75 research outputs found
Galois differential algebras and categorical discretization of dynamical systems
A categorical theory for the discretization of a large class of dynamical
systems with variable coefficients is proposed. It is based on the existence of
covariant functors between the Rota category of Galois differential algebras
and suitable categories of abstract dynamical systems. The integrable maps
obtained share with their continuous counterparts a large class of solutions
and, in the linear case, the Picard-Vessiot group.Comment: 19 pages (examples added
The Lazard formal group, universal congruences and special values of zeta functions
A connection between the theory of formal groups and arithmetic number theory
is established. In particular, it is shown how to construct general
Almkvist--Meurman--type congruences for the universal Bernoulli polynomials
that are related with the Lazard universal formal group
\cite{Tempesta1}-\cite{Tempesta3}. Their role in the theory of --genera for
multiplicative sequences is illustrated. As an application, sequences of
integer numbers are constructed. New congruences are also obtained, useful to
compute special values of a new class of Riemann--Hurwitz--type zeta functions.Comment: 16 pages in Transactions of the American Mathematical Society, 201
Beyond the Shannon-Khinchin Formulation: The Composability Axiom and the Universal Group Entropy
The notion of entropy is ubiquitous both in natural and social sciences. In
the last two decades, a considerable effort has been devoted to the study of
new entropic forms, which generalize the standard Boltzmann-Gibbs (BG) entropy
and are widely applicable in thermodynamics, quantum mechanics and information
theory. In [23], by extending previous ideas of Shannon [38], [39], Khinchin
proposed an axiomatic definition of the BG entropy, based on four requirements,
nowadays known as the Shannon-Khinchin (SK) axioms.
The purpose of this paper is twofold. First, we show that there exists an
intrinsic group-theoretical structure behind the notion of entropy. It comes
from the requirement of composability of an entropy with respect to the union
of two statistically independent subsystems, that we propose in an axiomatic
formulation. Second, we show that there exists a simple universal class of
admissible entropies. This class contains many well known examples of entropies
and infinitely many new ones, a priori multi-parametric. Due to its specific
relation with the universal formal group, the new family of entropies
introduced in this work will be called the universal-group entropy. A new
example of multi-parametric entropy is explicitly constructed.Comment: Extended version; 25 page
Formal Groups and -Entropies
We shall prove that the celebrated R\'enyi entropy is the first example of a
new family of infinitely many multi-parametric entropies. We shall call them
the -entropies. Each of them, under suitable hypotheses, generalizes the
celebrated entropies of Boltzmann and R\'enyi.
A crucial aspect is that every -entropy is composable [1]. 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 -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.Comment: 20 pages, no figure
A New family of higher-order Generalized Haantjes Tensors, Nilpotency and Integrability
We propose a new infinite class of generalized binary tensor fields, whose
first representative of is the known Fr\"olicher--Nijenhuis bracket. This new
family of tensors reduces to the generalized Nijenhuis torsions of level
recently introduced independently in \cite{KS2017} and \cite{TT2017} and
possesses many interesting algebro-geometric properties.
We prove that the vanishing of the generalized Nijenhuis torsion of level
of a nilpotent operator field over a manifold of dimension is
necessary for the existence of a local chart where the operator field takes a
an upper triangular form. Besides, the vanishing of a generalized torsion of
level provides us with a sufficient condition for the integrability of the
eigen-distributions of an operator field over an -dimensional manifold. This
condition does not require the knowledge of the spectrum and of the
eigen-distributions of the operator field. The latter result generalizes the
celebrated Haantjes theorem.Comment: 25 page
Haantjes Algebras of Classical Integrable Systems
A tensorial approach to the theory of classical Hamiltonian integrable
systems is proposed, based on the geometry of Haantjes tensors. We introduce
the class of symplectic-Haantjes manifolds (or manifolds),
as the natural setting where the notion of integrability can be formulated. We
prove that the existence of suitable Haantjes algebras of (1,1) tensor fields
with vanishing Haantjes torsion is a necessary and sufficient condition for a
Hamiltonian system to be integrable in the Liouville-Arnold sense. We also show
that new integrable models arise from the Haantjes geometry. Finally, we
present an application of our approach to the study of the Post-Winternitz
system and of a stationary flow of the KdV hierarchy.Comment: 31 page
Groups, Information Theory and Einstein's Likelihood Principle
We propose a unifying picture where the notion of generalized entropy is
related to information theory by means of a group-theoretical approach. The
group structure comes from the requirement that an entropy be well defined with
respect to the composition of independent systems, in the context of a recently
proposed generalization of the Shannon-Khinchin axioms. We associate to each
member of a large class of entropies a generalized information measure,
satisfying the additivity property on a set of independent systems as a
consequence of the underlying group law. At the same time, we also show that
Einstein's likelihood function naturally emerges as a byproduct of our
informational interpretation of (generally nonadditive) entropies. These
results confirm the adequacy of composable entropies both in physical and
social science contexts.Comment: 5 page
Multivariate Group Entropies, Super-exponentially Growing Complex Systems and Functional Equations
We define the class of multivariate group entropies as a novel set of
information - theoretical measures, which extends significantly the family of
group entropies. We propose new examples related to the "super-exponential"
universality class of complex systems; in particular, we introduce a general
entropy, representing a suitable information measure for this class. We also
show that the group-theoretical structure associated with our multivariate
entropies can be used to define a large family of exactly solvable discrete
dynamical models. The natural mathematical framework allowing us to formulate
this correspondence is offered by the theory of formal groups and rings.Comment: 16 page
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