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 ZZ-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 $Z$-entropies. Each of them, under suitable hypotheses, generalizes the celebrated entropies of Boltzmann and R\'enyi. A crucial aspect is that every $Z$-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 $Z$-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.Comment: 20 pages, no figure

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 $L$--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

Testing the difference between experts\u2019 and lay people\u2019s landscape preferences

The European Landscape Convention, ratified by 40 nations, has placed emphasis on the necessity that the value of the landscape is assessed by the population. However it is standard practice that a few experts decide which areas are of landscape interest and the transformations that are compatible with their conservation. To compare the landscape preferences of experts and lay people a study was done on the Po Delta Natural Park (Italy) using a psychophysical approach. In our case study the average scores of experts and lay people are not very different. However it was also ascertained that the experts evaluate the presence of some elements in a way that differs from lay people. As the responsibility for landscape policies is normally devolved to a few experts it would appear necessary that the preferences and opinions of lay people should always be carefully analysed

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 $m$ 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 $(n-1)$ of a nilpotent operator field $A$ over a manifold of dimension $n$ 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 $m$ provides us with a sufficient condition for the integrability of the eigen-distributions of an operator field over an $n$-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

Valuing the landscape benefits of rural policies actions in Veneto (Italy)

<p class="p1">This study addresses, with reference to the landscape, a precise request of the EU to quantify the benefits of public expenditure in agriculture. It analyses the implications on rural landscape of some measures of the Common Agricultural Policy at a regional level, taking the Rural Development Programme (RDP) 2007-2013 of the Veneto Region, in north-eastern Italy, as case study. A choice experiment (CE) is applied to value four measures of the RDP that directly affect the landscape characteristics.<br />The CE results point out that the landscape benefits of the measures in the Veneto RDP are higher than the subsidies paid to farmers for the provision of services that improve landscape quality. The CE results suggest the opportunity to rethink the distribution of the subsidies.</p

The Value of a Properly Maintained Hiking Trail Network and a Traditional Landscape for Mountain Recreation in the Dolomites

Alpine mountains represent one of the most important tourist destinations in the world, constituting approximately 3.1% of the global tourism market when considering the tourist flows coming from abroad. While there may be numerous factors that motivate tourists to choose rural areas, an important role is played by the opportunity to visit well-conserved landscapes and uncontaminated natural areas. The purpose of this study was to make a monetary valuation of the social benefits generated by the adoption of three measures of the Rural Development Plan (RDP) of Veneto (Italy) aimed specifically at enhancing the recreational usability of the mountain territory. In this regard, a discrete choice experiment (DCE) was applied for the economic valuation, and a qualitative survey was used to collect the opinion of respondents related to the measures to protect the meadows and mountain hiking trails. According to the DCE estimates, on average, the benefits due to the conservation of the existing meadows and pastures was equal to \u20ac851 per hectare, those due to the conservation and improvement of the trail network were \u20ac12,260 per km, and the benefits due to the recovery of the meadows and pastures of uncultivated and abandoned areas for naturalistic purposes amounted to \u20ac6,852 per hectare. Comparing the estimates obtained with the expenditure incurred by the RDP to finance the three actions considered in our DCE, it can be seen that the benefits are considerably higher than the costs, especially with regard to the conservation of paths and the recovery of abandoned areas for naturalistic purposes