This thesis is concerned with an application of the principle of inclusion and exclusion and with related approximation techniques. These procedures are extensively employed for developing test criteria based on statistics expressible as maxima. Upper percentage points of a number of such statistics have been tabulated by various special methods. However, the application of the principle of inclusion and exclusion, coupled with the Bonferroni inequalities, is often useful in providing good approximations. An extensive review of this method is presented in this report. This procedure allows one to establish upper and lower limits to upper percentage points, say λ_α, of statistics expressible as maxima. The upper bound approximation to λ_α requires only the knowledge of the distribution(s) of the variates under consideration. The lower bound, however, requires also the joint distribution(s) of pairs of the variates. Since the joint distribution is often difficult to calculate, an approximation technique may be necessary. A detailed discussion of such an approximation with guidelines for its applicability to statistics other than those discussed is presented. Two alternative methods for the determination of upper percentage points for statistics expressed as maxima are discussed: Whittle's lower bound approximation and the assumption of independence. It is pointed out that Whittle's lower bound is stronger than that of Bonferroni only under certain conditions. The assumption of independence leads to approximately the same result as Bonferroni. / Master of Science

When some population values are completely from observation, the distribution from which the observations came is said to be truncated. Estimation of the parameters from truncated distributions has been an open field for research. This thesis examines the developments which have taken place in this area, giving the major writers and the methods used by them to obtain estimators. A. C. Cohen is responsible for much work involving the maximum likelihood procedure. Using the method of moments and several methods which they have developed, Rider, Plackett, Samford, Moore, Des Raj, and Halperin have made significant contributions. The Poisson, Normal, Binomial, Negative Binomial, and Gamma distributions are included in the investigation and along with the estimators, in some cases, asymptotic variances are given. Though much work has been done, there are many things left to be investigated. Only a small number of distributions have been dealt with, with all multivariate distributions other than the normal lacking any investigation. It is not known how the estimators are affected by small sample sizes, and with the aid of the computer variances can be examined. A new problem arises when the points of truncation are not clearly defined and complicated equations often make estimators difficult to fine. / M.S.

The female of Spalangia drosophilae, Ashm, attack the puparia of Drosophila melanogaster, laying eggs between the puparial case and the pupa. The procedure is as follows: the parasite makes a careful examination of the puparia, tapping it with its antennae; then it stings the pupa to immobility and lays an egg. In a series of observations carried at the Imperial Institute of Entomology, Canada, using Spalangia as parasite and Drosophila as host, it was noted that the ovipositing female seemed to avoid superparasitism -- that is, laying eggs in already parasitized puparia. Naturally, if the number of non-parasitized insects still available was relatively small, the restraint exercised by the female tended to break down, and several eggs would be laid in some of the hosts. In order to determine the number of times the female Spalangia could exercise restraint before she was compelled to lay an egg, a series of three hundred and sixty separate experiments were performed. These involved the exposure of groups of five, ten and twenty-five Drosophila puparia to different numbers of parasites for various lengths of time, so that considerable variation in the number of eggs laid occurred. The observational results were expressed in terms of the number of puparia containing zero, one, two, etc., eggs. As the probability theory involved in this type of situation does not seem to be available, it is proposed to develop it in such a manner that experimental results may be analyzed to estimate the number of times the parasite could withhold an egg. The aspect of the experimental data which is of primary interest is the number of cases of superparasitism occurring for any given number of eggs laid and hosts exposed. / Master of Science

This research empirically tests three models that predict the expected intention change that results when people read and think about written advertising copy. The three models were derived from the principals of information processing 4 theory. This framework predicts that change in intention is a function of the discrepancy between a message and the person's initial stance before reading the message. From information processing theory, three models are derived and empirically tested against one another. The three models make specific predictions about behavioral and cognitive processing of the content in an advertisement. The three models tested in this research are termed the Presentation Model, the Random Model, and the Importance Model. The Presentation and Random Models assume that the way in which a multiple message is read determines the processing of the content. The Presentation Model assumes that people read down the page and change intention incrementally as they read. The Random Model assumes that people read copy by giving short consideration to each message and they cycle back through the copy, i.e., browsing several times. These two models were previously derived by Hunter, Danes and Cohen (1984). In addition to the aforementioned two models, this research proposed a third model, the Importance Model. This model assumes that people read the copy and enter it into short term memory. They then rank order the messages according to the importance of the attributes about which the message communicates a benefit. The processing then occurs in descending order of attribute importance. An experimental methodology was developed to test the assumptions made by each of these models and to examine the predictive validity of each model. Four treatments were employed to test the assumptions made in deriving the models. The presentation of the messages was computer controlled and the computer was used to simulate model assumptions. Also, magnitude scales were successfully employed to measure the independent and dependent variables of the three regression. The results of the research support the Importance Models as the appropriate models. This is based upon predictive validity and regression fit. This research expanded the information processing paradigm by offering a third processing model which is a more promising predictor of intention change. Future research in the area should expand upon these three models incorporating more homogeneous groups in terms of benefits. Also, future research should include allowances for logical distortions in the processing. Other topical areas for explorations are different delivery mechanisms such as verbal presentation and creative aspects of advertisements. / Ph. D.

A research report submitted to the Faculty of Engineering and the Built Environment, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Master of Science in Engineering, 2017 / The main problem to be investigated is how short duration data records can be augmented using existing data from nearby catchments with data with long periods of record. The purpose of the investigation is to establish a method of improving hydrological data using data from a gauged catchment to improve data from an ungauged catchment. The investigation is undertaken using rainfall data for catchments in Lesotho. Marginal distributions describing the annual maximum rainfall for the catchments, and a joint distribution of pairs of catchments were established. The parameters of these distributions were estimated using the Bayesian – Markov Chain Monte Carlo approach, and using both the single-site (univariate) estimation and the two-site (bivariate) estimations. The results of the analyses show that for catchments with data with short periods of record, the precision of the estimated location and scale parameters improved when the estimates were carried out using the two-site (bivariate) method. Rainfall events predicted using bivariate analyses parameters were generally higher than the univariate analyses parameters. From the results, it can be concluded that the two-site approach can be used to improve the precision of the rainfall predictions for catchments with data with short periods of record. This method can be used in practice by hydrologists and design engineers to enhance available data for use in designs and assessments. / CK2018

Orientador: Rafael Santos Mendes / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecânica / Made available in DSpace on 2018-08-18T03:24:20Z (GMT). No. of bitstreams: 1 Prado_Daniel_M.pdf: 1023728 bytes, checksum: 9342e53e6ff7f169478722c6bcd70fc0 (MD5) Previous issue date: 2011 / Resumo: Este projeto tem por objetivo estudar um problema de projeto para realizar o handoff vertical entre canais de comunicação GPRS e Satélite em um cenário que otimize o tempo e o custo através de equações probabilísticas. Através de um modelo de estados que descreve as diferentes possibilidades de comunicação entre sistemas GPRS e Satélite, o objetivo é determinar os tempos dos eventos controláveis que fazem a mudança entre os estados de modo a otimizar o tempo de comunicação num cenário probabilístico de handoff vertical entre canais de comunicação. Os resultados das simulações realizadas nesta dissertação em diferentes situações de qualidade dos sinais de transmissão GPRS e Satélite, mostram que através dos algoritmos de handoff desenvolvidos, a probabilidade estacionária de permanência nos estados de transmissão é aumentada / Abstract: This project aims to study a design problem to carry-out the handoff vertical of communication between communication channels GPRS and Satellite in a scenario that optimizes time and cost by probabilistic equations. Through a state model that describes the different possibilities of communication between GPRS and satellite systems, the goal is to set the times of the controllable events that do the changes between states in order to optimize the communication time in a probabilistic scenario of handoff vertical for communication channels. The results of the simulations in this work in different situations, quality of transmission signals GPRS and Satellite, show that through the handoff algorithms developed, the stationary probability of staying in the states of transmission is increased / Mestrado / Eletrônica / Mestre em Engenharia Automobilistica

Orientador: Antônio Carlos do Patrocinio / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-25T09:32:43Z (GMT). No. of bitstreams: 1 MastropauloNeto_Vicente_M.pdf: 1971257 bytes, checksum: 1c78e3e9085d370b2a40221cbcfe39a5 (MD5) Previous issue date: 2014 / Resumo: Este trabalho aborda o tema Combinatória e Probabilidade com aplicações no ensino de Geometria e tem como objetivo principal servir de apoio aos professores de Matemática da escola básica, fornecendo sugestões para a elaboração de problemas que reúnem conteúdos distintos do currículo, tomando Combinatória e Probabilidade como temas centrais. Os problemas aqui apresentados são voltados ao 3º ano do Ensino Médio e devem ser aplicados, preferencialmente, no quarto bimestre, no intuito de promover uma revisão geral, com ênfase em problemas de Geometria. Apresentamos inicialmente uma contextualização histórica da teoria das probabilidades, além da origem da probabilidade geométrica através do clássico problema da agulha de Buffon. Prosseguimos com uma fundamentação teórica e algumas aplicações dos temas centrais, Combinatória e Probabilidade, e concluímos com uma sequência didática aplicada em sala de aula com doze problemas que relacionam os princípios elementares de Combinatória e Probabilidade aos conceitos básicos de Geometria Plana, Geometria Espacial e Geometria Analítica / Abstract: This paper approaches the topic of Combinatorics and Probability with applications to the teaching of Geometry and has as its main objective to serve as support to Elementary School mathematics teachers, providing them with suggestions to elaborate problems which gather different contents of the curriculum, taking Combinatorics and Probability as their main topics. The problems presented here are thought for the 3rd grade of high school and must be preferably applied during the fourth bimester, aiming to promote a general review, with emphasis on Geometry problems. We initially present a historical contextualization of the probability theory besides the origin of geometric probability through Buffon's needle classic problem. Next we continue with a theoretical fundamentation and some applications of the central topics, Combinatorics and Probability, and then we conclude with a didactic sequence used in classroom with twelve problems which associate the main principles of Combinatorics and Probability with the basic concepts of Plane Geometry, Spatial Geometry and Analytical Geometry / Mestrado / Matemática em Rede Nacional - PROFMAT / Mestre em Matemática em Rede Nacional - PROFMAT

A research report submitted to the WITS SCHOOL OF EDUCATION, University of the Witwatersrand, Johannesburg, in partial fulfilment of the requirements for the degree of Master of Science (Science Education). Johannesburg 2015 / Using qualitative methods, this study reports on the selection and use examples in Probability by a teacher in a multilingual mathematics classroom where learners learn in a language which is not their first or home language. The study involved one teacher together with his Grade 11 multilingual class in a township school in Ekurhuleni South Johannesburg. Data was collected through audio-visual recording of four lessons. In addition two one-on-one semi-structured interviews were conducted with the teacher. Data was analysed using Rowland‘s (2008) categories of exemplification alongside Staples' (2007) conceptual model of collaborative inquiry mathematics practices. In the study it emerged that it is important for teachers to select examples by considering the context, ability of the example to be generalised, consistency in the use of symbols, syllabus requirements and accessibility. It also emerged that the selection of examples together with the accompanying mathematical practices has the potential to support or impede the learning of mathematics. In particular the findings revealed that the practice of ‗guiding the learners with the map‘ declines the cognitive level of examples and hence impedes learning. Code- switching and re-voicing were most frequently used practices seen in the findings with the use of code-switching encouraging full participation of the learners. The study recommends that methodology courses offered at tertiary institutions to pre-service teachers should include the selection, how to select or design and use examples in multilingual classrooms e.g. what constitutes a good example and how to maintain the cognitive level of an example. The study also recommends that more research needs to be done on effective mathematical practices that may be used to implement worked-out examples in multilingual classrooms.

yes / The orthocomplemented modular lattice of subspaces L[H(d)] , of a quantum system with d-dimensional Hilbert space H(d), is considered. A generalized additivity relation which holds for Kolmogorov probabilities is violated by quantum probabilities in the full lattice L[H(d)] (it is only valid within the Boolean subalgebras of L[H(d)] ). This suggests the use of more general (than Kolmogorov) probability theories, and here the Dempster-Shafer probability theory is adopted. An operator D(H1,H2) , which quantifies deviations from Kolmogorov probability theory is introduced, and it is shown to be intimately related to the commutator of the projectors P(H1),P(H2) , to the subspaces H 1, H 2. As an application, it is shown that the proof of the inequalities of Clauser, Horne, Shimony, and Holt for a system of two spin 1/2 particles is valid for Kolmogorov probabilities, but it is not valid for Dempster-Shafer probabilities. The violation of these inequalities in experiments supports the interpretation of quantum probabilities as Dempster-Shafer probabilities.

Quantum Markov processes are widely used models of the dynamics open quantum systems, a fundamental topic in theoretical and mathematical physics with important applications in experimental realisations of quantum systems such as ultracold atomic gases and new quantum information technologies such as quantum metrology and quantum control. In this thesis we present a mathematical framework which effectively characterises dynamical phase transitions in quantum Markov processes, using the theory of large deviations, by combining insights developed in non-equilibrium dynamics with techniques from quantum information and probability. We provide a natural decomposition for quantum Markov chains into phases, paving the way for the rigorous treatment of critical features of such systems such as phase transitions and phase purification. A full characterisation of dynamical phase transitions beyond properties of the steady state is described in terms of a dynamical perspective through critical behaviour of the quantum jump trajectories. We extend a fundamental result from large deviations for classical Markov chains, the Sanov theorem, to a quantum setting; we prove this Sanov theorem for the output of quantum Markov chains, a result which could be extended to a quantum Donsker-Varadhan theory. We perform an in-depth analysis of the atom maser, an infinite-dimensional quantum Markov process exhibiting various types of critical behaviour: for certain parameters it exhibits strong intermittency in the atom detection counts, and has a bistable stationary state. We show that the atom detection counts satisfy a large deviations principle, and therefore we deal with a phase cross-over rather than a genuine phase transition, although the latter occurs in the limit of infinite pumping rate. As a corollary, we obtain the Central Limit Theorem for the counting process.