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Diophantine approximation with prime variables
No full textThe first problem we are dealing with in Chapter 2 is a quaternary problem that can be seen as
a generalization of Languasco & Zaccagnini [2], Liu & Sun [4] and Wang & Yao [5].
The second quaternary problem outlined in Chapter 3 has a lower density and it leads to a
narrower range for k.
The last problem outlined in Chapter 4 of this dissertation deals with an improvement of the
result contained in Languasco & Zaccagnini [3]. Such improvements are contained in [1], due to
Languasco, Zaccagnini and the author of this dissertation.
All these problems were treated by combining Harman’s technique on the minor arc with a
suitable estimate for the L^4-norm of the relevant exponential sum over primes.Il primo problema trattato nel capitolo 2 è un problema quaternario che può essere visto come
una generalizzazione di Languasco & Zaccagnini [2], Liu & Sun [4] e Wang & Yao [5].
Il secondo problema quaternario descritto nel capitolo 3 ha una densitaà più bassa e conduce
a ottenere un intervallo più ristretto di valori di k.
L’ultimo problema trattato nel capitolo 4 di questa tesi è un miglioramento dei risultati ottenuti in Languasco & Zaccagnini [3]. Tali miglioramenti sono contenuti in [1], e dovuti a Languasco, Zaccagnini e l’autore di questa tesi.
Tutti i problemi descritti sono stati analizzati combinando la tecnica di Harman sull’arco
minore con una opportuna stima della norma L^4 delle attinenti somme esponenziali sui primi
Diophantine Approximation with a Quaternary Problem
Full text linkLet 1<7/6, and be non-zero real numbers, not all of the same sign such that is irrational and let be a real number. We prove that the inequality has infinitely many solutions in prime variables for any \varepsilon>0
La geometria non-Euclidea con la Sfera di Lénárt
No full textIl testo, scritto come supporto ai materiali di costruzione della Sfera di Lénárt, è composto da 43 attività progettate per gli studenti della scuola secondaria di primo e di secondo grado, alcune di esse adattabili anche alla scuola primaria.
Nelle schede iniziali, viene chiesto agli studenti di confrontare i principali concetti di geometria piana - come punti, rette, cerchi, distanze, angoli e aree - con i corrispondenti concetti di geometria sferica.
Nelle schede che seguono, si esplorano argomenti come la cartografia, i tassellamenti, le riflessioni sulla sfera, ed altri che riguardano esclusivamente la geometria serica come i triangoli polari ed i solidi platonici inscritti.
Ogni scheda è presentata sia come approccio aperto senza costrizioni, sia come una lezione più strutturata
The difficulties in geometry: A quantitative analysis based on results of mathematics competitions in Italy
Full text linkThis paper focuses on the difficulties encountered by Italian students in performing geometry tasks. A quantitative analysis, aimed at understanding the extent of the phenomenon, is carried out using the results of district competitions from the year 2018 to 2020, comparing the scores obtained in geometry questions with those in other areas of Olympic mathematics. In addition, the answers given by the students to a questionnaire administered at the end of the 2020 district competition are analyzed in order to better understand possible motivations behind the phenomenon in question. The results obtained need further confirmation through future research on the topic but represent clear trends worthy of further investigation
A Structural Approach to Gudermannian Functions
Full text linkThe Gudermannian function relates the circular angle to the hyperbolic one when their cosines are reciprocal.
Whereas both such angles are halved areas of circular and hyperbolic sectors, it is natural to develop similar considerations within the study of a class of curves images of maps with constant areal speed.
After a brief exposition of some use of the Gudermannian in applied sciences, we proceed to illustrate the class of curves, called \emph{Keplerian curves}, which can be parametrised by a map \mmm = (\cos_{\mmm}, \sin_{\mmm}) whose areal speed is 1.
In the next Sections, after a detailed study of p-circular and hyperbolic Fermat curves \crv F_p and \crv F^*_p, we define the \emph{p-Gudermannian} as the primitive of the derivative of the p-hyperbolic sine divided by the square of the p-hyperbolic cosine: all the analogues of the classical identities are proven.
Having realized that such curves correspond to each other using a homology, we extend our study to a wide class of Keplerian curves and their homologues; once again, defined the Gudermannian
in an identical manner, all the analogues of classical identities subsist. Below, three examples are detailed.
The last paragraph further extends this consideration, eliminating the hypothesis that the curves
are parametrised by maps with areal speed 1.
The Appendix illustrates integrating techniques for systems defining the Fermat curves, and the determination of the inverse of their tangent function
The Wallis Products for Fermat Curves
Full text linkAfter revisiting the properties of generalized trigonometric functions, i.e., the trigonometric function linked to the planar (Fermat) curve , using the tool of Keplerian trigonometry, introduced in (Gambini et al.: Monatsh. Math. 195, 55–72, 2021), we present the extension to this class of functions of the Wallis product, discovering connections with the representations of ordinary trigonometric functions by means of infinite products
-adic valuation of harmonic sums and their connections with Wolstenholme primes
Full text linkWe explore a conjecture posed by Eswarathasan and Levine on the distribution of -adic valuations of harmonic numbers that states that the set of the positive integers such that divides the numerator of is finite. We proved two results, using a modular-arithmetic approach, one for non-Wolstenholme primes and the other for Wolstenholme primes, on an anomalous asymptotic behaviour of the -adic valuation of when the -adic valuation of equals exactly 3
The relation between mathematical object/mathematical name: conceptual changes among designation, description, denotation, denomination and definition.
No full textThe actions of designation, description, denotation, denomination and definition are crucial in the didactic activity
in the classroom (D'Amore and Fandiño Pinilla, 2012) since they embody different interplays between objects,
representations, properties, names (in the sense of Duval (2008)). Switching from one action to the other may be the
result of a conceptual change (diSessa, 2006). We present the result of a teaching experiment in classes of grades from 2 to
4 where the relation object/name is investigated in the case of the circle. The experiment makes use of a particular
artefact, the Lénárt Spheres (Lénárt, 1996). Comparative geometry activities allow to deal with geometrical objects in a
learning environment where the relations between objects, representations and properties are different from the usual
ones, hence implying a restructuration of the interplays between them. As a result of the teaching experiment, as can be
seen, in particular, from the comparison of initial and conclusive questionnaires, children started a change of their way
of associating a name to an object. We argue that this is due also to a conceptual change and not only to “learning what
was taught”
Long-lasting bank relationships and growth of firms
Full text linkA puzzling but consistent result in the empirical literature on banking is that firms with close bank ties do not grow faster than bank-independent firms. In this paper, we reconsider the link between relationship lending and firms' growth, distinguishing firms by size and 'health'. The idea is that the beneficial effects of relationship lending on information asymmetries can be compensated by other negative capture, risk and externality effects which make relational banks reluctant to support long-term growth projects of client firms, and that the strength of these compensating effects varies with firm size and health status. We explore the influence of long-lasting bank relationships on employment and asset growth of a large sample of Italian firms. The main finding is that relationship lending hampers the efforts of small firms to increase their size, while it mitigates the negative growth of troubled, medium-large enterprises
Doubling the side, doubling the area: Managing representation with a Geoboard tool
No full textIn this paper we analysed the ability of high school students to handle the representation of the
squares and to construct new figures in Geoboard software. We proposed two questions to the
students, in which they were asked to construct squares under certain conditions. We analysed the
difficulties encountered and the strategies implemented framing them within Duval’s theoretical
framework. Moreover, we focused the attention on the presence of an intuitive rule that guided the
resolution process compromising the solution of the tasks and which processes are involved in the
handling of the representation of geometric figures
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