Hierarchical structure of operations defined in nonextensive algebra

Nivanen, Laurent; Wang, Q.A.; Méhauté, Alain Le; Kaabouchi, A. El; Basillais, P.; Donati, J.D.; Lacroix, A.; Paulet, J.; Perriau, S.; Chuisse, S. Sime; Kamdem, E. Simo; Théry, A.

doi:10.1016/s0034-4877(09)90004-0

Cited by 4 publications

(9 citation statements)

References 16 publications

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“…( 8) of Ref. [9]. Analytical extension into the real domain yields the non commutative ole-dot-multiplication:…”

Section: B Ole-arithmeticsmentioning

confidence: 99%

“…( 8) of Ref. [9]. We make an analytical extension from n ∈ N to y ∈ R, and the oel-power can also be written as…”

Section: Iel-arithmeticsmentioning

confidence: 99%

“…( 24)-( 28)]. In an extension of that work by the same authors with collaborators [9], the qmultiplication and the q-addition were identified as belonging to two different classes, and further operators were defined.…”

Section: Introductionmentioning

confidence: 99%

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Deformed mathematical objects stemming from the $q$-logarithm function

Borges¹,

Costa²

2021

Preprint

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Generalized numbers, arithmetic operators and derivative operators, grouped in four classes based on symmetry features, are introduced. Their building element is the pair of q-logarithm/q-exponential inverse functions. Some of the objects were previously described in the literature, while others are newly defined.Commutativity, associativity and distributivity, and also a pair of linear/nonlinear derivatives are observed within each class. Two entropic functionals emerge from the formalism, one of them is the nonadditive Tsallis entropy.

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“…( 8) of Ref. [9]. Analytical extension into the real domain yields the non commutative ole-dot-multiplication:…”

Section: B Ole-arithmeticsmentioning

confidence: 99%

“…( 8) of Ref. [9]. We make an analytical extension from n ∈ N to y ∈ R, and the oel-power can also be written as…”

Section: Iel-arithmeticsmentioning

confidence: 99%

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Deformed mathematical objects stemming from the $q$-logarithm function

Borges¹,

Costa²

2021

Preprint

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“…The q -logarithm function is defined [ 4 ] as follows

its inverse function being

when

; otherwise it vanishes. The definitions of the q -logarithm and q -exponential functions allow consistent generalizations of algebras [ 5 , 6 , 7 , 8 , 9 ], calculus [ 6 , 8 , 10 , 11 ] (see also [ 12 ]) and generalized numbers [ 8 , 13 ].…”

Section: Introductionmentioning

confidence: 99%

Along the Lines of Nonadditive Entropies: q-Prime Numbers and q-Zeta Functions

Borges

Kodama

Tsallis

2021

Entropy

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The rich history of prime numbers includes great names such as Euclid, who first analytically studied the prime numbers and proved that there is an infinite number of them, Euler, who introduced the function ζ(s)≡∑n=1∞n−s=∏pprime11−p−s, Gauss, who estimated the rate at which prime numbers increase, and Riemann, who extended ζ(s) to the complex plane z and conjectured that all nontrivial zeros are in the R(z)=1/2 axis. The nonadditive entropy Sq=k∑ipilnq(1/pi)(q∈R;S1=SBG≡−k∑ipilnpi, where BG stands for Boltzmann-Gibbs) on which nonextensive statistical mechanics is based, involves the function lnqz≡z1−q−11−q(ln1z=lnz). It is already known that this function paves the way for the emergence of a q-generalized algebra, using q-numbers defined as ⟨x⟩q≡elnqx, which recover the number x for q=1. The q-prime numbers are then defined as the q-natural numbers ⟨n⟩q≡elnqn(n=1,2,3,⋯), where n is a prime number p=2,3,5,7,⋯ We show that, for any value of q, infinitely many q-prime numbers exist; for q≤1 they diverge for increasing prime number, whereas they converge for q>1; the standard prime numbers are recovered for q=1. For q≤1, we generalize the ζ(s) function as follows: ζq(s)≡⟨ζ(s)⟩q (s∈R). We show that this function appears to diverge at s=1+0, ∀q. Also, we alternatively define, for q≤1, ζqΣ(s)≡∑n=1∞1⟨n⟩qs=1+1⟨2⟩qs+⋯ and ζqΠ(s)≡∏pprime11−⟨p⟩q−s=11−⟨2⟩q−s11−⟨3⟩q−s11−⟨5⟩q−s⋯, which, for q<1, generically satisfy ζqΣ(s)<ζqΠ(s), in variance with the q=1 case, where of course ζ1Σ(s)=ζ1Π(s).

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“…when [1 + (1 − q)z] ≥ 0; otherwise it vanishes. The definitions of the q-logarithm and qexponential functions allow consistent generalizations of algebras [5,6,7,8,9], calculus [6,10,11,8] (see also [12]) and generalized numbers [13,8].…”

Section: Introductionmentioning

confidence: 99%

Along the lines of nonadditive entropies: $q$-prime numbers and $q$-zeta functions

Borges,

Kodama,

Tsallis

2021

Preprint

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Hierarchical structure of operations defined in nonextensive algebra

Cited by 4 publications

References 16 publications

Deformed mathematical objects stemming from the $q$-logarithm function

Deformed mathematical objects stemming from the $q$-logarithm function

Along the Lines of Nonadditive Entropies: q-Prime Numbers and q-Zeta Functions

Along the lines of nonadditive entropies: $q$-prime numbers and $q$-zeta functions

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