Some Inverse Problems in Analysis and Geometry

Thesis (Ph.D.)--University of Washington, 2016-08

Some results on orbifold quotients and related objects

The quotient of a finite-dimensional Euclidean space by a finite linear group inherits different structures from the initial space, e.g. a topology, a metric and a piecewise linear structure. The question when such a quotient is a manifold leads to the study of finite groups generated by reflections and rotations, i.e. by orthogonal transformations whose fixed point subspace has codimension one or two. We classify such groups and thereby complete earlier results by M. A. Mikhaîlova from the 70s and 80s. Moreover, we show that a finite group is generated by reflections and) rotations if and only if the corresponding quotient is a Lipschitz-, or equivalently, a piecewise linear manifold (with boundary). For the proof of this statement we show in addition that each piecewise linear manifold of dimension up to four on which a finite group acts by piecewise linear homeomorphisms admits a compatible smooth structure with respect to which the group acts smoothly. This solves a challenge by Thurston and confirms a conjecture by Kwasik and Lee. In the topological category a counterexample to the above mentioned characterization is given by the binary icosahedral group. We show that this is the only counterexample up to products. In particular, we answer the question by Davis of when the underlying space of an orbifold is a topological manifold. As a corollary of our results we generalize a fixed point theorem by Steinberg on unitary reflection groups to finite groups generated by reflections and rotations. As an application thereof we answer a question by Petrunin on quotients of spheres.

Berufsausbildung und sozio-ökonomischer Wandel. Ursachen der "Krise des dualen Systems" der Berufsausbildung

Zu einem Zeitpunkt, zu dem zahlreiche Entwicklungs- und neuerdings auch Industrieländer das „duale System" der Berufsausbildung in Deutschland als eine Art Idealmodell der beruflichen Qualifizierung entdecken, wachsen die Zweifel, ob dieses Ausbildungssystem angesichts der anhaltenden Expansion von Gymnasium und Hochschule noch eine Zukunftsperspektive hat. Im vorliegenden Beitrag werden die zahlreichen Symptome der Krise des dualen Systems aufgezeigt, es wird versucht, die Ursachen dieser Krise unter sozialhistorischer Fragestellung zu deuten, und es werden die bislang aufgezeigten Vorschläge diskutiert, diese Krise zu überwinden. (DIPF/Orig.)

As vozes da quinta : uma abordagem lúdico-didática no ensino do canto a crianças

Neste projeto educativo pretende-se investigar e indicar as correntes e perspetivas atuais sobre o ensino-aprendizagem do canto a crianças, com destaque àquelas com idades aproximadamente entre os 8 e os 12 anos, bem como as estratégias a utilizar e conteúdos a lecionar. Faz-se neste trabalho um levantamento da realidade portuguesa em que se comprova a disseminação do ensino-aprendizagem do canto, principalmente a partir do 2º ciclo do ensino básico. Regista-se ainda que várias escolas oferecem já esta opção para alunos da Iniciação musical. Neste contexto destaca-se a importância do caráter lúdico do ensino de crianças e propõem-se uma abordagem lúdico-didática a partir da utilização de um conjunto de episódios narrativos criados pelo autor deste trabalho para facilitar a introdução ao canto por via de atividades a realizar anteriormente ao treino da voz cantada, a que se chamou As vozes da quinta. Estes foram construídos a partir de recomendações conhecidas após contacto com bibliografia dedicada a este tema, destacando-se autores como Phillips (2014), Williams (2012) e Mizener (2008), entre outros. As analogias assumem aqui papel extremamente relevante, destacando-se a utilização de gestos corporais, metáforas e imagens como forma de trabalhar e melhorar as competências fundamentais para o desenvolvimento vocal e performativo dos alunos a partir da interpretação dos episódios narrativos, em que porão em prática as suas capacidades dramáticas e comunicativas, tendo em vista o desenvolvimento das componentes essenciais ao desenvolvimento de um cantor: postura, respiração, fonação, ressonância, dicção-articulação e expressão. Os exercícios propostos assumir-se-ão, através das referidas analogias, como jogos, que estimularão uma prática mais motivada e significativa e que corroborarão na melhoria do desempenho das crianças nas várias competências mencionadas.

BET on Top Hat: challenges to improve language proficiency

Within the business context, communication and interaction tends to be considerably rooted in the use of English (as lingua franca), as well as in ICT use. Thus, professionals have to be able to speak the English language, resorting to specific, internationally recognised terminology and be proficient in the use of manifold ICT tools. In fact, the tendency is for the great majority of higher education (HE) students to own mobile devices (laptops, smartphones and/or tablets) and use them to access information and communicate/interact with content and other people. Bearing this in mind, a teaching and learning strategy was designed, in which m-learning (i.e. learning in which the delivery platform is a mobile device) was used to approach Business English Terminology (BET). The strategy was labelled as ‘BET on Top Hat’, once the selected application was Top Hat (https://tophat.com/) and the idea was for students to face it as if it were a game/challenge. In this scenario, the main goals of this exploratory study were to find evidence as to: i) the utility of m-learning activities for learning BET and ii) if and how m-learning activities can generate intrinsic motivation in students to learn BET. Participants (n=23) were enrolled in English II, a curricular unit of the 1st cycle degree in Retail Management offered at Águeda School of Technology and Management – University of Aveiro (2014/15 edition). The data gathered included the students’ results in quizzes and their answers to a short final evaluation questionnaire regarding their experience with BET on Top Hat. Consequently, data were treated and analysed resorting to descriptive statistical analysis, and, when considered pertinent, the teacher’s observation notes were also considered. The results unveil that, on the one hand, the strategy had a clear positive impact on the students’ intrinsic motivation and, on the other hand, the students’ performance as to BET use tended to improve over time.

The promotion of digital inclusion through MOOC design and use: a literature review

The use of Massive Open Online Courses (MOOCs) is being increasingly equated as a viable option by several educational shareholders in the scope of many scientific areas; nevertheless, research as to its potentialities in terms of digital (and consequently social) inclusion is still sparse and somehow atomised. Thus, this paper aims at putting forward the results of a thorough literature review focussed on the studies that bring together the concepts of MOOC and digital inclusion, published between January 2014 and January 2015. Thus, the main goal was to find out if there is evidence that MOOCS can be an important means for embracing digital inclusion, in particular, by promoting the development of soft skills (e.g., digital skills, communication skills, interaction skills). First and because the concept is becoming more and more polysemic (due to its manifold uses, theoretical frameworks, and application contexts), the MOOC’s main facets are depicted, considering its derivatives (e.g., cMOOC and xMOOC). Moreover, some critical aspects that stand out from the content analysis of the results of the literature review are also highlighted, namely as to: accessibility, employability and lifelong learning promoted through MOOC use. In general, results suggest that there is still a long way to go for MOOCs to fully address the digital inclusion challenge.

Immersions of Manifolds

Immersions of an m-manifold in an n-manifold, n>m, are classified up to regular homotopy by the homotopy classes of sections of a vector bundle E associated to the tangent bundle of M.  When N = Rn , the fibre of E is the Stiefel manifold of m-frames in n-space.

Immersions of almost parallizable

An almost parallelizable n-manifold M can be immersed in Euclidean q-space if 2q>3n.

On imbedding differentiable manifolds in Euclidean space

Assume n,k,m,q are positive integers. Let M^n denote a smooth differentiable n-manifold and R^k Euclidean k-space. (a) If M^n is open it imbeds smoothly in R^k, k=2n-1 (b) If M^n is open and parallelizable it immerses in R^n (c) Assume M^n is closed and (m-1)-connected, 1< 2m-n < n+1. If a neighborhood of the (n-m)-skeleton immerses in R^q, a>2n-2m, then the complement of a point of M^n imbeds smoothly in R^q.

Immersions in the stable range

Immersions of a differentiable m-manifold M in a differentiable n-manifold N, 2n > 3m+1, are classified up to regular homotopy by the homotopy classes of fibre maps F: T(M) ----> T(N) such that F(-X)=-F(X) and F(X) is nonzero of X is nonzero.

Smooth regular neighborhoods

If K is a subcomplex of a smooth triangulation of a manifold M and N is a regular neighborhood of K contained in an open neighborhood U of K in M, there is a piecewise regular homeomorphism f of M carrying N onto a smooth submanifold of U, and f reduces to the identity map on K and outside U.

On homotopy spheres of low dimension

Let M be a smooth compact manifold homotopy equivalent to the 4-sphere S^4. Then M x R^1 is homeomorphism co S^4 x R^1.

Perceção de emoções no canto: a importância dos estímulos auditivo e visual

Este estudo pretendeu examinar a importância dos estímulos auditivo (interpre-tação vocal do cantor) e visual (expressão facial do cantor) na perceção de emo-ções pelo público de uma performance de canto. Para tal, foram gravados, atra-vés de vídeo e áudio, dois cantores a interpretar pequenas frases melódicas com a intenção de expressar, isoladamente, as seis emoções básicas: alegria, tristeza, raiva, medo, surpresa e nojo. Para validar a expressividade dos canto-res, foi medida, através de eletromiografia, a atividade dos músculos faciais du-rante a performance da emoção e foram apresentadas as gravações áudio a um painel de especialistas que as caracterizaram em termos acústicos. Com base nas gravações audiovisuais dos cantores, foi criado um teste percetual no qual se pretendia que o ouvinte reconhecesse a emoção comunicada a partir apenas do áudio, apenas do vídeo, ou ambos. Comparando as respostas dadas, os re-sultados evidenciaram que o estímulo visual é mais eficaz do que o auditivo, e que a junção dos dois estímulos é a modalidade mais eficiente na perceção de emoções pelo público de uma performance de canto.

Strong shift equivalence, algebraic K-theory, and isolating zero-dimensional dynamics on manifolds

We study the relations of shift equivalence and strong shift equivalence for matrices over a ring $\mathcal{R}$, and establish a connection between these relations and algebraic K-theory. We utilize this connection to obtain results in two areas where the shift and strong shift equivalence relations play an important role: the study of finite group extensions of shifts of finite type, and the Generalized Spectral Conjectures of Boyle and Handelman for nonnegative matrices over subrings of the real numbers. We show the refinement of the shift equivalence class of a matrix $A$ over a ring $\mathcal{R}$ by strong shift equivalence classes over the ring is classified by a quotient $NK_{1}(\mathcal{R}) / E(A,\mathcal{R})$ of the algebraic K-group $NK_{1}(\calR)$. We use the K-theory of non-commutative localizations to show that in certain cases the subgroup $E(A,\mathcal{R})$ must vanish, including the case $A$ is invertible over $\mathcal{R}$. We use the K-theory connection to clarify the structure of algebraic invariants for finite group extensions of shifts of finite type. In particular, we give a strong negative answer to a question of Parry, who asked whether the dynamical zeta function determines up to finitely many topological conjugacy classes the extensions by $G$ of a fixed mixing shift of finite type. We apply the K-theory connection to prove the equivalence of a strong and weak form of the Generalized Spectral Conjecture of Boyle and Handelman for primitive matrices over subrings of $\mathbb{R}$. We construct explicit matrices whose class in the algebraic K-group $NK_{1}(\mathcal{R})$ is non-zero for certain rings $\mathcal{R}$ motivated by applications. We study the possible dynamics of the restriction of a homeomorphism of a compact manifold to an isolated zero-dimensional set. We prove that for $n \ge 3$ every compact zero-dimensional system can arise as an isolated invariant set for a homeomorphism of a compact $n$-manifold. In dimension two, we provide obstructions and examples.

Alternating surgeries

This thesis is concerned with the question of when the double branched cover of an alternating knot can arise by Dehn surgery on a knot in S^3. We approach this problem using a surgery obstruction, first developed by Greene, which combines Donaldson's Diagonalization Theorem with the $d$-invariants of Ozsvath and Szabo's Heegaard Floer homology. This obstruction shows that if the double branched cover of an alternating knot or link L arises by surgery on S^3, then for any alternating diagram the lattice associated to the Goeritz matrix takes the form of a changemaker lattice. By analyzing the structure of changemaker lattices, we show that the double branched cover of L arises by non-integer surgery on S^3 if and only if L has an alternating diagram which can be obtained by rational tangle replacement on an almost-alternating diagram of the unknot. When one considers half-integer surgery the resulting tangle replacement is simply a crossing change. This allows us to show that an alternating knot has unknotting number one if and only if it has an unknotting crossing in every alternating diagram. These techniques also produce several other interesting results: they have applications to characterizing slopes of torus knots; they produce a new proof for a theorem of Tsukamoto on the structure of almost-alternating diagrams of the unknot; and they provide several bounds on surgeries producing the double branched covers of alternating knots which are direct generalizations of results previously known for lens space surgeries. Here, a rational number p/q is said to be characterizing slope for K in S^3 if the oriented homeomorphism type of the manifold obtained by p/q-surgery on K determines K uniquely. The thesis begins with an exposition of the changemaker surgery obstruction, giving an amalgamation of results due to Gibbons, Greene and the author. It then gives background material on alternating knots and changemaker lattices. The latter part of the thesis is then taken up with the applications of this theory.