Matematicahttp://elea.unisa.it/xmlui/handle/10556/642026-04-14T14:04:51Z2026-04-14T14:04:51ZEstimates for the transition kernel for elliptic operators with unbounded coefficientsPorfido, Mariannahttp://elea.unisa.it/xmlui/handle/10556/72722025-04-30T17:38:08Z2023-02-09T00:00:00ZEstimates for the transition kernel for elliptic operators with unbounded coefficients
Porfido, Marianna
This manuscript is devoted to the study of the qualitative behaviour of the
solutions of evolution equations arising from elliptic and parabolic problems
on unbounded domains with unbounded coefficients. In particular, we deal
with the elliptic operator of the form
A = div(Q∇) + F · ∇ − V,
where the matrix Q(x) = (qij (x)) is symmetric and uniformly elliptic and the
coefficients qij , F and V are typically unbounded functions.
Since the classical semigroup theory does not apply in case of unbounded
coefficients, in Chapter 1 we illustrate how to construct the minimal semigroup
T(·) associated with A in Cb(R
d
). It provides a solution of the corresponding
parabolic Cauchy problem
(
∂tu(t, x) = Au(t, x), t > 0, x ∈ R
d
,
u(0, x) = f(x), x ∈ R
d
,
for f ∈ Cb(R
d
), that is given through an integral kernel p as follows
T(t)f(x) = Z
Rd
p(t, x, y)f(y) dy.
Moreover, such solution is unique if a Lyapunov function exists. Since an
explicit formula is in general not available, it is interesting to look for pointwise
estimates for the integral kernel p.
In Chapter 2 we consider a Schr ̈odinger type operator in divergence form,
namely the operator A when F = 0. We prove first that the minimal realiza-
tion Amin of A in L
2
(R
d
) with unbounded coefficients generates a symmetric
sub-Markovian and ultracontractive semigroup on L
2
(R
d
) which coincides on
L
2
(R
d
)∩Cb(R
d
) with the minimal semigroup generated by a realization of A on
Cb(R
d
). Moreover, using time dependent Lyapunov functions, we show point-
wise upper bounds for the heat kernel of A. We then improve such estimates
and deduce some spectral properties of Amin in concrete examples, such as in
the case of polynomial and exponential diffusion and potential coefficients.
Chapter 3 deals with the whole operator A. With appropriate modifi-
cations, similar kernel estimates described above are valid for this operator.
In addition, we prove global Sobolev regularity and pointwise upper bounds
for the gradient of p. We finally apply such estimates in case of polynomial
coefficients. [edited by Author]
2021 - 2022
2023-02-09T00:00:00ZA characterisation of mathematics education creativity inspired by active and popular pedagogyLo Sapio, Rosalia Mariahttp://elea.unisa.it/xmlui/handle/10556/72652025-04-30T17:36:09Z2023-05-30T00:00:00ZA characterisation of mathematics education creativity inspired by active and popular pedagogy
Lo Sapio, Rosalia Maria
The present PhD thesis has been developed through an interweaving of reflections and
experiences in the field of mathematics education that have, gradually, steered the
research interest in a specific direction. Initially, the focus on studies and research about
informal mathematics education (Nemirovsky, Kelton & Civil, 2017) and the field of critical
mathematics education (Skovsmose & Penteado, 2012) allowed for an in-depth study of
these two strands, with a view to a connection between mathematics education and ideals
of democracy and active citizenship. Contextually, particular educational experiences,
carried out during the PhD course, have stimulated a reflection in this same sense. In
particular, my involvement in two editions of the "Proud of You" 1 project fostered the
development of some issues concerning the creation of educational activities designed in
informal mathematics education contexts and the potential, through them, to mediate
mathematical content in students from socio-culturally disadvantaged backgrounds,
acting effectively and sustainably with respect to preventing and combating early school
drop-out. The curiosity about these two issues generated an opening of the horizon of
interests that led to explore the field of active and popular pedagogy. The literature
references taken into consideration turned out to be valuable and inspiring, and opened
the way to a reflection on the possible tools for implementing a motivating didactic design
and the possibility of intertwining it with the desire for social redemption and liberation from
any kind of oppression. With these assumptions, the research then moved in a more specific
and clear direction. In particular, the focus was on the design and implementation of
teaching activities that were meaningful, from a mathematical point of view, and inclusive
and attentive to the ideals of democracy, from a social point of view. The focus therefore
shifted to the process of designing and implementing mathematical teaching activities of
the kind made explicit, paving the way for the research and the characterisation of the
mathematics education creativity. With this in mind, studies and research into
mathematical creativity have been investigated in detail, enabling the object of
investigation to be framed. With the aim of investigating the kind of mathematical creativity
involved, effective methodologies were researched for data collection and subsequent
analysis. Inspired by the research (Hadamard, 1945; Liljedahl, 2004), interviews were
conducted, involving teachers who showed great creativity in teaching design, also with
reference to socially and culturally disadvantaged contexts. [edited by Author]
2021 - 2022
2023-05-30T00:00:00ZOn the regularity of solutions of free boundary problemsLamberti, Lorenzohttp://elea.unisa.it/xmlui/handle/10556/72632025-04-30T17:35:54Z2023-02-09T00:00:00ZOn the regularity of solutions of free boundary problems
Lamberti, Lorenzo
Optimal design problems have aroused particular interest in the scientific com-
munity over the past thirty years. In physics, for example, they find application
in the investigation of the minimal energy configurations of a mixture of two
materials in a bounded and connected open set.
The fascination of such problems derives from their variational formula-
tion, which involves not only the state function of a system, but also a shape,
that is a set. If a penalizing contribution of perimeter form, due to a surface
energy, is added to the integral mass energy, dependent on the configuration
state-shape, the problem becomes even more intriguing and inspiring.
It is not straightforward to investigate the regularity of minimizing pairs
because the two energies have different dimensions under commong scalings:
once a homothety of factor r is applied, the first energy “behaves” as a vol-
ume (rescaling with factor r
n
), the second as a perimeter (rescaling with factor
r
n1
). The coexistence of the two types of energies is managed using techniques
and tools of both the Calculus of Variations and the Geometric Measure The-
ory.
In the first part of this thesis we deal with two optimal design problems,
in which the integral functions that constitute the mass energy have different
growths.
If their growth is at most quadratic, we prove the C
1,μ regularity of the
interface of the shape that constitutes the optimal pair, up to a singular set of
Hausdorff dimension less than n 1. The technique used combines the regu-
larity theories of the Λ-minimizers of the perimeter and the minimizers of the
Mumford-Shah functional.
If the integrands have at most a polynomial growth of degree p, the anal-
ysis becomes more involved. The C
1,μ regularity of the interface remains an
open problem. However, it is proved that the optimal shape of the problem
is equivalent to an open set with a topological boundary that differs from its
reduced boundary for a set of Hausdorff dimension less than or equal to n 1.
In the second part of the thesis we address to a completely different varia-
tional problem, involving a frustrated spin system on a (one-dimensional and
two-dimensional) lattice confined in two magnetic anisotropy circles.
This topic is of significant scientific interest, as it is useful for understand-
ing the behavior of low-dimensional magnetic structures existing in nature.
The frustration parameter α ¡ 0 of the system averages the ferromagnetic
and antiferromagnetic interactions that coexist in the energy. The minimal
energy state of the system, for α ¤ 4, consists of a spin that “lives” within
only one of the two magnetic anisotropy circles and has a positive or negative
chirality.
We find the correct rescaling of the functional and prove the energy needed
to detect the two phenomena that break the rigid minimal symmetry described.
These are chirality transitions and magnetic anisotropy transitions of the spin. [edited by Author]
2021 - 2022
2023-02-09T00:00:00ZLa stampante 3D come mediatore semiotico per l’apprendimento della competenza geometrica nella scuola dell’infanziaEsposito, Antoniettahttp://elea.unisa.it/xmlui/handle/10556/64952025-04-30T16:37:44Z2021-06-09T00:00:00ZLa stampante 3D come mediatore semiotico per l’apprendimento della competenza geometrica nella scuola dell’infanzia
Esposito, Antonietta
This work is part of an experimental research project of INDIRE - National Institute
of Documentation, Innovation and Educational Research, about the didactical use
of the 3D printer in kindergarten.
Starting from the assumption that the evolution of geometric thought must be
sought from the first spatial experiences of the child and that his development
process does not depend exclusively on the age of the pupils but on the
"mathematical" education provided to them (Pierre and Dina van Hiele, 1986), the
introduction of CAD modeling with 3D printing in kindergarten can have an added
value, compared to all the artifacts already in use in kindergarten (plasticine, clay,
etc), that of enabling the child to identify and recognize the invariants of geometric
shapes, strengthening the entry skills in primary school, thus preventing those states
of geometric deprivation to which children are mistakenly subjected in the first
years of life.
The objective of this research work is therefore to verify whether the 3D printer can
be considered an artifact of semiotic mediation to contribute to the development of
geometric competence since the Kindergarten.
The focus was therefore on spatial skills, a group of processes that allow the
individual to interact correctly with the surrounding world. In fact, in the literature
it is now clear that spatial skills are the basis for a good learning of geometry.
Among these, spatial visualization was examined, that is the ability to understand,
mentally encode and manipulate 3D shapes (Carroll, 1993; Hegarty & Waller,
2004).
Therefore, a wide-ranging experimental research was designed which, starting from
the level of visual-spatial skills possessed by 5-year-olds, allowed to deepen the
classification, representation and dissection of solid figures in relation to the use of
a 3D printer using both qualitative and quantitative research methods.
Specifically, the research project envisaged the identification of standardized tests
suitably adapted to the purpose and the definition of an educational path codesigned with the teachers of the kindergarten, which starting from an integrating
background would include a series of didactic activities specific for the acquisition
of geometric concepts and the use of CAD software and which would lead to the creation of a narrative character with the 3D printer.
The trial, which took place in two preschools, involved about 80 children aged
about 5.
The analysis of the results of tests administered to pre-school children, before and
after an educational intervention that also includes the use of a 3D printer, showed
that the level of competence acquired by them for the classification and recognition
of the graphic representation of a solid figure as well as of its flat section, and
therefore in synthesis of the "spatial visualization", is superior in number of children
and quality of possession.
Furthermore, the use of this instrumentation within the didactic action favors on the
one hand, that of teachers, the planning of innovative activities, on the other, that
of children, a positive attitude towards learning: direct involvement of children in
the creation of concrete objects, makes them active protagonists and builders of
their own learning. [edited by Author]; Il lavoro di tesi si inserisce nell’ambito di un progetto di ricerca sperimentale
dell’INDIRE – Istituto Nazionale di Documentazione, Innovazione e Ricerca
Educativa, sull’introduzione e l’utilizzo didattico della stampante 3D nella scuola
dell’Infanzia.
A partire dall’assunto che l’evoluzione del pensiero geometrico va ricercata fin
dalle prime esperienze spaziali del bambino e che il suo processo di sviluppo non
dipende in modo esclusivo dall’età dell’allievo ma dall’educazione “matematica”
fornitagli (Pierre e Dina van Hiele, 1986), l’introduzione della modellazione CAD
con la stampa in 3D nella scuola dell’infanzia può avere come valore aggiunto,
rispetto a tutti gli artefatti già in uso nella scuola dell’Infanzia (plastilina, pongo,
etc), quello di mettere il bambino nella condizione di individuare e riconoscere le
invarianti delle forme geometriche, rafforzando le competenze di ingresso nella
scuola primaria, prevenendo così quegli stati di deprivazione geometrica cui i
bambini sono erroneamente sottoposti nei primi anni di vita.
L’obiettivo di questo lavoro di ricerca è pertanto verificare se la stampante 3D possa
essere considerata un artefatto di mediazione semiotica per contribuire allo sviluppo
della competenza geometrica sin dalla Scuola dell’Infanzia.
Si è pertanto posta l’attenzione sulle abilità spaziali, un gruppo di processi che
consentono la corretta interazione dell’individuo con il mondo circostante. In
letteratura, infatti, è ormai acclarato che le abilità spaziali costituiscono la base per
un buon apprendimento della geometria. Tra queste si è presa in esame la
visualizzazione spaziale, ovvero la capacità di comprendere, codificare
mentalmente e manipolare le forme 3D (Carroll, 1993; Hegarty & Waller, 2004).
È stata, pertanto, progettata una ricerca sperimentale ad ampio respiro che a partire
dal livello delle abilità visuo-spaziali possedute dai bambini di 5 anni, ha permesso
di approfondire la classificazione, la rappresentazione e il sezionamento delle figure
solide in relazione all’utilizzo di una stampante 3D utilizzando metodologie di
ricerca sia di tipo qualitativo che quantitativo.
Nello specifico il progetto di ricerca ha previsto l’individuazione di test standardizzati opportunamente adattati allo scopo e la definizione di un percorso
didattico co-progettato con le maestre della scuola dell’Infanzia, che a partire da
uno sfondo integratore prevedesse una serie di attività didattiche specifiche per
l’acquisizione di concetti geometrici e l’utilizzo del software CAD e che portasse
alla realizzazione di un personaggio della narrazione con la stampante 3D.
La sperimentazione, avvenuta in due scuole dell’Infanzia, ha coinvolto circa 80
bambini dell’età di 5 anni circa.
L’analisi dei risultati di test somministrati a bambini in età pre-scolare, prima e
dopo un intervento didattico che prevede al suo interno anche l’utilizzo di una
stampante 3D, hanno evidenziato che il livello di competenza, acquisito dagli stessi,
per la classificazione e il riconoscimento della rappresentazione grafica di una
figura solida nonché della sua sezione piana, e quindi in sintesi della
“visualizzazione spaziale”, è superiore in numero di bambini e qualità di possesso.
Inoltre, l’utilizzo di tale strumentazione all’interno dell’azione didattica favorisce
da un lato, quello dei docenti, la progettazione di attività innovative, dall’altro,
quello dei bambini, un atteggiamento positivo nei riguardi dell’apprendimento: il
coinvolgendo diretto dei bambini nella realizzazione di oggetti concreti, li rende
protagonisti attivi e costruttori del proprio apprendimento. [a cura dell'Autore]
2019 - 2020
2021-06-09T00:00:00Z