Formal schemes and formal groups

We shall prove that the celebrated Rényi entropy is the first example of a new family of infinitely many multi-parametric entropies. We shall call them the Z-entropies . Each of them, under suitable hypotheses, generalizes the celebrated entropies of Boltzmann and Rényi. A crucial aspect is that every Z -entropy is composable (Tempesta 2016 Ann. Phys. 365 , 180–197. ( doi:10.1016/j.aop.2015.08.013 )). This property means that the entropy of a system which is composed of two or more independent systems depends, in all the associated probability space, on the choice of the two systems only. Further properties are also required to describe the composition process in terms of a group law. The composability axiom, introduced as a generalization of the fourth Shannon–Khinchin axiom (postulating additivity), is a highly non-trivial requirement. Indeed, in the trace-form class, the Boltzmann entropy and Tsallis entropy are the only known composable cases. However, in the non-trace form class, the Z -entropies arise as new entropic functions possessing the mathematical properties necessary for information-theoretical applications, in both classical and quantum contexts. From a mathematical point of view, composability is intimately related to formal group theory of algebraic topology. The underlying group-theoretical structure determines crucially the statistical properties of the corresponding entropies.

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p-adic Measures and Formal Groups

Journal of Number Theory ◽

10.1006/jnth.1993.1058 ◽

1993 ◽

Vol 44 (3) ◽

pp. 340-351 ◽

Cited By ~ 1

Author(s):

N. Childress

Keyword(s):

Formal Groups

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Weak Formal Schemes

Nagoya Mathematical Journal ◽

10.1017/s0027763000014641 ◽

1972 ◽

Vol 45 ◽

pp. 1-38 ◽

Cited By ~ 16

Author(s):

David Meredith

Keyword(s):

Local Ring ◽

Maximal Ideal ◽

Noetherian Local Ring ◽

Formal Schemes

Throughout this paper, (R, m) denotes a (noetherian) local ring R with maximal ideal m.In [5], Monsky and Washnitzer define weakly complete R-algebras with respect to m. In brief, an R-algebra A† is weakly complete if

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The Indispensable Role of Systematic Philosophical Reflection in Scholarship

Phronimon ◽

10.25159/2413-3086/2847 ◽

2018 ◽

Vol 18 ◽

pp. 95-114

Author(s):

Daniel Francois Strauss

Keyword(s):

States Of Affairs ◽

Philosophical View ◽

Philosophical Reflection ◽

Formal Schemes ◽

Constant Change ◽

Theoretical Thinking ◽

Philosophical Orientations ◽

The Relationship ◽

Systematic Thinking

Sometimes systematic theoretical thinking is identified with abstract (formal) schemes. This opposition is also found in Malan and Goosenâ€™s dismissal of Dooyeweerdian reformational thinking. This article aims at making a contribution to this issue by analysing the indispensable role of systematic philosophical reflection within the world of scholarship. One way in which systematic thinking could be justified is to highlight the need for consistency and the role of logical principles in achieving it. It is argued that, since we are living in the same world, all philosophical orientations have to account for shared states of affairs. At this point attention is given to the question whether or not these â€œstates of affairsâ€ are â€œstatic or dynamic.â€ An alternative for the distinction between static and dynamic is proposed by alternatively considering the relationship between constancy and change. Von WeizsÃ¤cker articulates the problem aptly by pointing out that although our experience exhibits constant change, something exists that remains unchanged through all these changes. In conclusion it is pointed out that scholars have only two options: either they give an account of the philosophical presuppositions and systematic distinctions with which they workâ€”in which case they have a philosophical view of reality, or implicitly (and uncritically) they proceed from one or another philosophical view of realityâ€”in which case they are the victims of a philosophical view. The primary aim of this article is, therefore, to highlight the indispensability of systematic thought by referring to some of the main distinctions included in such a system of thought.

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