Let V( ) denote a local system of weight on X = A2;n(C), where X is the moduli space of principle polarized abelian varieties of genus 2 over C with xed n-level structure. The inner cohomology of X with coe cients in V( ), H3 ! (X;V( )), has a Hodge ltration of weight 3. Each term of this Hodge ltration can be presented as space of cuspidal automorphic representations of genus 2. We consider the purely non-holomorphic part of H3 ! (X;V( )) denoted by H3 Ends(X;V( )). First of all we show that there is a non-zero subspace of H3 Ends(X;V( )) denoted by V (K), where K is an open compact subgroup of GSp(4;A), such that elements of V (K) are obtained by the global theta lifting of cuspidal automorphic representations of GL(2) GL(2)=Gm. This means that there is a non-zero part of H3 Ends(X;V( )) which is endoscopic. Secondly, we consider the local theta correspondence and nd an explicit answer for the level of lifted cuspidal automorphic representations to GSp(4; F) over a non-archimedean local eld F. Therefore, we can present an explicit way for nding a basis for V (K) for a xed level structure K. ii There is a part of the Hodge structure that only contributes in H(3;0) ! (X;V( )) H(0;3) ! (X;V( )). This part is endoscopic and coming from the Yoshida lift from O(4). Finally, in the case X = A2, if eendo(A2;V( )) denotes the motive corresponded to the strict endoscopic part (the part that contributes only in non-holomorphic terms of the Hodge ltration), then we have eendo(A2;V( )) = s 1+ 2+4S[ 1 2 + 2]L 2+1; (1) where = ( 1; 2) and is far from walls. Here S[k] denotes the motive corresponded to Sk, the space of cuspidal automorphic forms of weight k and trivial level, and sk = dim(Sk). ii

Let V( ) denote a local system of weight on X = A2;n(C), where X is the moduli space of principle polarized abelian varieties of genus 2 over C with xed n-level structure. The inner cohomology of X with coe cients in V( ), H3 ! (X;V( )), has a Hodge ltration of weight 3. Each term of this Hodge ltration can be presented as space of cuspidal automorphic representations of genus 2. We consider the purely non-holomorphic part of H3 ! (X;V( )) denoted by H3 Ends(X;V( )). First of all we show that there is a non-zero subspace of H3 Ends(X;V( )) denoted by V (K), where K is an open compact subgroup of GSp(4;A), such that elements of V (K) are obtained by the global theta lifting of cuspidal automorphic representations of GL(2) GL(2)=Gm. This means that there is a non-zero part of H3 Ends(X;V( )) which is endoscopic. Secondly, we consider the local theta correspondence and nd an explicit answer for the level of lifted cuspidal automorphic representations to GSp(4; F) over a non-archimedean local eld F. Therefore, we can present an explicit way for nding a basis for V (K) for a xed level structure K. ii There is a part of the Hodge structure that only contributes in H(3;0) ! (X;V( )) H(0;3) ! (X;V( )). This part is endoscopic and coming from the Yoshida lift from O(4). Finally, in the case X = A2, if eendo(A2;V( )) denotes the motive corresponded to the strict endoscopic part (the part that contributes only in non-holomorphic terms of the Hodge ltration), then we have eendo(A2;V( )) = s 1+ 2+4S[ 1 2 + 2]L 2+1; (1) where = ( 1; 2) and is far from walls. Here S[k] denotes the motive corresponded to Sk, the space of cuspidal automorphic forms of weight k and trivial level, and sk = dim(Sk). ii

The study of zeta functions is one of the primary aspects of modern number theory. Hecke was the first to prove that the Dedekind zeta function of any algebraic number field has an analytic continuation over the whole plane and satisfies a simple functional equation. He soon realized that his method would work, not only for Dedekind zeta functions and L–series, but also for a zeta function formed with a new type of ideal character which, for principal ideals depends not only on the residue class of the number(representing the principal ideal) modulo the conductor, but also on the position of the conjugates of the number in the complex field. He then showed that these “Hecke” zeta functions satisfied the same type of functional equation as the Dedekind zeta function, but with a much more complicated factor. In his doctoral thesis, John Tate replaced the classical notion of zeta function, as a sum over integral ideals of a certain type of ideal character, by the integral over the idele group of a rather general weight function times an idele character which is trivial on field elements. He derived a Poisson Formula for general functions over the adeles, summed over the discrete subgroup of field elements. This was then used to give an analytic continuation for all of the generalized zeta functions and an elegant functional equation was established for them. The mention of the Poisson Summation Formula immediately reminds one of the Theta function and the proof of the functional equation for the Riemann zeta function. The two proofs share close analogues with the functional equation for the Theta function now replaced by the number theoretic Riemann–Roch Theorem. Translating the results back into classical terms one obtains the Hecke functional equation, together with an interpretation of the complicated factor in it as a product of certain local factors coming form the archimedean primes and the primes of the conductor. This understanding of Tate’s results in the classical framework essentially boils down to constructing the generalized weight function and idele group characters which are trivial on field elements. This is facilitated by the understanding of the local zeta functions. We explicitly compute in both cases, the local and the global, illustrating the working of the ideas in a concrete setup. I have closely followed Tate’s original thesis in this exposition.

The purpose of this study was to design a local theory explaining the relationship between metacognitive language and networks as constructs of a local instructional theory in the context of a fourth-year intermediate phase mathematics education methodology module. The local instructional theory was designed to facilitate an adapted lesson study through a problem-based learning instructional philosophy. A problem-based learning task was then designed outlining the education needs and resources of a South African primary school, characteristic of schools in a rural area. In particular the task describes a fictitious teacher’s concern for teaching a Grade 6 mathematics class the concept of place value. Two groups of students, who volunteered to participate in this research, collaboratively designed and presented research lessons across two educational design-based research cycles for two rural schools in North West, as a form of service learning. In implementing the local instructional theory phases, participants were required to follow the lesson study approach by investigating, planning, developing, presenting, reflecting, refining and re-presenting the research lesson and its resources. These design sessions were videorecorded, transcribed and then coded in Atlas.ti through interpretivistic and hermeneutic analysis. The coded data were then imported into NodeXL to illustrate embedded networks. Not only social network data but also metacognitive network data were visualised in terms of metacognitive networks. The results show that across the local instructional theory phases, constructs of metacognition, metacognitive language and networking emerged on a social (stratum 1), interpersonal (stratum 2) and social-metacognitive (stratum 3) level. Collectively, these strata form the architecture of the theory of metacognitive locale that explains the relationship between the constructs. The findings suggest that when students express their metacognitive processes through a metacognitive language (e.g. I am thinking or feeling), their interpersonal metacognitive networks develop into shared metacognitive experiences which foster their metacognitive locale, a dimension of their metacognitive language and networking. / PhD (Mathematics Education), North-West University, Potchefstroom Campus, 2015

The purpose of this study was to design a local theory explaining the relationship between metacognitive language and networks as constructs of a local instructional theory in the context of a fourth-year intermediate phase mathematics education methodology module. The local instructional theory was designed to facilitate an adapted lesson study through a problem-based learning instructional philosophy. A problem-based learning task was then designed outlining the education needs and resources of a South African primary school, characteristic of schools in a rural area. In particular the task describes a fictitious teacher’s concern for teaching a Grade 6 mathematics class the concept of place value. Two groups of students, who volunteered to participate in this research, collaboratively designed and presented research lessons across two educational design-based research cycles for two rural schools in North West, as a form of service learning. In implementing the local instructional theory phases, participants were required to follow the lesson study approach by investigating, planning, developing, presenting, reflecting, refining and re-presenting the research lesson and its resources. These design sessions were videorecorded, transcribed and then coded in Atlas.ti through interpretivistic and hermeneutic analysis. The coded data were then imported into NodeXL to illustrate embedded networks. Not only social network data but also metacognitive network data were visualised in terms of metacognitive networks. The results show that across the local instructional theory phases, constructs of metacognition, metacognitive language and networking emerged on a social (stratum 1), interpersonal (stratum 2) and social-metacognitive (stratum 3) level. Collectively, these strata form the architecture of the theory of metacognitive locale that explains the relationship between the constructs. The findings suggest that when students express their metacognitive processes through a metacognitive language (e.g. I am thinking or feeling), their interpersonal metacognitive networks develop into shared metacognitive experiences which foster their metacognitive locale, a dimension of their metacognitive language and networking. / PhD (Mathematics Education), North-West University, Potchefstroom Campus, 2015

Dans cet exercice, on essayera de produire une véritable théorie locale ou spéciale du précédent judiciaire pour Colombie. Compte tenu de la profonde transformation du rôle de la jurisprudence contentieuse administrative au cours des dernières années, abandonnant son rôle indicatif et assumant une valeur normative de précédent judiciaire, il est nécessaire d'analyser le précédent contentieux administratif d'après la culture des opérateurs juridiques en Colombie, les fondations doctrinales et jurisprudentielles de l'existence du précédent en Colombie et les objectifs du Code de Procédure Administratif.Ce travail a pour but de tester l'hypothèse selon laquelle en Colombie, les juges du contentieux administratifs et les autorités administratives, afin d'identifier et d'appliquer rationnellement le précédent judiciaire contenu dans les arrêts d'unification Jurisprudentielle du Conseil d'Etat pour résoudre les cas qui viennent à leur connaissance, devraient appliquer des critères théoriques et pratiques d'une théorie locale du précédent, jusque-là Inexistante. Pour cela, on souhaite montrer que cette théorie devrait aborder (i) les contributions théoriques à la doctrine transnationale du précédent judiciaire et les flux légaux ayant donné lieu à la figure de précédent en Colombie, (li) la base théorique et constitutionnelle du respect pour le précédent en Colombie ainsi que (iii) les objectifs de la création des « arrêts d’unification jurisprudentielle » dans la loi 1437 de 2011. / This dissertation produces a local or special theory of judicial precedents for Colombia. Public case law in Colombia has departed from its prior persuasive role and has acquired a normative value as judicial precedent. Given the deep transformation of the role of public case law in recent years, it is necessary to analyze the judicial precedents in light of the Colombian legal culture, the doctrinal and jurisprudential foundations of the existence of the precedent in Colombia and the objectives• pursued by the Code of Administrative Procedure. This work aims to test the hypothesis that public authorities and public law judges in Colombia should apply theoretical and practical criteria of a local theory of the precedent, hitherto nonexistent. The latter is necessary for them to identify and rationally apply the judicial precedents contained in the unification rulings of the Council of State (Conseil d'Etat) to solve the cases that come to their knowledge. This theory should address (i) the theoretical contributions to the transnational doctrine of the judicial precedent and the legal flows that gave rise to precedents in Colombia, (ii) the theoretical and constitutional basis for respecting precedents in Colombia as well as (iii) the objectives of law 1437 of 2011 when creating unification rulings of the Council of State.

Ce travail présente et valide une théorie nonlocale nouvelle et généralisée, de la propagation acoustique dans les milieux poreux à structure rigide, saturés par un fluide viscothermique. Cette théorie linéaire permet de dépasser les limites de la théorie classique basée sur la théorie de l'homogénéisation. Elle prend en compte non seulement les phénomènes de dispersion temporelle, mais aussi ceux de dispersion spatiale. Dans le cadre de la nouvelle approche, une nouvelle procédure d'homogénéisation est proposée, qui permet de trouver les propriétés acoustiques à l'échelle macroscopique, en résolvant deux problèmes d'action-réponse indépendants, posés à l'échelle microscopique de Navier-Stokes-Fourier. Contrairement à la méthode classique d'homogénéisation, aucune contrainte de séparation d'échelle n'est introduite. En l'absence de structure solide, la procédure redonne l'équation de dispersion de Kirchhoff-Langevin, qui décrit la propagation des ondes longitudinales dans les fluides viscothermiques. La nouvelle théorie et procédure d'homogénéisation nonlocale sont validées dans trois cas, portant sur des microgéométries significativement différentes. Dans le cas simple d'un tube circulaire rempli par un fluide viscothermique, on montre que les nombres d'ondes et les impédances prédits par la théorie nonlocale, coïncident avec ceux de la solution exacte de Kirchhoff, connue depuis longtemps. Au contraire, les résultats issus de la théorie locale (celle de Zwikker et Kosten, découlant de la théorie classique d'homogénéisation) ne donnent que le mode le plus attenué, et encore, seulement avec le petit désaccord existant entre la solution simplifiée de Zwikker et Kosten et celle exacte de Kirchhoff. Dans le cas où le milieu poreux est constitué d'un réseau carré de cylindres rigides parallèles, plongés dans le fluide, la propagation étant regardée dans une direction transverse, la vitesse de phase du mode le plus atténué peut être calculée en fonction de la fréquence en suivant les approches locale et nonlocale, résolues au moyen de simulations numériques par la méthode des Eléments Finis. Elle peut être calculée d'autre part par une méthode complètement différente et quasi-exacte, de diffusion multiple prenant en compte les effets viscothermiques. Ce dernier résultat quasi-exact montre un accord remarquable avec celui obtenu par la théorie nonlocale, sans restriction de longueur d'onde. Avec celui de la théorie locale, l'accord ne se produit que tant que la longueur d'onde reste assez grande.