Recent advances on Tsagas-Sourlas-Santilli isotopology

RECENT ADVANCES ON TSAGAS-SOURLAS-SANTILLI ISOTOPOLOGY R. M. FALC ´ ON and J. N ´ U˜ NEZ Departamento de Geometr´ıa y Topolog´ıa. Facultad de Matem´aticas. Universidad de Sevilla. Aptdo 1160. 41080-Sevilla (Espa˜na). [email protected] jnv[email protected] Abstract Because the Lie Theory solely applies to linear systems, in 1978 Santilli proposed the isotopic lifting of Lie’s theory for nonlinear systems, today known as the Lie-Santilli isotheory, via the reconstruction of linearity on the isotopic lifting of spaces and fields. In order to identify the proper mathematical background of the Lie-Santilli isotheory, Kadeisvili introduced in 1992 the notion of isocontinuity; Tsagas and Sourlas proposed in 1995 a for of isotopology defined over conventional fields; Santilli extended it in 1996 its formulation on isofields; and the authors conducted in 2003 a systematic study of the new isotopology. In this paper we outline the foundation of the new isotopology and present various advances. Dedication: This paper is dedicated to the memory of Prof. Gr. Tsagas in admiration of his studies on Lie-Santilli isotheory. 12000 Mathematics Subject Classification: Primary 03H05; Secondary: 08A05, 03C65 1 Nonlinear Funct. Anal. & Appl., Vol. 10, No. 5 (2005), pp. 873-883 1 Introduction Since the Lie theory solely applies to linear systems, while systems are generally nonlinear in reality, the physicist Santilli [7] proposed in 1978 the axiom-preserving isotopic lifting of Lie’s theory for nonlinear systems, today known ad the Lie-Santilli isotheory. The proposal was based on the lifting of enveloping associative algebras, Lie algebras, Lie group, representation theory, as well as the spaces on which they are defined. Consistency was achieved via the reconstruction of linearity on isospaces, with the nonlinearity emerging in the projection of the isotheory into ordinary spaces. Santilli then continued his studies on isotopies in monographs [5] [6] of 1978-1991, the lifting of fields in Ref. [8] of 1993, the introduction of the new isodifferential calculus in Ref. [9] of 1996, and other contributions. In order to identify the proper mathematical framework of the LieSantilli isotheory, the Russian mathematician Kadeisvili [4] introduced in 1992 the notion of isocontinuity. The Greek mathematicians Tsagas and Sourlas presented in monograph [7] a systematic study of the Lie-Santilli isotheory on isospaces over ordinary fields and in the subsequent papers [12] [13] of 1995 they introduced the notion of isomanifold as well as the first known form of isotopology formulated on conventional fields. Subsequently, Santilli [9] presented an extension of the isotopology to isofields and the new topology is today called the Tsagas-Sourlas-Santilli isotopology. We should also indicate that the Chinese mathematician Jiang conducted in monograph [3] of 2002 a comprehensive study of Santilli isonumber theory. More recently, in 2001, the authors have presented in monograph [1] a systematic study of the Lie-Santilli theory within a full isotopic context, including the isotopies of spaces and fields. In the subsequent memoir [2] of 2003, the authors have presented a systematic study of the TsagasSourlas-Santilli isotopology, with numerous advances. As a result of all these studies there has been the emergence of a new branch of mathematics that applies not only to nonlinear systems, but also to nonlocal and non-Hamiltonian systems occurring in physics, chemistry, biology and other fields. 2 In this paper we review the foundations of the Tsagas-Sourlas-Santilli isotopology and present various advances. The fundamental notion is that of Santilli isoreal isofield [8] (b Rm,b +, b ×) of dimension m, based on the isotopic lifting of the unit, called isounit, with expressions of the type b I=diag(n2 1, n2 2, ..., n2 m), with nk=nk(x, dx, d2x, τ, δ, ...)= 0, ∀k∈ {1, ..., m}. Particularly important for these studies is the classification of isofields into those of type I, occurring when the isounit is an arbitrary (non-null) element of the original field, and thsoe of type II, occurring when the isounit is not an element of the original field [8] (see also Jiang [3] for details). Next, we have the isotopic lifting of metric spaces, for the first time introduced by Santilli [5] and [6]: M(x, m, R)→c M(bx, bm, b RII ), where bx=x×b I,bm=T×bmand T=b I−1over the isoreal isofield b Rof the second type. This last construction is also important due to its practical applications. As an example, we have that the valence bond characterizing every molecule is, structurally, non local-integral, due to the penetration of the packages of waves of the valence electrons, which requires the use of a non local-integral topology to be studied in this paper. In fact, Santilli proved in [10] that the use of both a new integerdifferentiable isotopology and isotopic methods related for a Santilli’s isounit b I=O×exp−3r∫ψ†(r)×ψ(r), where Ois an operator and ψ(r) is the wave function of the degree electron has allowed, for the first time, to obtain an exact and invariant representation of all molecular characteristics. In this sense, it is necessary to use Santilli isospaces and isofields of the second type to get the results obtained in [7]. Tsagas and Sourlas [12] [13] introduced the isotopic lifting of topology over ordinary fields, subsequently extended by Santilli [9] to the isofields, by constructing an isotopy of the conventional space Rm, defined by T={∅,Rm,∪i∈IBi}, where each of Biis: Bi={P= (P1, ..., Pm) : αik< Pk< βik;αik, βik∈R,∀k∈ {1, ..., m}}, which can be considered as the Cartesian product of the topology of open 3 intervals on the straight line, mtimes. The topological space {Rm, T}is denoted by Tm(R) and it is called real Cartesian topological space. In the isospace b Rm, they defined the isotopic lifted of the topology T as: b T={∅,b Rm,∪i∈Ib Bi}, where each of b Biis b Bi={b P= ( b P1, ..., b Pm) : bαik<b Pk<b βik;bαik,b βik∈b Rn2 k,∀k∈ {1, ..., m}}. When b R=R, Tsagas and Sourlas pointed out that b Rn2 k≃Rand that b R m ≃Rm. For this reason, they called the pair {b Rm,b T}as real Cartesian isotopological space, and they denoted it by b Tm(b R). They also pointed out that Tm(R)≡b Tm(b R), which involves the coincidence between that new topology on b Rmand the conventional one on Rm, with the exception of b I, which incorporates integrals terms. The resulting structure is actually known as Tsagas-Sourlas Isotopology or Integro-differentiable topology. All the previous studies finally allowed to generalize in 2003 [2] the Tsagas-Sourlas-Santilli Isotopology for isofields of the types I and II, by making use of the isotopic construction model MCIM, introduced by the authors in 2001 [1], which generalizes in turn the model by Santilli in 1978 [7]. In particular, we provide an alternative formulation of Kadeisvili isocontinuity [4] from an analytic and a topological point of view. Note finally that some results appearing in this paper will not be proved, due to restrictions on length length. 2 Isotopology by using the MCIM isotopic model The generalization of the Tsagas - Sourlas Isotopology [11] to the case of isofields of the second type, proposed by Santilli in [9] was deeply analyzed by ourselves in a recent paper, appeared in 2003 (see [2]). Such an analysis is made by using the isotopic model named MCIM, which we also introduced in [1]. Every isotopy can be reduced to this 4 model and it is based on the use of so many isounits and ∗-laws as operations existing in the initial mathematical structure: Proposition 2.1. Fixed a mathematical structure (E, +,×,◦,•, ...), if we construct an isotopic lifting such that: a) Both primaries ∗,b Iand secondaries ⋆, b Selements of isotopy are used. b) (E, ⋆, ∗, ...)is a structure of the same type as the initial, which is endowed with isounits S, I, ..., with respect to ⋆, ∗, ..., respectively. c) Iis an unit with respect to ∗in the corresponding general set V, being T=b I−I∈Vthe associated isotopic element. So, by defining in the isotopic level the operations: bab +b b=d a ⋆ b;bab ×b b=d a∗b;... And being defined in the projection level: ba=a∗b I;αb +β= ((α∗T)⋆(β∗T)) ∗b I;αb ×β=α∗T∗β;... It is obtained that the isostructure (b E, b +,b ×, ...)is of the same type as the initial one. The study in [2] is made by taking into consideration both isotopic and projection levels. Equivalent results related to injective isotopies are also obtained. In the first place, it is verified Proposition 2.1 for topological spaces and for their elements and basic properties: isotopologies, isoclosed sets, isoopen sets, T2, etc: Atopological isospace is every isospace endowed with a topological space structure. If, besides, such an isospace is an isotopic projection of a topological space, it is called isotopological isospace. Similarly, they are defined concepts of (iso)boundary isopoint, closure of a set, closed set, isointerior isopoint, interior of a set, open set, (iso)Hausdorff isospace and second countable isospace, among others. 5 Proposition 2.2. The space from which any topological isospace in the isotopic level is obtained can be endowed with the final topology relative to the mapping I. The isotopic projection of a topological space is an isotopological isospace in the projection level. If such a projection is injective, then every topological isospace in such a level is, in fact, isotopological. Similar results are obtained for the concepts of (iso)boundary isopoint, isointerior isopoint and (iso)Hausdorff isospace. Next, we try to analyze the concept of isocontinuity of isofunctions, attempting to generalize the Kadeisvili isocontinuity [4]: Let b Ube a b R-isonormed vector isospace, where b Ris an isofield of the type I. Let ≤be the usual order in Rand b fan isofunction from b U on b R. We will say that b fis a Kadeisvili isocontinuous isofunction in b X∈b U, if for all bϵ > 0, there exists b δ > 0 such that, for all b Y∈b Uwith I((π◦I)−1(b X−b Y))<b δ, it is verified that: I((π◦I)−1(b f(b X)−b f(b Y)))<bϵ. We will say that b fis Kadeisvili isocontinuous in b Uif it is Kadeisvili isocontinuous in b X, for all b X∈b U. The Kadeisvili isocontinuity is defined for isofields of the type I obtained from R, which are endowed with the usual real order ≤. For this reason, it was proposed in [2] that the basic isofield can be endowed with an isoorder, according to: Let b Kbe an isofield associated with a field K, endowed with an order ≤, by using an isotopology which preserves the inverse element with respect to the addition. We define the isoorder b ≤as bab ≤b bif and only if a≤b. If the isotopy is injective, the isoorder b ≤en b Kis defined in the same way. 6 Proposition 2.3. The isoorders b ≤and b ≤are orders over b Kand b K, of the same type as ≤. The Kadeosvili isocontinuity was generalized in [2] of the following way: Let b Ube a b Risovectorspace with isonorm b ||.b || ≡ c ||.|| and isoorder b ≤, obtained from an isotopy compatible with respect to each one of the initial operations. It will be said that an isoreal isofunction b fof b Uis isocontinuous in b X∈b U, if for all bϵb >b S, there exists b δb >b Ssuch that for all b Y∈b Uwith b || b X−b Yb ||b <b δ, it is verified that b|b f(b X)−b f(b Y)b|b <bϵ. We will say that b fis isocontinuous in b Uif it is isocontinuous in b X, for all b X∈b U. Finally, when dealing with injective isotopies, the isocontinuity in the projection level is defined in a similar way. Proposition 2.4. The isocontinuity in b Uis equivalent to the continuity in U. In the case of injective isotopies, both ones are equivalent to the one in b U. We are going to observe in the following example that, indeed, the previous definition of isocontinuity really generalizes the Kadeisvili isocontinuity: Example 2.5. Let us consider an isoreal isofield of the type I obtained from an isotopy of the isotopic element ⋆≡+,b S= 0,∗ ≡ × and b I∈R+ non null. Such an isotopy is injective and allow to obtain the isofield b R≡R, due to fixed a∈Rit is a=[ a∗T, being T=b I−1. Such an isotopy preserves the inverse element and it is compatible with respect to +,◦,•y×. It is checked that b ≤ ≡≤,b +≡+and b ◦ ≡ ◦. The Kadeisvili isocontinuity is defined, in this case, of the following way (note that ≤≡ b ≤): Let b fbe an isofunction of b Uon b R. Then, b fis Kadeisvili isocontinuous in b X∈b Uif for all bϵ > 0, there exists b δ > 0such that, if b Y∈b Usatisfies b || b X−b Yb || <b δ, then b|b f(b X)−b f(b Y)b|<bϵ.▹ 7 Proposition 2.6. Under conditions of the Example 2.5, every (Kadeisvili) isocontinuous isofunction b fen b X∈b Uis conventionally continuous in such a point. Consequently, every Kadeisvili isocontinuous isofunction b fin b Uis conventionally continuous in b U. Proof. Let us suppose b X∈b Rand let b fbe an isocontinuous isofunction in b X. To see that b fis a conventionally continuous isofunction, we fix ϵ > 0. Let ϵ′>0 be such that b ϵ′=ϵ′∗b I=ϵ′×b I=ϵ. Due to the isocontinuity of b fthere exists b δ > 0 such that, if b Y∈b Uis such that b || b X−b Yb || <b δ, then b|b f(b X)−b f(b Y)b|<b ϵ′. But then, due to the compatibility with respect to ◦of the isotopy constructing b Uand due to b Iacts as a constant, which is positive, the previous condition is equivalent to the fact of if b Y∈b Uis such that || b X−b Y|| =|| \ X−Y|| =||(X−Y)∗b I|| =||X−Y||× b I=b || \ X−Yb || = b || b X−b Yb || <b δ, then |b f(b X)−b f(b Y)|=|[ f(X)−[ f(Y)|=|\ f(X)−f(Y)|= |(f(X)−f(Y)) ×b I|=|(f(X)−f(Y))| × b I=b|\ f(X)−f(Y)b|=b|b f(b X)− b f(b Y)b|<b ϵ′=ϵ. So, it implies that b fis conventionally continuous in X. The consequence of the assert is then evident.  A problem which appears when b Iis not constant, that is, it depends on external factors, is that the order is not equivalence with the isoorder in the model of Example 2.5. It involves that isocontinuity of [2] in not a particular case of the Kadeisvili isocontinuty. We will see it in the following: Example 2.7. Under conditions of Example 2.5, let us consider (U, ◦,•) = (R,+,×), although we take now as an isounit to: b I=b I(x) = {1, if x= 0 1 x2, if x= 0 }. Then, b Ris now given by the lifting: 8 x→bx=x×b I={0, if x= 0 1 x, if x= 0 }. So, we have b R≡R, and the isotopic lifting used is injective. Moreover, as it preserves the inverse element with respect to the addition, it has a perfect sense to consider the isoorder b ≤, which is not equivalent to the usual one. Indeed, as an example, we have that b 2b ≤b 3, due to 2≤3, but on the opposite b 2 = 1 2≥1 3=b 3. It cannot be also said that b ≤is equivalent to the inverse order ≤, because b 0b ≤b 2, due to 0≤2, being b 0 = 0 ≤1 2=b 2. ▹ Example 2.8. Under conditions of Example 2.5 let us consider: b I=b I(x) =        1, if x < 1 x+1 x,if x∈[1,3) x−2 x, if x∈[3,4) 1, if x≥4        . Therefore, b Iis so positive defined, non singular and invertible, whose inverse is: T=T(x) =        1, if x < 1 x+2 x, if x∈[1,2) x−1 x, if x∈[2,4) 1, if x≥4        . Then, the injective isotopic lifting from Rto b R=R, is defined by: x→bx=       x, if x < 1 x+ 1, if x∈[1,3) x−2, if x∈[3,4) x, if x≥4        . Let us consider the function f(x) = x−1, which is conventionally continuous. We have then the isofunction: 9