arXiv subjects

Paolo Giordano

Publications and source records attributed to Paolo Giordano.

At least 19 recordsLinked to original sources

arXiv2026

Complex hyper-power series and generalized complex analytic functions

This paper studies the equivalence between generalized holomorphic functions (GHF) and complex analytic functions in the framework of Robinson-Colombeau generalized numbers. In every non-Archimedean ring, the use of ordinary series is severely restricted by the topological property that a series converges (in a topology of infinitesimal neighborhoods) if and only if its general term is infinitesimal. Consequently, classical Taylor series representations for generalized functions are limited to infinitesimal neighborhoods. To overcome this drawback, we introduce and develop the theory of hyperpower series, defined by summation over the set of hyperfinite natural numbers. We establish the foundational algebraic and topological properties of hyperpower series, including their radii of convergence and sets of convergence. Building on this, we define generalized complex analytic functions and extend several fundamental theorems of complex analysis to the GHF setting, specifically, providing generalizations of Goursat's theorem, Lioville's theorem, the identity theorem, and a Paley-Wiener type theorem.

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arXiv2026

Classical finite dimensional fixed point methods for generalized functions

We prove Banach, Newton-Raphson and Brouwer fixed point theorems in the framework of generalized smooth functions, a minimal extension of Colombeau's theory (and hence of classical distribution theory) which makes it possible to model nonlinear singular problems, while at the same time sharing a number of fundamental properties with ordinary smooth functions, such as the closure with respect to composition and several non trivial classical theorems of the calculus. The proved results allows one to deal with equations of the form F(x)=0, where F is a generalized smooth function, in particular, a Sobolev-Schwartz distribution. We consider examples with singularities that are not included in the classical version of these theorems.

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arXiv2025

Dirac delta as a generalized holomorphic function

The definition of a non-trivial space of generalized functions of a complex variable allowing to consider derivatives of continuous functions is a non-obvious task, e.g. because of Morera theorem, because distributional Cauchy-Riemann equations implies holomorphicity and of course because including Dirac delta seems incompatible with the identity theorem. Surprisingly , these results can be achieved if we consider a suitable non-Archimedean extension of the complex field, i.e. a ring where infinitesimal and infinite numbers return to be available. In this first paper, we set the definition of generalized holomorphic function and prove the extension of several classical theorems, such as Cauchy-Riemann equations, Goursat, Looman-Menchoff and Montel theorems, generalized differentiability implies smoothness, intrinsic embedding of compactly supported distributions, closure with respect to composition and hence non-linear operations on these generalized functions. The theory hence addresses several limitations of Colombeau theory of generalized holomorphic functions. The final aim of this series of papers is to prove the Cauchy-Kowalevski theorem including also distributional PDE or singular boundary conditions and nonlinear operations.

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arXiv2024

The Hahn-Banach theorem in spaces of nonlinear generalized functions

In this paper, we establish a suitable version of the Hahn-Banach theorem within the framework of Colombeau spaces, a class of spaces used to model generalized functions. Our approach addresses the case where maps are defined $\varepsilon$-wise, which simplifies the framework and makes the extension of linear functionals more manageable. As an application of our main result, we demonstrate the separation of convex sets in Colombeau spaces.

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arXiv2024

Interaction spaces: towards a universal mathematical theory of complex systems

We present the first steps of interaction spaces theory, a universal mathematical theory of complex systems which is able to embed cellular automata, agent based models, master equation based models, stochastic or deterministic, continuous or discrete dynamical systems, networked dynamical models, artificial neural networks and genetic algorithms in a single notion. Therefore, interaction spaces represent a common mathematical language that can be used to describe several complex systems modeling frameworks. This is the first step to start a mathematical theory of complex systems. Every notion is introduced both using an intuitive description by listing lots of examples, and using a modern mathematical language.

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arXiv2024

A mathematical definition of complex adaptive system as interaction space

We define a mathematical notion of complex adaptive system by following the original intuition of G.K. Zipf about the principle of least effort, an intuitive idea which is nowadays informally widespread in complex systems modeling. We call generalized evolution principle this mathematical notion of interaction spaces theory. Formalizing and generalizing Mandelbrot's ideas, we also prove that a large class of these systems satisfy a power law. We finally illustrate the notion of complex adaptive system with theorems describing a Von Thünen-like model. The latter can be easily generalized to other complex systems and describes the appearance of emergent patterns. Every notion is introduced both using an intuitive description with lots of examples, and using a modern mathematical language.

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arXiv2024

Universal properties of spaces of generalized functions

By means of several examples, we motivate that universal properties are the simplest way to solve a given mathematical problem, explaining in this way why they appear everywhere in mathematics. In particular, we present the co-universal property of Schwartz distributions, as the simplest way to have derivatives of continuous functions, Colombeau algebra as the simplest quotient algebra where representatives of zero are infinitesimal, and generalized smooth functions as the universal way to associate set-theoretical maps of non-Archimedean numbers defined by nets of smooth functions (e.g. regularizations of distributions) and having arbitrary derivatives. Each one of these properties yields a characterization up to isomorphisms of the corresponding space. The paper requires only the notions of category, functor, natural transformation and Schwartz's distributions, and introduces the notion of universal solution using a simple and non-abstract language.

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arXiv2024

Infinitesimal and infinite numbers in applied mathematics

The need to describe abrupt changes or response of nonlinear systems to impulsive stimuli is ubiquitous in applications. Also the informal use of infinitesimal and infinite quantities is still a method used to construct idealized but tractable models within the famous J. von Neumann reasonably wide area of applicability. We review the theory of generalized smooth functions as a candidate to address both these needs: a rigorous but simple language of infinitesimal and infinite quantities, and the possibility to deal with continuous and generalized function as if they were smooth maps: with pointwise values, free composition and hence nonlinear operations, all the classical theorems of calculus, a good integration theory, and new existence results for differential equations. We exemplify the applications of this theory through several models of singular dynamical systems: deduction of the heat and wave equations extended to generalized functions, a singular variable length pendulum wrapping on a parallelepiped, the oscillation of a pendulum damped by different media, a nonlinear stress-strain model of steel, singular Lagrangians as used in optics, and some examples from quantum mechanics.

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arXiv2022

Hyper-power series and generalized real analytic functions

This article is a natural continuation of the paper Tiwari, D., Giordano, P., Hyperseries in the non-Archimedean ring of Colombeau generalized numbers in this journal. We study one variable hyper-power series by analyzing the notion of radius of convergence and proving classical results such as algebraic operations, composition and reciprocal of hyper-power series. We then define and study one variable generalized real analytic functions, considering their derivation, integration, a suitable formulation of the identity theorem and the characterization by uniform upper bounds of derivatives on functionally compact sets. On the contrary with respect to the classical use of series in the theory of Colombeau real analytic functions, we can recover several classical examples in a non-infinitesimal set of convergence. The notion of generalized real analytic function reveals to be less rigid both with respect to the classical one and to Colombeau theory, e.g. including classical non-analytic smooth functions with flat points and several distributions, such as the Dirac delta. On the other hand, each Colombeau real analytic function is also a generalized real analytic function.

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arXiv2022

A Picard-Lindelöf theorem for smooth PDE

We prove that Picard-Lindelöf iterations for an arbitrary smooth normal Cauchy problem for PDE converge if we assume a suitable Weissinger-like sufficient condition. This condition includes both a large class of non-analytic PDE or initial conditions, and more classical real analytic functions. The proof is based on a Banach fixed point theorem for contractions with loss of derivatives. From the latter, we also prove an inverse function theorem for locally Lipschitz maps with loss of derivatives in arbitrary graded Fréchet spaces.

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arXiv2022

A Fourier transform for all generalized functions

Using the existence of infinite numbers $k$ in the non-Archimedean ring of Robinson-Colombeau, we define the hyperfinite Fourier transform (HFT) by considering integration extended to $[-k,k]^{n}$ instead of $(-\infty,\infty)^{n}$. In order to realize this idea, the space of generalized functions we consider is that of generalized smooth functions (GSF), an extension of classical distribution theory sharing many nonlinear properties with ordinary smooth functions, like the closure with respect to composition, a good integration theory, and several classical theorems of calculus. Even if the final transform depends on $k$, we obtain a new notion that applies to all GSF, in particular to all Schwartz's distributions and to all Colombeau generalized functions, without growth restrictions. We prove that this FT generalizes several classical properties of the ordinary FT, and in this way we also overcome the difficulties of FT in Colombeau's settings. Differences in some formulas, such as in the transform of derivatives, reveal to be meaningful since allow to obtain also non-tempered global unique solutions of differential equations.

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arXiv2022

Hyperseries in the non-Archimedean ring of Colombeau generalized numbers

This article is the natural continuation of the paper: Mukhammadiev A.~et al Supremum, infimum and hyperlimits of Colombeau generalized numbers in this journal. Since the ring $\tilde{R}$ of Robinson-Colombeau is non-Archimedean, a classical series $\sum_{n=0}^{+\infty}a_{n}$ of generalized numbers $a_{n}\in\tilde{R}$ is convergent if and only if $a_{n}\to0$ in the sharp topology. Therefore, this property does not permit us to generalize several classical results, mainly in the study of analytic generalized functions (as well as, e.g., in the study of sigma-additivity in integration of generalized functions). Introducing the notion of hyperseries, we solve this problem recovering classical examples of analytic functions as well as several classical results.

math.FA↗

arXiv2021

Calculus of variations and optimal control for generalized functions

We present an extension of some results of higher order calculus of variations and optimal control to generalized functions. The framework is the category of generalized smooth functions, which includes Schwartz distributions, while sharing many nonlinear properties with ordinary smooth functions. We prove the higher order Euler-Lagrange equations, the D'Alembert principle in differential form, the du Bois-Reymond optimality condition and the Noether's theorem. We start the theory of optimal control proving a weak form of the Pontryagin maximum principle and the Noether's theorem for optimal control. We close with a study of a singularly variable length pendulum, oscillations damped by two media and the Pais-Uhlenbeck oscillator with singular frequencies.

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arXiv2021

A Grothendieck topos of generalized functions I: basic theory

The main aim of the present work is to arrive at a mathematical theory close to the historically original conception of generalized functions, i.e. set theoretical functions defined on, and with values in, a suitable ring of scalars and sharing a number of fundamental properties with smooth functions, in particular with respect to composition and nonlinear operations. This is how they are still used in informal calculations in Physics. We introduce a category of generalized functions as smooth set-theoretical maps on (multidimensional) points of a ring of scalars containing infinitesimals and infinities. This category extends Schwartz distributions. The calculus of these generalized functions is closely related to classical analysis, with point values, composition, non-linear operations and the generalization of several classical theorems of calculus. Finally, we extend this category of generalized functions into a Grothendieck topos of sheaves over a concrete site. This topos hence provides a suitable framework for the study of spaces and functions with singularities. In this first paper, we present the basic theory; subsequent ones will be devoted to the resulting theory of ODE and PDE.

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arXiv2016

The classical theory of calculus of variations for generalized functions

We present an extension of the classical theory of calculus of variations to generalized functions. The framework is the category of generalized smooth functions, which includes Schwartz distributions while sharing many nonlinear properties with ordinary smooth functions. We prove full connections between extremals and Euler-Lagrange equations, classical necessary and sufficient conditions to have a minimizer, the necessary Legendre condition, Jacobi's theorem on conjugate points and Noether's theorem. We close with an application to low regularity Riemannian geometry.

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arXiv2016

Inverse Function Theorems for Generalized Smooth Functions

Generalized smooth functions are a possible formalization of the original historical approach followed by Cauchy, Poisson, Kirchhoff, Helmholtz, Kelvin, Heaviside, and Dirac to deal with generalized functions. They are set-theoretical functions defined on a natural non-Archimedean ring, and include Colombeau generalized functions (and hence also Schwartz distributions) as a particular case. One of their key property is the closure with respect to composition. We review the theory of generalized smooth functions and prove both the local and some global inverse function theorems.

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arXiv2016

A convenient notion of compact set for generalized functions

We introduce the notion of functionally compact sets into the theory of nonlinear generalized functions in the sense of Colombeau. The motivation behind our construction is to transfer, as far as possible, properties enjoyed by standard smooth functions on compact sets into the framework of generalized functions. Based on this concept, we introduce spaces of compactly supported generalized smooth functions that are close analogues to the test function spaces of distribution theory. We then develop the topological and functional analytic foundations of these spaces.

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arXiv2015

Calculus in the ring of Fermat reals Part I: Integral calculus

We develop the integral calculus for quasi-standard smooth functions defined on the ring of Fermat reals. The approach is by proving the existence and uniqueness of primitives. Besides the classical integral formulas, we show the flexibility of the Cartesian closed framework of Fermat spaces to deal with infinite dimensional integral operators. The total order relation between scalars permits to prove several classical order properties of these integrals and to study multiple integrals on Peano-Jordan-like integration domains.

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