For explanation and prediction of the evolution of infectious diseases in populations, researchers often use simplified mathematical models for simulation. We believe that the results from these models are often questionable when the epidemic dynamics becomes more complex, and that developing more realistic models is intractable. In this dissertation we propose to simulate infectious disease propagation using dynamic and complex networks. We present the Simulator of Epidemic Evolution using Complex Networks (SEECN), an expressive and high-performance framework that combines algorithms for graph generation and various operators for modeling temporal dynamics. For graph generation we use the Kronecker algorithm, derive its underlying statistical structure and exploit it for a variety of purposes. Then the epidemic is evolved over the network by simulating the dynamics of the population and the epidemic simultaneously, where each type of dynamics is performed by a separate operator. All dynamics operators can be fully and independently parameterized, facilitating incremental model development and enabling different influences to be toggled for differential analysis. As a prototype, we simulate two relatively complex models for the HIV epidemic and find a remarkable fit to reported data for AIDS incidence and prevalence. Our most important conclusion is that the mere dynamics of the HIV epidemic is sufficient to produce rather complex trends in the incidence and prevalence statistics, e.g. without the introduction of particularly effective treatments at specific times. We show that this invalidates assumptions and conclusions made previously in the literature, and argue that simulations used for explanation and prediction of trends should incorporate more realistic models for both the population and the epidemic than is currently done. In addition, we substantiate a previously predicted paradox that the availability of Highly Active Anti-Retroviral Treatment likely causes an increased HIV incidence.

Thesis (M. Sc.)--Carleton University, 2005. / Includes bibliographical references (p. 111-114). Also available in electronic format on the Internet.

Thesis advisor: Cowles, Mary K. Includes bibliographic references (p. 167-172).

OLIVEIRA, P. M. R. PARATUCK-N semi-blind receivers for multi-hop cooperative MIMO relay systems. 2017. 51 f. Dissertação (Mestrado em Engenharia de Teleinformática)–Centro de Tecnologia, Universidade Federal do Ceará, Fortaleza, 2017. / Submitted by Renato Vasconcelos (ppgeti@ufc.br) on 2017-09-09T02:17:00Z No. of bitstreams: 1 2017_dis_pmroliveira.pdf: 1188957 bytes, checksum: 745420c86668fea09707dfcc7364adf8 (MD5) / Rejected by Marlene Sousa (mmarlene@ufc.br), reason: Prezado Pedro: Existe uma orientação para que normalizemos as dissertações e teses da UFC, em suas paginas pré-textuais e lista de referencias, pelas regras da ABNT. Por esse motivo, sugerimos consultar o modelo de template, para ajudá-lo nesta tarefa, disponível em: http://www.biblioteca.ufc.br/educacao-de-usuarios/templates/ Vamos agora as correções sempre de acordo com o template: 1. 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Se vc optar por abreviar os nomes em toda a lista eles devem ser abreviados. Se vc optar em colocar por extenso em toda e lista deve ficar por extenso para haver uma padronização. Os títulos das publicações devem ser destacadas com o uso do negrito (não use aspas). Com exceção dos artigos de periódicos que devem ser destacados o nome das revistas e não o titulo do artigo. Ao citar os artigos coloque as abreviações v. n. p. e não vol. no e pp. Após essas correções, enviaremos o nada consta por e-mail. Att. Marlene Rocha mmarlene@ufc.br on 2017-09-11T12:55:01Z (GMT) / Submitted by Renato Vasconcelos (ppgeti@ufc.br) on 2017-09-14T13:32:11Z No. of bitstreams: 1 2017_dis_pmroliveira.pdf: 1155061 bytes, checksum: 0da92fa912cabfa49b279fbe8c199877 (MD5) / Rejected by Marlene Sousa (mmarlene@ufc.br), reason: Prezado Pedro e Renato: Será que foi enviada a versão certa? Tenho a impressão que já fiz as recomendações de correção e o Pedro afirma que corrigiu, mas a versão que abri no Repositório ainda esta sem as alterações que solicitei. Marlene on 2017-09-14T16:38:31Z (GMT) / Submitted by Renato Vasconcelos (ppgeti@ufc.br) on 2017-10-03T13:47:30Z No. of bitstreams: 1 2017_dis_pmroliveira.pdf: 1357256 bytes, checksum: 308dc86c85e412f7947531cc8692dd60 (MD5) / Approved for entry into archive by Marlene Sousa (mmarlene@ufc.br) on 2017-11-08T17:23:03Z (GMT) No. of bitstreams: 1 2017_dis_pmroliveira.pdf: 1357256 bytes, checksum: 308dc86c85e412f7947531cc8692dd60 (MD5) / Made available in DSpace on 2017-11-08T17:23:03Z (GMT). No. of bitstreams: 1 2017_dis_pmroliveira.pdf: 1357256 bytes, checksum: 308dc86c85e412f7947531cc8692dd60 (MD5) Previous issue date: 2017-07-28 / Cooperative communication systems is a promising research field nowadays due to its advantages, like the increase of the received power, better quality of signal, and spatial diversity gains. Specifically, multi-hop systems are a very important part of cooperative communications, since they have the advantage of needing less transmission power than the direct systems. Also, multiple-input multiple-output (MIMO) systems are present in several standards of communications, providing some advantages, like the spatial multiplexing gains. Based on this scenario, this work proposes two semi-blind receivers based on the Kronecker product, that jointly estimate the symbol and the channels in a multi-hop Amplify-and-Forward (AF) MIMO relay-assisted system (AF protocol is widely used, due to its great performance and easy implementation). We consider a transmission scheme using a simplified Khatri-Rao space-time (KRST) coding at the source node, combined with an AF scheme at the relay nodes. We show that the third-order tensor of signals received by the destination node satisfies a PARATUCK-(K+1) decomposition, where K is the number of relays. This tensorial modeling enables a semi-blind estimation of symbols and channels with the use of a few pilot symbols. The first receiver called Least-Squares Kronecker-Factorization (LS-KF) is based on a factorization of the Kronecker product matrix between the symbols and the channel matrices. The other receiver called Least-Squares Kronecker-Rearrangement (LS-KR) is based on a rearrangement of this Kronecker product matrix, in order to achieve a rank-1 matrix. The performance of these receivers is evaluated by means of computational simulation results proving their efficiency in estimating the channels and, hence, providing a low Symbol Error Rate (SER). The proposed algorithms in the multi-hop scenario were compared to each other and with the two-hop and three-hop scenarios. The proposed algorithms in the multi-hop scenario were better at all simulations, providing a lower SER and better channels estimations. / Sistemas de comunicações cooperativas são um importante campo de pesquisa atualmente devido às vantagens que oferecem, como o aumento da potência recebida, melhor qualidade de sinal, e ganhos de diversidade espacial. Particularmente, sitemas multi-saltos são uma importante parte das comunicações cooperativas, uma vez que tais sistemas precisam de menos potência de transmissão se comparados a sistemas diretos. Por outro lado, sistemas MIMO (Multiple-Input Multiple-Output) estão presentes em diversos padrões de comunicação, provendo algumas vantagens, como ganhos de multiplexação espacial. Com base nesse cenário, o presente trabalho propõe dois receptores semi-cegos baseados no pro- duto de Kronecker, que estimam conjuntamente os símbolos e os canais de um sistema AF (Amplify- and-Forward) MIMO cooperativo com multiplos saltos (O protocolo AF é bastante usado, devido a sua grande performance e fácil implementação). É considerado um esquema de transmissão usando uma codificação simplificada KRST (Khatri-Rao Space-Time) na fonte, combinada com um esquema AF no relays. É mostrado que o tensor de terceira ordem dos sinais recebidos pelo destino satisfazem uma decomposição tensorial PARATUCK-(K+1), em que K é o número de relays. Esse modelo tensorial permite uma estimação semi-cega dos símbolos e canais, com o uso de alguns símbolos pilotos. O primeiro algoritmo, chamado de Least-Squares Kronecker-Factorization (LS-KF), é baseado em uma fatorização da matrix do produto de Kronecker entre as matrizes de símbolos e canal. Já o outro algoritmo, chamado de Least-Squares Kronecker-Rearrangement (LS-KR), é baseado no re- arranjo dessa matrix do produto de Kronecker, com o objetivo de conseguir uma matrix com rank 1. A performance desses receptores é avaliada através do resultado de simulações computacionais, provando sua eficiência em estimar os canais e prover uma baixa SER (Symbol Error Rate). Os algo- ritmos propostos no cenário multi-hop são comparados um com o outro e com os cenários two-hop e three-hop. Os algoritmos propostos no cenário multi-hop tiveram melhores performance em todas as simulações, resultando numa menor SER e melhores estimações dos canais.

Este trabalho consiste de três partes: Teoremas de coordenatização de Wedderburn e de Zorn, O problema de Nathan Jacobson e Teoremas de Fatorização de Kronecker para as superálgebras alternativas. Na primeira parte apresentamos os teoremas de coordenatização de Wedderburn e de Zorn e suas aplicações na teoria de representações das álgebras associativas e alternativas. Na segunda parte resolvemos um problema de longa data que foi anunciado por Nathan Jacobson sobre a descrição das álgebras alternativas que contém M&#8322(F ) (álgebra associativa de matrizes 2 × 2) com o mesmo elemento identidade. Na terceira parte damos uma prova independente que é válida em qualquer característica do clássico Teorema de Fatorização de Kronecker de Nathan Jacobson. Generalizamos esse resultado e provamos um teorema de Fatorização de Kronecker para as superálgebras alternativas cuja parte par contém O com o mesmo elemento identidade. Além disso, provamos um Teorema de Fatorização de Kronecker para as superálgebras alternativas que contêm a superálgebra associativa M(1|1)(F ) com o mesmo elemento identidade. Como Corolário desse resultado, respondemos a um análogo do problema de Jacobson para as superálgebras alternativas, isto é, descrevemos as superálgebras alternativas que contêm à superálgebra associativa M(1|1)(F ) com o mesmo elemento identidade. Finalmente, estudamos as representações das superálgebras alternativa simples O(4,4) e O[u]. Classificamos os bimodules sobre essas superálgebras e provamos alguns análogos do Teorema de Fatorização de Kronecker para as superálgebras alternativas que contenham O(4|4) ou O[u] com o mesmo elemento identidade / This work consists of three parts: Wedderburn and Zorn coordinatizations theorems, Nathan Jacobsons problem and Kroneckers Factorization theorems for alternative superalgebras. In the first part we present Wedderburn and Zorn coordinatizations theorems and their applications in the theory of representations of associative and alternative algebras. In the second part we solve a long standing problem that was announced by Nathan Jacobson on the description of alternative algebras containing M&#8322(F ) (associative matrix algebra 2 × 2) with the same identity element. In the third part we give an independent proof that is valid in any characteristic of Nathan Jacobsons classic Kronecker Factorization Theorem. We generalize this result and prove a Kronecker Factorization Theorem for alternative superalgebras whose even part contains O with the same identity element. In addition, we prove a Kronecker Factorization Theorem for alternative superalgebras containing the associative superalgebra M(1|1)(F ) with the same identity element. As a corollary of this result, we respond to an analogue of Jacobsons problem for alternative superalgebras, that is, we describe the alternative superalgebras containing the associative superalgebra M(1|1)(F ) with the same identity element. Finally, we study the representations of the simple alternative superalgebras O(4|4) e O[u]. We classify the bimodules on these superalgebras and prove some analogues of the Kronecker Factorization Theorem for alternative superalgebras containing O(4|4) or O[u] with the same identity element

Numerical techniques for linear systems arising from discretization of partial differential equations are nowadays essential for understanding the physical world. Among these techniques, iterative methods and the accompanying preconditioning techniques have become increasingly popular due to their great potential on large scale computation. In this work, we present preconditioning techniques for linear systems built with tensor product basis functions. Efficient algorithms are designed for various problems by exploiting the Kronecker product structure in the matrices, inherited from tensor product basis functions. Specifically, we design preconditioners for mass matrices to remove the complexity from the basis functions used in isogeometric analysis, obtaining numerical performance independent of mesh size, polynomial order and continuity order; we also present a compound iteration preconditioner for stiffness matrices in two dimensions, obtaining fast convergence speed; lastly, for the Helmholtz problem, we present a strategy to `hide' its indefiniteness from Krylov subspace methods by eliminating the part of initial error that corresponds to those negative generalized eigenvalues. For all three cases, the Kronecker product structure in the matrices is exploited to achieve high computational efficiency.

Likelihood-based marginalized models using random effects have become popular for analyzing longitudinal categorical data. These models permit direct interpretation of marginal mean parameters and characterize the serial dependence of longitudinal outcomes using random effects [12,22]. In this paper, we propose model that expands the use of previous models to accommodate longitudinal nominal data. Random effects using a new covariance matrix with a Kronecker product composition are used to explain serial and categorical dependence. The Quasi-Newton algorithm is developed for estimation. These proposed methods are illustrated with a real data set and compared with other standard methods.

Alan Turing explored how patterns emerged in embryos. The modelsexplained by Turing describe how two morphogens interact with eachother and develop certain patterns though different conditions. Usingthe differential equation expressed by Turing gives a lot of freedom inthe choice of variables depending on how the variables are chosen, whichwill be explored and simulated in this thesis. Later, a systems theoreticformulation will be used to explore the stability and controllability ofthe homogenous diffusion problem and Turing’s Reaction and Diffusionproblem. It will be found that under the assumptions made in this thesisboth models will be stable and controllable.

No / Recent work by Billinton et al. has highlighted the importance of employing more than one adverse weather state when modelling transmission line failures by Markov processes. In the present work the structure of the modelling Markov process is identified, allowing the rate matrix to be written in a closed form using Kronecker matrix operations. This approach allows larger models to be handled safely and with ease. The MAXIMA implementation of two asymptotic reliability indices for such systems is addressed, exemplifying the combination of symbolic and numerical steps, perhaps not seen in this context before. It is also indicated how the three-state weather model can be extended to a multi-state model, while retaining the scope of the proposed closed-form expression for the rate matrix. Some possible future work is discussed.

Les coefficients de Kronecker, qui sont indexés par des triplets de partitions et décrivent la décomposition du produit tensoriel de deux représentations irréductibles d'un groupe symétrique en somme directe de telles représentations, ont été introduits par Francis Murnaghan dans les années 1930. Il a notamment remarqué un comportement particulier de ces coefficients : à partir de n'importe quel triplet de partitions, on peut construire une certaine suite de coefficients de Kronecker qui est stationnaire.Afin de généraliser cette propriété, John Stembridge a introduit en 2014 une notion de stabilité pour les triplets de partitions, ainsi qu'une autre notion -- celle de triplet faiblement stable -- dont il a conjecturé qu'elle serait équivalente à la précédente. Cette conjecture a été démontrée peu après par Steven Sam et Andrew Snowden, par des méthodes algébriques.Dans cette thèse, on donne notamment une autre démonstration -- cette fois géométrique -- de cette équivalence grâce à l'interprétation classique des coefficients de Kronecker comme dimensions d'espaces de sections de fibrés en droites sur des variétés de drapeaux. Ces méthodes permettent également de s'intéresser à quelques questions plus précises : la stabilité dont on parle consiste en le fait que certaines suites de coefficients sont stationnaires, et on se demande à partir de quand ces suites deviennent constantes.On applique ensuite ces techniques à d'autres exemples de coefficients de branchement, puis on s'intéresse à un autre problème : celui de produire des triplets stables de partitions. On généralise ainsi un résultat obtenu indépendamment par Laurent Manivel et Ernesto Vallejo sur ce sujet / The Kronecker coefficients, which are indexed by triples of partitions and describe how the tensor product of two irreducible representations of the symmetric group decomposes as a direct sum of such representations, were introduced by Francis Murnaghan in the 1930s. He notably noticed a remarkable behaviour of these coefficients: from any triple of partitions, one can construct a particular sequence of Kronecker coefficients which eventually stabilises.In order to generalise this property, John Stembridge introduced in 2014 a notion of stability for triples of partitions, as well as another notion -- of weakly stable triple -- about which he conjectured that it should be equivalent to the previous one. This conjecture was proven shortly after by Steven Sam and Andrew Snowden, with algebraic methods.In this thesis we especially give another proof -- this time geometric -- of this equivalence, using the classical expression of the Kronecker coefficients as dimensions of spaces of sections of line bundles on flag varieties. With these methods we can also be interested in more specific questions: since the stability which we discuss means that some sequences of coefficients stabilise, one can wonder at which point these sequences become constant.We then apply these techniques to other examples of branching coefficients, and are also interested in another problem: how can we produce stable triples of partitions? We thus generalise a result obtained independently by Laurent Manivel and Ernesto Vallejo on this subject