The Lazard formal group, universal congruences and special values of zeta functions

Tempesta, Piergiulio

doi:10.1090/tran/6234

Cited by 19 publications

(23 citation statements)

References 31 publications

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“…The theory of formal groups was introduced by Bochner in the seminal paper [ 21 ] and developed in algebraic topology, analysis, and other branches of pure and applied mathematics by G. Faltings, S. P. Novikov, D. Quillen, J. P. Serre and many others [ 20 , 22 ]. For recent applications in number theory, see also [ 23 , 24 ].…”

Section: Basic Results On Group Entropiesmentioning

confidence: 99%

Group Entropies: From Phase Space Geometry to Entropy Functionals via Group Theory

Tempesta

2018

Entropy

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The entropy of Boltzmann-Gibbs, as proved by Shannon and Khinchin, is based on four axioms, where the fourth one concerns additivity. The group theoretic entropies make use of formal group theory to replace this axiom with a more general composability axiom. As has been pointed out before, generalised entropies crucially depend on the number of allowed degrees of freedom N. The functional form of group entropies is restricted (though not uniquely determined) by assuming extensivity on the equal probability ensemble, which leads to classes of functionals corresponding to sub-exponential, exponential or super-exponential dependence of the phase space volume W on N. We review the ensuing entropies, discuss the composability axiom and explain why group entropies may be particularly relevant from an information-theoretical perspective.

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Section: Basic Results On Group Entropiesmentioning

confidence: 99%

Group Entropies: From Phase Space Geometry to Entropy Functionals via Group Theory

Tempesta

2018

Entropy

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“…The theory of formal groups was introduced by Bochner in the seminal paper [20] and developed in algebraic topology, analysis, and other branches of pure and applied mathematics by G. Faltings, S. P. Novikov, D. Quillen, J. P. Serre and many others [19], [21]. For recent applications in number theory, see also [22], [23]. A property crucial for the subsequent discussion is the following: given a one-dimensional formal group law φ(x, y), there exist a series G(t…”

Section: Basic Results On Group Entropiesmentioning

confidence: 99%

Group entropies: from phase space geometry to entropy functionals via group theory

Tempesta¹

2018

Preprint

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The entropy of Boltzmann-Gibbs, as proved by Shannon and Khinchin, is based on four axioms, where the fourth one concerns additivity. The group theoretic entropies make use of formal group theory to replace this axiom with a more general composability axiom. As has been pointed out before, generalized entropies crucially depend on the number of allowed number degrees of freedom $N$. The functional form of group entropies is restricted (though not uniquely determined) by assuming extensivity on the equal probability ensemble, which leads to classes of functionals corresponding to sub-exponential, exponential or super-exponential dependence of the phase space volume $W$ on $N$. We review the ensuing entropies, discuss the composability axiom, relate to the Gibbs' paradox discussion and explain why group entropies may be particularly relevant from an information theoretic perspective.

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“…( The universal higher-order Bernoulli polynomials , ( ) introduced in [13] and later discussed in [11] and [14] are defined as…”

Section: Introductionmentioning

confidence: 99%

“…The universal Bernoulli numbers provide extensions of the celebrated Kummer and Clausenvon Staudt congruences [1] and general Almkvist-Meurman-type congruences [14]. In [12], it was shown that interesting realizations of polynomials (1.1) can be constructed by means of finite operator theory introduced by G.C.…”

Section: Introductionmentioning

confidence: 99%

Some relations for universal Bernoulli polynomials

Dağli

2019

Ufimsk. Mat. Zh.

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In this paper, we consider a generalization of the Bernoulli polynomials, which we call universal Bernoulli polynomials. They are related to the Lazard universal formal group. The corresponding numbers by construction coincide with the universal Bernoulli numbers. They turn out to have an important role in complex cobordism theory. They also obey generalizations of the celebrated Kummer and Clausen-von Staudt congruences. We derive a formula on the integral of products of higher-order universal Bernoulli polynomials. As an application of this formula, the Laplace transform of periodic universal Bernoulli polynomials is presented. Moreover, we obtain the Fourier series expansion of higher-order universal Bernoulli function.

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The Lazard formal group, universal congruences and special values of zeta functions

Cited by 19 publications

References 31 publications

Group Entropies: From Phase Space Geometry to Entropy Functionals via Group Theory

Group Entropies: From Phase Space Geometry to Entropy Functionals via Group Theory

Group entropies: from phase space geometry to entropy functionals via group theory

Some relations for universal Bernoulli polynomials

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