A generalization of the classical Lipschitz summation formula is proposed. It involves new polylogarithmic rational functions constructed via the Fourier expansion of certain sequences of Bernoulli--type polynomials. Related families of one--dimensional hyperfunctions are also constructed.

Stefano Marmi Piergiulio Tempesta

In this work, we introduce the notion of polarization of generalized Nijenhuis torsions and establish several algebraic identities. We prove that these polarizations are relevant in the characterization of Haantjes $C^{\infty}$(M)-modules of operator fields.

Piergiulio Tempesta Giorgio Tondo

In this paper, we develop a wide class Mirror Descent (MD) algorithms, which play a key role in machine learning. For this purpose we formulated the constrained optimization problem, in which we exploits the Bregman divergence with the Tempesta multi-parametric deformation logarithm as a link function. This link function called also mirror function defines the mapping between the primal and dual spaces and is associated with a very-wide (in fact, theoretically infinite) class of generalized trace-form entropies. In order to derive novel MD updates, we estimate generalized exponential function, which closely approximates the inverse of the multi-parametric Tempesta generalized logarithm. The shape and properties of the Tempesta logarithm and its inverse-deformed exponential functions can be tuned by several hyperparameters. By learning these hyperparameters, we can adapt to distribution or geometry of training data, and we can adjust them to achieve desired properties of MD algorithms. The concept of applying multi-parametric logarithms allow us to generate a new wide and flexible family of MD and mirror-less MD updates.

Andrzej Cichocki

We study a pair of canonoid (fouled) Hamiltonians of the harmonic oscillator which provide, at the classical level, the same equation of motion as the conventional Hamiltonian. These Hamiltonians, say $K_{1}$ and $K_{2}$, result to be explicitly time-dependent and can be expressed as a formal rotation of two cubic polynomial functions, $H_{1}$ and $H_{2}$, of the canonical variables (q,p). We investigate the role of these fouled Hamiltonians at the quantum level. Adopting a canonical quantization procedure, we construct some quantum models and analyze the related eigenvalue equations. One of these models is described by a Hamiltonian admitting infinite self-adjoint extensions, each of them has a discrete spectrum on the real line. A self-adjoint extension is fixed by choosing the spectral parameter $\epsilon$ of the associated eigenvalue equation equal to zero. The spectral problem is discussed in the context of three different representations. For $\epsilon =0$, the eigenvalue equation is exactly solved in all these representations, in which square-integrable solutions are explicity found. A set of constants of motion corresponding to these quantum models is also obtained. Furthermore, the algebraic structure underlying the quantum models is explored. This turns out to be a nonlinear (quadratic) algebra, which could be applied for the determination of approximate solutions to the eigenvalue equations.

P. Tempesta E. Alfinito R. A. Leo G. Soliani

The classical Smorodinsky-Winternitz systems on the ND sphere, Euclidean and hyperbolic spaces S^N, E^N and H^N are simultaneously approached starting from the Lie algebras so_k(N+1), which include a parametric dependence on the curvature k. General expressions for the Hamiltonian and its integrals of motion are given in terms of intrinsic geodesic coordinate systems. Each Lie algebra generator gives rise to an integral of motion, so that a set of N(N+1)/2 integrals is obtained. Furthermore, 2N-1 functionally independent ones are identified which, in turn, shows that the well known maximal superintegrability of the Smorodinsky-Winternitz system on E^N is preserved when curvature arises. On both S^N and H^N, the resulting system can be interpreted as a superposition of an "actual" oscillator and N "ideal" oscillators (for the sphere, these are alike the actual ones), which can also be understood as N "centrifugal terms"; this is the form seen in the Euclidean limiting case.

Francisco J. Herranz Angel Ballesteros Mariano Santander Teresa Sanz-Gil

We present an algebraic formulation of the notion of integrability of dynamical systems, based on a nilpotency property of its flow: it can be explicitly described as a polynomial on its evolution parameter. Such a property is established in a purely geometric--algebraic language, in terms both of the algebra of all higher-order constants of the motion (named the nilpotent algebra of the dynamics), and of a maximal Abelian algebra of symmetries (called a Cartan subalgebra of the dynamics). It is shown that this notion of integrability amounts to the annihilator of the nilpotent algebra being contained in a Cartan subalgebra of the dynamics. Systems exhibiting this property will be said to be nilpotent integrable. Our notion of nilpotent integrability offers a new insight into the intrinsic dynamical properties of a system, which is independent of any auxiliary geometric structure defined on its phase space. At the same time, it extends in a natural way the classical concept of complete integrability for Hamiltonian systems. An algebraic reduction procedure valid for nilpotent integrable systems, generalizing the well-known reduction procedures for symplectic and/or Poisson systems on appropriate quotient spaces, is also discussed. In particular, it is shown that a large class of nilpotent integrable systems can be obtained by reduction of higher-order free systems. The case of the third-order free system is analyzed and a nontrivial set of third-order Calogero-Moser-like nilpotent integrable equations is obtained.

A. Ibort G. Marmo M. A. Rodriguez P. Tempesta

This is a paper in the intersection of time series analysis and complexity theory that presents new results on permutation complexity in general and permutation entropy in particular. In this context, permutation complexity refers to the characterization of time series by means of ordinal patterns (permutations), entropic measures, decay rates of missing ordinal patterns, and more. Since the inception of this \textquotedblleft ordinal\textquotedblright\ methodology, its practical application to any type of scalar time series and real-valued processes have proven to be simple and useful. However, the theoretical aspects have remained limited to noiseless deterministic series and dynamical systems, the main obstacle being the super-exponential growth of visible permutations with length when randomness (also in form of observational noise) is present in the data. To overcome this difficulty, we take a new approach through complexity classes, which are precisely defined by the growth of visible permutations with length, regardless of the deterministic or noisy nature of the data. We consider three major classes: exponential, sub-factorial and factorial. The next step is to adapt the concept of Z-entropy to each of those classes, which we call permutation entropy because it coincides with the conventional permutation entropy on the exponential class. Z-entropies are a family of group entropies, each of them extensive on a given complexity class. The result is a unified approach to the ordinal analysis of deterministic and random processes, from dynamical systems to white noise, with new concepts and tools. Numerical simulations show that permutation entropy discriminates time series from all complexity classes.

J. M. Amigó R. Dale P. Tempesta

A construction of new sequences of generalized Bernoulli polynomials of first and second kind is proposed. These sequences share with the classical Bernoulli polynomials many algebraic and number--theoretical properties. A new class of Euler--type polynomials is also presented.

Piergiulio Tempesta

We show that the notion of generalized Lenard chains naturally allows formulation of the theory of multi-separable and superintegrable systems in the context of bi-Hamiltonian geometry. We prove that the existence of generalized Lenard chains generated by a Hamiltonian function defined on a four-dimensional \omega N manifold guarantees the separation of variables. As an application, we construct such chains for the H\'enon-Heiles systems and for the classical Smorodinsky-Winternitz systems. New bi-Hamiltonian structures for the Kepler potential are found.

Piergiulio Tempesta Giorgio Tondo

We construct a new class of directed and bipartite random graphs whose topology is governed by the analytic properties of L-functions. The bipartite L-graphs and the multiplicative zeta graphs are relevant examples of the proposed construction. Phase transitions and percolation thresholds for our models are determined.

Piergiulio Tempesta