Piergiulio Tempesta

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.

Group entropies, correlation laws and zeta functions

The notion of group entropy is proposed. It enables the unification and generaliztion of many different definitions of entropy known in the literature, such as those of Boltzmann-Gibbs, Tsallis, Abe, and Kaniadakis. Other entropic functionals are introduced, related to nontrivial correlation laws characterizing universality classes of systems out of equilibrium when the dynamics is weakly chaotic. The associated thermostatistics are discussed. The mathematical structure underlying our construction is that of formal group theory, which provides the general structure of the correlations among particles and dictates the associated entropic functionals. As an example of application, the role of group entropies in information theory is illustrated and generalizations of the Kullback-Leibler divergence are proposed. A new connection between statistical mechanics and zeta functions is established. In particular, Tsallis entropy is related to the classical Riemann zeta function.

Superintegrable systems in quantum mechanics and classical Lie theory

The relation is established between some concepts of quantum mechanics and those of soliton theory. In particular, superintegrable systems in two-dimensional quantum mechanics are shown to be invariant under generalized Lie symmetries and to allow recursion operators.

Reduction of superintegrable systems: the anisotropic harmonic oscillator.

We introduce a 2N-parametric family of maximally superintegrable systems in N dimensions, obtained as a reduction of an anisotropic harmonic oscillator in a 2N-dimensional configuration space. These systems possess closed bounded orbits and integrals of motion which are polynomial in the momenta. They generalize known examples of superintegrable models in the Euclidean plane.

Weak transversality and partially invariant solutions

New exact solutions are obtained for several nonlinear physical equations, namely the Navier–Stokes and Euler systems, an isentropic compressible fluid system and a vector nonlinear Schrodinger equation. The solution method makes use of the symmetry group of the system in situations when the standard Lie method of symmetry reduction is not applicable.

Finding frequent items in parallel

We present a deterministic parallel algorithm for the k‐majority problem, that can be used to find in parallel frequent items, i.e. those whose multiplicity is greater than a given threshold, and is therefore useful to process iceberg queries and in many other different contexts of applied mathematics and information theory. The algorithm can be used both in the online (stream) context and in the offline setting, the difference being that in the former case we are restricted to a single scan of the input elements, so that verifying the frequent items that have been determined is not allowed (e.g. network traffic streams passing through internet routers), while in the latter a parallel scan of the input can be used to determine the actual k‐majority elements. To the best of our knowledge, this is the first parallel algorithm solving the proposed problem. Copyright

Symmetry reduction and superintegrable Hamiltonian systems

We construct complete sets of of invariant quantities that are integrals of motion for two Hamiltonian systems obtained through a reduction procedure, thus proving that these systems are maximally superintegrable. We also discuss the reduction method used in this article and its possible generalization to other maximally superintegrable systems.

Thermostatistics in the neighbourhood of the π-mode solution for the Fermi–Pasta–Ulam β system: from weak to strong chaos

We consider a π-mode solution of the Fermi–Pasta–Ulam β system. By perturbing it, we study the system as a function of the energy density from a regime where the solution is stable to a regime where it is unstable, first weakly and then strongly chaotic. We introduce, as an indicator of stochasticity, the ratio ρ (when it is defined) between the second and the first moment of a given probability distribution. We will show numerically that the transition between weak and strong chaos can be interpreted as the symmetry breaking of a set of suitable dynamical variables. Moreover, we show that in the region of weak chaos there is numerical evidence that the thermostatistic is governed by the Tsallis distribution.

A parallel space saving algorithm for frequent items and the Hurwitz zeta distribution

We design a parallel algorithm for frequent items based on the sequential Space Saving algorithm.We show that the algorithm is cost-optimal for k = O ( 1 ) and therefore extremely fast and useful in a wide range of applications.We experimentally validate the algorithm on synthetic data distributed using a Hurwitz zeta function and a Zipf one, and also on real datasets, confirming the theoretical cost-optimality and that the error committed is very low and close to zero.We compare the performances and the error committed by our algorithm against a parallel algorithm recently proposed by Agarwal et?al. for merging datasets derived by the Space Saving or Frequent algorithms. We present a message-passing based parallel version of the Space Saving algorithm designed to solve the k-majority problem. The algorithm determines in parallel frequent items, i.e., those whose frequency is greater than a given threshold, and is therefore useful for iceberg queries and many other different contexts. We apply our algorithm to the detection of frequent items in both real and synthetic datasets whose probability distribution functions are a Hurwitz and a Zipf distribution respectively. Also, we compare its parallel performances and accuracy against a parallel algorithm recently proposed for merging summaries derived by the Space Saving or Frequent algorithms.

Formal groups and Z-entropies

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.