Consider distinct latin squares, L and M, of order n. Then the pair (T1, T2) where T1 = L \M and T2 = M \ L is called a latin bitrade. Furthermore T1 and T2 are referred to as latin trades, in which T2 is said to be a disjoint mate of T1 (and vice versa). Drápal (1991) showed that, under certain conditions, a partition of an equilateral triangle of side length n, where n is some integer, into smaller, integer length sided equilateral triangles gives rise to a latin trade within the latin square based on the addition table for the integers modulo n. A partial latin square P of order n is said to be completable if there exists a latin square L of order n such that P ⊆ L. If there is only one such possible latin square, L, of order n then P is said to be uniquely completable and P is called a defining set of L. Furthermore, if C is a uniquely completable partial latin square such that no proper subset of C is uniquely completable, C is said to be a critical set or a minimal defining set. These concepts, namely latin trades and defining sets in latin squares, are intimately connected by the following observation. If L is a latin square and D ⊆ L is a defining set, then D intersects all latin bitrades for which one mate is contained in L. In Part I of this thesis Dr´apal’s result is applied to investigate the structure of certain defining sets in latin squares. The results that are obtained are interesting in themselves; furthermore, the geometric approach to the problem yields additional appealing results. These geometric results are discussed in Chapters 3, 4, 5 and 6. They pertain to partitioning regions (polygons in R2 that satisfy certain obvious conditions) into equilateral, integer length sided, triangles, such that no point, in the region, is a corner of more than three distinct triangles. In Chapter 2 one of the main two theorems on defining sets is established, as is a method for using the above geometric results to prove the nonexistence of certain types of defining sets. In Part II of this thesis, intersection problems, for latin squares and Steiner triple systems, are considered. The seminal works, for problems of these types, are due to Lindner and Rosa (1975) and Fu (1980). A natural progression, from the established literature, for intersection problems between elements in a pair of latin squares or Steiner triple systems is to problems in which the intersection is composed of a number of disjoint configurations (isomorphic copies of some specified partial triple system). In this thesis solutions to two intersection problems for disjoint configurations are detailed. An m-flower, (F,F), is a partial triple system/configuration, such that: F = {{x, yi, zi} | {yi, zi} ∩ {yj , zj} = ∅, for 0 ≤ i, j ≤ m − 1, i 6= j}; and F = UX∈FX. The first such problem considered in this thesis asks for necessary and sufficient conditions for integers k and m ≥ 2 such that a pair of latin squares of order n exists that intersect precisely in k disjoint m-flowers. The necessary terminology, constructions, lemmas and proof for this result are contained in Chapters 7, 8 and 9. The second such problem considered in this thesis asks for necessary and sufficient conditions for integers k such that a pair of Steiner triple systems of order u exists that intersect precisely in k disjoint 2-flowers. This result relies on the solution to the latin square problem and an additional result from Chapter 9. The further constructions and lemmas used to prove this result are detailed in Chapter 10.

The theory of relational complexity (RC) developed by Halford and his associates (Halford et al., 1998a) proposes that, in addition to the number of unique entities that can be processed in parallel, it is the structure (complexity) of the relations between these entities that most appropriately captures the essence of processing capacity limitations. Halford et al. propose that the relational complexity metric forms an ordinal scale along which both task complexity and an individual’s processing capacity can be ranked. However, the underlying quantitative structure of the RC metric is largely unknown. It is argued that an assessment of the measurement properties of the RC metric is necessary to first demonstrate that the scale is able to rank order task complexity and cognitive capacity in adults. If in addition to ordinal ranking, it can be demonstrated that a continuous monotonic scale underlies the ranking of capacity (the natural extension of the complexity classification), then the potential to improve our understanding of adult cognition is further realised. Using a combination of cognitive psychology and individual differences methodologies, this thesis explores the psychometric properties of RC in three high level reasoning tasks. The Knight-Knave Task and the Sentence Comprehension Task come from the psychological literature. The third task, the Latin Square Task, was developed especially for this project to test the RC theory. An extensive RC analysis of the Knight-Knave Task is conducted using the Method for Analysis of Relational Complexity (MARC). Processing in the Knight-Knave Task has been previously explored using deduction-rules and mental models. We have taken this work as the basis for applying MARC and attempted to model the substantial demands these problems make on limited working memory resources in terms of their relational structure. The RC of the Sentence Comprehension Task has been reported in the literature and we further review and extend the empirically evidence for this task. The primary criterion imposed for developing the Latin Square Task was to minimize confounds that might weaken the identification and interpretation of a RC effect. Factors such as storage load and prior experience were minimized by specifying that the task should be novel, have a small number of general rules that could be mastered quickly by people of differing ages and abilities, and have no rules that are complexity level specific. The strength of MARC lies in using RC to explicitly link the cognitive demand of a task with the capacity of the individual. The cognitive psychology approach predicts performance decrements with increased task complexity and primarily deals with aggregated data across task condition (comparison of means). It is argued however that to minimise the subtle circularity created by validating a task’s complexity using the same information that is used to validate the individual’s processing capacity, an integration of the individual differences approach is necessary. The first major empirical study of the project evaluates the utility of the traditional dual-task approach to analyse the influence of the RC manipulation on the dual-task deficit. The Easy-to-Hard paradigm, a modification of the dual-task methodology, is used to explore the influence of individual differences in processing capacity as a function of RC. The second major empirical study explores the psychometric approach to cognitive complexity. The basic premise is that if RC is a manipulation of cognitive complexity in the traditional psychometric sense, then it should display similar psychometric properties. That is, increasing RC should result in an increasing monotonic relationship between task performance and Fluid Intelligence (Gf) – the complexity-Gf effect. Results from the comparison of means approach indicates that as expected, mean accuracy and response times differed reliably as a function of RC. An interaction between RC and Gf on task performance was also observed. The pattern of correlations was generally not consistent across RC tasks and is qualitatively different in important ways to the complexity-Gf effect. It is concluded that the Latin Square Task has sufficient measurement properties to allows us to discuss (i) how RC differs from complexity in tasks in which expected patterns of correlations are observed, (ii) what additional information needs to be considered to assist with the a priori identification of task characteristics that impose high cognitive demand, and (iii) the implications for understanding reasoning in dynamic and unconstrained environments outside the laboratory. We conclude that relational complexity theory provides a strong foundation from which to explore the influence of individual differences in performance further.

L’amélioration de la qualité et de l’efficacité en santé est un réel enjeu sociétal. Elle implique la surveillance continue des paramètres vitaux ou de l’état mental du sujet. Les champs d’applications sont vastes : l’application la plus importante est la surveillance des patients à distance. Les avancées en micro-électronique, capteurs et réseaux sans-fil permettent aujourd’hui le développement de systèmes ambulatoires performants pour le monitoring de paramètres physiologiques, capables de prendre en compte d’importantes contraintes techniques : forte intégration pour la réduction de la taille et faible consommation pour une plus grande autonomie [1]. Cependant, la conception de ce type de réseaux de capteurs médicaux WBANs (Wireles Body Area Networks) se heurte à un certain nombre de difficultés techniques, provenant des contraintes imposées par les capacités réduites des capteurs individuels : basse puissance, énergie limitée et faible capacité de stockage. Ces difficultés requièrent des solutions différentes, encore très embryonnaires, selon l’application visée (monitoring à but médical). La forte mobilité et le changement rapide de la topologie du réseau dévoilent un verrou scientifique et social. En outre, l’interférence de différents capteurs constituant le WBAN augmente la difficulté de la mise en place de ce type de réseaux. De nombreuses solutions dans la littérature ont été étudiées, comme nous allons illustrer dans ce manuscrit, néanmoins elles restent limitées. Nous nous intéresserons tout particulièrement à la gestion des interférences Intra- et Inter-WBAN, leur impacte sur la fiabilité des transmissions (des liens) et la durée de vie de ce type de réseaux. Plus précisément, nous abordons ces problématiques en se basant sur des modélisations théoriques et analytiques et avec une conception pratique des solutions proposées. Afin d’atteindre les objectifs cités ci-dessous, nous abordons quatre solutions : • Une gestion des interférences intra-WBAN • Une gestion coopérative des interférences Inter-WBAN • Une gestion non coopérative des interférences, Inter-WBAN • Une gestion des interférences WBAN dans un contexte IoT Dans la première partie de cette thèse et afin de répondre en partie aux problèmes de gestion des interférences Intra-WBAN. Nous présentons deux mécanismes pour le WBAN : (a) CFTIM qui alloue dynamiquement des slots et des canaux dit- stables (avec un taux d’interférences le bas possible dans le temps) pour réduire les interférences intra-WBAN. (b) IAA ajuste dynamiquement la taille du superframe et limite le nombre de canaux à 2 pour abaisser les interférences Intra-WBAN et ainsi économiser l’énergie. Une validation avec un model probabiliste est proposé afin de valider théoriquement l’efficacité de notre solution. Les résultats de la simulation démontrent l’efficacité du CFTIM et de l’IAA en termes de réduction de la probabilité d’interférence, l’extension de la durée de vie du réseau et l’amélioration du débit et de la fiabilité des transmissions. Notre seconde contribution, propose une gestion coopératives des interférences Inter-WBAN en utilisant des codes orthogonaux. Motivé par un approvisionnement temporel distribué basé sur la norme [2] IEEE 802.15.6, nous proposons deux solutions. (a) DTRC qui fournit à chaque WBAN les connaissances sur les superframes qui se chevauchent. Le second, (b) OCAIM qui attribue des codes orthogonaux aux capteurs appartenant à deux listes de groupe de capteur en interférences de deux WBAN différents (SIL). Les résultats démontrent qu’OCAIM diminue les interférences, améliore le débit et préserve la ressources énergétiques. La troisième partie nous a permis d’aborder la gestion des interférences, mais cette fois ci d’une manière non-coopérative en se basant sur l’affectation couple Slot/Canal. Plus précisément, nous proposons deux schémas basés sur les carrés latins. (...) / A Wireless Body Area Network (WBAN) is a short-range network that consists of a coordinator (Crd) and a collection of low-power sensors that can be implanted in or attached to the human body. Basically, WBANs can provide real-time patient monitoring and serve in various applications such as ubiquitous health-care, consumer electronics, military, sports, etc. [1]. As the license-free 2.4 GHz ISM band is widely used among WBANs and across other wireless technologies, the fundamental problem is to mitigate the resulting co-channel interference. Other serious problems are to extend the network lifetime and to ensure reliable transmission within WBANs, which is an urgent requirement for health-care applications. Therefore, in this thesis, we conduct a systematic research on a few number of research problems related to radio co-channel interference, energy consumption, and network reliability. Specifically, we address the following problems ranging from theoretical modeling and analysis to practical protocol design: • Intra-WBAN interference mitigation and avoidance • Cooperative inter-WBAN interference mitigation and avoidance • Non-cooperative inter-WBAN interference mitigation and avoidance • Interference mitigation and avoidance in WBANs with IoT Firstly, to mitigate the intra-WBAN interference, we present two mechanisms for a WBAN. The first is called CSMA to Flexible TDMA combination for Interference Mitigation, namely, CFTIM, which dynamically allocates time-slots and stable channels to lower the intra-WBAN interference. The second is called Interference Avoidance Algorithm, namely IAA that dynamically adjusts the superframe length and limits the number of channels to 2 to lower the intra-WBAN interference and save energy. Theoretically, we derive a probabilistic model that proves the SINR outage probability is lowered. Simulation results demonstrate the effectiveness and the efficiency of CFTIM and IAA in terms of lowering the probability of interference, extending network lifetime, improving throughput and reliability. Secondly, we address the problem of interference among cooperative WBANs through using orthogonal codes. Motivated by distributed time provisioning supported in IEEE 802.15.6 standard [2], we propose two schemes. The first is called Distributed Time Correlation Reference, namely, DTRC that provides each WBAN with the knowledge about which superframes overlap with each other. The second is called Orthogonal Code Allocation Algorithm for Interference Mitigation, namely, OCAIM, that allocates orthogonal codes to interfering sensors belonging to sensor interference lists (SILs), which are generated based on the exchange of power-based information among WBANs. Mathematically, we derive the successful and collision probabilities of frames transmissions. Extensive simulations are conducted and the results demonstrate that OCAIM can diminish the interference, improve the throughput and save the power resource. Thirdly, we address the problem of co-channel interference among non-cooperative WBANs through time-slot and channel hopping. Specifically, we propose two schemes that are based on Latin rectangles. The first is called Distributed Algorithm for Interference mitigation using Latin rectangles, namely, DAIL that allocates a single channel to a timeslot combination to each sensor to diminish inter-WBAN interference and to yield better schedules of the medium access within each WBAN. The second is called Channel Hopping for Interference Mitigation, namely, CHIM, which generates a predictable interference free transmission schedule for all sensors within a WBAN. CHIM applies the channel switching only when a sensor experiences interference to save the power resource. Furthermore, we present an analytical model that derives bounds on collision probability and throughput for sensors transmissions. (...)