Renormalization group analysis is a useful tool for studying critical behaviour of stochastic systems. In this thesis, field-theoretic renormalization group will be applied to the scalar model representing directed percolation, known as Gribov model, in presence of the random velocity field. Turbulent mixing will be modelled by the compressible form of stochastic Navier-Stokes equation where the compressibility is described by an additional field related to the density. The task will be to find corresponding scaling properties.

The American vaccine pricing market has many actors, making it a complex system to model. Because of this, previous papers have chosen to model only vaccine manufacturers while leaving out the government. However, the government is also an important actor in the market, since it buys over half of vaccines produced. In this work, we aim to introduce the government into vaccine pricing models to better recommend pricing strategies to the Centers for Disease Control and Prevention.

In this dissertation, I introduce some number theoretic methods and discuss their intriguing applications to a variety of problems in computer science, information theory, combinatorics, and geometry. First, using properties of Ramanujan sums and of the discrete Fourier transform of arithmetic functions, we give an explicit formula for the number of solutions of restricted linear congruences in their `most general case'. As a consequence, we derive necessary and su cient conditions under which these congruences have no solutions. The number of solutions of this kind of congruence was rst considered by Rademacher in 1925 and Brauer in 1926, in a special case. Since then, this problem has been studied, in several other special cases, in many papers. The problem is very well-motivated and has found intriguing applications in several areas of mathematics, computer science, and physics, and there is promise for more applications/implications in these or other directions. Universal hash functions, discovered by Carter and Wegman in 1979, have many important applications in computer science. Applying our results we construct an almost-universal hash function family which is used to give a generalization of a recent authentication code with secrecy scheme. As another application of our results, we prove an explicit and practical formula for the number of surface-kernel epimorphisms from a co-compact Fuchsian group to iv a cyclic group. This problem has important applications in combinatorics, geometry, string theory, and quantum eld theory (QFT). As a consequence, we obtain an `equivalent' form of Harvey's famous theorem on the cyclic groups of automorphisms of compact Riemann surfaces. We also consider the number of solutions of linear congruences with distinct coordinates, and using a graph theoretic method, generalize a result of Sch onemann from 1839. Also, we give explicit formulas for the number of solutions of unweighted linear congruences with distinct coordinates. Our main tools are properties of Ramanujan sums and of the discrete Fourier transform of arithmetic functions. Then, as an application, we derive an explicit formula for the number of codewords in the Varshamov{Tenengolts code V Tb(n) with Hamming weight k, that is, with exactly k 1's. The Varshamov{Tenengolts codes are an important class of codes that are capable of correcting asymmetric errors on a Z-channel. As another application, we derive Ginzburg's formula for the number of codewords in V Tb(n), that is, jV Tb(n)j. We even go further and discuss applications to several other combinatorial problems, some of which have appeared in seemingly unrelated contexts. This provides a general framework and gives new insight into these problems which might lead to further work. Finally, we bring a very deep result of Pierre Deligne into the area of coding theory we connect Lee codes to Ramanujan graphs by showing that the Cayley graphs associated with some quasi-perfect Lee codes are Ramanujan graphs (this solves a recent conjecture). Our main tools are Deligne's bound from 1977 for estimating a particular kind of trigonometric sum and a result of Lov asz from 1975 (or of Babai from 1979) which gives the eigenvalues of Cayley graphs of nite Abelian groups. Our proof techniques may motivate more work in the interactions between spectral graph theory, character theory, and coding theory, and may provide new ideas towards the long-standing Golomb{Welch conjecture. / Graduate / 0984

Nesta tese é apresentado um novo método de reconstrução de imagens, por tomografia de transmissão, de projeções sujeitas a ruído na contagem de fótons. O método de reconstrução selecionado utiliza a técnica POCS (Projections Onto Convex Sets). A estimação das projeções originais a partir das observações ruidosas se dá por meio de quatro métodos: a) estimação utilizando o critério MAP (Maximum a Posteriori); b) filtragem nos coeficientes wavelets das projeções ruidosas; c) aplicação do filtro de Wiener pontual e d) aplicação do filtro de Goodman-Belsher. É apresentado o resultado da reconstrução após a estimação das projeções, mostrando o ISNR (Improvement Signal-to-Noise Ratio) entre as imagens reconstruídas, a partir das projeções ruidosas, com a técnica POCS, estimando as projeções e sem a realização da estimação. Foram utilizados, para reconstrução tomográfica, projeções de corpos de prova obtidos por meio de simulação e também projeções obtidas experimentalmente no minitomógrafo do CNPDIA - EMBRAPA. O uso de estimação sobre as projeções ruidosas mostrou-se eficaz para melhorar a relação sinal-ruído na imagem final, pois esse pré-processamento faz com que os conjuntos impostos pelas projeções sejam mais restritivos. Deve-se observar que a melhoria das imagens obtidas com o uso de filtragem das projeções é obtida com uma relação custo-benefício bastante baixa, pois a maior parte do custo computacional está na fase de reconstrução das imagens. / In this thesis is pesented a new method for image reconstruction, by transmission tomography, for projections under noise in the counting of photons. The selected method of reconstruction uses the POCS (Projections Onto Convex Sets) technique. The estimation of the original projections from the noisy projections observed is performed through four methods: a) estimation using the MAP (Maximum a Posteriori) criteria; b) through of filtering of the wavelets coefficients of the noisy projections; c) using the pointwise Wiener filter and d) using the Goodman-Belsher filter. We present the result of reconstruction after projection estimation, showing the ISNR (Improvement Signal-to-Noise Ratio) between the reconstructed images on noisy projections, using POCS technique after the estimated projections and without this estimation. We use, for tomographic reconstruction, test body projections obtained through simulation and also projections obtained experimentally in the minitomograph scanner of CNPDIA-EMBRAPA. The use of estimation on noisy projections demonstrated to be efficient in improving the signalnoise ratio in the final image, since this pre-processing makes the sets that projections more restrictive. We should observe that the use of projection filtering is obtained with a cost-benefit ratio rather low, since the largest part of the computational effort is in the image reconstruction phase.

Glaucoma is the second leading cause of blindness worldwide. The clinical standard for monitoring the functional deficits in the retina that are caused by glaucoma is the visual field test. In addition to monitoring the functional loss, evaluating the disease-related structural changes in the human retina also helps with diagnosis and management of this progressive disease. The characteristic changes of retinal structures such as the optic nerve head (ONH) are monitored utilizing imaging modalities such as color (stereo) fundus photography and, more recently, spectral-domain optical coherence tomography (SD-OCT). With the inherent subjectivity and time required for manually segmenting retinal structures, there has been a great interest in automated approaches. Since both fundus and SD-OCT images are often acquired for the assessment of glaucoma, the automated segmentation approaches can benefit from combining the multimodal complementary information from both sources. The goal of the current work is to automatically segment the retinal structures and extract the proper parameters of the optic nerve head related to the diagnosis and management of glaucoma. The structural parameters include the cup-to-disc ratio (CDR) which is a 2D parameter and is obtainable from both fundus and SD-OCT modalities. Bruch's membrane opening-minimum rim width (BMO-MRW) is a recent 3D structural parameter that is obtainable from the SD-OCT modality only. We propose to use the complementary information from both fundus and SD-OCT modalities in order to enhance the segmentation of structures of interest. In order to enable combining information from different modalities, a feature-based registration method is proposed for aligning the fundus and OCT images. In addition, our goal is to incorporate the machine-learning techniques into the graph-theoretic approach that is used for segmenting the structures of interest. Thus, the major contributions of this work include: 1) use of complementary information from SD-OCT and fundus images for segmenting the optic disc and cup boundaries in both modalities, 2) identifying the extent that accounting for the presence of externally oblique border tissue and retinal vessels in rim-width-based parameters affects structure-structure correlations, 3) designing a feature-based registration approach for registering multimodal images of the retina, and 4) developing a multimodal graph-based approach to segment the optic nerve head (ONH) structures such as Internal Limiting Membrane (ILM) surface and Bruch's membrane surface's opening.

Roughgarden et al. (2006) proposed a theory called social selection as a behavioral game theoretic model for sexual reproduction that incorporates both competition and cooperation in 2006. Players oscillate between playing competitively to maximize their individual fitness, leading to a Nash Competitive Equilibrium, and playing cooperatively to maximize a team fitness function, leading to a Nash Bargaining Solution. Roughgarden et al. (2006) gives rates of change for both the competitive state and the cooperative state, but does not explain her rates or how to switch between the states in sufficient detail. We test and rederive the rates, critiquing an assumption that the derivation of such a rate must make, as well as create a probabilistic model that switches between the two states. We test our model on the reproductive behaviors of Symphodus tinca, the peacock wrasse. The results follow the trajectory of the reproductive strategies for the wrasse throughout the breeding system, suggesting that cooperation could be a mechanism through which wrasse change their reproductive behaviors. However, the inputs to the model need to be analyzed more critically. Future work could include deriving rates for competitive play and cooperative play that do not rely on assumptions of being able to quantify strategy allocation proportion and refining the model and drawing generalized conclusions about it.

Two methods of analyzing multi-dimensional frequency data are detailed. The Second Order Exponential (SOE) model is applicable for dichotomous classifications. The distribution has two sets of parameters, ϴi's and ϴj's. The ϴi's are interpreted as the log of the odds of the marginal probabilities if no two factor relationships exist. Or if all ϴij are not zero, then the ϴi's are analogous to a main effect in a 2m factorial analysis, (m = number of factors or classifications). The ϴif's may be interpreted as a measure and direction of the two factor relationships. These ϴij are analogous to partial or adjusted phi-coefficients. The second method discussed assumes a multinomial distribution and the statistics are developed from an Information Theoretic Approach. Each hypothesis is tested using twice the minimum discrimination information statistic (m.d.i.s), 2I. From the null hypothesis it is possible to estimate unique cell probabilities by an iterative metod. Then 2 is equal to 2 (sample frequencies) log (sample frequencies) - 2 (expected frequencies) log (expected frequencies). (141 pages)

<p>Problem:</p><p>The problem under consideration is the World Trade Organization’s (WTO) agreement called Trade-Related Aspects of Intellectual Property Rights (TRIPS) and its impact on equal access to essential drugs in the least developed countries. Especially the countries of sub-Saharan Africa lack such access. Moreover, these countries are the ones where severe diseases like AIDS/HIV, tuberculosis and malaria are widely spread over the population. The authors focus also on patents and their obligatory length imposed through the articles of TRIPS agreement.</p><p>Purpose:</p><p>The purpose of the thesis is to describe and analyse the impact of global trade regulations (TRIPS in particular) on the accessibility of essential drugs in developing countries, and to come up with a possible solution as the way of coping with the problem is concerned. The investigation includes detailed description of solutions accomplished by Brazil and India, and their importance for the least developed countries, in terms of importing generic pharmaceuticals from these.</p><p>Method:</p><p>Qualitative method was used in order to obtain data from interviews with citizens of Botswana, Ghana, Ethiopia and South Africa for better understanding of the situation in these countries. Furthermore, the theories included in the theoretical background of this paper were gathered through deep research in the field of studies regarding Intellectual Property protection and World Trade Organization’s agreements and other legal acts.</p><p>Results:</p><p>The result of the analysis is a model developed from the Game-Theoretic Model, and called Developed Game-Theoretic Model. It is a tool which the least developed countries can use while negotiating prices of medicines with pharmaceutical companies, having the possibility of importing the pharmaceuticals from other countries manufacturing the patented product under compulsory licensing.</p>

For coupled systems of differential equations on networks, a graph-theoretic approach to the construction of Lyapunov functions is systematically developed in this thesis. Kirchhoffs Matrix-Tree Theorem in graph theory plays an essential role in the approachs development. The approach is successfully applied to several coupled systems well-known in the literature to demonstrate its applicability and effectiveness. / Applied Mathematics

Problem: The problem under consideration is the World Trade Organization’s (WTO) agreement called Trade-Related Aspects of Intellectual Property Rights (TRIPS) and its impact on equal access to essential drugs in the least developed countries. Especially the countries of sub-Saharan Africa lack such access. Moreover, these countries are the ones where severe diseases like AIDS/HIV, tuberculosis and malaria are widely spread over the population. The authors focus also on patents and their obligatory length imposed through the articles of TRIPS agreement. Purpose: The purpose of the thesis is to describe and analyse the impact of global trade regulations (TRIPS in particular) on the accessibility of essential drugs in developing countries, and to come up with a possible solution as the way of coping with the problem is concerned. The investigation includes detailed description of solutions accomplished by Brazil and India, and their importance for the least developed countries, in terms of importing generic pharmaceuticals from these. Method: Qualitative method was used in order to obtain data from interviews with citizens of Botswana, Ghana, Ethiopia and South Africa for better understanding of the situation in these countries. Furthermore, the theories included in the theoretical background of this paper were gathered through deep research in the field of studies regarding Intellectual Property protection and World Trade Organization’s agreements and other legal acts. Results: The result of the analysis is a model developed from the Game-Theoretic Model, and called Developed Game-Theoretic Model. It is a tool which the least developed countries can use while negotiating prices of medicines with pharmaceutical companies, having the possibility of importing the pharmaceuticals from other countries manufacturing the patented product under compulsory licensing.