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Piergiulio Tempesta

Publications and source records attributed to Piergiulio Tempesta.

At least 19 recordsLinked to original sources
arXiv2026

Group Entropies and Mirror Duality: A Class of Flexible Mirror Descent Updates for Machine Learning

We introduce a comprehensive theoretical and algorithmic framework that bridges formal group theory and group entropies with modern machine learning, paving the way for an infinite, flexible family of Mirror Descent (MD) optimization algorithms. Our approach exploits the rich structure of group entropies, which are generalized entropic functionals governed by group composition laws, encompassing and significantly extending all trace-form entropies such as the Shannon, Tsallis, and Kaniadakis families. By leveraging group-theoretical mirror maps (or link functions) in MD, expressed via multi-parametric generalized logarithms and their inverses (group exponentials), we achieve highly flexible and adaptable MD updates that can be tailored to diverse data geometries and statistical distributions. To this end, we introduce the notion of \textit{mirror duality}, which allows us to seamlessly switch or interchange group-theoretical link functions with their inverses, subject to specific learning rate constraints. By tuning or learning the hyperparameters of the group logarithms enables us to adapt the model to the statistical properties of the training distribution, while simultaneously ensuring desirable convergence characteristics via fine-tuning. This generality not only provides greater flexibility and improved convergence properties, but also opens new perspectives for applications in machine learning and deep learning by expanding the design of regularizers and natural gradient algorithms. We extensively evaluate the validity, robustness, and performance of the proposed updates on large-scale, simplex-constrained quadratic programming problems.

cs.LG↗
arXiv2026

Universality classes, Thermodynamics of Group Entropies, and Black Holes

Conventional Boltzmann--Gibbs statistical mechanics successfully describes systems with weak to moderate correlations, where the number of accessible configurations $W(N)$ grows exponentially with the number of degrees of freedom~$N$. However, this framework breaks down for systems with strong correlations or long-range interactions, for which the configuration space exhibits non-exponential growth. While numerous generalized entropies have been proposed to address this limitation, a coherent link to classical thermodynamic laws has remained elusive. Here, we propose group entropies as a unifying framework, defining universality classes of entropies through the asymptotic scaling of $W(N)$, each yielding an extensive entropy. We show that this approach provides the basis for a consistent thermodynamic formulation beyond the Boltzmann--Gibbs paradigm. In particular, by expressing these entropies in terms of thermodynamic state variables and taking the thermodynamic limit, we demonstrate their consistency with classical thermodynamics, in close analogy to the emergence of the Clausius entropy from the Boltzmann--Gibbs formalism. Focusing on the zeroth thermodynamic law, we identify the empirical temperature and, by using Carath\'{e}odory's formulation of the second law, we derive the associated absolute temperature. As an application of the thermodynamic framework obtained, we analyze black-hole thermodynamics using the group entropy class corresponding to stretched-exponential behavior of $W(N)$. In particular, we show that a hallmark property of black holes -- their negative specific heat -- emerges naturally within this framework while the entropy remains extensive. This result holds for the stretched-exponential entropies associated with both the Bekenstein--Hawking and Barrow entropy scalings.

cond-mat.stat-mech↗
arXiv2025

Laurent Sequences, Extended Rota Algebras and Categorical Discretization of Dynamical Systems

We introduce a novel integrability-preserving discretization for a broad class of differential equations with variable coefficients, encompassing both linear and nonlinear cases. The construction is achieved via a categorical approach that enables a unified treatment of continuous and discrete dynamical systems. Our theoretical framework is grounded on a generalization of G. C. Rota's finite operator calculus, which enables us to extend the theory of basic sequence of polynomials to the setting of Laurent polynomials. Accordingly, we introduce the notion of an \textit{extended Rota algebra}, defined as a Galois differential algebra in which all difference operators act as derivations on the space of Laurent power series with respect to a suitably defined functional product. The core of our theory relies on the existence of covariant functors between the newly proposed Rota category of Galois differential algebras and suitable categories of abstract dynamical systems. In this setting, under certain regularity assumptions, a differential equation and its discrete analogues are naturally interpreted as objects of the same category. This perspective enables the construction of a vast class of integrable maps that share with their continuous analogues a wide set of exact solutions, \textit{regular} or \textit{singular} and, in the linear case, the Picard-Vessiot group.

math-ph↗
arXiv2025

St\"ackel and Eisenhart lifts, Haantjes geometry and Gravitation

We study lifts of integrable systems by means of generalized St\"ackel geometry. To this end, we present the notion of St\"ackel lift as a unified setting for the construction of new classes of integrable Hamiltonian systems of physical interest. The St\"ackel lift extends the geometric framework underlying both the Riemannian and the Lorentzian-type classical Eisenhart lifts. Moreover, we prove that Hamiltonian systems constructed through momentum-dependent St\"ackel matrices are naturally endowed with a non-trivial symplectic-Haantjes structure. We further illustrate applications to magnetic systems separable in cylindrical coordinates; we describe them within the St\"ackel framework by means of modified St\"ackel bases. Finally, we show that explicitly momentum-dependent lifting matrices generate Platonic-wave geometries with potential applications in modified gravity theories, or momentum-dependent metrics of Hamilton and Finsler geometries.

math-ph↗
arXiv2025

Jacobi-Haantjes manifolds, integrability and dissipative mechanical systems

The notion of Jacobi-Haantjes manifold, consisting of a Jacobi manifold endowed with an algebra of extended Haantjes operator fields, is proposed as a natural geometric framework which allows us to define the notion of integrability of both conservative and dissipative Hamiltonian systems, in a unified way. As a reduction, contact-Haantjes manifolds are defined. We prove that the integrability of a contact Hamiltonian system is equivalent to the existence of a suitable Abelian extended Haantjes algebra associated with the system. This result allows us to define a large class of new, completely integrable contact Hamiltonian systems from a given extended Haantjes algebra. Moreover, we propose a theory of separation of variables for dissipative systems. This result is achieved by lifting a dissipative system into a higher-dimensional manifold, obtained as the symplectization of the Jacobi-Haantjes structure associated with the system. This new manifold naturally acquires the structure of a symplectic-Haantjes manifold. We prove that the Darboux-Haantjes coordinates which separate the Hamilton-Jacobi equation of the higher-dimensional symplectic-Haantjes manifold are in fact separation variables for the Hamilton equations associated with the original dissipative system.

math-ph↗
arXiv2025

Group structure as a foundation for entropies

Entropy can signify different things: For instance, heat transfer in thermodynamics or a measure of information in data analysis. Many entropies have been introduced and it can be difficult to ascertain their different importance and merits. Here we consider entropy in an abstract sense, as a functional on a probability space and we review how being able to handle the trivial case of non-interacting systems together with the subtle requirement of extensivity allows a systematic classification of the functional form.

math-ph↗
arXiv2025

A new discretization of the Euler equation via the finite operator theory

We propose a novel discretization procedure for the classical Euler equation based on the theory of Galois differential algebras and the finite operator calculus developed by G.C. Rota and collaborators. This procedure allows us to define algorithmically a new discrete model that inherits from the continuous Euler equation a class of exact solutions.

math-ph↗
arXiv2024

A Frobenius-type theory for discrete systems

We develop an approach analogous to classical Frobenius theory for the analysis of singularities of ODEs in the case of discrete dynamical systems. Our methodology is based on the Roman-Rota theory of finite operators and relies crucially on the idea of preserving the Leibniz rule on a lattice of points by means of the notion of Rota algebras. The relevant cases of the Bessel, Hermite and Airy equations are discussed.

math-ph↗
arXiv2024

Hamiltonian integrable systems in a magnetic field and Symplectic-Haantjes geometry

We investigate the geometry of classical Hamiltonian systems immersed in a magnetic field in three-dimensional Riemannian configuration spaces. We prove that these systems admit non-trivial symplectic-Haantjes manifolds, which are symplectic manifolds endowed with an algebra of Haantjes (1,1)-tensors. These geometric structures allow us to determine separation variables for known systems algorithmically; besides, the underlying St\"ackel geometry is used to construct new families of integrable Hamiltonian models immersed in a magnetic field.

math-ph↗
arXiv2024

Permutation group entropy: a new route to complexity for real-valued processes

This is a review of group entropy and its application to permutation complexity. Specifically we revisit a new approach to the notion of complexity in time serie analysis, based on both permutation entropy and group entropy. As a result, the permutation entropy rate can be extended from deterministic dynamics to random processes. More generally, our approach provides a unified framework to discuss chaotic and random behaviours.

math-ph↗
arXiv2023

Partial separability and symplectic-Haantjes manifolds

A theory of partial separability for classical Hamiltonian systems is proposed in the context of Haantjes geometry. As a general result, we show that the knowledge of a non-semisimple symplectic-Haantjes manifold for a given Hamiltonian system is sufficient to construct sets of coordinates (called Darboux-Haantjes coordinates) which allow both the partial separability of the associated Hamilton-Jacobi equations and the block-diagonalization of the operators of the corresponding Haantjes algebra. We also introduce a novel class of Hamiltonian systems, characterized by the existence of a generalized St\"ackel matrix, which by construction are partially separable. They widely generalize the known families of partially separable Hamiltonian systems. Our systems can be described in terms of semisimple but non-maximal-rank symplectic-Haantjes manifolds.

math-ph↗
arXiv2023

Polarization of generalized Nijenhuis torsions

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.

math-ph↗
arXiv2022

On higher-dimensional superintegrable systems: A new family of classical and quantum Hamiltonian models

We introduce a family of $n$-dimensional Hamiltonian systems which, contain, as special reductions, several superintegrable systems as the Tremblay-Turbiner-Winternitz system, a generalized Kepler potential and the anisotropic harmonic oscillator with Rosochatius terms. We conjecture that there exist special values in the space of parameters, apart from those leading to known cases, for which this new Hamiltonian family is superintegrable.

math-ph↗
arXiv2022

Generalized Nijenhuis Torsions and block-diagonalization of operator fields

The theory of generalized Nijenhuis torsions, which extends the classical notions due to Nijenhuis and Haantjes, offers new tools for the study of normal forms of operator fields. We propose a general result ensuring that, given a family of commuting operator fields whose generalized Nijenhuis torsion of level $l$ vanishes, there exists a local chart where all operators can be simultaneously block-diagonalized. We also introduce the notion of generalized Haantjes algebra, consisting of operators with a vanishing higher-level torsion, as a new algebraic structure naturally generalizing standard Haantjes algebras.

math-ph↗
arXiv2022

Classical Multiseparable Hamiltonian Systems, Superintegrability and Haantjes Geometry

We show that the theory of classical Hamiltonian systems admitting separating variables can be formulated in the context of ($ω, \mathscr{H}$) structures. They are symplectic manifolds endowed with a compatible Haantjes algebra $\mathscr{H}$, namely an algebra of (1,1)-tensor fields with vanishing Haantjes torsion. A special class of coordinates, called Darboux-Haantjes coordinates, will be constructed from the Haantjes algebras associated with a separable system. These coordinates enable the additive separation of variables of the corresponding Hamilton-Jacobi equation. We shall prove that a multiseparable system admits as many $ω\mathscr{H}$ structures as separation coordinate systems. In particular, we will show that a large class of multiseparable, superintegrable systems, including the Smorodinsky-Winternitz systems and some physically relevant systems with three degrees of freedom, possesses multiple Haantjes structures.

math-ph↗
arXiv2021

On Appell sequences of polynomials of Bernoulli and Euler type

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.

math.NT↗
arXiv2021

Bipartite and directed scale-free complex networks arising from zeta functions

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.

math-ph↗