Orientador: Jamil Viana Pereira / Resumo: Neste trabalho, será abordada a teoria matemática utilizada no estudo de quatro modelos clássicos de crescimento populacional: Malthus, Verhulst, Gompertz e Montroll. Serão apresentadas e/ou discutidas algumas de suas características, propriedades, diferenças e diferentes métodos para obtenção de seus parâmetros. Posteriormente, estes modelos e métodos serão aplicados a um conjunto de dados relativos ao crescimento populacional do Brasil. Também foi elaborada uma atividade para ser aplicada no Ensino Médio, na qual se explora, por tabelas e gráficos, a forma como os alunos observam certos fenômenos de crescimento, bem como, construtivamente e com auxílio de dados, a forma como ocorrem esses crescimentos de fato. / Abstract: In this work, the mathematical theory of four classic population growth models (Malthus, Verhulst, Gompertz and Montroll) will be addressed. Some of their characteristics, properties, differences and different methods for obtaining their parameters will be presented and/or discussed. Subsequently, these models and methods will be applied to a Brazilian population growth data set. An activity was also elaborated to be applied in high school. It explores, by tables and graphs, how students observe some growth phenomena, as well as, constructively and with the help of data, how these growths really occur. / Mestre

As current educational policies continue to emphasize the importance of college readiness and growth, it is essential to understand the degree to which test scores collected throughout middle school and high school can provide information to make valid inferences about students' college readiness. This thesis sought to summarize the college readiness of Iowa students, describe the nature of student growth, and clarify the relationship between student growth and college readiness. Together, the results support the validity argument that scores from a general achievement test can be used for measuring student growth and making on-track interpretations about college readiness. Results of analyses on the use of benchmarks as indicators of college readiness are presented first. The analyses showed that the state's general achievement test was just as accurate as the ACT when the criterion was defined by grades in domain-specific, credit-bearing courses. Next, latent growth models and growth mixture models were used to summarize and evaluate longitudinal changes in student achievement and their relationship with college outcomes. A calibration sample representing potential college-bound students was used to set the growth trajectories. Then a cohort of students representing the full student population was used to provide validity evidence in support of the growth trajectories. It was shown that students in the highest-performing group could be considered college ready. Several applications of the growth models are also presented. The typical performance on a variety of college outcomes for each developmental group was presented for the validation sample. A second application illustrated how individual patterns of growth in Grade 8 could be used to predict future class membership in Grade 11. This thesis was predicated on the notion that understanding and documenting the nature of student growth, the college readiness of Iowa students, and the relationship between the two is an important step in improving the college readiness of Iowa students and meeting the future needs of an aligned K-16 educational system. As this study is among the first to examine the relationship between college readiness and student growth using modern latent variable modeling techniques with actual college outcomes, guidelines for future research are presented.

Dans le domaine de l'anatomie, à l'investissement massif dans la constitution de base de données collectant des données d'imagerie médicale, doit répondre le développement de techniques numériques modernes pour une quantification de la façon dont les pathologies affectent et modifie les structures biologiques. Le développement d'approches géométriques via les espaces homogènes et la géométrie riemannienne en dimension infinie, initialisé il y a une dizaine d'années par Christensen et Miller, et simultanément Trouvé et Younes, et mettant en œuvre des idées originales de d'Arcy Thompson, a permis de construire ces dernières années un cadre conceptuel extrêmement efficace pour attaquer le problème de la modélisation et de l'analyse de la variabilité de populations de formes. Néanmoins, à l'intégration de l'analyse longitudinale des données, ont émergé des phénomènes biologiques de croissance ou de dégénérescence se manifestant via des déformations spécifiques de nature non difféomorphique. On peut en effet observer lors de la croissance d'un composant organique, une apparition progressive de matière qui ne s'apparente pas à un simple étirement du tissu initial. Face à cette observation, nous proposons de garder l'esprit géométrique qui fait la puissance des approches difféomorphiques dans les espaces de formes mais en introduisant un concept assez général de déploiement où l'on modélise les phénomènes de croissance comme le déploiement optimal progressif d'un modèle préalablement replié dans une région de l'espace. Nous présentons donc une généralisation des méthodes difféomorphiques classiques pour modéliser plus fidèlement l'évolution de chaque individu d'une population et saisir l'ensemble de la dynamique de croissance. Nous nous appuyons sur l'exemple concret de la croissance des cornes animales. La considération d'un a priori sur la dynamique de croissance de la corne, nous permet de construire un chemin continu dans un espace de formes, modélisant l'évolution de la corne de sa naissance, d'un état réduit à un point (comme l'état d'embryon pour un humain ou de graine pour une plante) à un âge adulte quelconque de corne bien déployée. Au lieu d'étirer la corne, nous anticipons l'arrivée matière nouvelle en des endroits prédéfinis. Pour cela, nous définissons une forme mère indépendante du temps dans un espace virtuel, qui est progressivement plongée dans l'espace ambiant en fonction d'un marqueur temporel prédéfini sur la forme mère. Finalement, nous aboutissons à un nouveau problème de contrôle optimal pour l'assimilation de données de surfaces évoluant dans le temps, conduisant à un problème intéressant dans le domaine du calcul des variations où le choix pour la représentation des données, courant ou varifold, joue un rôle inattendu. / The Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework has proved to be highly efficient for addressing the problem of modelling and analysis of the variability of populations of shapes, allowing for the direct comparison and quantization of diffeomorphic morphometric changes. However, the analysis of medical imaging data also requires the processing of more complex changes, which especially appear during growth or aging phenomena. The observed organisms are subject to transformations over the time which are no longer diffeomorphic, at least in a biological sense. One reason might be a gradual creation of new material uncorrelated to the preexisting one. For this purpose, we offer to extend the LDDMM framework to address the problem of non diffeomorphic structural variations in longitudinal scenarios during a growth or degenerative process. We keep the geometric central concept of a group of deformations acting on a shape space. However, the shapes will be encoded by a new enriched mathematical object allowing through partial mappings an intrinsic evolution dissociated from external deformations. We focus on the specific case of the growth of animal horns.Ultimately, we integrate these growth priors into a new optimal control problem for assimilation of time-varying surface data, leading to an interesting problem in the field of the calculus of variations where the choice of the attachment term on the data, current or varifold, plays an unexpected role.

The Mississippi Institute for Forest Inventory (MIFI) multi-product forest inventory divides Mississippi into five inventory regions with one region inventoried each year on a rotating basis. Resource analyses that overlap these temporally separated regions require adjustment to a common comparative time base by applying appropriate forest stand growth and harvest allocation models to the portions of a selected area not inventoried at the desired common time base. Currently the Mississippi Dynamic Inventory Reporter (MDIR) does not make adjustments to temporally synchronize portions of user selected working circles, polygons, or counties that occur in separate inventory regions. Separate inventory reports for each overlap area must be prepared to which growth and harvest are manually allocated to bring each area to the same point in time. The study objective was to provide an automated solution to temporal reconciliation by developing a growth and yield system that reconciles modeled timber volume growth, mortality, and harvests with known values from previous successive inventories and state tax records of harvested volumes at the county level. The modeling effort focused on constructing an optimized system for the Southwest MIFI 2004 and 2012 inventories. Species group specific, distance independent, tree-list models, including probability of survival and diameter growth equations, were developed through logistic and linear regressions, respectively. Probability of survival models were assessed for model performance using logistic regression concordant/discordant pairs. R2 and parameter p-values were used to evaluate diameter growth model performance. As the 2004 and 2012 datasets are each composed of randomly selected plots within the Southwest region, county totals were used for temporal comparison. County level Doyle volume calibration was within 150 units of tolerance for all counties in the Southwest region. The resulting growth and yield system represents a successful effort to develop a methodology for bridging temporally separated MIFI inventory analyses, while providing newly developed diameter and mortality equations for the state. The accompanying computer application allows the addition of both enhanced growth and yield and stand table projection models. System implementation will greatly reduce the time required for producing multi-temporal analyses and, thus, increase their usability and functionality.

No description available.

The changing topology of a network is driven by the need to maintain or optimize network function. As this function is often related to moving quantities such as traffic, information, etc., efficiently through the network, the structure of the network and the dynamics on the network directly depend on the other. To model this interplay of network structure and dynamics we use the dynamics on the network, or the dynamical processes the network models, to influence the dynamics of the network structure, i.e., to determine where and when to modify the network structure. We model the dynamics on the network using Jackson network dynamics and the dynamics of the network structure using minimal specialization, a variant of the more general network growth model known as specialization. The resulting model, which we refer to as the integrated specialization model, coevolves both the structure and the dynamics of the network. We show this model produces networks with real-world properties, such as right-skewed degree distributions, sparsity, the small-world property, and non-trivial equitable partitions. Additionally, when compared to other growth models, the integrated specialization model creates networks with small diameter, minimizing distances across the network. Along with producing these structural features, this model also sequentially removes the network's largest bottlenecks. The result are networks that have both dynamic and structural features that allow quantities to more efficiently move through the network.

This study examined the effects of government expenditure on different components of economic growth in South Africa using quarterly data from the period 1970Q1 to 2016Q4. The six key policy variables employed in the analysis were derived from the Ram (1986) production model and the New Growth Path (NGP), a macroeconomic framework designed to address the main challenges (unemployment, poverty and inequality) facing the economy as a result of its political past. The analysis of the relationship was carried out using the VECM while the findings from the analysis revealed that though there exists a long-run equilibrium relationship among the variables. The long-run estimates showed that aggregate private consumption expenditure and employment-to-population ratio are significant but negatively, related to economic growth. However, the net inflows of foreign direct investment and gross fixed capital formation are negatively related to gross government expenditure. This implies that excessive public capital expenditure might reduce the positive impact of the two variables on economic growth. The study therefore suggests that government should consider increasing its expenditure on the significant variables that support labour and capital development, in order to enhance economic growth in South Africa. / Economics

Rational expectations provide people or economic agents making future decision with available information and past experiences. The first approach to the idea of rational expectations was given approximately fifty years ago by John F. Muth. Many models in economics have been studied using the rational expectations idea. The most familiar one among them is the rational expectations version of the Cagans hyperination model where the expectation for tomorrow is formed using all the information available today. This model was reinterpreted by Thomas J. Sargent and Neil Wallace in 1973. After that time, many solution techniques were suggested to solve the Cagan type rational expectations (CTRE) model. Some economists such as Muth [13], Taylor [26] and Shiller [27] consider the solutions admitting an infinite moving-average representation. Blanchard and Kahn [28] find solutions by using a recursive procedure. A general characterization of the solution was obtained using the martingale approach by Broze, Gourieroux and Szafarz in [22], [23]. We choose to study martingale solution of CTRE model. This thesis is comprised of five chapters where the main aim is to study the CTRE model on isolated time scales. Most of the models studied in economics are continuous or discrete. Discrete models are more preferable by economists since they give more meaningful and accurate results. Discrete models only contain uniform time domains. Time scale calculus enables us to study on m-periodic time domains as well as non periodic time domains. In the first chapter, we give basics of time scales calculus and stochastic calculus. The second chapter is the brief introduction to rational expectations and the CTRE model. Moreover, many other solution techniques are examined in this chapter. After we introduce the necessary background, in the third chapter we construct the CTRE Model on isolated time scales. Then we give the general solution of this model in terms of martingales. We continue our work with defining the linear system and higher order CTRE on isolated time scales. We use Putzer Algorithm to solve the system of the CTRE Model. Then, we examine the existence and uniqueness of the solution of the CTRE model. In the fourth chapter, we apply our solution algorithm developed in the previous chapter to models in Finance and stochastic growth models in Economics.

No description available.

The criticality of the applications modeled by the real-time software places stringent requirements on software quality before deploying into real use. Though automated test tools can be used to run a large number of tests efficiently, the functionality of any test tool is not complege without providing a means for analyzing the test results to determine potential problem sub-domains and sub-domains that need to be covered, and estimating the reliability of the modeled system. This thesis outlines a solution strategy and implementation of that strategy for deriving quantitative metrics from domain testing of real-time control software tested via simulation. The key portion of this thesis addresses the combinatorial problems involved with effective evaluation of test coverage and provides the developer with reliability metrics from testing of the software to gain confidence in the test phase of development. The two approaches for reliability analysis- time domain and input domain approaches are studied and a hybrid approach that combines the strengths of both these approaches is proposed. A Reliability analysis Test Tool (RATT) has been developed to implement the proposed strategies. The results show that the metrics are practically feasible to compute and can be applied to most real-time software.