In dieser Doktorarbeit untersuchen wir einige Modulräume der Prym-Paaren, Prym-Varietäten und Spin-Kurven. Nachdem der passende theoretische Rahmen eingeführt wird, erhalten wir neue Ergebnisse zu zwei verschiedenen Aspekten ihrer Geometrie, die wir in zwei entsprechenden Kapiteln beschreiben. In Kapitel 1 betrachten wir die universelle Prym-Varietät über dem Modulraum R_g der Prym-Paaren vom Geschlecht g und bestimmen ihre Unirationalität für g=3. Dazu bilden wir eine explizite rationale Parametrisierung der universellen 2-fachen Prym-Kurve über R_3, die die universelle Prym-Varietät durch die globale Version der Abel-Prym-Abbildung dominiert. Darüber hinaus passen wir den Beweis an den Rahmen von Nikulin-Flächen an und zeigen, dass die universelle doppelte Nikulin-Fläche ebenfalls unirational ist. In Kapitel 2 untersuchen wir die Wechselwirkung zwischen R_g und dem Modulraum S_g der (stabilen) Spin-Kurven vom Geschlecht g. Wenn man den Divisor der Kurven, die mit einem verschwindenden Thetanull ausgestattet sind, von S_g^+ nach R_g versetzt, erhält man zwei geometrische Divisoren der (stabilen) Prym-Kurven mit einem verschwindenden Thetanull. Wir verwenden Testkurventechniken, um die Klassen dieser (Prym-Null-)Divisoren für g>=5 zu berechnen, und werten die Prymnull-Klassen auf einigen weiteren Familien von Kurven aus, um ihre verschwindenden Thetanulls zu analysieren. Darüber hinaus diskutieren wir am Ende von Kapitel 2 eine mögliche Kompaktifizierung des Modulraums der Kurven, die eine doppelte Quadratwurzel tragen. Anschließend untersuchen wir den Rand des Modulraums RS_g der (stabilen) Prym-Spin-Kurven vom Geschlecht g und überprüfen die Prymnull-Klassen anhand des Diagramms R_g<--RS_g-->S_g. Zum Schluss schlagen wir eine Erweiterung des Produkts von Wurzeln, das über glatten Kurven durch das Tensorprodukt definiert ist, zu einer Operation auf stabilen Doppelwurzeln vor. / In this thesis, we study several moduli spaces of Prym pairs, Prym varieties, and spin curves. After the appropriate theoretical framework is introduced, we obtain new results concerning two different aspects of their geometry, which we describe across two corresponding chapters. In Chapter 1, we consider the universal Prym variety over the moduli space R_g of Prym pairs of genus g, and determine its unirationality for g=3. To do this, we build an explicit rational parametrization of the universal 2-fold Prym curve over R_3, which dominates the universal Prym variety through the global version of the Abel-Prym map. Furthermore, we adapt the proof to the setting of Nikulin surfaces and show that the universal double Nikulin surface is also unirational. In Chapter 2, we explore the interaction between R_g and the moduli space S_g of (stable) spin curves of genus g. When the divisor of curves equipped with a vanishing theta-null is moved from S_g^+ to R_g, it yields two geometric divisors of (stable) Prym curves with a vanishing theta-null. We use test curve techniques to compute the classes of these (Prym-null) divisors for g>=5, and evaluate the Prym-null classes on some more families of curves in order to analyse their vanishing theta-nulls. In addition, at the end of Chapter 2 we discuss a potential compactification of the moduli space of curves carrying a double square root. We then examine the boundary of the moduli space RS_g of (stable) Prym-spin curves of genus g and check the Prym-null classes against the diagram R_g<--RS_g-->S_g. Finally, we propose an extension of the product of roots, defined over smooth curves by the tensor product, to an operation on stable double roots.

We define and study the fundamental pro-finite 2-groupoid of varieties X defined over a field k. This is a higher algebraic invariant of a scheme X, analogous to the higher fundamental path 2-groupoids as defined for topological spaces. This invariant is related to previously defined invariants, for example the absolute Galois group of a field, and Grothendieck’s étale fundamental group. The special case of Brauer-Severi varieties is considered, in which case a “sections conjecture” type theorem is proved. It is shown that a Brauer-Severi variety X has a rational point if and only if its étale fundamental 2-groupoid has a special sort of section.

This thesis discusses three distinct topics. A topological space X is said to be self- Tietze if for every closed C eX, every continuous f: C -+ X admits a continuous extension F: X -+ X. We show that every disconnected, self- Tietze space is ultranormal. The Tychonoff Plank is an example of a compact self- Tietze space which is not completely normal, and we establish that a completely normal, zero- dimensional, homogeneous space need not be self- Tietze. A subset of the plane is a two-point set if it meets every straight line in exactly two points. We show that a two-point set cannot contain a dense G8 subset of an arc. We also show that the complement of a two-point set is necessarily path-connected. Finally, we construct a zero-dimensional subset of the plane of which the complement is simply-connected. For A E lR, the tent-map with slope A is the function f: [0, 1] -+ lR such that f(x) = AX for x :=:; ~ and f(x) = A(l - x) for x ~ ~. Properties of w-limit sets of tent-maps, i.e. sets of the form n {fn+k(x) I kEN} nEN for x E [0,1], are examined, and an example of a tent-map and a closed, invariant, nonempty, internally chain transitive subset of [0, 1] which is not an w-limit set is given.

This research examines the understandings and practices of inclusive minded principals toward facilitating the development of inclusive school environments for intermediate level (Grade 7 and 8) students who are experiencing behavioural problems in their schools. Qualitative interviews with 16 principals across 4 school districts were conducted to explore how these inclusive minded principals conceptualize and understand the needs of this particular group of students, and what they consider to be their roles and responsibilities as principals in meeting these needs. The data suggest that despite the number of barriers that serve to hamper principals’ efforts to develop the ideal inclusive school, there are a great many strategies principals intentionally use to facilitate change toward more inclusive school cultures and pedagogy. These strategies emanate from, and are reflective of, an inclusive philosophy that is common among participants. Principals’ individual philosophies and ideologies serve as a compass in guiding decision-making and actions that affect staff, students, and the wider school community. In an inclusive school, these ideologies are reflective of the principles of inclusion, such as the need to create a culture of care wherein all students feel valued, supported, and experience a sense of belonging and individual self worth. The implications of this research toward improving the schooling experiences of students with behavioural problems as well as other marginalized groups of learners are discussed in the context of the call for a re-culturing of schools toward more inclusive environments.

This research examines the understandings and practices of inclusive minded principals toward facilitating the development of inclusive school environments for intermediate level (Grade 7 and 8) students who are experiencing behavioural problems in their schools. Qualitative interviews with 16 principals across 4 school districts were conducted to explore how these inclusive minded principals conceptualize and understand the needs of this particular group of students, and what they consider to be their roles and responsibilities as principals in meeting these needs. The data suggest that despite the number of barriers that serve to hamper principals’ efforts to develop the ideal inclusive school, there are a great many strategies principals intentionally use to facilitate change toward more inclusive school cultures and pedagogy. These strategies emanate from, and are reflective of, an inclusive philosophy that is common among participants. Principals’ individual philosophies and ideologies serve as a compass in guiding decision-making and actions that affect staff, students, and the wider school community. In an inclusive school, these ideologies are reflective of the principles of inclusion, such as the need to create a culture of care wherein all students feel valued, supported, and experience a sense of belonging and individual self worth. The implications of this research toward improving the schooling experiences of students with behavioural problems as well as other marginalized groups of learners are discussed in the context of the call for a re-culturing of schools toward more inclusive environments.

Esta Tesis Doctoral versa sobre el aspecto sagrado de la Arquitectura, tanto de la Obra en sí como del proceso proyectual. En este sentido, se ha dicho y demostrado que la Arquitectura, para poder ser considerada como tal, es una física desprendida de una metafísica. Y se ha visto que en absoluto esto debe considerarse como una forma de pasado sino, por el contrario, una posibilidad siempre actual en la medida que son los símbolos, ritos y mitos los soportes para que la Arquitectura exprese aquellas ideas universales que, en virtud misma de su universalidad, lo son para todas las generalidades. Fundamental ha sido la distinción entre símbolo y signo, viendo que, si bien desde Aristóteles, y Hegel como último exponente, el símbolo se ha considerado un modo convencional de significación, en realidad y por el contrario, debe tomarse al símbolo como un código revelado y que es precisamente a este código al que hace referencia la Arquitectura o la Simbólica arquitectónica, que es lo mismo.Al considerar la Arquitectura como un hecho simbólico en sí mismo me ha interesado los invariantes, aquella Forma que, siguiendo a Kahn, no tiene figura ni dimensión. El objeto de la Tesis responde, pues, a una pregunta: ¿qué es (tì estín) la Arquitectura? -es decir, ha sido temático el Orden-. No ha sido temático cómo es (poiòn estín) la Arquitectura -es decir, el Diseño-. Esta distinción, prestada de Platón supone preguntarse por la sustancia (naturaleza o esencia) de la cosa no por su accidente (cualidades, propiedades o atributos).Luego, todo Edificio proyectado y necesariamente construido según criterios estrictamente tradicionales conforma un paradigma cosmológico y, en tanto que tal, un "mapa de conocimiento", una "piedra viva", luego un símbolo. Dado que la Metafísica a la cual me he referido es revelación, hay un componente mistérico inherente a la propia práctica del Oficio; luego, el individuo capaz de desarrollar el Oficio requiere de una iniciación. Y es precisamente la cuestión iniciática lo que me ha llevado a considerar la Arquitectura como una propedéutica, en igualdad de condiciones con aquella Filosofía no racional esgrimida por Pitágoras y Platón. Entonces, se ha visto que si la Filosofía no racional es un deseo erótico para con Sophia la Arquitectura participa de este eros, siendo su objeto la Luz, otra de sus figuraciones simbólicas. Además, se ha demostrado que la Simbólica arquitectónica ha sido un instrumento eficaz, diría que imprescindible, para los filósofos prearistotélicos y para aquellos que después del estarigita participaron de aquella corriente intelectual. Se ha visto, aunque entiendo que para muchos esto es harto discutible, acaso incomprensible y extravagante, que contemplar la Arquitectura como un hecho a-histórico ha desvelado su trabazón con una corriente intelectual que sitúa la Simbólica arquitectónica como un lenguaje excepcional para revelar y transmitir las ideas metafísicas.EL DOCUMENT "TRIPTIC" TÉ FORMAT AUTOCAD

Esta tesis se divide en dos partes. La primera parte contiene el estudio de tres pesos o profundidades, asociados a conjuntos finitos de puntos en el plano: el peso definido por las capas convexas, convex depth (introducido por Hubert (72) y Barnett (76)), la separabilidad lineal, también conocido por location, halfspace o Tukey depth (Tukey 75) y el peso Delaunay (Green 81). De la noción de peso, se obtiene una estratificación de los conjuntos de puntos en el plano en capas y una partición del plano en regiones o niveles, cuyas fronteras son conocidas por depth contours. Se definen los conceptos de capa y nivel en los tres pesos señalados y se estudian sus propiedades y complejidades. Chazelle obtuvo métodos para hallar en tiempo óptimo las capas convexas, que coinciden con las fronteras de los niveles convexos. En esta tesis, para los pesos de separabilidad lineal y Delaunay, se proporcionan algoritmos de obtención, tanto de capas como de niveles, y de cálculo del peso de un punto nuevo que se incorpore a la nube. De forma independiente, han sido obtenidos para el peso de la separabilidad lineal los algoritmos de construcción de los niveles, location depth contours, y el de cálculo del peso de un punto nuevo, por Miller et al. (01). Para los tres pesos mencionados, se analizan árboles generadores, poligonizaciones o triangulaciones, con peso mínimo, donde el peso se ha considerado como la suma de los pesos de las aristas de dichas estructuras. Se obtienen propiedades generales entorno a la caracterización de tales estructuras y algoritmos de obtención para alguna de ellas. Se definen dos pesos relacionados con la separabilidad mediante cuñas: el peso según dominación isotética y la separabilidad &#61537;. En ambos, se dan algoritmos para el cálculo de los pesos de los puntos de un conjunto dado. La separabilidad &#61537; está estrechamente relacionada con la enumeración eficiente de (&#61537;,k)-sets. Se realiza un estudio combinatorio del conjunto de (&#61537;,k)-sets para nubes de puntos en el plano y se describen algoritmos de construcción de todos los (&#61537;,k)-sets en cada uno de los cuatro casos posibles, según sean, &#61537; o k, fijos o variables. En la segunda parte, se tratan diversos problemas de transversalidad. Se obtienen resultados acerca de la caracterización de las permutaciones realizables, tanto como polígonos simples, como convexos, sobre arreglos de rectas. Para colecciones de segmentos en el plano, se definen cuña y círculo transversales separadores. Se realiza un análisis del orden de estos elementos transversales separadores y se obtienen diversos algoritmos de decisión de existencia de los mismos y construcción de todos ellos. Para colecciones de círculos, también se define el círculo transversal separador y se obtiene un algoritmo de existencia y construcción de dichos círculos para círculos con el mismo radio. / This thesis can be divided into two parts. The first part contains the study of three weights or depths associated to finite point sets in the plane: the convex depth convex hull peeling depth (introduced by Hubert (72) and Barnett (76)), the location depth (also known by halfspace or Tukey depth (Tukey (75)), and the Delaunay depth (Green (81)).From any notion of depth, a stratification of the point sets of the plane into layers and a partition of the plane into regions or levels are obtained. The boundaries of the levels are known by depth contours. We define the concepts of layers and levels for all three depths and we study their properties and their complexities. Chazelle obtained methods to find the layers, which are the boundaries of the convex levels, with an optimal time algorithm. We present the algorithms for constructing the layers and levels, in location and Delaunay depths. Also, for both depths, we show algorithms to calculate the depth of a new point joining the cloud. In an independent way, the algorithms to obtain the levels (location depth contours) and to calculate the location depth of a new point, are obtained by Miller et al. (01).For each one of the three mentioned depths, we study the geometric structures (spanning trees, polygonizations and triangulations) with minimum weight, where this weight has been considered as t-weight (the addition of the weight of their edges). We obtain general properties about the characterization of such structures and some algorithms to obtain them. We define two depths related with the separability by wedges: the isothetic-domination and the &#61537;-separability which generalizes the location depth. We develop the algorithms in order to obtain the depths of all points of a given set in both cases. The &#61537;-separability (in particular the location depth) is closely related with the efficient enumeration of the (&#61537;,k)-sets. We make a combinatorial study of the (&#61537;,k)-sets for point sets in the plane. We give lower and upper bounds for the maximum number of the (&#61537;,k)-sets and we give algorithms for constructing all them, in each one of the four cases according to the case where &#61537; or k are fixed or variable.In the second part, we consider some transversality problems. We obtain results about the characterization of the realizable permutations both as simple and as convex polygons, over arrangements of lines. We also study some transversality problems with wedges and circles. We have defined the separating transversal wedge and the separating transversal circle for sets of segments. We analyze the size of the set of the transversal elements. Furthermore, we obtain some decision algorithms on the existence and construction of all of them. Finally, we define also the separating transversal circle for sets of circles and we obtain an algorithm for sets of circles with the same radius.

This study aims to determine the recharge, discharge and the mixing mechanisms of a complex aquifer system located above the Kazan trona ore field using the environmental isotopes of deuterium, oxygen-18, carbon-13 and carbon-14, chlorofluorocarbons (CFC- 11, CFC-12 and CFC-113) and the noble gas isotopes (He, Ne, Ar, Kr and Xe). The groundwater system consists of three different aquifers: shallow, middle and deep. The Akpinar formation lying between deep and middle systems acts as an aquitard. Oxygen-18 and deuterium data showed an isotopic contrast between the shallow and deeper aquifer systems and even between the unconfined and confined parts of the middle and deep aquifers. The Noble gas temperatures indicated the average yearly air temperatures in shallow aquifer system whereas the recharge temperatures came out to be lower than todays in deep groundwater system. This finding is also supported by the dissolved inorganic carbon&rsquo / s radiocarbon activities being close to the detection limits in the same system. These activities together with the stable isotope data revealed there might be evidence of recharge to the middle and deep aquifer systems under colder climate conditions during the late Pleistocene. CFC concentrations indicated modern recharge to the shallow aquifer system, whereas the concentrations were close to the detection limits therefore CFC&rsquo / s were unable to date the middle and deep aquifer systems however proved the existence of modern recharge to this system. Mantle-He escape to shallow aquifer system is believed to be along a deep buried fault system located in downgradient areas.

This dissertation revolves around the idea of linear interference alignment (IA) for a network consisting of several mutually interfering transmitter-receiver pairs, which is commonly known as interference channel. In particular, we consider the case where nodes are equipped with multiple antennas and exploit the spatial dimension to perform interference alignment. This work explores the problem of linear spatial domain interference alignment in three different facets. Our first contribution is to analyze the conditions, i.e., number of antennas, users and streams, under which IA is feasible. For this task, we distinguish between systems in which each user transmits a single stream of information (single-beam systems) and those in which multiple streams per user are transmitted (multi-beam systems). For single-beam systems, we show that the problem admits a closed-form solution with a time-complexity that is linear in the number of users. For multi-beam systems, we propose a numerical feasibility test that completely settles the question of IA feasibility for arbitrary networks and is shown to belong to the bounded-error probabilistic polynomial time (BPP) complexity class. The second contribution consists in generalizing the aforementioned feasibility results to characterize the number of existing IA solutions. We show that different IA solutions can exhibit dramatically different performances and, consequently, the number of solutions turns out to be an important metric to evaluate the ability of a system to improve its performance in terms of sum-rate or robustness while maintaining perfect IA. Finally, our contributions conclude with the design of two algorithms for the computation of IA solutions. / Esta tesis gira en torno a la idea de alineado de interferencias (interference alignment, IA) lineal en redes donde varios pares transmisor-receptor se comunican simultáneamente; escenario conocido como canal de interferencia. En particular, se considera el caso en el que cada nodo (ya sea transmisor o receptor) está equipado con varias antenas y hace uso de la dimensión espacial para llevar a cabo el citado alineado de interferencias. En esta tesis se explora el problema del alineado de interferencias en el dominio espacial desde tres puntos de vista diferentes. En primer lugar, se analizan las condiciones (número de antenas, usuarios y flujos de información) bajo las cuales el alineado de interferencias es posible. Para esta tarea, se distingue entre sistemas en los que cada usuario envía uno o múltiples flujos de información. En el primer caso, se demuestra que el problema admite una solución cerrada que puede ser evaluada con complejidad lineal en el número de usuarios. En el segundo caso, se propone un test numérico que da una respuesta concluyente al problema y muestra que el problema pertenece a la clase de complejidad BPP. En segundo lugar, los resultados anteriores son generalizados para calcular el número de soluciones existentes. En ocasiones, no sólo interesa determinar si el problema de alineado de interferencias tiene solución o no, sino que es interesante conocer cuántas soluciones existen. En esta tesis se muestra que diferentes soluciones pueden exhibir resultados dramáticamente diferentes. Por consiguiente, el número de soluciones actúa como una métrica de diversidad que refleja la capacidad de una red para mejorar su rendimiento en términos de tasa suma, robustez o cualquier otra métrica. Por último, se proponen dos algoritmos para la obtención de soluciones de alineado de interferencias.

We investigate an explicit tamed Euler scheme of stochastic differential equation with random coefficients driven by Lévy noise, which has super-linear drift coefficient. The strong convergence property of the tamed Euler scheme is proved when drift coefficient satisfies one-sided local Lipschitz condition whereas diffusion and jump coefficients satisfy local Lipschitz conditions. A rate of convergence for the tamed Euler scheme is recovered when local Lipschitz conditions are replaced by global Lipschitz conditions and drift satisfies polynomial Lipschitz condition. These findings are consistent with those of the classical Euler scheme. New methodologies are developed to overcome challenges arising due to the jumps and the randomness of the coefficients. Moreover, as an application of these findings, a tamed Euler scheme is proposed for the stochastic delay differential equation driven by Lévy noise with drift coefficient that grows super-linearly in both delay and non-delay variables. The strong convergence property of the tamed Euler scheme for such SDDE driven by Lévy noise is studied and rate of convergence is shown to be consistent with that of the classical Euler scheme. Finally, an explicit tamed Milstein scheme with rate of convergence arbitrarily close to one is developed to approximate the stochastic differential equation driven by Lévy noise (without random coefficients) that has super-linearly growing drift coefficient.