Impulsive differential equations are useful for modelling certain biological events. We present three biological applications showing the use of impulsive differential equations in real-world problems. We also look at the effects of stability on a reduced two-dimensional impulsive HIV system. The first application is a system describing HIV induction-maintenance therapy, which shows how the solution to an impulsive system is used in order to find biological results (adherence, etc). A second application is an HIV system describing the interaction between T-cells, virus and drugs. Stability of the system is determined for a fixed drug level in three specific regions: low, intermediate and high drug levels. Numerical simulations show the effects of varying drug levels on the stability of a system by including an impulse. We reduce these two models to a two-dimensional impulsive model. We show analytically the existence and uniqueness of T-periodic solutions, and show how stability changes when varying the immune response rate, the impulses and a certain nonlinear infection term. The third application shows how seasonal changes can be incorporated into an impulsive differential system of Rift Valley Fever, and looks at how stability may differ when impulses are included. The analysis of impulsive differential systems is crucial in developing more realistic mathematical models for infectious diseases.

abstract: The primary focus of this dissertation lies in extremal combinatorics, in particular intersection theorems in finite set theory. A seminal result in the area is the theorem of Erdos, Ko and Rado which finds the upper bound on the size of an intersecting family of subsets of an n-element set and characterizes the structure of families which attain this upper bound. A major portion of this dissertation focuses on a recent generalization of the Erdos--Ko--Rado theorem which considers intersecting families of independent sets in graphs. An intersection theorem is proved for a large class of graphs, namely chordal graphs which satisfy an additional condition and similar problems are considered for trees, bipartite graphs and other special classes. A similar extension is also formulated for cross-intersecting families and results are proved for chordal graphs and cycles. A well-known generalization of the EKR theorem for k-wise intersecting families due to Frankl is also considered. A stability version of Frankl's theorem is proved, which provides additional structural information about k-wise intersecting families which have size close to the maximum upper bound. A graph-theoretic generalization of Frankl's theorem is also formulated and proved for perfect matching graphs. Finally, a long-standing conjecture of Chvatal regarding structure of maximum intersecting families in hereditary systems is considered. An intersection theorem is proved for hereditary families which have rank 3 using a powerful tool of Erdos and Rado which is called the Sunflower Lemma. / Dissertation/Thesis / Ph.D. Mathematics 2011

[pt] Em países de clima tropical, como o Brasil, é comum a formação de solos residuais, que, por estarem expostos a períodos de seca após as chuvas, costumam encontrar-se não saturados em campo. Portanto, uma análise de estabilidade de um talude considerando o solo saturado não seria a mais condizente com a realidade em um caso destes, devendo ser considerado o acréscimo na resistência ao cisalhamento por conta da sucção na determinação do fator de segurança. Este trabalho teve como objetivo analisar a estabilidade de um talude composto por solo residual de granulito localizado no bairro do Santo Antônio Além do Carmo, Salvador, BA. O programa experimental contou com a coleta de material em campo para realização de ensaios de caracterização química, física e mineralógica, resistência e permeabilidade. A envoltória não saturada, de onde se obteve a relação entre o fluxo magnético e sucção mártrica, foi determinada através da função hiperbólica proposta por Vilar (2006). Com os parâmetros obtidos, foram realizadas as análises de infiltração e estabilidade. Os resultados mostraram que o mecanismo de ruptura provável associou-se à elevação do nivel de água provocado pela infiltração das águas provenientes das fortes chuvas dos dias 8 e 9 de novembro, bem como das águas servidas lançadas no talude pela população. A ação antrópica também se fez presente através de sobrecarga no talude provocada pelo peso das edificações. Conclui-se, então, que seria necessária a ação conjunta da chuva e fatores humanos para que haja uma situação de instabilidade no local estudado. / [en] In countries with a tropical climate, such as Brazil, the formation of residual soils is common, and because they are exposed to periods of drought after rainfall, they are often found unsaturated. Therefore, a stability analysis of a slope considering saturated soil would not be the most consistent in these case, and the increase in shear strength due to suction should be considered on the determination of the the safety factor. The objective of this work was to analyze the stability of a slope composed of residual granulite soil located in the district of Santo Antônio Além do Carmo, Salvador, BA. The experimental program included the collection of material in the field to perform chemical, physical and mineralogical characterization, resistance and permeability tests. The unsaturated envelope from which the relationship between magnetic flow and matric suction was determined by the hyperbolic function proposed by Vilar (2006). With the obtained parameters, infiltration and stability analyzes were performed. The results showed that the probable failure mechanism was associated to the elevation of the water level caused by the infiltration of the water from the heavy rains of November 8 and 9, as well as the wastewater thrown into the slope by the population. The anthropic action was also present through overloading the slope, caused by the weight of the houses. It can be concluded that it would be necessary the rainfall to work together with the human factors so it will happen an instabilization situation in the study site.

This thesis proposes novel methods for identification and control of multivariable systems including multiple delays and describes their application to control of general anesthesia administration. First, an identification method for multivariable systems whose input and output paths have different time delays is presented. Second, a state predictor for multivariable systems whose input and output paths have different time delays is proposed. Third, the state predictor is used for constructing a state-predictive servo control system for controlled processes whose output paths have different time delays. A robust stability analysis method of the state-predictive servo control system is also examined. Furthermore, based on results of these theoretical studies, control systems for use in general anesthesia administration are developed. First, an identification method for multivariable systems whose input and output paths have different time delays is proposed. This method comprises two steps. In the first step, the delay lengths are estimated from the impulse response matrix identified from input and output (I/O) sequences using a subspace identification algorithm. In the second step, I/O sequences of a delay-free part are constructed from the original sequences and the delay estimates, and the system matrices of the delay-free part are identified. The proposed method is numerically stable and efficient. Moreover, it requires no complex optimization to obtain the delay estimates, nor does it require an assumption about the structure of the system matrices. Second, a state predictor is proposed for multivariable systems whose input and output paths have different time delays. The predictor consists of a full-order observer and a prediction mechanism. The former estimates a vector consisting of past states from the output. The latter predicts the current state from the estimated vector. The prediction error converges to zero at an arbitrary rate, which can be determined using pole assignment method, etc. In the proposed predictor, the interval length of the finite interval integration fed to the observer is shorter than that of an existing delay-compensating observer. Consequently, the proposed predictor is more numerically accurate than the delay-compensating observer. Using the proposed state predictor, a design method of a state-predictive servo controller is described for multivariable systems whose output paths have different time delays. Furthermore, a sufficient stability condition of the state-predictive servo control system against parameter mismatches is derived. Using a characteristic equation of the perturbed closed-loop system, a stability margin can be given on a plane whose axes correspond to the magnitudes of the mismatches on system matrices and on delay lengths. In the remainder of this thesis, development of anesthesia control systems is described to illustrate an application of the theoretical results described above. First, a hypnosis control system is presented. This system administers an intravenous hypnotic drug to regulate an electroencephalogram-derived index reflecting the patient’s hypnosis. The system comprises three functions: i) a model predictive controller that can take into account effects of time delay adequately, ii) an estimation function of individual parameters, and iii) a risk-control function for preventing undesirable states such as drug over-infusion or intra-operative arousal. Results of 79 clinical trials show that the system can reduce the total amount of drug infusion and maintain hypnosis more accurately than an anesthesiologist’s manual adjustment. Second, a simultaneous control system of hypnosis and muscle relaxation is described. For development of this system, a multivariable model of hypnosis and muscle relaxation is identified using the method proposed in this thesis. Then a state-predictive servo control system is designed for controlling hypnosis and muscle relaxation. Finally, the control system’s performance is evaluated through simulation. The resultant simultaneous control system satisfies the performance specifications of settling time, disturbance rejection ability, and a robust stability range. Although this system is not fully developed, the procedure of constructing this control system demonstrates the effectiveness of the proposed methods: the identification method for systems whose input and output paths have different time delays and the design and stability analysis methods of the state-predictive servo control system. / Kyoto University (京都大学) / 0048 / 新制・課程博士 / 博士(工学) / 甲第13820号 / 工博第2924号 / 新制||工||1432(附属図書館) / 26036 / UT51-2008-C736 / 京都大学大学院工学研究科電気工学専攻 / (主査)教授 小林 哲生, 教授 萩原 朋道, 准教授 古谷 栄光 / 学位規則第4条第1項該当

Philosophiae Doctor - PhD / Malaria is most prevalent in tropical climates, where there are sufficient rainfall for mosquitoes to breed and conducive temperatures for both the mosquito and protozoa to live. A slight change in temperature can drastically affect the lifespan and patterns of mosquitoes, and moreover, the protozoan itself can only survive in a certain temperature range. With higher temperatures, mosquitoes can mature faster, and thus have more time to spread the disease. The malaria parasite also matures more quickly at warmer temperatures. However, if temperatures become too high, neither mosquitoes nor the malaria pathogen can survive. In addition, stagnant water is also a major contributor to the spread of malaria, since most mosquito species breed in small pools of water. The correct amount and distribution of rainfall increases the possible breeding sites for mosquito larvae, which eventually results in more vectors to spread the disease. With little rainfall, there are few places for the mosquitoes to breed. For these reasons, and in order to control mosquito population, it is important to examine the weather parameters such as temperature and rainfall which are imperative in determining the disease epidemics. Accurate seasonal climate forecasts of these variables, together with malaria models should be able to drive an early warning system in endemic regions. These models can also be used to evaluate the possible change in regions under climate change scenarios, and the spread of malaria to new regions. In this study, we develop and analyse a mosquito model to study the population dynamics of mosquitoes. Ignoring the impact of climate, the model is further developed by introducing human compartments into the model. We perform both analytical and numerical analyses on the two models and verify that both models are epidemiological and mathematical well-posed. Using the next generation matrix method, the basic reproduction number of each system is calculated. Results from both analyses confirm that mosquito- and disease-free equilibria are locally asymptotically stable whenever R0 < 1 and unstable whenever R0 > 1. We further establish the global stability of the mosquito-free equilibrium using a Lyapunov function. In order to examine the effectiveness of control measures, we calculate the sensitivity coefficients of the reproductive number of the mosquito-human malaria model and highlight the importance of mosquito biting rate on malaria transmission. In addition, we introduce climate dependent parameters of Anopheles gambiae and climate data of Limpopo province into the malaria model to study malaria transmission over the province. Climate variables and puddle dynamics are further incorporated into the mosquito model to study the dynamics of Anopheles arabiensis. The climatedependent functions are derived from the laboratory experiments in the study of Maharaj [114], and we further verify the sensitivity of the model to parameters through sensitivity analysis. Running the climate data of Dondotha village in Kwazulu-Natal province over the mosquito model, it is used to simulate the impact of climate variables on the population dynamics of Anopheles arabiensis over the village. Furthermore, we incorporate human compartments into the climate-based mosquito model to explore the impact of climate variability on malaria incidence over KwaZulu-Natal province over the period 1970-2005. The outputs of the climate-based mosquito-human malaria model are further analysed with Principal Component Analysis (PCA), Wavelet Power Spectrum (WPS) and Wavelet Cross-coherence Analysis (WCA) to investigate the relationship between the climate variables and malaria transmission over the province. The results from the mosquito model fairly accurately quantify the seasonality of the population of Anopheles arabiensis over the study region and also demonstrate the influence of climatic factors on the vector population dynamics. The model simulates the population dynamics of both immature and adult Anopheles arabiensis and increases our understanding on the importance of mosquito biology in malaria models. Also, the simulated larval density produces a curve which is similar to observed data obtained from another study. In addition, the mosquito-malaria models produce reasonable fits with the observed data over Limpopo and KwaZulu Natal provinces. In particular, they capture all the spikes in malaria prevalence. Our results further highlight the importance of climate factors on malaria transmission and show the seasonality of malaria epidemics over the provinces. The results of the PCA on the model outputs suggest that there are two major process in the model simulation. One of the processes indicate high loadings on the population of Susceptible, Exposed and Infected humans, while the other is more correlated with Susceptible and Recovered humans. However, both processes reveal the inverse correlation between Susceptible-Infected and Susceptible-Recovered humans respectively. Through spectrum analysis, we notice a strong annual cycle of malaria incidence over the province and ascertain a dominant periodicity of one year. Consequently, our findings indicate that an average of 0 to 120-day lag is generally noted over the study period, but the 120-day lag is more associated with temperature than rainfall. The findings of this study would be useful in an early warning system or forecasting of malaria transmission over the study areas.

[pt] O presente trabalho de dissertação de mestrado tem por finalidade contribuir para uma maior dinamização do ensino de engenharia de sistemas de potência através da elaboração de um conjunto de algoritmos interativos para microcomputadores, onde o aluno poderá realizar um processo contínuo de experimentação, questionamento e revisão de conceitos. Para um bom desenvolvimento deste processo de aprendizagem é criado um ambiente amigável onde o usuário tem suas ações amparadas por mensagens explicativas e de monitoração de erros, resultando na redução do tempo gasto em cada seção de estudo e estimulando o mesmo a realizar um número maior de simulações. Foram elaboradas quatro programas de computadores. O primeiro, responsável pela estrutura conversacional e gerenciamento, foi escrito em PASCAL. Os demais foram escritos em FORTRAN77 e se restringem à execução dos cálculos necessários aos estudos de: fluxo de potência, curto-circuito e estabilidade transitória. Estes três últimos programas utilizam técnicas especiais para armazenagem de matrizes esparsas e obtenção do elementos de interesse de suas inversas. Três sistemas de potência foram escolhidos para estar imediatamente disponíveis ao usuário, sendo que um deles é utilizado no texto durante a explanação da estrutura conversacional para os estudos disponíveis. / [en] The present Dissertation aims to provide more dynamism to the teaching and learning process in power system engineering through a set of interactive personal computer algorithms. The student may carry out a continuous process of experimenting, questioning and revision of concepts. In order to better develop the leaining process, a frindly environment is created where the user has his actions supported by explanatory and error monitoring messages. This results in a reduction of the time spent in each study section and, consequently, it estimulates the student to perform a greater number of simulations. Four computer programs have been developed. There is one written in PASCAL language, responsible for the following studies: load flow, slort-circuit and transient stability. These programs use special sparsity techniques and programming. Three power systems have been choosen to be immediately available to the user. One of them is used in the text during the explanation process of the conversational structure of the power system studies.

Les amplifications optimales d'énergie de structures quasiment alignées dans le sens de l'écoulement sont calculées dans le cas d'un sillage parallèle, d'un sillage synthétique faiblement non-parallèle et du sillage d'un cylindre. Il a été observé que de très grandes amplifications d'énergie peuvent être supportés par ces sillages. L'amplification d' énergie s'accroît avec la longueur d'onde des perturbations en envergure à l'exception du sillage du cylindre pour lequel l'accroissement d'énergie est maximal pour λz ≈ 5 − 7 D. Les structures amplifiées de manière optimale sont les streaks fluctuant dans le sens de l’écoulement. Il est montré que ces streaks sont capables de supprimer complètement l'instabilité absolue d'un sillage parallèle lorsqu'ils sont déclenchés avec une amplitude finie. L'instabilité globale d'un sillage faiblement non-parallèle et celle du sillage d'un cylindre peuvent être complètement supprimées par des streaks d'amplitude modeste. L'énergie de contrôle requise pour stabiliser le sillage est très faible lorsque les perturbations optimales sont utilisées, et il est montré qu'elle est toujours plus faible que celle qui devrait être utilisée pour un contrôle uniforme en envergure (2D). Il est aussi montré que la dépendance du taux de croissance est quadratique et que, par conséquent, les classiques analyses de sensibilité au premier ordre ne permettent pas de prédire la grande efficacité de la technique de contrôle par streaks. La dernière partie de ce travail livre des résultats préliminaires sur l'étude expérimentale du contrôle par streaks dans le cas du sillage turbulent d'un corps 3D. Il est montré que les streaks forcés artificiellement dans la zone d'instabilité absolue de l'écoulement sont capables de modifier la dynamique du sillage. / We compute optimal energy growths leading to streamwise streaks in parallel, weakly non-parallel and the circular cylinder wakes. We find that very large energy amplifications can be sustained by these wakes. The energy amplifications increase with the spanwise wavelength of the perturbations except in the circular cylinder wake where maximum energy growths are reached for λz ≈ 5 − 7 D. The optimally amplified structures are streamwise streaks. When forced with finite amplitudes these streaks are shown, in parallel wakes, to be able to completely suppress the absolute instability. The global instability of the weakly non-parallel and the circular cylinder wakes can be completely suppressed with moderate streaks amplitudes. The energy required to stabilize the wake is much reduced when optimal perturbations are used, and it is shown to be always smaller than the one that would be required if a 2D control was used. It is also shown that the sensitivity of the global mode growth rate is quadratic and that therefore usual first order sensitivity analyses are unable to predict the high efficiency of the control-by-streaks strategy.

abstract: Recently, a novel non-iterative power flow (PF) method known as the Holomorphic Embedding Method (HEM) was applied to the power-flow problem. Its superiority over other traditional iterative methods such as Gauss-Seidel (GS), Newton-Raphson (NR), Fast Decoupled Load Flow (FDLF) and their variants is that it is theoretically guaranteed to find the operable solution, if one exists, and will unequivocally signal if no solution exists. However, while theoretical convergence is guaranteed by Stahl’s theorem, numerical convergence is not. Numerically, the HEM may require extended precision to converge, especially for heavily-loaded and ill-conditioned power system models. In light of the advantages and disadvantages of the HEM, this report focuses on three topics: 1. Exploring the effect of double and extended precision on the performance of HEM, 2. Investigating the performance of different embedding formulations of HEM, and 3. Estimating the saddle-node bifurcation point (SNBP) from HEM-based Thévenin-like networks using pseudo-measurements. The HEM algorithm consists of three distinct procedures that might accumulate roundoff error and cause precision loss during the calculations: the matrix equation solution calculation, the power series inversion calculation and the Padé approximant calculation. Numerical experiments have been performed to investigate which aspect of the HEM algorithm causes the most precision loss and needs extended precision. It is shown that extended precision must be used for the entire algorithm to improve numerical performance. A comparison of two common embedding formulations, a scalable formulation and a non-scalable formulation, is conducted and it is shown that these two formulations could have extremely different numerical properties on some power systems. The application of HEM to the SNBP estimation using local-measurements is explored. The maximum power transfer theorem (MPTT) obtained for nonlinear Thévenin-like networks is validated with high precision. Different numerical methods based on MPTT are investigated. Numerical results show that the MPTT method works reasonably well for weak buses in the system. The roots method, as an alternative, is also studied. It is shown to be less effective than the MPTT method but the roots of the Padé approximant can be used as a research tool for determining the effects of noisy measurements on the accuracy of SNBP prediction. / Dissertation/Thesis / Masters Thesis Electrical Engineering 2018

Mathematical modelling is widely used in the field of biomechanics. The traditional approach to investigate spine related injuries is to check the strength of the components of the spine. Spinal stability approach focuses on the force polygons formed by the body weight, muscle forces, ligament forces and external load. This force polygon is expected to stay within the boundaries of the spine to ensure stability. Proving the possibility of one force polygon within the spine boundaries proves the stability of the spine. This study focuses on the full curvature of the spine for spinal stability investigations in a lifting activity. An experiment has been designed to investigate the postural differences in males and females by measuring the full spinal curvature with a skin surface device. Distributed body weight force, with increased detail of muscle and ligament forces acting on the spine have been modelled by writing a code in Visual Basic, while lifting a load from the boot of a car in the sagittal plane. This model is flexible enough to reflect changes in body weight parameter. Results show that there is a difference between male and female postures during the full span of lifting activities. Application of individual muscle forces provides greater control of stability at each vertebral level. By considering the elongation of the ligaments and the force requirements of the muscle groups, it is possible to diagnose soft tissue failure. The differences in posture result in different moment arms for muscles and ligaments causing different loading on the spine. Most critical postures have been identified as the fully flexed postures with external load acting on the spine. Conceptual design ideas have been proposed to assist lifting a load from the boot of a car to eliminate the excessive flexion and loading on the spine.

The study of geological and geotechnical characteristics of the slopes and indication of eviction those considered high risk geotechnically, can prevent tragedies caused by landslides materials. This dissertation was the main objective of evaluate the landslide risk due to urban occupation on the slope of strand west hill Cechela located in the Bela Vista village of the municipality of Santa Maria-RS. The methodology applied in this work consisted in review of the literature, field investigations, laboratory tests for characterization and determination of the resistance parameters of the soil layers, survey report of the intervention area, registration of housing under conditions of high risk to landslides, stability analysis of the slope of strand west hill Cechela and comparison of risk observed with that presented in the Municipal Plan of Reducing Risks of Santa Maria (PMRRSM). The results of inspection report are: the hillside with slopes greater than 30% is located in area not suited to urban occupation; there are deposits of small thickness colluviums situated directly on the substrate sandstone bedrock of the formation Botucatu; the most significant processes of instability on the slope are the landslides planar shallow colluvial soils. The register of homes indicates that if not done containment works two houses should be removed. The results of direct shear tests indicated that the increase of humidity and saturation degree cancels the cohesive intercept and significantly reduces the angle of internal friction of colluvium and sandstone. However, under natural conditions no saturation occurs due to the local geological conditions. The stability analysis with the use of software obtained safety factors (FS) of 1.40 and 2.57 for profiles of 07 and 17 HOUSES, showing that the slopes of the hill are stable. The landslide risk observed in the study area is less than that indicated in the PMRRSM. This work shows that must be constructed surface drainage devices and/or containment of the soil near the House 07 and 17 sites and suggests that today's families are kept in the slope of strand west hill Cechela. It is noteworthy that the village community Bela Vista should be instructed with relevant information to the identification of risks and the municipal plan risks should be reviewed. / O estudo das características geológicas e geotécnicas de encostas e indicação de desocupação daquelas consideradas geotecnicamente de alto risco, podem evitar tragédias causadas por deslizamentos de materiais. Esta dissertação de mestrado teve o objetivo principal de avaliar o risco a escorregamentos na encosta da vertente oeste do morro Cechela devido à ocupação urbana localizada na vila Bela Vista do município de Santa Maria-RS. A metodologia aplicada neste trabalho consistiu no levantamento bibliográfico, investigações de campo, ensaios laboratoriais para a caracterização e determinação dos parâmetros de resistência das camadas dos solos, laudo de vistoria da área de intervenção, cadastro das moradias em condições de risco elevado a escorregamentos, análise de estabilidade da encosta da vertente oeste do morro Cechela e comparação do risco observado com aquele apresentado no Plano Municipal de Redução de Riscos de Santa Maria (PMRRSM). Os resultados do laudo de vistoria são: a encosta com declividades superiores a 30% está localizada em área não adequada à ocupação urbana; existem depósitos de colúvios de pequena espessura situados diretamente sobre substrato rochoso arenítico da formação Botucatu; os processos de instabilização mais significativos na encosta são os escorregamentos planares de solos coluvionares rasos. O cadastro das moradias indica que se não forem executadas obras de contenção duas casas deverão ser removidas. Os resultados dos ensaios de cisalhamento direto indicaram que o aumento da umidade e grau de saturação anula o intercepto coesivo e reduz significativamente o ângulo de atrito interno do colúvio e do arenito. No entanto, nas condições naturais não ocorre a saturação devido a condições geológicas do local. As análises de estabilidade com o uso de software obtiveram fatores de segurança (FS) de 1,40 e de 2,57 para os perfis das CASAS 07 e 17, evidenciando que os taludes da encosta são estáveis. O risco a escorregamentos observado na área em estudo é menor do que aquele indicado no PMRRSM. Este trabalho evidencia que devem ser construídos dispositivos de drenagem superficial e/ou de contenção do solo junto aos locais da CASA 07 e 17 e sugere que as atuais famílias sejam mantidas na encosta da vertente oeste do morro Cechela. Destaca-se que a comunidade da vila Bela Vista deve ser instruída com informações relevantes para a identificação de riscos e que o plano municipal de riscos deve ser revisto.