct [A, B]* is fully invariant. For instance, by considering again the operator realization, the originally nonunitary structure of G* is turned into identical isounitary forms, i.e., we can write W = (W*)x(square root of T) for which WxWÝ = (W*)*(W*Ý) = (W*Ý)*(W*) = E, in which vase the base isounit E of the isofield is invariant, E -> E’ = (W*)*E*(W*Ý) = (W*)*(W*Ý) = E, and the isoproduct is consequently invariant, (W*)*[A, B]*x(W*Ý) = A’xTxB’ - B’xTxA’, where one should note that T is numerically preserved. The above occurrence illustrate the necessity of using Santilli’s isonumbers and isospaces for meaningful applications.

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PROBLEM V-4/V-5: STUDIES ON LIE-SANTILLI GENOTHEORY AND ITS ISODUAL HISTORICAL NOTE: Lie algebras are "nonassociative" in the sense that their product [A, B] = AxB - BxA is nonassociative. Yet their representations are reduced to those of their universal enveloping associative algebras U(L) with product AxB because of the homeomorphism between L and the attached antisymmetric algebra U(L)-. In 1948 the American mathematician A. A. Albert (Trans. Amer. Mat. Soc. Vol. 64, p. 552, 1948) introduced the notion of Lie-admissible algebras (presented in Sect. I and called the First Condition of Lie-admissibility). The formulation was done within the context of nonassociative algebras, in which context they have been studied by various mathematicians until recently. Also, Albert was primarily interested in the "Jordan content" of a given "nonassociative" algebra and, for this reason, he studied the product (a, b) = pxaxb + (1-p)xbxa, where p is a real parameter, which admits the commutative Jordan product for p = 1/2.

In 1967 Santilli (Nuovo Cimento Vol 51, p. 570, 1967) was the first physicist to study Lie-admissible algebras. He noted that Albert's definition did not admit Lie algebras in their classification and, for this reason, the algebras had limitations in physical applications. He therefore introduced a new notion of Lie-admissibility which is Albert's definition plus the condition of admitting Lie algebras in their classification (this is called the Second Condition of Lie-admissibility). In particular, Santilli studied the product (a,b) = pxaxb - qxbxa, where p, q, and p+or-q are non-null parameters, which does indeed admit the conventional Lie product as a (nondegenerate) particular case, and which constitutes the first formulation in scientific record of the "deformations" of Lie algebras of the contemporary physical literature (see the next Web Page 19).

In 1978 Santilli (Hadronic J. Vol. 1, p. 574, 1978) notes his Second Condition of Lie-admissibility was still insufficient for physical applications because Lie-admissibility implies "nonunitary" time evolutions under which the "parameters" p and q become "operators". He therefore introduced a more general definition of Lie-admissibility which is Albert's definition plus the conditions that the attached antisymmetric algebra is Lie-isotopic, rather than Lie, and the algebras admit conventional Lie algebras in their classification (this is the Third Condition of Lie-admissibility, also called Albert-Santilli Lie-admissibility, or General Lie-admissibility).

In this way, Santilli introced the product (A, B) = AxRxB - BxSxA, Eq.s (5b), where R, S, and R+or-S are fixed and nonull vector-fields, matrices, operators, etc. for which the attached antisymmetric algebra is the isotopic form [A,B] = (A, B) - (B, A) = AxTxB - BxTxA, T = R+S, while admitting of the conventional Lie product for R+S = 1. The product (A, B) is also the first on scientific records of the so-called "quantum groups" of the contemporary physical literature. The same product, being neither totally antisymetric nor totally symmetric, includes as particular cases supersymmetric and other generalizations of the Lie product (see the next Web Page 19).

In the same memoir of 1978, Santilli reduced the study of the Lie-admissible product (A, B) = AxRxB - BxSxA to its two isoassociative envelopes AxRxB and BxSxA, that is, he reduced the representation theory of the nonassociative product (A, B) to that of its two, right and left envelopes with "isoassociative" product AxRxB and BxSxA [V-5], in essentially the same way as the study of the Lie product [A, B] = AxB - BxA is reduced to that of the associative ones AxB and BxA.

The terms Santilli's Lie-admissible theory or genotheory are referred to the latter context, that is, to a dual left and right lifting of Lie's theory (enveloping associative lagebras, Lie algebras, Lie groups, representation theory, etc.).

The tool which permitted this formulation is that of a bi-representation (split-null extension) [V.5]. The main point is that bi-modular Lie-admissible structures are contained in the structure of CONVENTIONAL Lie’s groups. In fact, Eq.s (13) can be written [V-1]

(18a) Q(w) = {exp(ixXxw)} > Q(0) < {exp(-ixwxX)} (18b) i [Q(dw) - Q(0) ] / dw = W < X - X > W

where > means conventional modular-associative “action to the right” and < “action to the left”. The bi-modular character is trivial in Lie’s case because the action of a conventional Lie group from the left is minus the transpose action from the right. An important observation of Ref. [V-1] is that group structure (18) can also be written in a non-trivial bi-modular form characterized, first, by the isotopic modular actions to the right and to the left and, then, their differentiation into genotopic forms. To put it bluntly, a bimodular Lie-admissible structure is already contained in the conventional structure of Lie groups. It merely remained un-noticed until 1978. In fact, the modular associative product to the right can be realized via the right genoassociative algebra U> with product A>B = AxSxB and that to the left via the left genoassociative algebra <U with product A = 1/S and <E = 1/R. Eq.s (18) then yield Santilli’s Lie admissible theory [V-1]

(19) Q(w) = {exp>(ixXxw)} > Q(0) < {exp<(-ixwxX)} =

# {[exp(ixXxSxw)]xE>} x S x Q(0) x R x { (19 i [Q(dw) - Q(0) ] / dw = Q < X - X > Q

= Q x R x X - X x S x Q = (Q, X)

In this way the representation theory of the Lie-admissible algebra with nonassociative product (A, B) is first reduced to a bi-representation theory on {<U, U>} and then shown to admit a Lie-admissible group structure in a way fully parallel to the conventional Lie case. In an evening seminar delivered at ICM94 Santilli completed his Lie-admissible theory by showing that the algebra with Product (A, B) = AxRxB - BxSxA does indeed verify the Lie axioms (antisymmetry and Jacobi law), provided that the terms AA are represented in their respective genoenvelopes over corresponding genofields <F and .

To understand better how the Lie-admissible product (A, B) = AxRxB - BxSxA, with R different than S, can be antisymmetric, recall that conventional Lie algebra admit one single realization, that on conventional spaces and fields (read: with respect to the trivial unit I = Diag.(1, 1, ...1)); the isoalgebras admit instead a dual realization, that on isospaces over isofield (read: with respect to the isounit E) as well as the projection on conventional spaces over conventional fields (read: with respect to the conventional unit I); for the genoalgebras we have essentially the a similar occurrence, namely, they can be computed on the right and left genospaces over right and left genofields (read: right and left genounits E> and <E), in which case the product (A, B) verifies the Lie axioms, or it can be computed in its projection in conventional spaces and fields (read: with respect to the conventional unit I), in which case the product (A, B) is manifestly non-Lie.

Equivalently, the Lie character of the product (A, B) = AA on genospaces over genofields can be seen from the fact that the lifting of the associative envelope AxB -> A>B = AxSxB is compensated by an INVERSE lifting of the unit I -> E> = 1/S, thus preserving the original structure (i.e., U and U> are isomorphic), and the same occurs for the right product (i.e., U and <U are also isomorphic). Thus, at the abstract, realization-free level, the product (A, B) verifies the anti-commutative law and the Jacobi law

(20a) (A, B){<S,S>} = -(B, A){<S, S>} (20b) ((A, B), C){<S, S>} + ((B, C), A){<S, S>} + ((C, A), B){<S, S>} = 0.

PROPOSED RESEARCH V-4 : Study the Lie-admissible theory and its isodual with particular reference to: the genostructure theory, the transition from the genoalgebras to related genogroups; the representation theory; etc. SIGNIFICANCE: Lie algebras are the algebraic counterpart of conventional geometries; Lie-Santilli isoalgebras are the algebraic counterpart of the isogeometries; and, along similar lines, genoalgebras are the algebraic counterpart of the genogeometries. The conventional, modular representation theory of Lie algebras characterize particles in linear, local, canonical and reversible conditions; the isomodular representation theory of Lie-Santilli isoalgebras characterize particles in nonlinear, nonlocal and noncanonical but still reversible conditions; the bi-modular representation theory of the genoalgebras characterizes particles in nonlinear, nonlocal, noncanonical as well as irreversible conditions, such as a neutron in the core of a neutron star. The most advanced definition of “particle” in physics, admitting all other as particular case, including those characterized by string and supersymmetric theories, is a bi-isomodular representation of the Lie-admissible covering of the Poincare’ symmetry.

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PROBLEMS V-6/V-7: STUDIES ON THE LIE-SANTILLI HYPERTHEORY AND ITS ISODUAL A further advance of the recent memoir [I-1] is that the genounits E> and <E can be ordered sets of nonhermitean quantities under ordinary operations, in which case the Lie-admissible theory becomes multivalued yet still preserving the original Lie axioms at the abstract level. A form of hyper-Lie theory defined via hyperoperations was also introduced by Santilli and Vougiouklis in ref. [II-2]. PROPOSED RESEARCH V-5: Study the multivalued realization of the Lie-admissible theory, first, with ordinary operations, and then with hyperoperations.

SIGNIFICANCE: Besides the evident mathematical significance, multi-valued spaces have already seen their appearance in cosmology, and their need in biology is now established in view of the complexity of biological systems.

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REFERENCES FOR SECT. V: [V-1] R. M. Santilli, Foundations of Theoretical Mechanics, Vol. II, Springer-Verlag, 1983; [V-2] D. S. Sourlas and Gr. Tsagas, Mathematical Foundations of the Lie-Santilli Theory, Ukraine Academy of Sciences, Kiev, 1993. [V-3] J. V. Kadeisvili, An introduction to the Lie-Santilli isotheory, Rendiconti Circolo Matematico Palermo, Suppl. No. 42, pages 83-136, 1996. [V-4] R. M. Santilli, Nonlinear, nonlocal and noncanonical, axiom-preserving isotopies of the Poincare’ symmetry, J. Moscow Phys. Soc. Vol. 3, pages 255-297, 1993. [V-5] R. M. Santilli, Initiation of the representation theory of Lie-admissible algebras on a bimodular Hilbert space, Hadronic J. Vol. 3, pages 440-506, 1979.

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