Commutative formal groups of dimension one

AbstractA method for obtaining the exact solution for the total variation denoising problem of piecewise constant images in dimension one is presented. The validity of the algorithm relies on some results concerning the behavior of the solution when the parameter λ in front of the fidelity term varies. Albeit some of them are well-known in the community, here they are proved with simple techniques based on qualitative geometrical properties of the solutions.

Download Full-text

Nonlinear Conditions for Ultradifferentiability

Journal of Geometric Analysis ◽

10.1007/s12220-021-00718-w ◽

2021 ◽

Author(s):

David Nicolas Nenning ◽

Armin Rainer ◽

Gerhard Schindl

Keyword(s):

General Validity ◽

Dimension One ◽

Analogous Theorem

AbstractA remarkable theorem of Joris states that a function f is $$C^\infty $$ C ∞ if two relatively prime powers of f are $$C^\infty $$ C ∞ . Recently, Thilliez showed that an analogous theorem holds in Denjoy–Carleman classes of Roumieu type. We prove that a division property, equivalent to Joris’s result, is valid in a wide variety of ultradifferentiable classes. Generally speaking, it holds in all dimensions for non-quasianalytic classes. In the quasianalytic case we have general validity in dimension one, but we also get validity in all dimensions for certain quasianalytic classes.

Download Full-text

Elementary abelian Hopf Galois structures and polynomial formal groups

Journal of Algebra ◽

10.1016/j.jalgebra.2004.07.009 ◽

2005 ◽

Vol 283 (1) ◽

pp. 292-316 ◽

Cited By ~ 6

Author(s):

Lindsay N. Childs

Keyword(s):

Formal Groups

Download Full-text

General Theory of Saturation and of Saturated Local Rings I: Saturation of Complete Local Domains of Dimension One Having Arbitrary Coefficient Fields (of Characteristic Zero)

American Journal of Mathematics ◽

10.2307/2373462 ◽

1971 ◽

Vol 93 (3) ◽

pp. 573 ◽

Cited By ~ 9

Author(s):

Oscar Zariski

Keyword(s):

General Theory ◽

Characteristic Zero ◽

Local Rings ◽

Dimension One

Download Full-text

Formal groups and Z -entropies

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2016.0143 ◽

2016 ◽

Vol 472 (2195) ◽

pp. 20160143 ◽

Cited By ~ 13

Author(s):

Piergiulio Tempesta

Keyword(s):

Algebraic Topology ◽

Formal Group ◽

Point Of View ◽

Trace Form ◽

Formal Groups ◽

Form Class ◽

New Family ◽

Mathematical Properties ◽

Composition Process ◽

Group Law

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.

Download Full-text