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A mathematical structure for the generalization of conventional algebra

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A mathematical structure for the generalization of conventional algebra
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  Cent. Eur. J. Phys. • 7(3) • 2009 • 549-554DOI: 10.2478/s11534-009-0046-4 Central European Journal of   Physics A mathematical structure for the generalization ofconventional algebra Research Article Aziz El Kaabouchi 1 ∗ , Laurent Nivanen 1 , Qiuping Alexandre Wang 1 , Jean Pierre Badiali 2 , Alain LeMéhauté 1 1 Institut Supérieur des Matériaux et Mécaniques Avancés du Mans, 44 Avenue Bartholdi, 72000 Le Mans, France 2 UMR 7575 LECA ENSCP-UPMC, 11 rue P. et M. Curie, 75231 Cedex 05, Paris, France Received 1 December 2008; accepted 23 February 2009 Abstract:  An abstract mathematical framework is presented in this paper as a unification of several deformed orgeneralized algebra proposed recently in the context of generalized statistical theories intended to treatcertain complex thermodynamic or statistical systems. It is shown that, from a mathematical point of view,any bijective function can in principle be used to formulate an algebra in which the conventional algebraicrules are generalized. PACS (2008):  02.; 02.10.-v; 02.50.-r Keywords:  deformed algebra • generalized algebra • statistical systems©  Versita Warsaw and Springer-Verlag Berlin Heidelberg. 1. Introduction In the last two decades, statistical physics has ex-perienced several extensions from the conventionalBoltzmann-Gibbs-Shannon (BGS) formalism to otherseemingly more general formalisms. Among these ex-tensions, there are two called: non-extensive statisti-cal mechanics (NSM) [1, 2]; and  κ  -statistical mechan-ics (KSM) [4–6]. Both of them  overcome , among others,the limitative rule of additivity of energy and entropy,a paradigm governing conventional statistical mechan-ics. Due to the strong relationship between non-extensivestatistical physics and fractal geometry [7] it has been ∗ E-mail: aek@ismans.fr claimed that these statistics can be used to describe com-plex systems whose anomalous behaviors cannot be in-terpreted within BGS statistics. Since the proposition of these extensions, many pros and cons have been writtenfrom the physical point of view. We will not enter intothe physical debate here. The present work is limited tothe pure mathematical study of some algebraic aspectsrelated to the mathematical functionals used in extendedstatistics.The first functional has been used in NSM and called − exponential e    = (1 +   ) 1/   and  − logarithmln     =    − 1   where    = 1 −  or    =  − 1. They are inverse func-tions of one another. When    tends toward 1, or when   tends toward 0, the  − exponential tends toward the usualexponential function and the − logarithmtends toward theusual logarithm. From a mathematical point of view, thesefunctions have interesting properties that extend partially 549  A mathematical structure for the generalization of conventional algebra to those of the corresponding usual functions. For exam-ple, the usual exponential and logarithm entail the usualalgebraic operators such as addition, subtraction, multipli-cation and division. In mimicking the related morphisms,an extended algebra has been defined from e    and ln    using generalized operators [8, 9]. Further development shows application to the derivation of    −  factorial and − multinomial coefficient [10].The second functional called the  κ− exponential is used inKSM. In that case exp {κ} (   ) = κ   + √  1 + κ  2   2  1/ κ  and κ − logarithmln κ      =    κ   −  −κ  2 κ   . They are also inverse func-tions of one another. In the case of   κ   tends toward 0, thesegeneralized functions recover the usual ones. An extended κ − algebra is also developed from these functions [4–6]. Other deformations in [3–6] or extensions of conventional statistics with extended algebra [11, 12] were also pro- posed. A common character of extended algebra is to usea generalization of the exponential function or its inverseto expand the morphism of, say, a usual algebraic operator.For example, using e    = (1 +   ) 1/  , if we calculate theusual product of two exponentials, we find e   × e   = e   +   where the operator +   is the generalized   -addition givenby     +     =     +  +  .The present work is an extension of this logic within amore general mathematical framework [4–6]. It will be shown that the same methodology can be carried out notonly with the generalized exponentials, but also at leastwith any bijection. The concomitant mathematical struc-ture can yield different algebraic rules according to thechoice of different functions and different field of theirfunctional definition. We think that this formulation of extended algebra may be beneficial for the further devel-opment and understanding of the new mathematical ten-dency stimulated by the development of physics. 2. Structure of generalized groupand ring 2.1. Preliminaries Let  X   and  Y   be two nonempty sets, and let  ∗  and  ⊥  be twobinary operations on  X   and  Y   respectively. We denote by1 X   and 1 Y   the neutral element (when this exists) of  ( X∗ )and ( Y⊥ ) respectively, and by    − 1  the inverse of      if      isinvertible in ( X∗ ) or in ( Y⊥ ). We mean by    a bijectionof  ( X∗ ) in ( Y⊥ ) and by  ψ   the inverse function of    . 2.2. Definitions and theorem a) The functions + ( ⊥ )  and  − ( ⊥ )  defined on  X ×X   by : ∀    ′    ∈ X ×X    + ( ⊥ )    ′   =  ψ    (   ) ⊥ (   ′  ) ∀    ′    ∈ X ×X − ( ⊥ )    ′   =  ψ    (   ) ⊥   (   ′  ) − 1  (when this exists)are binary operations (laws) on  X  .b) The functions  × (  ∗ )  and  ÷ (  ∗ )  defined on  Y   × Y   by ∀   ′    ∈  Y   × Y  × (  ∗ )   ′   =     ψ  (  ) ∗ ψ  (  ′  )  ∀   ′    ∈  Y   × Y ÷ (  ∗ )   ′   =     ψ  (  ) ∗  ψ  (  ′  ) − 1  (when this exists)are binary operations (laws) on  Y  .c)     is an homomorphism of  X + (  ⊥ )  in ( Y ⊥ ), and  ψ  is an homomorphism of  Y × (  ∗ )  in ( X ∗ ) .d) When     is an homomorphism of  ( X ∗ ) in ( Y ⊥ ) then+ (  ⊥ )  =  ∗  and  × (  ∗ )  =  ⊥ . In particular,     becomes anisomorphism of  X + (  ⊥ )  into Y × (  ∗ ) . 2.3. Examples a) When ( X ∗ ) = ( R  +), ( Y ⊥ ) = R ∗ +  ×  and     = exp,then  ψ   = ln.We have for all (    ′  )  ∈  X   × X   and (  ′  )  ∈  Y   × Y     + (  ⊥ )    ′   =  ψ     (    ) ×   (    ′  ) = lnexp     × exp    ′   =     +    ′      − (  ⊥ )    ′   =  ψ     (    ) ×    (    ′  ) − 1  = ln exp    exp    ′   =     −    ′   × (  ∗ )   ′   =     ψ  (  ) +  ψ  (  ′  ) = expln   + ln  ′   =   ×  ′   ÷ (  ∗ )   ′   =     ψ  (  ) − ψ  (  ′  ) = expln  − ln  ′   =   ÷  ′   we thus find the ordinary operations.b) Let    ∈  R ∗ +  and we consider the set  X   = − 1   + ∞ with the additive law  ∗  = + and ( Y ⊥ ) = R ∗ +  × . Wemean by     the function of   X   in ( Y ⊥ ) defined in [7, 8] by   (    ) = (1 +    ) 1/  = e     It’s easy to check that     is a bijection of   X   into  Y  , andthat the inverse function of      is defined by ∀     ∈  Y ψ  (    ) = ln      =     − 1   In this case, we have for all (    ′  )  ∈  X   × X   and (  ′  )  ∈ Y   × Y   according to generalized operations [8, 9] 550  Aziz El Kaabouchi, Laurent Nivanen, Qiuping Alexandre Wang, Jean PierreBadiali, Alain Le Méhauté    + ( ⊥ )    ′   =  ψ    (   ) × (   ′  ) =     +   ′   +   ′   =     +     ′   − ( ⊥ )    ′   =  ψ    (   ) ×   (   ′  ) − 1  =   −  ′  1 +   ′   =   −     ′  × ( ∗ )   ′   =    ψ  (  ) + ψ  (  ′  )=    +  ′  − 1 1/   =  ×    ′  ÷ ( ∗ )   ′   =    ψ  (  ) + ψ  (  ′  ) − 1 =    − ′  − 1 1/   =  ÷    ′   c) Let  κ ∈  R ∗ +  and we consider ( X∗ ) = ( R  +), ( Y⊥ ) = R ∗ + × . Let    be the function of   X   in ( Y⊥ ) defined by ∀ ∈ X  (   ) = κ   + √  1 + κ  2   2  1/ κ  = exp {κ} (   )   is a bijective of   X   into  Y  , and that the inverse functionof     is defined by ∀ ∈ Y ψ  (   ) = ln {κ} (   ) =    κ   −  −κ  2 κ   In this case, we have all (   ′  )  ∈ X×X   and (  ′  )  ∈ Y×Y    + ( ⊥ )    ′   =  ψ    (   ) × (   ′  )=   √  1 + κ  2   2  +   ′  √  1 + κ  2   ′  2  =    κ  ⊕  ′   − ( ⊥ )    ′   =  ψ    (   ) ×   (   ′  ) − 1 = √  1 + κ  2   2  −  ′  √  1 + κ  2   ′  2  =    κ  ⊖  ′  × ( ∗ )   ′   =    ψ  (  ) + ψ  (  ′  )= 1 κ  shargsh( κ ) + argsh κ ′   =   κ  ⊗ ′   we thus find the operations introduced by Kaniadakis [4–6, 11]. d) Let  κ ∈  R ∗ +  and we consider     ∈  [ − κκ  ], ( X ∗ ) =( R  +), ( Y ⊥ ) = R ∗ +  × . Let  ψ   be the function of   Y   in( X ∗ ) defined by ∀     ∈  Y ψ  (    ) = ln { κ  } (    ) =      + κ  −      − κ  2 κ   We confirm easily that  ψ   is a bijective of   Y   into  X   ( ψ   iscontinuous and strictly increasing on  Y  ). We denote by    :     →    (    ) = e { κ  } (    ) the inverse function of   ψ  .We note that  ψ   verifies the following property ∀  ′   ∈  Yψ  (  ′  ) =   κ  +   ψ  (  ′  ) +   ′  κ  +   ψ  (  ) − 2 κψ  (  ) ψ  (  ′  )  It follows that for all (    ′  )  ∈  X   × X  , we have     + (  ⊥ )    ′   =  ψ     (    ) ×   (    ′  ) =    κ  ⊕    ′   See [11].e) Let  γ   ∈  R ∗ +  and we consider  κ   = 3 γ  2 ,     =  γ  2 , ( X ∗ ) =( R  +), ( Y ⊥ ) = R ∗ +  × . Let  ψ   be the function of   Y   in( X ∗ ) defined by ∀     ∈  Yψ  (    ) = ln γ  (    ) = ln { κ  } (    ) =      + κ  −      − κ  2 κ   =    2 γ  −    − γ  3 γ   We denote by     :     →    (    ) = e γ  (    ) the inverse functionof   ψ  , then     is defined by ∀     ∈  X  (    ) = e γ  (    ) =  1 +   1 − 4 γ  3    3 2  1 / 3 +  1 −   1 − 4 γ  3    3 2  1 / 3  1 /γ   We note that  ψ   verifies the following property ∀  ′   ∈  Y ψ  (  ′  ) =  ψ  (  ′  ) +   ′  ψ  (  ) − 3 γψ  (  ) ψ  (  ′  )  It follows that for all (    ′  )  ∈  X   × X  , we have     + (  ⊥ )    ′   =  ψ     (    ) ×   (    ′  )   =    γ  ⊕    ′   See [11].f) Let  γ   ∈  R ∗ +  and we consider ( X ∗ ) = ( R  +), ( Y ⊥ ) =(]0  1]  × ). Let     be the function of  ( X +) into ( Y ⊥ ) de-fined in [13–16] by ∀     ∈  X   (    ) = exp( −    γ  )The inverse function  ψ   of      is defined by ∀     ∈  Y ψ  (    ) =  ln  1     1 γ   It is easy to check that for all (    ′  )  ∈  X  × X   and (  ′  )  ∈ Y   × Y  , we have     + (  ⊥ )    ′   =  ψ     (    ) ×   (    ′  )   =     γ  +    ′  γ   1 γ   551  A mathematical structure for the generalization of conventional algebra × ( ∗ )   ′   = e xp  −  ln  1   1 γ  +  ln   1  ′   1 γ   γ    As an e xample, see the following operation for  γ   = 2: ∀    ′    ∈ R ∗ +  × R ∗ +     + ( ⊥ )    ′   = √   2  +   ′  2  We observe that the Gauss distribution is related to thePythagoras relation on the circle. This observation is keyto fully understanding the relation between the diffusionprocess and 2D fractal Dynamics [7]. 2.4. Remarks a) Let   >  0. We show that    :   →  (   ) = e    is the unique  function defined on  X   =  − 1   + ∞  , differentiableat 0 with   ′  (0) = 1 and verifies for all (   ′  )  ∈ X ×X  (    +     ′  ) =   (   ) × (   ′  )  b) Let  κ >  0. We show that    :   →  (   ) = exp {κ} (   ) isthe  unique  function defined on  R , differentiable at 0 with  ′  (0) = 1 and verifies for all (   ′  )  ∈ X ×X  (   κ  ⊕  ′  ) =   (   ) × (   ′  )  Proposition 2.1. With the  previous notations, we have + ( × ( ∗ ) )  =  ∗  and   × (  + ( ⊥ ) )  =  ⊥ Proposition 2.2. Let     ′    ′′   ∈ X   , we have the following properties    ∗    ′   + ( ⊥ )    ′′    =   (   ) × ( ∗ )   (   ′  ) ⊥ (   ′′  )      ′   + ( ⊥ )    ′′   ∗    =   (   ′  ) ⊥ (   ′′  )  × ( ∗ )   (   )  2.5. Remarks a) The commutativity of the law  ∗  (respectively  ⊥ ) im-plies the commutativity of the law  × ( ∗ )  (respec-tively + ( ⊥ ) ).b) The associativity of the law  ∗  (respectively  ⊥ ) im-plies the associativity of the law  × ( ∗ )  (respectively+ ( ⊥ ) ).c) The existence of a neutral element of  ( X∗ ) (respec-tively of  ( Y⊥ )), implies the existence of a neutralelement  Y× ( ∗ )   (respectively of   X + ( ⊥ )  ).d) Due to the fact that the two laws are not definedin the same set there is no reason to consider thedistributivity of   × ( ∗ )  with respect to + ( ⊥ )  or con-versely.We can therefore make the following proposition. Proposition 2.3. With the previous notations, we havea)  ( Y⊥ )  is an abelian group if and only if   X + ( ⊥ )  is an abelian group.b)  ( X∗ )  is an abelian group if and only if   Y× ( ∗ )  is an abelian group.c)  ( Y⊥◦ )  is a ring if and only if   X + ( ⊥ )  + ( ◦ )   isa ring.d)  Y⊥× ( ∗ )   is a ring if and only if   X + ( ⊥ ) ∗  is a ring. 3. Structure of generalized vectorspace 3.1. Definition and theorem Let  X   and  Y   be two nonempty sets,  • a external law on  Y  , We mean by    a bijection of  X   in  Y   and by  ψ   the inverse function of    .The function  • ( • )  defined on  K ×X   by ∀ ( λ  )  ∈ R ×X • ( • ) ( λ  ) =  ψ  ( λ• (   ))is an external law on  X  .We denote  • ( • ) ( λ  ) by  λ• ( • )    . 3.2. Applications a) When  K  = R ,  X   = R ,  Y   = R ∗ +  and  •  is the law definedon  Y   by ∀ ( λ  )  ∈ R ×Y λ•   =    λ  If we consider the function    of   X   in  Y   defined by ∀ ∈ X  (   ) = exp   Then for all ( λ  )  ∈ R ×X  , λ• ( • )     = ln( λ• (   )) =  λ In this case we denote  • (exp • )  instead of   • ( • ) .We thus find the ordinary multiplication. 552  Aziz El Kaabouchi, Laurent Nivanen, Qiuping Alexandre Wang, Jean PierreBadiali, Alain Le Méhauté b) Let    ∈  R ∗ + . When  X   = − 1   + ∞  ,  K  =  R ,  Y   =  R ∗ + and  •  is the law defined on  Y   by ∀ ( λ  )  ∈ R × Y λ •    =    λ  If we consider the function    from  X   in  Y   defined by ∀    ∈  X  (   ) = (1 +   ) 1/   =  e    Then for all  ( λ  )  ∈ R × X  , λ • (  • )     =  ψ  ( λ •  (   )) = (1 +   ) λ  −  1   In this case we denote  • (  • )  instead of   • (  • ) .We thus find the operation defined in [7, 8]. c) Let  κ   ∈  R ∗ + . When  K  =  R ,  X   =  R ,  Y   =  R ∗ +  and  •  isthe law defined on  Y   by ∀ ( λ  )  ∈ R × Y λ •    =    λ  If we consider the function    from  X   in  Y   defined by ∀    ∈  X  (   ) =  κ   + √  1 + κ  2   2  1/ κ  = exp {κ} (   )  Then for all ( λ  )  ∈ R ×X  , λ• ( • )     = ln {κ}  ( λ• (   ))=  κ   + √  1 + κ  2   2  λ −  κ   + √  1 + κ  2   2  −λ 2 κ   In this case we denote  • ( {κ}• )  instead of   • ( • ) . 3.3. Remarks a) We show that    :   →  (   ) = exp     is the unique  function defined on  R , differentiable at 0with   ′  (0) = 1 and verifies ∀ ( λ  )  ∈ R × R    λ• (exp • )      = (  (   )) λ  b) Let   >  0. We show that    :   →  (   ) = e   is the  unique  function defined on  X   =  − 1   + ∞  ,differentiable at 0 with   ′  (0) = 1 and verifies ∀ ( λ  )  ∈ R ×X   λ• ( • )      = (  (   )) λ  c) Let  κ >  0We show that    :   →  (   ) = exp {κ} (   ) isthe  unique  function defined on  R , differentiable at0 with   ′  (0) = 1 and verifies ∀ ( λ  )  ∈ R ×X   λ• ( {κ}• )      = (  (   )) λ  Theorem 3.1. If   ( Y⊥• )  is a vector space on  K  , then  X + ( ⊥ ) • ( • )  is a vector space on  K . 4. Conclusion This paper reports a general framework that unifies all theextended algebra recently proposed from physical consid-erations. Our main conclusion is that (at least) any bijec-tive function can be the characteristic function of an alge-bra that may generalize the conventional algebra charac-terized by the usual exponential function.It is for algebraic and functional reasons that the exponen-tial function plays a major role both in mathematics and inphysics. This role is essentially related with elementarymathematical manipulations and fitting the behaviour of separable physical phenomena. Nevertheless the physicsof complex media breaks this relevance in different ways:nonextensivity, correlation and coupling, scaling proper-ties  etc  . [13, 14]. In that case the exponential function and all related properties lose their physical accuracy andare transformed into power laws. The present mathemat-ical analysis shows not only how this accuracy is lost,but which key factors must be reconstructed to cope withcomplex problems.It must be observed that the generalization of exponen-tial analysis suggested in the present paper is based onthe link between functional definitions and their algebraicand functional fields of validity. Work on this link, whichis a key factor in understanding the dynamics of physi-cal phenomena in complex geometry, is currently still inprogress. Another work relating to the generalization of usual differential calculus with bijective and non bijectivefunctions will be presented in the near future. References [1] C. Tsallis, J. Stat. Phys. 52, 479 (1988)[2] C. Tsallis, R. S. Mendes, A. R. Plastino, Physica A261, 534 (1999)[3] S. Abe, Phys Lett. A 224, 326 (1997)[4] G. Kaniadakis, Physica A 296, 405 (2001) 553
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