1,721,042 research outputs found
The Alexander polynomial of (1,1)-knots
No full textIn this paper we investigate the Alexander polynomial of (1, 1)-knots, which are knots
lying in a 3-manifold with genus one at most, admitting a particular decomposition. More
precisely, we study the connections between the Alexander polynomial and a polynomial
associated to a cyclic presentation of the fundamental group of an n-fold strongly-cyclic
covering branched over the knot K, which we call the n-cyclic polynomial of K. In
this way, we generalize to all (1, 1)-knots, with the only exception of those lying in
S^2 × S^1, a result obtained by Minkus for 2-bridge knots and extended by the author
and M. Mulazzani to the case of (1, 1)-knots in S^3. As corollaries some properties of
the Alexander polynomial of knots in S^3 are extended to the case of (1, 1)-knots in lens
spaces
Diffeomorphic vs Isotopic Links in Lens Spaces
Full text linkLinks in lens spaces may be defined to be equivalent by ambient isotopy or by diffeomorphism of pairs. In the first case, for all the combinatorial representations of links, there is a set of Reidemeister-type moves on diagrams connecting isotopy equivalent links. In this paper, we provide a set of moves on disk, band and grid diagrams that connects diffeo-equivalent links: there are up to four isotopy equivalent links in each diffeo-equivalence class. Moreover, we investigate how the diffeo-equivalence relates to the lift of the link in the 3-sphere: in the particular case of oriented primitive-homologous knots, the lift completely determines the knot class in L(p, q) up to diffeo-equivalence, and thus only four possible knots up to isotopy equivalence can have the same lift
A Markov theorem for generalized plat decomposition
Full text linkWe prove a Markov theorem for tame links in a connected closed orientable 3-manifold with respect to a plat-like representation. More precisely, given a genus Heegaard surface for we represent each link in as the plat closure of a braid in the surface braid group and analyze how to translate the equivalence of links in under ambient isotopy into an algebraic equivalence in . First, we study the equivalence problem in , and then, to obtain the equivalence in , we investigate how isotopies corresponding to ``sliding'' along meridian discs change the braid representative. At the end we provide explicit constructions for Heegaard genus 1 manifolds, i.e. lens spaces and
Virtual quandle for links in lens spaces
Full text linkWe construct a virtual quandle for links in lens spaces L(p, 1). This invariant has two valuable advantages over an ordinary fundamental quandle for links in lens spaces: the virtual quandle is an essential invariant and the presentation of the virtual quandle can be easily written from the band diagram of a link
TRAINING SECONDARY SCHOOL TEACHERS TO DEAL WITH INTERDISCIPLINARITY BETWEEN PHYSICS AND MATHEMATICS: THE CASE OF THE BLACK BODY MODEL
Full text linkBoth the most recent EU trends and the current Italian reform advocate for new interdisciplinary
approaches that integrate the STEM disciplines. In this talk we report the results of a research concerning
the "interdisciplinary skills" necessary for secondary school teachers to implement didactical activities that
authentically integrate mathematics and physics. The empirical results that will support our argumentations
have been collected in three experimentations: a pilot study with 5 mathematics and physics studentteachers
and two experiences in in-service training courses (44 teachers). In this work we consider the case
of the blackbody model, since mathematics is particularly relevant, the logic underlying the process of
modeling is complex but, at the same time, the search for "oversimplification" usually leads authors of
textbooks to reproduce rituals that mortify any intellectually satisfaction (Viennot, 2006). We firstly carried
out an educational reconstruction (Duit et al., 2012) of the historical path that led Planck to “open the
secret door” of the new quantum world, starting from a mathematical conjecture. Then, we developed a
tutorial, where teams of teachers are asked to fill-in some missing steps in the reconstruction of Planck’s
modeling and to recognize where and how discreteness was introduced in physics to interpret the modeling
process of interaction between matter and radiation (as an exchange of discrete packets of energy). In this
paper we focus on teachers’ difficulties with the different uses of mathematics in modeling blackbody, in
particular in “zooming in and out” and keeping control on the whole process
Interplay between mathematics and physics to catch the nature of a scientific breakthrough: The case of the blackbody
Full text linkThis paper aims to provide a contribution to the research in physics education regarding the interplay between mathematics and physics in teaching and learning physics at the university level. The argument is developed through a study focused on the historical case study of the blackbody that led Planck to make one of the most significant scientific breakthroughs in physics: the introduction of discreteness and quantization into physical processes. The study is methodologically guided by the model that Udhen, Karam, Pietrocola, and Pospiech elaborated to highlight the interplay between physics and mathematics within teaching and learning practices [O. Uhden, R. Karam, M. Pietrocola, and G. Pospiech, Modelling mathematical reasoning in physics education, Sci. Educ. Netherlands 21, 485 (2012).]. The model emphasizes the distinction between the technical and structural roles of mathematics in physics, with the latter role being argued to correspond to processes of mathematization and interpretation. We used this model to analyze Planck’s original papers and to reconstruct the reasoning that, thanks to the structural role played by mathematics, paved the way for the quantistic scientific breakthrough. The results of the analysis led us to design a teaching tutorial that we implemented with mathematics and physics university students. Students’ reactions are reported to discuss the educational potential of the approach beyond the specific case and to argue for its potential general application to other similar physics topics
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Didattica tra Problem Solving e Intelligenza Artificiale
Full text linkNegli ultimi decenni, l'educazione ha subito trasformazioni significative grazie all'avanzamento tecnologico e alla crescente importanza delle competenze trasversali. In questo contesto, il problem solving è riconosciuto come una competenza fondamentale non solo per il successo accademico ma anche per l'apprendimento permanente e l'inclusione sociale. La tesi esplora come l'intelligenza artificiale (IA), in particolare i large language models (LLM) come ChatGPT, possa supportare e migliorare l'insegnamento del problem solving nella scuola secondaria in Italia.
Il Capitolo 1 introduce i fondamenti teorici del problem solving e dell'IA nell'educazione matematica, evidenziando contributi storici di psicologi e matematici come Poincaré e Polya. Vengono analizzate le euristiche di risoluzione dei problemi e l'interazione tra pensiero conscio e subconscio, oltre al ruolo delle tecnologie digitali nella didattica.
Il Capitolo 2 presenta un questionario somministrato a docenti di matematica e fisica per raccogliere dati sulle loro opinioni e pratiche didattiche riguardanti il problem solving e l'IA. L'analisi dei dati rivela che, sebbene molti docenti riconoscano l'importanza del problem solving, esistono ancora sfide significative nell'integrazione efficace delle tecnologie digitali in aula.
Il Capitolo 3 approfondisce queste tematiche attraverso un focus group con alcuni dei docenti che hanno partecipato al questionario, esplorando in dettaglio le loro esperienze, opinioni e strategie didattiche.
Infine, il Capitolo 4 esamina casi specifici di risoluzione di problemi matematici utilizzando chatbot come ChatGPT, valutando punti di forza e criticità delle risposte generate.
La tesi conclude che, sebbene l'IA presenti potenziali benefici per l'insegnamento del problem solving, è necessaria una maggiore formazione e supporto per i docenti per sfruttare appieno queste tecnologie, garantendo al contempo un'educazione equilibrata e inclusiva
Concatenazioni e singolarità di curve piane
No full textIn questa tesi vengono enunciate alcune nozioni di teoria dei nodi. Sono definiti i concetti di concatenazione e di equivalenza di due concatenazioni e viene introdotto il calcolo del gruppo fondamentale con l’esempio particolare dei nodi torici. Questi concetti sono poi applicati allo studio delle singolarità di curve complesse piane mediante l’intersezione di queste con una sfera di raggio sufficientemente piccolo centrata nella singolarità che si sta considerando
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