This practice-based project explores the photographic phenomenon of ‘moment beyond moment’, which refers to the combined representations of an existing image in an environment, together with the real-life situation at the moment the photograph is taken. I call this photograph an ‘extended photograph’. Employing practical works of extended photographs and focusing on interactions between the moment in real-life and the moment in an existing image, the research explores the transformation of meanings caused by the interactions of these moments in an extended photograph. The research owes its approach to grounded theory, contrary thinking and Chinese Buddhist ‘Sudden Enlightenment’ to further its aim of exploring the unpredictable interaction of these moments, and to disclose the potentials of meaning transformation. My research outcome intends to initiate a discourse with photographic practitioners and theorists on the phenomenon of moment beyond moment in a working environment that is encaged by the excessive existence of displayed images. The thesis is composed as a creative work that consists of a series of photographic images accompanied by an exegesis component. The images represent a nominal 80%, and the exegesis 20% of the final submission.

Une partie de cette thèse étudie des produits tensoriels de deux représentations irreductibles d'un groupes algébrique simple. Il s'agit de comprendre les tenseurs pures dans la componente de Cartan du produit.<br />Une partie étudie la propriété de séparation d'un sous-ensemble dans un espace vectoriel complex.

Soit p un nombre premier et G un groupe classique de type B, C ou D defini sur la cloture algebrique K du corps a p elements (si G est de type B ou D, p est impair). A l'aide de paires duales de groupes et de modules basculants, on trouve le caractere de certaines representations rationnelles irreductibles de G sur K. On obtient tout d'abord des formules en termes de tableaux semi-standards, non couvertes par la conjecture de Lusztig. Puis on determine la dimension et/ou le caractere des representations irreductibles de plus haut poids un poids fondamental, ou une somme de deux poids fondamentaux, suivant G. On en deduit notamment le comportement asymptotique de leur dimension, a p fixe, quand le rang du groupe tend vers l'infini. On dresse enfin la liste des modules de Weyl simples de plus haut poids un poids fondamental quand G est un groupe symplectique, ou de plus haut poids la somme d'un poids fondamental et du plus haut poids de la representation spin quand G est un groupe spin.

When studying the molecular aspect of the life sciences, learners must be introduced to somewhat inaccessible phenomena that occur at the sub-micro scale. Despite the difficulties, students need to be familiar with and understand the highly dynamic nature of molecular processes. Thus, external representations1 (ERs) can be considered unavoidable and essential tools for student learning. Besides meeting the challenge of interpreting external representations, learners also encounter a large array of abstract concepts2, which are challenging to understand (Orgill &amp; Bodner, 2004). Both teachers and learners use metaphorical language as a way to relate these abstract phenomena to more familiar ones from everyday life. Scientific papers, as well as textbooks and popular science articles, are packed with metaphors, analogies and intentional expressions. Like ERs, the use of metaphors and analogies is inevitable and necessary when communicating knowledge concerning molecular phenomena. Therefore, a large body of published research related to metaphors concerns science teachers’ and textbook writers’ interpretation and use of metaphors (Harrison &amp; Treagust, 2006). In this paper we present a theoretical framework for examining metaphorical language use in relation to abstract phenomena and external representations. The framework was verified by using it to analyse students’ meaning-making in relation to an animation representing the sub-microscopic and abstract process of ATP-synthesis in Oxidative Phosphorylation. We seek to discover the animator’s intentions while designing the animation and to identify the metaphors that students use while interacting with the animation. Two of these metaphors serve as examples of a metaphor analysis, in which the characteristics of metaphors are outlined. To our knowledge,  no strategies to identify and understand the characteristics, benefits, and potential pitfalls of particular metaphors have, to date, been presented in science education research. Our aspiration is to contribute valuable insights into metaphorical language use at the interface between external representations, molecular processes, and student learning.

A novel framework based on cluster co-occurrence and graph theory for structure discovery is applied to audio to find new types of audio objects which enable the compression of an input signal. These new objects differ from those found in current object coding schemes as their shape is not restricted by any a priori psychoacoustic knowledge. The framework is novel from an application perspective, as it marks the first time that graph theory is applied to audio, and with regards to theoretical developments, as it involves new extensions to the areas of unsupervised learning algorithms and frequent subgraph mining methods. Tests are performed using a corpus of audio files spanning a wide range of sounds. Results show that the framework discovers new types of audio objects which yield average respective overall and relative compression gains of 15.90% and 23.53% while maintaining a very good average audio quality with imperceptible changes.

W. D. Munn proved that a finite dimensional representation of an inverse semigroup is equivalent to a ⋆-representation if and only if it is bounded. The first goal of this thesis will be to give new analytic proof that every finite dimensional representation of a compact inverse semigroup is equivalent to a ⋆-representation. The second goal is to parameterize all finite dimensional irreducible representations of a compact inverse semigroup in terms of maximal subgroups and order theoretic properties of the idempotent set. As a consequence, we obtain a new and simpler proof of the following theorem of Shneperman: a compact inverse semigroup has enough finite dimensional irreducible representations to separate points if and only if its idempotent set is totally disconnected. Our last theorem is the following: every norm continuous irreducible ∗-representation of a compact inverse semigroup on a Hilbert space is finite dimensional.

Around 1990, Arthur proved a local (ordinary) trace formula for real or p-adic connected reductive groups. The local trace formula is a powerful tool in the local harmonic analysis of reductive groups. One of the aims of this thesis is to establish a local twisted trace formula for certain non-connected reductive groups, which is a twisted version of Arthur’s local trace formula. As an application of the local twisted trace formula, we will prove some twisted orthogonality relations, which are generalizations of Arthur’s results about orthogonality relations for tempered elliptic characters. To establish these relations, we will also give a classification of twisted elliptic representations.

Around 1990, Arthur proved a local (ordinary) trace formula for real or p-adic connected reductive groups. The local trace formula is a powerful tool in the local harmonic analysis of reductive groups. One of the aims of this thesis is to establish a local twisted trace formula for certain non-connected reductive groups, which is a twisted version of Arthur’s local trace formula. As an application of the local twisted trace formula, we will prove some twisted orthogonality relations, which are generalizations of Arthur’s results about orthogonality relations for tempered elliptic characters. To establish these relations, we will also give a classification of twisted elliptic representations.

W. D. Munn proved that a finite dimensional representation of an inverse semigroup is equivalent to a ⋆-representation if and only if it is bounded. The first goal of this thesis will be to give new analytic proof that every finite dimensional representation of a compact inverse semigroup is equivalent to a ⋆-representation. The second goal is to parameterize all finite dimensional irreducible representations of a compact inverse semigroup in terms of maximal subgroups and order theoretic properties of the idempotent set. As a consequence, we obtain a new and simpler proof of the following theorem of Shneperman: a compact inverse semigroup has enough finite dimensional irreducible representations to separate points if and only if its idempotent set is totally disconnected. Our last theorem is the following: every norm continuous irreducible ∗-representation of a compact inverse semigroup on a Hilbert space is finite dimensional.

El reconeixement de patrons és la tasca que pretén distingir objectes entre diferents classes. Quan aquesta tasca es vol solucionar de forma automàtica un pas crucial és el com representar formalment els patrons a l'ordinador. En funció d'aquests formalismes, podem distingir entre el reconeixement estadístic i l'estructural. El primer descriu objectes com un conjunt de mesures col·locats en forma del que s'anomena un vector de característiques. El segon assumeix que hi ha relacions entre parts dels objectes que han de quedar explícitament representades i per tant fa servir estructures relacionals com els grafs per codificar la seva informació inherent. Els espais vectorials són una estructura matemàtica molt flexible que ha permès definir diverses maneres eficients d'analitzar patrons sota la forma de vectors de característiques. De totes maneres, la representació vectorial no és capaç d'expressar explícitament relacions binàries entre parts dels objectes i està restrigida a mesurar sempre, independentment de la complexitat dels patrons, el mateix nombre de característiques per cadascun d'ells. Les representacions en forma de graf presenten la situació contrària. Poden adaptar-se fàcilment a la complexitat inherent dels patrons però introdueixen un problema d'alta complexitat computational, dificultant el disseny d'eines eficients per al procés i l'anàlisis de patrons. Resoldre aquesta paradoxa és el principal objectiu d'aquesta tesi. La situació ideal per resoldre problemes de reconeixement de patrons seria el representar-los fent servir estructures relacionals com els grafs, i a l'hora, poder fer ús del ric repositori d'eines pel processament de dades del reconeixement estadístic. Una solució elegant a aquest problema és la de transformar el domini dels grafs en el domini dels vectors, on podem aplicar qualsevol algorisme de processament de dades. En altres paraules, assignant a cada graf un punt en un espai vectorial, automàticament tenim accés al conjunt d'algorismes del món estadístic per aplicar-los al domini dels grafs. Aquesta metodologia s'anomena graph embedding. En aquesta tesi proposem de fer una associació de grafs a vectors de característiques de forma simple i eficient fixant l'atenció en la informació d'etiquetatge dels grafs. En particular, comptem les freqüències de les etiquetes dels nodes així com de les aretes entre etiquetes determinades. Tot i la seva localitat, aquestes característiques donen una representació prou robusta de les propietats globals dels grafs. Primer tractem el cas de grafs amb etiquetes discretes, on les característiques són sencilles de calcular. El cas continu és abordat com una generalització del cas discret, on enlloc de comptar freqüències d'etiquetes, ho fem de representants d'aquestes. Ens trobem que les representacions vectorials que proposem pateixen d'alta dimensionalitat i correlació entre components, i tractem aquests problems mitjançant algorismes de selecció de característiques. També estudiem com la diversitat de diferents representacions pot ser explotada per tal de millorar el rendiment de classificadors base en el marc d'un sistema de múltiples classificadors. Finalment, amb una extensa evaluació experimental mostrem com la metodologia proposada pot ser calculada de forma eficient i com aquesta pot competir amb altres metodologies per a la comparació de grafs. / Pattern recognition is the task that aims at distinguishing objects among different classes. When such a task wants to be solved in an automatic way a crucial step is how to formally represent such patterns to the computer. Based on the different representational formalisms, we may distinguish between statistical and structural pattern recognition. The former describes objects as a set of measurements arranged in the form of what is called a feature vector. The latter assumes that relations between parts of the underlying objects need to be explicitly represented and thus it uses relational structures such as graphs for encoding their inherent information. Vector spaces are a very flexible mathematical structure that has allowed to come up with several efficient ways for the analysis of patterns under the form of feature vectors. Nevertheless, such a representation cannot explicitly cope with binary relations between parts of the objects and it is restricted to measure the exact same number of features for each pattern under study regardless of their complexity. Graph-based representations present the contrary situation. They can easily adapt to the inherent complexity of the patterns but introduce a problem of high computational complexity, hindering the design of efficient tools to process and analyze patterns. Solving this paradox is the main goal of this thesis. The ideal situation for solving pattern recognition problems would be to represent the patterns using relational structures such as graphs, and to be able to use the wealthy repository of data processing tools from the statistical pattern recognition domain. An elegant solution to this problem is to transform the graph domain into a vector domain where any processing algorithm can be applied. In other words, by mapping each graph to a point in a vector space we automatically get access to the rich set of algorithms from the statistical domain to be applied in the graph domain. Such methodology is called graph embedding. In this thesis we propose to associate feature vectors to graphs in a simple and very efficient way by just putting attention on the labelling information that graphs store. In particular, we count frequencies of node labels and of edges between labels. Although their locality, these features are able to robustly represent structurally global properties of graphs, when considered together in the form of a vector. We initially deal with the case of discrete attributed graphs, where features are easy to compute. The continuous case is tackled as a natural generalization of the discrete one, where rather than counting node and edge labelling instances, we count statistics of some representatives of them. We encounter how the proposed vectorial representations of graphs suffer from high dimensionality and correlation among components and we face these problems by feature selection algorithms. We also explore how the diversity of different embedding representations can be exploited in order to boost the performance of base classifiers in a multiple classifier systems framework. An extensive experimental evaluation finally shows how the methodology we propose can be efficiently computed and compete with other graph matching and embedding methodologies.