This thesis provides a general methodology for classifying and describing many combinatoric problems, systematising and finding theoretical expressions for quantities of interest, and investigating their feasible numerical evaluation. Unifying notation and definitions are provided. Our knowledge of random allocations is also extended. This is achieved by investigating new processes, generalising known processes, and by providing a formal structure and innovative techniques for analysing them. The random allocation models described in this thesis can be classified as occupancy urn models, in which we have a sequence of urns and throw balls into them, and investigate static, waiting-time and dynamic processes. Various structures are placed on the relationship(s) between cells, balls, and the selection of items being distributed, including varieties, batch arrivals, taboo sets and blocking sets. Static, waiting-time and dynamic processes are investigated. Both without-replacement and with-replacement sampling types are considered. Emphasis is placed on the distributions of waiting-times for one or more events to occur measured from the time a particular event occurs; this begins as an abstraction and generalisation of a model of departures of cars parked in lanes. One of several additional determinations is the platoon size distribution. Models are analysed using combinatorial analysis and Markov Chains. Global attributes are measured, including maximum waits, maximum room required, moments and the clustering of completions. Various conversion formulae have been devised to reduce calculation times by several orders of magnitude. New and extended applications include Queueing in Lanes, Cake Displays, Coupon Collector's Problem, Sock-Sorting, Matching Dependent Sets (including Genetic Code Attribute Matching and the game SET), the Zig-Zag Problem, Testing for Randomness (including the Cake Display Test, which is a without-replacement test similar to the standard Empty Cell test), Waiting for Luggage at an Airport, Breakdowns in a Network, Learning Theory and Estimating the Number of Skeletons at an Archaeological Dig. Fundamental, reduction and covering theorems provide ways to reduce the number of calculations required. New combinatorial identities are discovered and a well-known one is proved in a combinatorial way for the first time. Some known results are derived from simple cases of the general models. / http://proxy.library.adelaide.edu.au/login?url= http://library.adelaide.edu.au/cgi-bin/Pwebrecon.cgi?BBID=1309598 / Thesis (Ph.D.) -- University of Adelaide, School of Mathematical Sciences, 2007

This thesis provides a general methodology for classifying and describing many combinatoric problems, systematising and finding theoretical expressions for quantities of interest, and investigating their feasible numerical evaluation. Unifying notation and definitions are provided. Our knowledge of random allocations is also extended. This is achieved by investigating new processes, generalising known processes, and by providing a formal structure and innovative techniques for analysing them. The random allocation models described in this thesis can be classified as occupancy urn models, in which we have a sequence of urns and throw balls into them, and investigate static, waiting-time and dynamic processes. Various structures are placed on the relationship(s) between cells, balls, and the selection of items being distributed, including varieties, batch arrivals, taboo sets and blocking sets. Static, waiting-time and dynamic processes are investigated. Both without-replacement and with-replacement sampling types are considered. Emphasis is placed on the distributions of waiting-times for one or more events to occur measured from the time a particular event occurs; this begins as an abstraction and generalisation of a model of departures of cars parked in lanes. One of several additional determinations is the platoon size distribution. Models are analysed using combinatorial analysis and Markov Chains. Global attributes are measured, including maximum waits, maximum room required, moments and the clustering of completions. Various conversion formulae have been devised to reduce calculation times by several orders of magnitude. New and extended applications include Queueing in Lanes, Cake Displays, Coupon Collector's Problem, Sock-Sorting, Matching Dependent Sets (including Genetic Code Attribute Matching and the game SET), the Zig-Zag Problem, Testing for Randomness (including the Cake Display Test, which is a without-replacement test similar to the standard Empty Cell test), Waiting for Luggage at an Airport, Breakdowns in a Network, Learning Theory and Estimating the Number of Skeletons at an Archaeological Dig. Fundamental, reduction and covering theorems provide ways to reduce the number of calculations required. New combinatorial identities are discovered and a well-known one is proved in a combinatorial way for the first time. Some known results are derived from simple cases of the general models. / http://proxy.library.adelaide.edu.au/login?url= http://library.adelaide.edu.au/cgi-bin/Pwebrecon.cgi?BBID=1309598 / Thesis (Ph.D.) -- University of Adelaide, School of Mathematical Sciences, 2007

La règlementation des jeux semble frappée de désuétude, à l’image des articles 1965 à 1967 du Code civil qui n’ont jamais été modifiés depuis 1804. Cependant, l’avènement successif de la télévision, des logiciels informatiques et d’Internet a métamorphosé les problématiques ludiques. Les juristes s’étonnent, après s’être désintéressés du jeu pendant de longues années, de le voir ressurgir dans toutes les composantes de leur discipline. À l’heure où le jeu sportif échappe encore à l’obsession réparatrice du droit de la responsabilité civile, où le jeu de hasard sacrifie un monopole étatique sur l’autel de la libre concurrence, et où le jeu télévisé redéfinit les frontières de la prestation de travail, le moment semblait propice à l’élaboration d’une réflexion globale sur les problèmes juridiques du jeu. Dans le cadre d’une démarche systémique, le jeu sera le point de départ de nos travaux et la règle sa pierre angulaire, permettant ainsi d’écarter l’amusement libre au profit du jeu réglé. Ce dernier, lorsqu’il prend naissance dans la rencontre de consentement entre un organisateur, qui émet une offre de jeu, et un ou plusieurs joueurs qui l’acceptent, soulève des problèmes juridiques similaires dans toutes les disciplines ludiques. L’organisateur d’un jeu est-il responsable des équipements de jeu qu’il fournit ? Doit-il règlementer l’accès à son jeu ? A-t-il l’obligation de délivrer un enjeu au gagnant d’une partie ? Quelles sont les conséquences des différentes fautes de jeu ? La licéité du but fixé est-elle une condition de validité du jeu ? En nous efforçant d'approfondir ces interrogations, nous chercherons à poser les premiers jalons d’une théorie juridique du jeu. / Gaming regulations appear to have become obsolete as articles 1965 to 1967 of the Civil Code which has not been modified since1804 demonstrate. However, the successive arrival of television, computer software and internet has metamorphosed recreational activities, including gaming. Lawyers, long disinterested by gaming issues, are now surprised to see these reemerging throughout their profession. At a time when sports games still escape the “obsessive rectification” the rights of civil responsibility grant, where principles of free competition have removed the state monopoly concerning the games of chance and where televised games redefine the characteristics of work contracts, it seems to be appropriate to elaborate a global discussion on the judicial aspect of gaming. Within a framework of a general approach gaming will be studied but from the point of view of its rules in order to lay aside unregulated amusement activities to favor regulated gaming. The latter, whenever it takes place under an agreement between an organizer presenting an offer to participate and one or several players who accept, raises similar juridical questions for all types of gaming activities. Is the organizer responsible for the equipment he provides? Should he regulate access to his game? Does he have an obligation to offer a prize to the winner of a game? What are the consequences concerning the various violations of the game? Is the licitly of the established target a valid condition of the game? By seeking to elaborate on these questions, this dissertation aims to pave the way for a more appropriate judicial framework for gaming.