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32 results on '"Caldarola, Fabio"'

1. Unimaginable Numbers and Infinity Computing at School: An Experimentation in Northern Italy

Author

Antoniotti, Luigi, Astorino, Annabella, Caldarola, Fabio, Goos, Gerhard, Series Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Sergeyev, Yaroslav D., editor, Kvasov, Dmitri E., editor, and Astorino, Annabella, editor

Published
2025
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2. A Simple Mathematical Solution to an Invariance Problem in Water Networks

Author

Caldarola, Fabio, Carini, Manuela, Maiolo, Mario, Goos, Gerhard, Series Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Sergeyev, Yaroslav D., editor, Kvasov, Dmitri E., editor, and Astorino, Annabella, editor

Published
2025
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4. The notion of 'Unimaginable Numbers' in computational number theory

Author

Leonardis, Antonino, D'Atri, Gianfranco, and Caldarola, Fabio

Subjects
Computer Science - Logic in Computer Science, 03D80, F.4.1, G.2.2
Abstract

Literature considers under the name \emph{unimaginable numbers} any positive integer going beyond any physical application, with this being more of a vague description of what we are talking about rather than an actual mathematical definition. This simply means that research in this topic must always consider shortened representations, usually involving \emph{recursion}, to even being able to describe such numbers.\par\medskip One of the most known methodologies to conceive such numbers is using \emph{hyperoperations}, that is a sequence of binary functions defined recursively starting from the usual chain: addition - multiplication - exponentiation. The most important notations to represent such hyperoperations have been considered by Knuth, Goodstein, Ackermann and Conway as described in this work's introduction.\par\medskip Within this work we will give an axiomatic set for this topic, and then try to find on one hand other ways to represent unimaginable numbers, as well as on the other hand applications to computer science, where the algorithmic nature of representations and the increased computation capabilities of computers give the perfect field to develop further the topic.\par\medskip After the introduction, we will give axioms and generalizations for the up-arrow notation; in the subsequent section we consider a representation via rooted trees of the \emph{hereditary base-$n$ notation} involved in Goodstein's theorem, which can be used efficiently to represent some defective unimaginable numbers, and in the last section we will analyse some methods to compare big numbers, proving specifically a theorem about approximation using scientific notation., Comment: 15 pages, 5 figures. Related to the conference "The First Symposium of the International Pythagorean School -- da Pitagora a Sch\"utzenberger: numeri inimmaginabil\^i\^i\^i"

Published
2019

7. Numerical Experimentations for a New Set of Local Indices of a Water Network

Author

Bonora, Marco Amos, Caldarola, Fabio, Muranho, Joao, Sousa, Joaquim, Maiolo, Mario, Goos, Gerhard, Founding Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Woeginger, Gerhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Sergeyev, Yaroslav D., editor, and Kvasov, Dmitri E., editor

Published
2020
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8. Algebraic Tools and New Local Indices for Water Networks:Some Numerical Examples

Author

Caldarola, Fabio, Maiolo, Mario, Goos, Gerhard, Founding Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Woeginger, Gerhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Sergeyev, Yaroslav D., editor, and Kvasov, Dmitri E., editor

Published
2020
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9. Combinatorics on n-sets: Arithmetic Properties and Numerical Results

Author

Caldarola, Fabio, d’Atri, Gianfranco, Pellegrini, Marco, Goos, Gerhard, Founding Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Woeginger, Gerhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Sergeyev, Yaroslav D., editor, and Kvasov, Dmitri E., editor

Published
2020
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10. On the Arithmetic of Knuth’s Powers and Some Computational Results About Their Density

Author

Caldarola, Fabio, d’Atri, Gianfranco, Mercuri, Pietro, Talamanca, Valerio, Goos, Gerhard, Founding Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Woeginger, Gerhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Sergeyev, Yaroslav D., editor, and Kvasov, Dmitri E., editor

Published
2020
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11. Paradoxes of the Infinite and Ontological Dilemmas Between Ancient Philosophy and Modern Mathematical Solutions

Author

Caldarola, Fabio, Cortese, Domenico, d’Atri, Gianfranco, Maiolo, Mario, Goos, Gerhard, Founding Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Woeginger, Gerhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Sergeyev, Yaroslav D., editor, and Kvasov, Dmitri E., editor

Published
2020
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12. The Sequence of Carboncettus Octagons

Author

Caldarola, Fabio, d’Atri, Gianfranco, Maiolo, Mario, Pirillo, Giuseppe, Goos, Gerhard, Founding Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Woeginger, Gerhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Sergeyev, Yaroslav D., editor, and Kvasov, Dmitri E., editor

Published
2020
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13. New Approaches to Basic Calculus: An Experimentation via Numerical Computation

Author

Antoniotti, Luigi, Caldarola, Fabio, d’Atri, Gianfranco, Pellegrini, Marco, Goos, Gerhard, Founding Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Woeginger, Gerhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Sergeyev, Yaroslav D., editor, and Kvasov, Dmitri E., editor

Published
2020
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17. The Role of Computer Simulations in Learning Analytic Mechanics towards Chaos Theory: A Course Experimentation

Author

Bertacchini, Francesca, Bilotta, Eleonora, Caldarola, Fabio, and Pantano, Pietro

Abstract

We present the results of an experimental project, made at the University of Calabria, which uses computer simulations and scientific computational systems in teaching and learning the fundamentals of analytic mechanics, in particular the classical Lagrangian and Hamiltonian formulation of the matter. Starting with examples of classical dynamical systems we also introduced students to appreciate chaos concepts going step by step to more complex settings. The success obtained with the use of computer-based methodologies has been very clear and the final students' tests confirm it. In this paper, in particular, we compare the results obtained by four groups of students: the first two attended such an experimental course and employed computer support for their individual work at home, while the learning environment of the second two groups (the control ones) was based only on traditional classroom lectures with paper and pencil home activities. If on the one hand a better overall performance of the first two groups was evident, it is very interesting (and not so easy) to investigate how much in quantity and quality, the learning of the experimental groups differs from that of the other groups, and why. For example, we illustrate some relevant differences between them in the comprehension of the studied phenomena, in dealing with and solving the proposed problems, and also in the individual development of further personal ideas.

Published
2019
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23. Beyond Knuth's notation for unimaginable numbers within computational number theory

Author

D&#039, LEONARDIS, Antonino, and CALDAROLA, Fabio

Subjects
Matematik, Computational number theory,unimaginable numbers,Knuth,big data,number representation,Goodstein, Mathematics
Abstract

Literature considers under the name "unimaginable numbers" anypositive integer going beyond any physical application. One of the most knownmethodologies to conceive such numbers is using hyper-operations, that is asequence of binary functions dened recursively starting from the usual chain:addition - multiplication - exponentiation. The most important notations torepresent such hyper-operations have been considered by Knuth, Goodstein,Ackermann and Conway as described in this work's introduction. Within thiswork we will give an axiomatic setup for this topic, and then try to nd on onehand other ways to represent unimaginable numbers, as well as on the otherhand applications to computer science, where the algorithmic nature of representations and the increased computation capabilities of computers give theperfect eld to develop further the topic, exploring some possibilities to effectively operate with such big numbers. In particular, we will give some axiomsand generalizations for the up-arrow notation and, considering a representation via rooted trees of the hereditary base-n notation, we will determine insome cases an effective bound related to "Goodstein sequences" using Knuthsnotation. Finally, we will also analyze some methods to compare big numbers,proving specically a theorem about approximation using scientic notationand a theorem on hyperoperation bounds for Steinhaus-Moser notation.

Published
2020

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