A new entropy based on a group-theoretical structure

A multi-parametric version of the nonadditive entropy $S_{q}$ is introduced. This new entropic form, denoted by $S_{a,b,r}$, possesses many interesting statistical properties, and it reduces to the entropy $S_q$ for $b=0$, $a=r:=1-q$ (hence Boltzmann-Gibbs entropy $S_{BG}$ for $b=0$, $a=r \to 0$). The construction of the entropy $S_{a,b,r}$ is based on a general group-theoretical approach recently proposed by one of us \cite{Tempesta2}. Indeed, essentially all the properties of this new entropy are obtained as a consequence of the existence of a rational group law, which expresses the structure of $S_{a,b,r}$ with respect to the composition of statistically independent subsystems. Depending on the choice of the parameters, the entropy $S_{a,b,r}$ can be used to cover a wide range of physical situations, in which the measure of the accessible phase space increases say exponentially with the number of particles $N$ of the system, or even stabilizes, by increasing $N$, to a limiting value. This paves the way to the use of this entropy in contexts where a system "freezes" some or many of its degrees of freedom by increasing the number of its constituting particles or subsystems.Comment: 12 pages including 1 figur

Second order superintegrable systems in conformally flat spaces. IV. The classical 3D Stäckel transform and 3D classification theory

This article is one of a series that lays the groundwork for a structure and classification theory of second order superintegrable systems, both classical and quantum, in conformally flat spaces. In the first part of the article we study the Stäckel transform (or coupling constant metamorphosis) as an invertible mapping between classical superintegrable systems on different three-dimensional spaces. We show first that all superintegrable systems with nondegenerate potentials are multiseparable and then that each such system on any conformally flat space is Stäckel equivalent to a system on a constant curvature space. In the second part of the article we classify all the superintegrable systems that admit separation in generic coordinates. We find that there are eight families of these systems

Families of classical subgroup separable superintegrable systems

We describe a method for determining a complete set of integrals for a classical Hamiltonian that separates in orthogonal subgroup coordinates. As examples, we use it to determine complete sets of integrals, polynomial in the momenta, for some families of generalized oscillator and Kepler-Coulomb systems, hence demonstrating their superintegrability. The latter generalizes recent results of Verrier and Evans, and Rodriguez, Tempesta and Winternitz. Another example is given of a superintegrable system on a non-conformally flat space.Comment: 9 page

Morphological and molecular identification of a strain of the unicellular green alga Dunaliella sp. isolated from Tarquinia Salterns

1 - Algae of the genus Dunaliella are among the most studied micro-algae. They are used for the production of feed, for nutritional reinforcement as a vitamin A precursor and for pharmaceuticals and fine chemicals. 2 - The current taxonomy of the genus is based on morphological and physiological attributes including the ability of some species to grow over wide salinity ranges and at extreme salinities, as well as the accumulation of high levels of â-carotene. The taxonomic status of the genus Dunaliella involves some uncertainty, moreover it is very difficult to compare results from different authors, owing to uncertainty on names and species. 3 - In this work, we compare morphological and molecular analysis to characterize a strain of Dunaliella isolated from Tarquinia salt ponds. Samples of natural populations of the unicellular green alga, were collected at various times during the study period to detail the vegetative motile cells and the different stages of its life cycle microscopically. The ITS1 and ITS2 regions were used for the molecular identification analysis. Conserved oligonucleotides of 18S rDNAs (MA3) and speciesspecific primers (DSs), designed from variable sequences, were used to corroborate the identification. 4 - Blast results indicated that our sequences matched at the 100% level with Dunaliella salina Teod reported in Gen Bank. Consequently, based on comparative cell morphology and molecular analysis, the new Dunaliella isolate from Tarquinia salt ponds was classified as D. salina

Integrable and superintegrable systems with spin

A system of two particles with spin s=0 and s=1/2 respectively, moving in a plane is considered. It is shown that such a system with a nontrivial spin-orbit interaction can allow an 8 dimensional Lie algebra of first-order integrals of motion. The Pauli equation is solved in this superintegrable case and reduced to a system of ordinary differential equations when only one first-order integral exists.Comment: 12 page

Exact Solvability of Superintegrable Systems

It is shown that all four superintegrable quantum systems on the Euclidean plane possess the same underlying hidden algebra $sl(3)$. The gauge-rotated Hamiltonians, as well as their integrals of motion, once rewritten in appropriate coordinates, preserve a flag of polynomials. This flag corresponds to highest-weight finite-dimensional representations of the $sl(3)$-algebra, realized by first order differential operators.Comment: 14 pages, AMS LaTe