Dissertations / Theses on the topic 'Function extrema'

In this paper, we use the following scheme to construct a continuous, nowhere-differentiable function 𝑓 which is the uniform limit of a sequence of sawtooth functions 𝑓ₙ : [0, 1] → [0, 1] with increasingly sharp teeth. Let 𝑋 = [0, 1] x [0, 1] and 𝐹(𝑋) be the Hausdorff metric space determined by 𝑋. We define contraction maps 𝑤₁ , 𝑤₂ , 𝑤₃ on 𝑋. These maps define a contraction map 𝑤 on 𝐹(𝑋) via 𝑤(𝐴) = 𝑤₁(𝐴) ⋃ 𝑤₂(𝐴) ⋃ 𝑤₃(𝐴). The iteration under 𝑤 of the diagonal in 𝑋 defines a sequence of graphs of continuous functions 𝑓ₙ. Since 𝑤 is a contraction map in the compact metric space 𝐹(𝑋), 𝑤 has a unique fixed point. Hence, these iterations converge to the fixed point-which turns out to be the graph of our continuous, nowhere-differentiable function 𝑓. Chapter 2 contains the background we will need to engage our task. Chapter 3 includes two results from the Baire Category Theorem. The first is the well known fact that the set of continuous, nowhere-differentiable functions on [0,1] is a residual set in 𝐶[0,1]. The second fact is that the set of continuous functions on [0,1] which have a dense set of proper local extrema is residual in 𝐶[0,1]. In the fourth and last chapter we actually construct our function and prove it is continuous, nowhere-differentiable and has a dense set of proper local extrema. Lastly we iterate the set {(0,0), (1,1)} under 𝑤 and plot its points. Any terms not defined in Chapters 2 through 4 may be found in [2,4]. The same applies to the basic properties of metric spaces which have not been explicitly stated. Throughout, we will let 𝒩 and 𝕽 denote the natural numbers and the real numbers, respectively.

The thesis deals with the mathematical modelling of economic processes, particulary production. Basic approach, models and methods of modelling concerning the mentioned area are described. Further, specific characterizations are investigated. Application of selected methods to typical examples is presented. In the final part models on concrete data are constructed.

Orientador: Rodolfo Bode de Moraes
Dissertaçao (mestrado) - Universidade Federal do Paraná, Setor de Ciencias Humanas, Letras e Artes, Programa de Pós-Graduaçao em Sociologia. Defesa: Curitiba, 2007
Inclui bibliografia

This study is based on the notion that the most widely cited and used instrument for assessing volunteer motivations, the Volunteer Function Inventory (VFI), might not be sufficient in its aspiration to aid, streamline and benefit the recruitment and retention process within volunteer organizations. Understanding volunteers’ motivation is a crucial component in the process of securing future volunteer engagement in a world where their efforts are desperately needed. This study aims to explain motives and motivation behind volunteer efforts in extreme situations as well as to examine the dependability of the theoretical framework in relation to the forthcoming result, based on the following research questions: What motivated Swedish medical personnel to volunteer during the 2014 Ebola outbreak in West Africa? How could the motives and motivation of Swedish medical personnel be categorized in accordance with the Volunteer Function Inventory (VFI)? Is the Volunteer Function Inventory (VFI) a sufficient tool for analyzing the motives and motivation of Swedish medical personnel in the given case? The presented research questions will be answer through a qualitative theory-consuming study, based mainly on interviews. The interviews are, however, complimented by various written sources in the field of research. The result and subsequent analysis confirms the authors’ notion on the VFI and highlights certain deficits in the tool’s execution and its succeeding result. The result of the study shows that the respondents were motivated by a great variety of motivational elements, a variety that the VFI cannot properly account for. Essential information is therefore left unexposed by the VFI, leaving the volunteer organizations in an undesirable situation where they, in their pursuit to recruit and retain the volunteers, are forced to evaluate and accommodate candidates without knowing their complete motivational process. Therefore, the authors deem the tool as applicable but not sufficient. Concludingly, the authors call for additional and more distinct functions in the VFI; an instrument for assessing if the volunteers are intrinsically or extrinsically motivated as well as a sub-section within the tool for determining the presence and character of inhibiting factors affecting the volunteers motivation.
Att fastställa och möta en volontärarbetares motiv och motivation är en avgörande del i arbetet med att säkerställa ett framtida engagemang. I en värld med återkommande naturkatastrofer, väpnade konflikter och epidemier är betydelsen av volontärinsatser större än någonsin. Studien ämnar således undersöka om Volunteer Function Inventory (VFI) – det mest citerade och använda verktyget inom forskningsområdet för att undersöka och fastställa volontärarbetares motiv och motivation – är tillräckligt tillförlitligt i dess strävan att underlätta, effektivisera och gynna rekryterings- och retentionsprocessen inom volontärorganisationer. Undersökningen bygger på en kvalitativ teorikonsumerande studie vilken baseras på djupintervjuer med svensk sjukvårdspersonal kring deras volontärinsatser under ebolautbrottet i Västafrika 2014. Studiens resultat och efterföljande analys påvisar att VFI som verktyg uppvisar vissa brister i dess nuvarande utformning och i den data som genereras. Studien visar att respondenterna motiverades av en stor variation motivationsdrivande element som VFI inte kan redogöra för tillräckligt väl. Detta lämnar volontärorganisationer i en oönskad situation där de i sin strävan att rekrytera och behålla volontärarbetare tvingas utvärdera personer utan att känna till deras fullständiga motivationsbild. Sammanfattningsvis efterfrågar författarna fler och mer distinkta funktioner i VFI; ett instrument för att avgöra om volontärerna drivs av inre eller yttre motivation samt ett tillägg för att kunna fastställa förekomsten, och således karaktären, av motivationshämmande faktorer hos en volontärarbetare.

In this dissertation, a problem related to Hadwiger's conjecture has been studied. We first proved a conjecture of Jakobsen from 1983 which states that every simple graphs on $n$ vertices and at least (11n-35)/2 edges either has a minor isomorphic to K_8 with one edge deleted or is isomorphic to a graph obtained from disjoint copies of K_{1, 2, 2, 2, 2} and/or K_7 by identifying cliques of size five. We then studied the extremal functions for complete minors. We proved that every simple graph on nge9 vertices and at least 7n-27 edges either has a minor, or is isomorphic to K_{2, 2, 2, 3, 3}, or is isomorphic to a graph obtained from disjoint copies of K_{1, 2, 2, 2, 2, 2} by identifying cliques of size six. This result extends Mader's theorem on the extremal function for K_p minors, where ple7. We discussed the possibilities of extending our methods to K_{10} and K_{11} minors. We have also found the extremal function for K_7 plus a vertex minor.

In this paper we analyze the behavior of the Gamma function at its critical points and points of discontinuity on the negative side of the x-axis. We will also explain the bluntness of the gamma function on this negative side.

ATP activates P2X receptors in the apical surface of kidney epithelial cells. However, tubular fluid is unusual - with low sodium levels, high calcium levels and acidic in nature. This study investigated how P2X receptors operate in extreme ionic environments. Xenopus oocytes easily tolerated extreme ionic environments. Their membrane properties and pharmacological profile were examined under normal and extreme ionic conditions. It was confirmed that oocytes lack endogenous P2X receptors, and were unaffected by P2X receptor antagonists and allosteric modulators. Also, oocytes lacked muscarinic receptors. The pharmacological profile of P2X2 and P2X4 receptors expressed in oocytes were examined under normal and extreme ionic conditions. Each P2X subtype was activated by ATP and CTP and this activity was affected by extracellular ions. At rat P2X2, reduced extracellular sodium had an inhibitory effect on ATP signalling, which was reversed by acidifying of the medium or intensified by adding calcium to the medium. At rat P2X4, reduced extracellular sodium had a biphasic action (potentiation, then inhibition) on ATP signalling, which was insensitive to acidification but was intensified by adding calcium. A unique electrophysiological property of P2X2-expressing oocytes was used to investigate an ATP-releasing mechanism. An inward current (Ix) was identified and this current was blocked by the P2 receptor antagonist, suramin, and inhibited by the ATP-degrading enzyme, apyrase. The amplitude of IX was affected by the flow rate of the superfusate, confirming that ATP was released by a mechanosensory process. The present results show that ATP may activate P2X receptors present in the kidney, even in the extreme ionic environment of the distal nephron. Salt and water reclamation in the kidney may depend on flow rate, since a common mechanosensory mechanism can release enough ATP to activate those P2X receptors regulating ENaC and AQP2 channel function in the distal nephron.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 63-66).
Extremal functions of forbidden sequences and 0 - 1 matrices have applications to many problems in discrete geometry and enumerative combinatorics. We present a new computational method for deriving upper bounds on extremal functions of forbidden sequences. Then we use this method to prove tight bounds on the extremal functions of sequences of the form (12 ... 1)t for 1 >/= 2 and t >/= 1, abc(acb)t for t >/= 0, and avav'a, such that a is a letter, v is a nonempty sequence excluding a with no repeated letters and v' is obtained from v by only moving the first letter of v to another place in v. We also prove the existence of infinitely many forbidden 0 - 1 matrices P with non-linear extremal functions for which every strict submatrix of P has a linear extremal function. Then we show that for every d-dimensional permutation matrix P with k ones, the maximum number of ones in a d-dimensional matrix of sidelength n that avoids P is 20(k) nd-1
by Jesse Geneson.
Ph. D.

The study of conformal symmetry is motivated through an example in statistical mechanics and then rigorously developed in quantum field theories in general spatial dimensions. In particular, primary fields are introduced as the fundamental objects of such theories and then studied in the formalism of radial quantization. The implications of conformal invariance on the functional form of correlation functions are studied in detail. Conformal blocks are defined and various approaches to their analytical and numerical calculation are presented with a special emphasis on the one-dimensional case. Building on these preliminaries, a modern formulation of the conformal bootstrap program and its various extensions are discussed. Examples are given in which bounds on the scaling dimensions in a one-dimensional theory are derived numerically. Using these results I motivate the technique of using the extremal functional bootstrap which I then develop in more detail. Many technical details are discussed and examples shown. After a brief discussion of conformal field theories with a boundary I apply numerical methods to find constraints on the spectrum of the 3D Ising model. Another application is presented in which I study the 4-point function on the boundary of a particular theory in Anti-de-Sitter space in order to approximate the mass spectrum of the theory.
O estudo da simetria conforme é motivado através de um exemplo em mecânica estatística e em seguida rigorosamente desenvolvido em teorias de campos quânticos em dimensões espaciais gerais. Em particular, os campos primários são introduzidos como os objetos fundamentais de tais teorias e então estudados através do formalismo de quantização radial. As implicações da invariância conforme na forma funcional das funções de correlação são estudadas em detalhe. Blocos conformes são definidos e várias abordagens para seu cálculo analítico e numérico são apresentadas com uma ênfase especial no caso unidimensional. Com base nessas preliminares, uma formulação moderna do programa de bootstrap conforme e suas várias extensões são discutidas. Exemplos são dados em que limites nas dimensões de escala em uma teoria unidimensional são derivados numericamente. Usando esses resultados, motivei a técnica de usar o bootstrap funcional extremo, que depois desenvolvo em mais detalhes. Diversos detalhes técnicos são discutidos e exemplos são apresentados. Após uma breve discussão das teorias de campo conformes com fronteiras, eu aplico métodos numéricos para encontrar restrições no espectro do modelo de Ising em 3D. Outra aplicação é apresentada em que eu estudo a função de 4 pontos na fronteira de uma teoria particular no espaço Anti-de-Sitter, a fim de aproximar o espectro de massa da teoria.

Extremal Functions for Graph Linkages and Rooted Minors Paul Wollan 137 pages Directed by: Robin Thomas A graph G is k-linked if for any 2k distinct vertices s_1,..., s_k,t_1,..., t_k there exist k vertex disjoint paths P_1,...,P_k such that the endpoints of P_i are s_i and t_i. Determining the existence of graph linkages is a classic problem in graph theory with numerous applications. In this thesis, we examine sufficient conditions that guarantee a graph to be k-linked and give the following theorems. (A) Every 2k-connected graph on n vertices with 5kn edges is k-linked. (B) Every 6-connected graph on n vertices with 5n-14 edges is 3-linked. The proof method for Theorem (A) can also be used to give an elementary proof of the weaker bound that 8kn edges suffice. Theorem (A) improves upon the previously best known bound due to Bollobas and Thomason stating that 11kn edges suffice. The edge bound in Theorem (B) is optimal in that there exist 6-connected graphs on n vertices with 5n-15 edges that are not 3-linked. The methods used prove Theorems (A) and (B) extend to a more general structure than graph linkages called rooted minors. We generalize the proof methods for Theorems (A) and (B) to find edge bounds for general rooted minors, as well as finding the optimal edge bound for a specific family of bipartite rooted minors. We conclude with two graph theoretical applications of graph linkages. The first is to the problem of determining when a small number of vertices can be used to cover all the odd cycles in a graph. The second is a simpler proof of a result of Boehme, Maharry and Mohar on complete minors in huge graphs of bounded tree-width.

This thesis is concerned with extreme problems for certain classes of analytic functions. In many cases, the classes of functions considered form proper subclasses of the class S of normalised analytic functions which are univalent in the unit disc D. In Chapter 1, we present some definitions and known results which are required in subsequent chapters. In Chapter 2, we state some known results concerning the so-called Fekete-Szegö Theorem. We give some extensions and new results in the case of close-to-convex functions. Chapter 3 contains some miscellaneous Fekete-Szegö Theorems. In this chapter, we introduce a new class of analytic functions, which we call logarithmically convex. These functions are a natural analogue to the so-called α-convex functions, studied extensively over the last decade or so. Some extreme coefficient problems are solved for logarithmically convex functions. The final chapter deals with subordination. We apply a lemma of Miller and Mocanu to obtain a series of best possible subordination theorems when the super-ordinate function lies in a sector, rather than the usual half-plane. A consequence of one such result is that the logarithmically convex functions defined in Chapter 3 form a subset of the starlike functions and are thus univalent in D.

This study concentrates on pair distribution function (PDF) analysis of various related systems under extreme conditions using variable temperatures and pressures. The local structure of BaTiO\(_3\) (BTO) was investigated using x-rays (xPDF) between 0 and 8.78 GPa and x-rays and neutrons (nPDF) between 100 and 500K. Evidence is presented that indicates that the “Comes model” of Ti disorder is too simplistic to describe the local structure of BTO at variable temperature but results from the high pressure xPDF study are inconclusive due to the inability to observe the very weak first Ti-O peak. The ambient structure of related Ba\(_1\)\(_-\)\(_x\)Bi\(_x\)Ti\(_1\)\(_-\)\(_x\)Yb\(_x\)O\(_3\) was also investigated using both nPDF and xPDF between x = 0 and 0.15, with PDFs indicating significant local disorder present on both A and B sites. Cubic ZrMo\(_2\)O\(_7\) undergoes pressure induced amorphisation at ~0.7 GPa and this was investigated up to 11.02 GPa using xPDF which showed an increase in Mo coordination with pressure but no significant change around Zr. Iron undergoes a bcc-hcp phase transition between 10 and 20 GPa and xPDF was used to investigate this phase transition and the local structure of the hcp phase up to 50.07 GPa, but texture in the sample at high pressures hindered analysis.

Dans cette thèse, nous abordons l'estimation non paramétrique de la fonction de dépendance des distributions multivariées à valeurs extrêmes. Dans une première partie, on adopte l’hypothèse classique stipulant que les variables aléatoires sont indépendantes et identiquement distribuées (i.i.d). Plusieurs estimateurs non paramétriques sont comparés pour une fonction de dépendance trivariée de type logistique dans deux différents cas. Dans le premier cas, on suppose que les fonctions marginales sont des distributions généralisées à valeurs extrêmes. La distribution marginale est remplacée par la fonction de répartition empirique dans le deuxième cas. Les résultats des simulations Monte Carlo montrent que l'estimateur Gudendorf-Segers (Gudendorf et Segers, 2011) est plus efficient que les autres estimateurs pour différentes tailles de l’échantillon. Dans une deuxième partie, on ignore l’hypothèse i.i.d vue qu’elle n'est pas vérifiée dans l'analyse des séries temporelles. Dans le cadre univarié, on examine le comportement extrêmal d'un modèle autorégressif Gaussien stationnaire. Dans le cadre multivarié, on développe un nouveau théorème qui porte sur la convergence asymptotique de l'estimateur de Pickands vers la fonction de dépendance théorique. Ce fondement théorique est vérifié empiriquement dans les cas d’indépendance et de dépendance asymptotique. Dans la dernière partie de la thèse, l'estimateur Gudendorf-Segers est utilisé pour modéliser la structure de dépendance des concentrations extrêmes d’ozone observées dans les stations qui enregistrent des dépassements de la valeur guide et limite de la norme Tunisienne de la qualité d'air NT.106.04
In this thesis, we investigate the nonparametric estimation of the dependence function for multivariate extreme value distributions. Firstly, we assume independent and identically distributed random variables (i.i.d). Several nonparametric estimators are compared for a trivariate dependence function of logistic type in two different cases. In a first analysis, we suppose that marginal functions are generalized extreme value distributions. In a second investigation, we substitute the marginal function by the empirical distribution function. Monte Carlo simulations show that the Gudendorf-Segers (Gudendorf and Segers, 2011) estimator outperforms the other estimators for different sample sizes. Secondly, we drop the i.i.d assumption as it’s not verified in time series analysis. Considering the univariate framework, we examine the extremal behavior of a stationary Gaussian autoregressive process. In the multivariate setting, we prove the asymptotic consistency of the Pickands dependence function estimator. This theoretical finding is confirmed by empirical investigations in the asymptotic independence case as well as the asymptotic dependence case. Finally, the Gudendorf-Segers estimator is used to model the dependence structure of extreme ozone concentrations in locations that record several exceedances for both guideline and limit values of the Tunisian air quality standard NT.106.04

In this dissertation separation and selection theorems are presented. It begins by presenting a detailed proof of the Inductive Definability Theorem of D. Cenzer and R.D. Mauldin, including their boundedness principle for monotone coanalytic operators.

Inverse problems naturally occur in many branches of science and mathematics. An inverse problem entails finding the values of one or more parameters using the values obtained from observed data. A typical example of an inverse problem is the inversion of the Radon transform. Here a function (for example of two variables) is deduced from its integrals along all possible lines. This problem is intimately connected with image reconstruction for X-ray computerized tomography. Moment problems are a special class of inverse problems. While the classical theory of moments dates back to the beginning of the 20th century, the systematic study of truncated moment problems began only a few years ago. In this dissertation we will first survey the elementary theory of truncated moment problems, and then focus on those problems with cubic column relations. For a degree 2n real d-dimensional multisequence β ≡ β (2n) ={β i}i∈Zd+,|i|≤2n to have a representing measure μ, it is necessary for the associated moment matrix Μ(n) to be positive semidefinite, and for the algebraic variety associated to β, Vβ, to satisfy rank Μ(n)≤ card Vβ as well as the following consistency condition: if a polynomial p(x)≡ ∑|i|≤2naixi vanishes on Vβ, then Λ(p):=∑|i|≤2naiβi=0. In 2005, Professor Raúl Curto collaborated with L. Fialkow and M. Möller to prove that for the extremal case (Μ(n)= Vβ), positivity and consistency are sufficient for the existence of a (unique, rank Μ(n)-atomic) representing measure. In joint work with Professor Raúl Curto we have considered cubic column relations in M(3) of the form (in complex notation) Z3=itZ+ubar Z, where u and t are real numbers. For (u,t) in the interior of a real cone, we prove that the algebraic variety Vβ consists of exactly 7 points, and we then apply the above mentioned solution of the extremal moment problem to obtain a necessary and sufficient condition for the existence of a representing measure. This requires a new representation theorem for sextic polynomials in Z and bar Z which vanish in the 7-point set Vβ. Our proof of this representation theorem relies on two successive applications of the Fundamental Theorem of Linear Algebra. Finally, we use the Division Algorithm from algebraic geometry to extend this result to other situations involving cubic column relations.

Polymorphism is the existence of multiple phenotypes within a given species in a population. In social insects, worker polymorphism plays a key role in maximizing the effectiveness of the division of labor. Some ants have an extreme form of worker polymorphism where workers have a broad size range and multiple size modes. Pheidole ants, for example, consist of both highly polymorphic species as well as those with relatively low polymorphism. Here, we examined why different Pheidole species have different degrees of polymorphism and how polymorphism is produced. We thoroughly characterized the worker caste systems of P. spadonia, P. rhea, P. obtusospinosa, and P. tepicana. There were significant differences among the four species in size range, number of size modes, caste ratios, allometry, and caste biomass allocation. An examination of worker caste traits of P. spadonia, P. rhea, and P. obtusospinosa revealed that as head size increased for all three species: (1) mandibles became broader and less serrated, (2) head muscle volume increased, and (3) bite force increased. These traits of large supersoldiers are likely adapted for crushing while those of small minors are likely for cutting. Foraging experiments showed that P. spadonia, P. rhea, and P. obtusospinosa used their workforce in different ways for food processing outside the nest. For P. rhea, the frequency of supersoldiers involved in food processing increased as the processing level required increased. However, P. obtusospinosa supersoldiers were rarely found processing food outside the nest and P. spadonia soldiers assisted in processing dead prey but did not help at all in processing live prey. P. obtusospinosa and P. spadonia workers may be more involved with other colony tasks. This hypothesis was confirmed when field observations of P. obtusospinosa showed supersoldiers participating in head-blocking at their entrance to fend off invading army ants; no other castes exhibited this behavior. Lastly, we tested genetic influences on worker polymorphism. We found that as colony genetic diversity increased (via polyandry), the degree of polymorphism increased. We also showed evidence of paternal genes influences on the development of worker castes in the highly polymorphic P. rhea.

Este trabalho tem por objetivo estudar algumas variações do teorema de Banach- Stone. Elas podem ser encontradas no artigo Variations on the Banach- Stone Theorem, [14]. Além disso, apresentamos um resultado, provado por D. Amir em [1], que generaliza a versão clássica do Teorema de Banach- Stone. Consideramos os espaços C(K) e C(L), que representam os espaços de funções contínuas de K em R e de L em R respectivamente, onde K e L são espaços Hausdor compactos. O enunciado da versão clássica do teorema de Banach- Stone é a seguinte: \"Sejam K e L espaços Hausdor compactos. Então C(K) é isométrico a C(L) se e somente se, K e L são homeomorfos\". Apresentamos a primeira das variações que considera isomorfismo entre álgebras e foi feita por Gelfand e Kolmogoro em [15], no ano de 1939. A segunda versão apresentada trata de isomorfismo isométrico e a demonstração é originalmente devida a Arens e Kelley e é encontrada em [2]. Finalmente, estudamos o teorema provado por D. Amir e apresentado em [1]. Este teorema generaliza o teorema clássico de Banach- Stone e tem o seguinte enunciado: Se K e L são espaços Hausdor compactos e T é um isomorfismo linear de C(K) sobre C(L), com ||T||.||T^||< 2 então K e L são homeomorfos
This work aims to study some variations of the Banach- Stone theorem. They can be found in the article Variations on the Banach- Stone Theorem, [14]. In addition, we present a result, proved by D. Amir in [1], that generalizes the classic version of the Theorem Banach- Stone. We consider the spacesC(K) andC(L), representing the spaces of continuous functions from K into R and from L into R respectively, where K and L are compact Hausdor spaces. The wording of the classic version of the Banach- Stone theorem is as follows: \"Let K e L be compact Haudor spaces. Then C(K) isisometrictoC(L) if,andonlyif, K and L are homeomorphic\".Here the first of the variations that considers isomorphism between algebras and was made by Gelfand and Kolmogoro in [15], in 1939. The second version presented is about isometric isomorphisms and the demonstration is originally due to Arens and Kelley and it is found in [2]. Finally, we study the theorem proved by D. Amir and presented in [1]. This theorem generalizes the classical theorem Banach- Stone and states the following: \"Let K e L be compact Haudor spaces and let T be a linear isomorphism from C(K) into C(L), with ||T||.||T^||< 2. Then K and L are homeomorphic\".

We study the structure of the set of extreme points of the compact convex set of matrix-valued holomorphic functions with positive real part on a finitely-connected planar domain R normalized to have value equal to the identity matrix at some prescribed point t0 in R. This leads to an integral representation for such functions more general than what would be expected from the result for the scalar-valued case. After Cayley transformation, this leads to a integral Agler decomposition for the matrix Schur class over R (holomorphic contractive matrix-valued functions over R). Application of a general theory of abstract Schur-class generated by a collection of test functions leads to a transfer-function realization for the matrix Schur-class over R, extending results known up to now only for the scalar case. We also explain how these results provide a new perspective for the dilation theory for Hilbert space operators having R as a spectral set.
Ph. D.

No
Given a prescribed boundary of a Bezier surface we compare the Bezier surfaces generated by two different methods, i.e. the Bezier surface minimising the Biharmonic functional and the unique Bezier surface solution of the Biharmonic equation with prescribed boundary. Although often the two types of surfaces look visually the same, we show that they are indeed different. In this paper we provide a theoretical argument showing why the two types of surfaces are not always the same.

La théorie spatiale des valeurs extrêmes permet de modéliser et prédire la fréquence d'évènements extrêmes ayant une étendue spatiale comme, par exemple, des pluies ou des températures extrêmes, ou encore de fortes concentrations de pollution atmosphérique. Elle s'adapte bien aux données temporelles, lorsque le phénomène spatial étudié est observé plusieurs fois dans le temps. Cependant, nous n'avons parfois pas accès à de telles données: seulement un ou quelques enregistrements sont disponibles. C'est le cas, par exemple, des études sur l'estimation des ressources minières ou sur l'évaluation de la pollution des sols et plus généralement de toute recherche dont l'objet d'étude varie très peu au cours du temps ou pour lequel le coût d'échantillonnage est trop élevé. Ce cas de figure est très peu abordé par la communauté des extrêmes. Au contraire, c'est un cadre d'analyse auquel la Géostatistique s'intéresse particulièrement. Les travaux réalisés au cours de cette thèse ont pour objectif d'établir des connexions mathématiques entre ces deux disciplines afin de mieux appréhender les évènements extrêmes, lorsque le phénomène spatial sous-jacent n'est observé qu'une seule fois.Nous nous intéressons, dans un premier temps, au concept de portée intégrale. Intrinsèquement lié aux propriétés d'ergodicité et de mélange, ce paramètre issu de la théorie géostatistique caractérise les fluctuations statistiques, à large échelle, d'un champ aléatoire stationnaire. Lorsque ce dernier est un champ max-stable, nous montrons que sa fonction coefficient extrémal (ECF) est fortement liée à la portée intégrale du champ des excès, au dessus d'un certain seuil, correspondant. Cette approche permet de retrouver et de compléter des résultats précédemment établis dans un contexte de risque spatialisé. Elle met également en évidence une nouvelle expression de la fonction coefficient extrémal qui dépend du variogramme du champ des excès.À partir de cette formule, nous proposons un nouvel estimateur non-paramétrique de l'ECF. Ses propriétés asymptotiques sont établies lorsqu'il est évalué à partir d'une unique réalisation, partiellement observée, d'un champ stationnaire max-stable. En particulier, lorsque le nombre d'observations se densifie en même temps que le champ d'observation grandit, et sous certaines hypothèses concernant la portée intégrale susmentionnée, nous montrons qu'il est consistent et asymptotiquement normal. Il est donc pertinent d'utiliser les outils géostatistiques pour enrichir l'analyse des valeurs extrêmes.Finalement, nous développons un nouvel algorithme permettant de simuler, en continu, des processus aléatoires tempête pour lesquels la fonction de forme est déterministe. Il se distingue donc de la plupart des algorithmes existants qui s'utilisent exclusivement lorsque le domaine de simulation est composé d'un nombre fini de points. À cet égard, il permet d'étudier plus facilement la géométrie des réalisations de tels processus. Cela est particulièrement intéressant quand la caractéristique géométrique étudiée mêle différentes échelles d'observation
Spatial extreme value theory helps model and predict the frequency of extreme events in a spatial context like, for instance, extreme precipitations, extreme temperatures. It is well adapted to time series. However, in some cases, such types of data cannot be accessed: only one or just a few records are made available. This is the case, for instance in soil contamination evaluation. This situation is rarely addressed in the spatial extremes community, contrary to Geostatistics,which typically deals with such issues. The aim of this thesis is to make some connections between both disciplines,in order to better handle the study of spatial extreme events, and especially their spatial dependence structure, when having only one set of spatial observations. A link is first established through the concept of integral range. It is a geostatistical parameter that characterizes the statistical fluctuations of a stationary random field at large scale. When the latter is max-stable, we show that its extremal coefficient function (ECF), which is a measure of spatial dependence, is closely related to the integral range of the corresponding exceedance field above a threshold. From this, we move toproposing a new nonparametric estimator of the ECF. Its asymptotic properties are derived when it is computed from a single and partially observed realization of a stationary max-stable random field. Specifically, under some assumptions on the aforementioned integral range, we prove that it is consistent and asymptotically normal. Finally, we develop a novel algorithm to perform exact simulations in a continuous domain of storm processes with deterministic shape function. It distinguishes itself from most existing procedures, which apply to simulation domains made of a finite number of points. Most part of the algorithm are designed to be parallelizable

The biodiversity of two distinct marine environments was observed to describe the biogeocomplexity of these extreme ecological systems. A shallow-water hydrothermal vent in Papua New Guinea served as a study of a thermophilic ecosystem influenced by arsenic rich vent fluids while a 60 m deep offshore primarily anoxic karst sink served as a study of an anaerobic sulfur-influenced habitat. Both environments support unique biological communities that are influenced by the physical and chemical pressures imposed on them by the harsh conditions of these systems. In Tutum Bay, Ambitle Isle, Papua New Guinea, a transect was created from a shallow hydrothermal vent that extended 120 m away from the vent. Previous studies have shown that the geochemistry of the system is heavily influenced by arsenic which is toxic to most organisms. In this study, macro- and meiofauna were collected and scored and combined with bacterial sequence data collected along the length of the transect. It was found that near vent sites harbored biological communities more similar than sites further from the vent. Many species were found only at sites near the hydrothermal vent. Near-vent communities were less diverse than those away from the vent, and biodiversity generally increased as distance from the vent increased. Distinct correlations between thermophilic organisms and temperature were observed. The metabolic repertoire of the microbial communities suggests that many strategies are used to obtain energy and carbon. The relative abundance of bacteria containing genes to reduce arsenic was comparable to those able to reduce sulfur compounds. Primary production appeared to be a mix of chemo- and phototrophy. Food webs and association analysis suggest a complex interplay between macrofaunal, meiofaunal and bacterial communities. While the system is heavily influenced by arsenic, no specific correlation between the relative abundance of arsenic metabolizing organisms and the amount of arsenic in the system could be drawn. This is likely due to the fact that most of the arsenic produced by the system is readily adsorbed onto iron oxyhydroxides, reducing the arsenic's bioavailability. The anoxic conditions at Jewfish sink provide a different hurdle than the hot arsenic conditions found in Papua New Guinea. The anoxic conditions are shared by other pit features found in karst geography, but the metabolic processes between Jewfish sink and these other karst habitats are different. The blue holes and black holes of the Bahamas are some of the most well-studied of these karstic pits. In these features, which are large circular pits with diameters of over 300 m, light and sulfur are used as a means of energy acquisition. Jewfish sink, having an opening only 6 m in diameter, is light restricted compared to these systems. As a result, the strategy of organisms dwelling in the anoxic conditions of the sink is different than those found at the well-studied holes in the Bahamas. Geochemical measurements were recorded over two time periods spanning a combined total of 6 years. The anoxic bottom waters of Jewfish sink remain stable and contained high levels of sulfide throughout most of the seasons studies. Sequence analysis of prokaryotes within the sink showed that sulfur reducers had the highest relative abundance compared to other functional guilds. To monitor the changes of the microbial communities within the sink, bacterial communities were examined at 4 depths within the sink at 9 different intervals over a period of 685 days. Denaturing Gradient Gel Electrophoresis (DGGE) was used to fingerprint 16s rRNA bacterial communities and dissimilatory sulfite reducing communities by targeting the 16s rRNA bacterial gene and the dsr gene associated with dissimilatory sulfite reducing bacteria and archaea. The lowest depth studied within the sink (40 m) remained stable chemically and biologically until a turnover event occurred within the second winter of the study. This turnover event disrupted the biological communities at 40 m and led to a reestablished community comprised of different species that those found prior to the event. Upper waters within the sink show that clines establish themselves seasonally and partition zones that confine bacterial communities that are more similar to each other within these zones while excluding bacterial communities that are outside of these zones. Oxygenated water was shown to not contain prokaryotes containing the dsr gene. As the oxycline changed seasonally, dissimilatory sulfite reducing prokaryotes containing the dsr gene remained in the anoxic zone and required time to reestablish themselves whenever oxygenated water displaced them.

In this thesis, we consider linear extremal problems in the Hp spaces. For many of these extremal problems, a unique solution can be guaranteed. We will examine some of the classical examples of extremal problems in these spaces. With this framework in place we will then consider a particular problem which does not always have a unique solution.

Transcranial magnetic stimulation (TMS) is a widely-used investigative technique in motor cortical evaluation. TMS is now being used in the investigation of fatigue to help partition the effects of central fatigue. Few studies have utilized this technique to evaluate the effects of locomotor exercise and none in conditions of extreme exercise. Therefore, the purpose of this thesis was twofold; first, to answer methodological questions pertaining to the use of TMS in fatigue evaluation, particularly of the quadriceps, and second, to investigate the effects of extreme exercise conditions on the development of central and supraspinal fatigue and corticospinal excitability and inhibition. In Studies 1 and 2, the effect of approaching a target force in different ways before the delivery a TMS pulse and the difference between commonly-employed methods of determining TMS intensity on the selection of optimal TMS intensity were investigated. In Study 3, the effect of one night sleep deprivation on cognitive and exercise performance and central parameters was investigated. The effect of a 110-km ultra-trail on the supraspinal component of central fatigue was evaluated in Study 4. The principal findings from this thesis are that during TMS evaluation during brief voluntary contractions, it is essential to deliver the TMS pulse once the force has stabilized at the target and that a stimulus-response curve at 20% MVC is appropriate for determining optimal TMS intensity in exercise and fatigue studies. Furthermore, while sleep deprivation negatively-impacted cognitive and exercise performance, it did not influence neuromuscular parameters nor result in greater central fatigue. Supraspinal fatigue develops and corticospinal excitability increases during endurance/ultra-endurance running and cycling, while the effects on inhibitory corticospinal mechanisms are equivocal and probably depend on exercise characteristics and TMS intensity

Today's e-commerce systems have evolved from traditional middleware technologies such as CORBA and DCOM to a multi-tiered infrastructure. Java 2 Enterprise Edition (J2EE) and Microsoft .NET are two main competitors building on such architecture. Usually a multi-tiered architecture involves distinct architecture layers, components, distributed applications and enterprise resources. This complexity poses considerable risks to e-commerce systems building on such infrastructure. To address the current problems of high testing cost and non-formal test specification, we propose a new process called FTEXT (Functional Testing of E-commerce systems with eXtreme programming and TTCN-3). FTEXT combines an agile software development process, cost-effective test tools and a formal test specification language that serve a common purpose and work together to support a cost-effective approach to test J2EE based e-commerce systems. To facilitate TTCN-3 test execution, we have designed and implemented a parser that translates TTCN-3 test script to Java source code that conforms to the selected open source test tool. The source code is then compiled into executable test code. In the context of J2EE based system development and testing, this thesis offers the following contributions: (1) Methodology: Proposed a new cost-effective model (FTEXT) for testing J2EE based e-commerce systems. (2) Parser: Designed and implemented a parser to translate TTCN-3 test script and generate Java test code that conforms to the selected open source test tool for functional testing of Web-based applications. (3) Case Study: Implemented our approach in a case study that tests a typical J2EE ecommerce application. The case study demonstrates the working mechanism of the FTEXT method, relevant open source tools (e.g. HttpUnit and Apache Ant) and the capability of our TTCN-3 to Java parser.

[ES] Los espacios libres Lipschitz F(M) son linearizaciones canónicas de espacios métricos M cualesquiera. Más concretamente, F(M) es el único espacio de Banach que contiene una copia isométrica de M que es linearmente densa, y tal que toda aplicación Lipschitz de M en cualquier espacio de Banach X puede extenderse a un operador linear continuo de F(M) en X. Estos espacios suponen una herramienta muy potente para el estudio de la geometría no lineal de espacios de Banach, al permitir la aplicación de las técnicas lineales clásicas, bien conocidas, a problemas no lineales. Pero este esfuerzo sólo merece la pena si se dispone de un conocimiento lo bastante detallado de la estructura de F(M). El estudio sistemático de los espacios libres Lipschitz es bastante reciente y, por ello, dicho conocimiento es todavía más bien limitado. Esta tesis se enmarca en el programa general de estudio de la estructura espacios libres Lipschitz genéricos. Empezamos nuestro estudio desarrollando algunas herramientas básicas para la teoría general de espacios libres Lipschitz. Primero definimos operadores de ponderación en espacios Lipschitz y los usamos para demostrar la conjetura de Weaver de que todos los funcionales normales del bidual F(M)** son débil* continuos. A continuación demostramos el teorema de la intersección, que en esencia dice que la intersección de espacios libres Lipschitz es de nuevo un espacio libre Lipschitz. Este resultado nos permite desarrollar el concepto de soporte de un elemento de F(M), análogo al de soporte de una medida. Además, extendemos el uso de estas herramientas al bidual F(M) y las usamos para establecer una descomposición del bidual en espacios de funcionales que están "concentrados en el infinito" y "separados del infinito", respectivamente. Con estas herramientas en nuestro poder, emprendemos el estudio de dos aspectos concretos de los espacios libres Lipschitz. En primer lugar analizamos la relación entre F(M) y los espacios de medidas sobre M. En particular, obtenemos caracterizaciones de los elementos de F(M) que pueden representarse como la integración con respecto a una medida de Borel (no necesariamente finita) sobre M y viceversa, y probamos que el soporte coincide con el de la medida asociada. También identificamos los espacios métricos M en los cuales todo elemento de F(M) puede ser representado como una medida de Borel. Este análisis se generaliza al bidual F(M)**, utilizando en este caso medidas sobre la compactificación uniforme de M y llegando a resultados similares. Obtenemos también algunas consecuencias para los elementos de F(M) y F(M)** que pueden expresarse como diferencia de dos elementos positivos, como la existencia de un análogo de la descomposición de Jordan para medidas. En segundo lugar, estudiamos la estructura extremal de la bola unidad de F(M) y hacemos algunas contribuciones al programa general consistente en encontrar caracterizaciones puramente geométricas de todos sus elementos extremales. Concretamente, caracterizamos los puntos extremos preservados de la bola, así como aquellos puntos extremos y expuestos que tienen soporte finito. Además damos una descripción completa de la estructura extremal de la parte positiva de la bola unidad. La teoría de los soportes en F(M) desarrollada anteriormente juega un papel crucial en las demostraciones de estos resultados.
[CA] Els espais lliures Lipschitz F(M) són linearitzacions canòniques d'espais mètrics M qualssevol. Més concretament, F(M) és l'únic espai de Banach que conté una còpia isomètrica de M que és linealment densa, i tal que tota aplicació Lipschitz de M en qualsevol espai de Banach X pot ser estesa a un operador lineal continu de F(M) en X. Aquests espais són una eina molt potent per a l'estudi de la geometria no lineal d'espais de Banach, ja que permeten l'aplicació de les tècniques lineals clàssiques, ben conegudes, a problemes no lineals. Però aquest esforç nomes val la pena si es disposa d'un coneixement bastant detallat de l'estructura de F(M). L'estudi sistemàtic dels espais lliures Lipschitz és bastant recent i, per això, aquest coneixement és encara prou limitat. Aquesta tesi s'emmarca en el programa general d'estudi de l'estructura dels espais lliures Lipschitz genèrics. Comencem el nostre estudi desenvolupant algunes eines bàsiques per a la teoria general d'espais lliures Lipschitz. Primer definim operadors de ponderació en espais Lipschitz i els fem servir per demostrar la conjectura de Weaver que tots els funcionals normals del bidual F(M)** son feble* continus. A continuació demostrem el teorema de la intersecció, que en essència diu que la intersecció d'espais lliures Lipschitz és de nou un espai lliure Lipschitz. Aquest resultat ens permet desenvolupar el concepte de suport d'un element de F(M), anàleg al de suport d'una mesura. A més, estenem l'ús d'aquestes eines al bidual F(M)** i les fem servir per establir una descomposició del bidual en espais de funcionals que estan "concentrats a l'infinit" i "separats de l'infinit", respectivament. Amb aquestes eines al nostre abast, emprenem l'estudi de dos aspectes concrets dels espais lliures Lipschitz. En primer lloc, analitzem la relació entre F(M) i els espais de mesures sobre M. En particular, obtenim caracteritzacions dels elements de F(M) que poden representar-se com la integració respecte a una mesura de Borel (no necessàriament finita) sobre M i viceversa, i provem que el suport coincideix amb el de la mesura associada. També identifiquem els espais mètrics M on tot element de F(M) pot ser representat com una mesura de Borel. Aquesta anàlisi es generalitza al bidual F(M)**, utilitzant en aquest cas mesures sobre la compactificació uniforme de M i arribant a resultats similars. També obtenim algunes conseqüències per als elements de F(M) i F(M)** que poden expressar-se com a diferència de dos elements positius, com ara l'existència d'un anàleg de la descomposició de Jordan per a mesures. En segon lloc, estudiem l'estructura extremal de la bola unitat de F(M) i fem algunes contribucions al programa general consistent en trobar caracteritzacions purament geomètriques de tots els seus elements extremals. Concretament, caracteritzem els punts extrems preservats de la bola, així com aquells punts extrems i exposats que tenen suport finit. A més fem una descripció completa de l'estructura extremal de la part positiva de la bola unitat. La teoria dels suports en F(M) desenvolupada anteriorment juga un paper crucial en les demostracions d'aquests resultats.
[EN] Lipschitz-free spaces F(M) are canonical linearizations of arbitrary complete metric spaces M. More specifically, F(M) is the unique Banach space that contains an isometric copy of M that is linearly dense, and such that any Lipschitz mapping from M into some Banach space X extends to a bounded linear operator from F(M) into X. Those spaces are a very powerful tool for studies of the nonlinear geometry of Banach spaces, as they allow the application of well-known classical linear techniques to nonlinear problems. But this effort is only worthwhile if we have sufficient knowledge about the structure of F(M). The systematic study of Lipschitz-free spaces is rather recent and so the current understanding of their structure is still quite limited. This thesis is framed within the general program of studying the structure of general Lipschitz-free spaces. We start our study by developing some basic tools for the general theory of Lipschitz-free spaces. First we introduce weighting operators and use them to solve Weaver's conjecture that all normal functionals in the bidual F(M)** are weak* continuous. Next we prove the intersection theorem, which essentially says that the intersection of Lipschitz-free spaces is again a Lipschitz-free space. That result allows us to develop the concept of support of an element of F(M), analogous to the support of a measure. Furthermore, we extend the use of these tools to the bidual F(M)** and apply them to establish a decomposition of the bidual into spaces of functionals that are "concentrated at infinity" and "separated from infinity", respectively. With these tools at our disposal, we undertake the study of two particular aspects of Lipschitz-free spaces. First we analyze the relationship between F(M) and spaces of measures on M. In particular, we obtain characterizations of those elements of F(M) that can be represented as integration against a (not necessarily finite) Borel measure on M and vice versa, and we show that their supports agree. We also identify those metric spaces such that every element of F(M) can be represented by a Borel measure. This analysis is generalized to the bidual F(M)**, using measures on the uniform compactification of M in that case and obtaining similar results. We also derive some consequences for those elements of F(M) and F(M)** that can be expressed as the difference between two positive elements, such as the existence of an analog of the Jordan decomposition for measures. Secondly, we study the extremal structure of the unit ball of F(M) and provide some contributions to the general program of finding purely geometric characterizations of all of its extremal elements. Namely, we characterize all of its preserved extreme points, and its extreme and exposed points of finite support. We also give a full description of the extremal structure of the positive unit ball. The theory of supports developed previously plays a crucial role in the proofs of these results
The author would like to thank Marek Cúth, Michal Doucha, Antonio José Guirao, Gilles Lancien and Eva Pernecká for their careful reading and correction of this document or parts of it. Some activities related to this thesis were partially supported by the Spanish Ministry of Economy, Industry and Competitiveness under Grant MTM2017-83262-C2-2-P, and by a travel grant of the Institute of Mathematics (IEMath-GR) of the University of Granada. Part of this research was conducted during visits to the Czech Technical University in Prague in 2018 and 2020, the Laboratoire de Mathématiques de Besançon in 2019, and the University of Granada in 2020. The author wishes to express his gratitude for the hospitality and the excellent working conditions during his visits.
Aliaga Varea, RJ. (2020). Geometry and structure of Lipschitz-free spaces and their biduals [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/159256
TESIS

The event of a default for low-default portfolios, such as sovereign debt or banks, have received much attention as a result of the increasing instabilities in financial markets. The lack of sufficient default information on low-default portfolios complicates the protection of such portfolios. Default protections have typically, in the past, relied on extreme value theory and reporting the value at risk. The focus here, is the application of an engineering concept, accelerated test techniques, to the problem of insufficient data on low-default portfolios. In the application, high-default portfolios serve as stressed cases of low-default portfolios. Since high-default portfolios have more data available, viewing it as a stressed case of a low-default portfolio enables us to extrapolate the data to the low-default portfolio environment, and do estimation such as estimating the default probability for a low-default portfolio. The flexible framework through which the above is achieved, is provided.
Dissertation (MSc)--University of Pretoria, 2016.
Statistics
MSc
Unrestricted

Almost one-half of the annual precipitation in the southwestern United States occurs during the North American monsoon (NAM). Given favorable synoptic-scale conditions, organized monsoon thunderstorms may affect relatively large geographic areas. Through an objective analysis of atmospheric reanalysis and observational data, the dominant synoptic patterns associated with NAM extreme events are determined for the period from 1993 to 2010. Thermodynamically favorable extreme-weather-event days are selected on the basis of atmospheric instability and precipitable water vapor from Tucson, Arizona, rawinsonde data. The atmospheric circulation patterns at 500 hPa associated with the extreme events are objectively characterized using principal component analysis. The first two dominant modes of 500-hPa geopotential-height anomalies of the severe-weather-event days correspond to type-I and type-II severe-weather-event patterns previously subjectively identified by Maddox et al. These patterns reflect a positioning of the monsoon ridge to the north and east or north and west, respectively, from its position in the "Four Corners" region during the period of the climatological maximum of monsoon precipitation from mid-July to mid-August. An hourly radar gauge precipitation product shows evidence of organized, westward-propagating convection in Arizona during the type-I and type-II severe weather events. This new methodological approach for objectively identifying severe weather events may be easily adapted to inform operational forecasting or analysis of gridded climate data.

The interaction of embedded systems with their environments and their resource limitations make it important to take into account properties such as timing, security, and resource consumption in designing such systems. These so-called Extra-Functional Properties (EFPs) capture and describe the quality and characteristics of a system, and they need to be taken into account from early phases of development and throughout the system's lifecycle. An important challenge in this context is to ensure that the EFPs that are defined at early design phases are actually preserved throughout detailed design phases as well as during the execution of the system on its platform. In this thesis, we provide solutions to help with the preservation of EFPs; targeting both system design phases and system execution on the platform. Starting from requirements, which form the constraints of EFPs, we propose an approach for modeling Non-Functional Requirements (NFRs) and evaluating different design alternatives with respect to the satisfaction of the NFRs. Considering the relationship and trade-off among EFPs, an approach for balancing timing versus security properties is introduced. Our approach enables balancing in two ways: in a static way resulting in a fixed set of components in the design model that are analyzed and thus verified to be balanced with respect to the timing and security properties, and also in a dynamic way during the execution of the system through runtime adaptation. Considering the role of the platform in preservation of EFPs and mitigating possible violations of them, an approach is suggested to enrich the platform with necessary mechanisms to enable monitoring and enforcement of timing properties. In the thesis, we also identify and demonstrate the issues related to accuracy in monitoring EFPs, how accuracy can affect the decisions that are made based on the collected information, and propose a technique to tackle this problem. As another contribution, we also show how runtime monitoring information collected about EFPs can be used to fine-tune design models until a desired set of EFPs are achieved. We have also developed a testing framework which enables automatic generation of test cases in order verify the actual behavior of a system against its desired behavior. On a high level, the contributions of the thesis are thus twofold: proposing methods and techniques to 1) improve maintenance of EFPs within their correct range of values during system design, 2) identify and mitigate possible violations of EFPs at runtime.
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In this thesis, we tried to solve those problems by adapting agile development methods with Non-functional requirements-framework (NFR-Framework). In this thesis, we have inspected many research papers, and we have met industrial experts for feedback regarding our theoretical results. As a result of the inspection, we have been able to adapt agile development methods (extreme programming (XP)) with NFR-framework. We use XP since it is more practically oriented process than other agile development methods. In the first try for this process model, we got three alternatives for applying it. The first one is based on collecting all NFRs from the beginning of the development process. The second one is based on updating the SIG (software interdependency graph) every time we have new functional requirements (FR) and the third one is based on the incremental nature of agile development methods. Each one of these alternatives has it is own advantages and disadvantages. We tried to extract those advantages and disadvantages by brainstorming and reading research papers. The most important issue in all of the three alternatives is the applicability. Finally we got industrial feedback regarding all of them. As a result of the industrial feedback, we were able to find another alternative of how to apply the process model which is presented in 7.2.

The purpose of this thesis is to investigate a class of four left-invariant optimal control problems on the special orthogonal group SO(3). The set of all control-affine left-invariant control systems on SO(3) can, without loss, be reduced to a class of four typical controllable left-invariant control systems on SO(3) . The left-invariant optimal control problem on SO(3) involves finding a trajectory-control pair on SO (3), which minimizes a cost functional, and satisfies the given dynamical constraints and boundary conditions in a fixed time. The problem is lifted to the cotangent bundle T*SO(3) = SO(3) x so (3)* using the optimal Hamiltonian on so(3)*, where the maximum principle yields the optimal control. In a contribution to the study of this class of optimal control problems on SO(3), the extremal equations on so(3)* (ident ified with JR3) are integrated via elliptic functions to obtain explicit expressions for the solution curves in each typical case. The energy-Casimir method is used to give sufficient conditions for non-linear stability of the equilibrium states.
KMBT_363
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The current mantra of development advocates macro level, top down approaches which largely ignore the potential for change inherent within existing indigenous grass root social structures. One such indigenous social institution prevalent in the Islamic World is that of the religious school locally known as the madrassa. After 9-11, madaris have been prominently featured in the international media as a potential breeding ground for terrorists and fundamentalists and has become a policy concern for both the American and Pakistani governments. Recent policy interventions have included steps to centralize madaris, curriculum revisions along with a renewed interest in improving existing public and private schools as an alternative to religious schooling. Most of these steps have been rejected by the madrassa community and there is ongoing antagonism between madrassa officials and government authorities. This research examines how madrassa mission and mandate intersect with community development goals as well as with the national project of development and modernity, and how they create linkages and multiplier effects within the local culture and institutions particularly in the absence of social support services. It utilizes a mixed method approach based on ethnographic interviews, mapping, content and discourse analyses and participant observations to uncover the daily patterns of the research participants’ lives in two selected research sites in Islamabad, Pakistan. The research develops recommendations based on the perspective of madrassa students, alumni, administrators, teachers and local community members. The study argues that madaris have the ability to reach out to Pakistan’s marginalized and disenfranchised. It is important for the madaris community to get recognized by the state to gain access to state funding and aid agencies. However the government needs to be very flexible with its policy of reforms and needs to allow the madaris to operate in the academic spaces they deem fit. The purpose and intent of madaris should be preserved and allowed to remain intact. This is where planners and development practioners fit in. They need to mobilize a participatory dialogue between the state and the madaris community and exact this crucial integration of indigenous institutions such as the madrassa into mainstream development policy.

Les transistors de puissance grands gaps tels que les MOSFETs SiC et HEMT GaN repoussent les compromis classiques en électronique de puissance. Brièvement, des gains significatifs ont été démontrés par les transistors SiC et GaN: meilleurs rendements, couplés à une augmentation des densités de puissance offertes par la montée en fréquence de découpage. Les MOSFET SiC à haute tension présentent des spécificités telles qu'une faible tenue en court-circuit (SC) par rapport aux IGBT Si et un oxyde de grille aminci, et une tension de commande rapprochée grillesource élevée. La polarisation négative sur la grille à l'état bloqué crée un stress supplémentaire qui réduit la fiabilité du MOSFET SiC. La forte polarisation positive de la grille provoque un courant de saturation de drain important en cas de SC. Ainsi, cette technologie fait émerger des besoins spécifiques de surveillance et de protection ultra-rapides. Pour cela, le travail de cette thèse se focalise sur deux études pour surmonter ces contraintes toute en gardant un bon compromis de performances entre « niveau d’intégration technologique ‘CMS/ASIC-CMOS’–rapidité–robustesse ». La première, regroupe un ensemble de solutions nouvelles permettant une détection du courtcircuit sur le cycle de commutation, sur la base d'une architecture conventionnelle de commande rapprochée dite à 2 niveaux de tension. La deuxième étude est plus exploratoire et basée sur une nouvelle architecture de gate–driver, dite multi-niveaux, à faible niveau de stress pour le MOSFET SiC tout en maintenant les performances dynamiques. Les travaux portent tout d’abord sur l’environnement du SiC MOSFET, (caractérisation et propriétés de comportement en SC par simulations orientées "circuit" de type PLECS™ et LTSpice™), puis présentent une étude bibliographique sur les commandes rapprochées dites Gate Driver, une étude approfondie a été réalisée sur les court-circuits type I & II (Hard switch fault) (Fault under Load) ; regroupés dans un premier chapitre du manuscrit. Un banc de test réalisé antérieurement au sein du laboratoire, a permis de compléter et de valider l’étude d'analyse-simulation et de préparer des stimuli test pour l'étape de conception des nouvelles solutions. Inspirée par la méthode de Gate charge apparue pour les IGBTs en silicium et évoquée pour les MOSFETs SiC, cette première approche fait l'objet d'un travail de conception, de dimensionnement et de prototypage. Cette méthode de référence permet une détection de type HSF en moins de 200ns sous 0-600V avec des composants 1,2kV allant de 80 mOhm à 120mOhm. S'agissant des nouvelles méthodes de détection rapides et intégrées, les travaux de cette thèse se focalisent particulièrement sur la conception d’un circuit ASIC CMOS. Pour cela, la conception d’un gate driver adapté est essentiel. Un ASIC est conçu en technologie X-Fab XT-0,18μm SOICMOS sous Cadence™, et puis mis en boitier et assemblé sur PCB conçu pour les besoins de tests et adaptable au banc principal. La conception du gate driver a considéré de nombreuses fonctions (détection du SC, SSD Soft shut down, buffer segmenté, AMC Active Miller Clamp", …). Du point de vue de la détection du SC, les fonctions nouvelles de surveillance intégrées concernent la méthode de dérivation temporelle de VGS qui est basée sur une détection par un circuit dérivateur analogique RC sur la séquence de plateau avec deux variantes. Une deuxième méthode nouvelle partiellement intégrée dans l'ASIC a été conçu, non développé dans ce mémoire dans le but d’une valorisation. En marge de cette étude principale, une étude exploratoire a porté sur une nouvelle architecture modulaire de commande rapprochée à plusieurs niveaux de tension de polarisation tirant profit de l'isolation SOI et des transistors CMOS à basse tension pour piloter le MOSFETs SiC et améliorer leur fiabilité grâce à une sélection active et dynamique à plusieurs niveaux sur les séquences de commutation et les états marche/arrêt
Wide bandgap power transistors such as SiC MOSFETs and HEMTs GaN push furthermore the classical compromises in power electronics. Briefly, significant gains have been demonstrated: better efficiency, coupled with an increase in power densities offered by the increase in switching frequency. HV SiC MOSFETs have specific features such as a low short-circuit SC withstand time capability compared to Si IGBTs and thinner gate oxide, and a high gate-to-source switching control voltage. The negative bias on the gate at the off-state creates additional stress which reduces the reliability of the SiC MOSFET. The high positive bias on the gate causes a large drain saturation current in the event of a SC. Thus, this technology gives rise to specific needs for ultrafast monitoring and protection. For this reason, the work of this thesis focuses on two studies to overcome these constraints, with the objective of reaching a good performance compromise between “CMS/ASIC-CMOS technological integration level-speed–robustness”. The first one, gathers a set of new solutions allowing a detection of the SC on the switching cycle, based on a conventional switch control architecture with two voltage levels. The second study is more exploratory and is based on a new gate-driver architecture, called multi-level, with low stress level for the SiC MOSFET while maintaining dynamic performances. The manuscript covers firstly the SiC MOSFET environment, (characterization and properties of SC behavior by simulation using PLECS and LTSpice software) and covers secondly a bibliographical study on the Gate drivers. And last, an in-depth study was carried out on SC type I & II (hard switch fault) (Fault under Load) and their respective detection circuits. A test bench, previously carried out in the laboratory, was used to complete and validate the analysis-simulation study and to prepare test stimuli for the design stage of new solutions. Inspired by the Gate charge method that appeared for Si IGBTs and evoked for SiC MOSFETs, this method has therefore been the subject of design, dimensioning and prototyping work, as a reference. This reference allows an HSF type detection in less than 200ns under 400V with 1.2kV components ranging from 80 to 120mOhm. Regarding new rapid and integrated detection methods, the work of this thesis focuses particularly on the design of a CMOS ASIC circuit. For this, the design of an adapted gate driver is essential. An ASIC is designed in X-Fab XT-0.18 SOICMOS technology under Cadence, and then packaged and assembled on a PCB. The PCB is designed for test needs and adaptable to the main bench. The design of the gate driver considered many functions (SC detection, SSD, segmented buffer, an "AMC", ...). From the SC detection point of view, the new integrated monitoring functions concern the VGS time derivative method which is based on a detection by an RC analog shunt circuit on the plateau sequence with two approaches: the first approach is based on a dip detection, i.e. the presence or not of the Miller plateau. The second approach is based on slope detection, i.e. the variability of the input capacitance of the power transistor under SC-HSF compared to normal operation. These methods are compared in the third chapter of the thesis, and demonstrate fault detection times between 40ns and 80ns, and preliminary robustness studies and critical cases are presented. A second new method is partially integrated in the ASIC, was designed. This method is not developed in the manuscript for valorization purposes. In addition to the main study, an exploratory study has focused on a modular architecture for close control at several bias voltage levels taking advantage of SOI isolation and low voltage CMOS transistors to drive SiC MOSFETs and improve their reliability through active and dynamic multi-level selection of switching sequences and on/off states

Orientador: Prof. Dr. Roldão da Rocha Jr.
Tese (doutorado) - Universidade Federal do ABC, Programa de Pós-Graduação em Física, 2017.
Neste trabalho foram investigadas algumas conseguências da física de buracos negros em teorias cujo domínio está além do domínio da relatividade geral, em especial em teorias efetivos com dimensões extras. A investigação foi em substancialmente conduzida baseando-se em três efeitos gravitacionais, a saber, a radiação Hawking, o regime de deflexão forte de lentes gravitacionais e a formação de buracos negros quânticos. Uma solução de modelo cosmológico imerso em uma brana espessa foi também investigada. Modelos e teorias efetivas fornecem meios para testar os limites de validade de teorias conhecidas e indicam o que deveríamos esperar além desses limites. Baseado nessa ideia foram usados alguns modelos efetivos para estudar efeitos não previstos pela relatividade geral, associados a cada um dos fenômenos mencionados.
This work is devoted to investigate some consequences of black holes physics beyond the domain of general relativity, mainly in effective extra dimensional models. The investigation is carried along three gravitational effects, namely the Hawking radiation, the strong deflection of gravitational lensing and the formation of quantum black holes. A cosmological thick brane solution is also investigated. Effective theories and models provide a prominent approach for testing the limits of known theories and show what would be expected beyond that. Based on such idea we have used effective models for finding deviations of general relativity associated to each of the mentioned phenomena.

We employ the formalism of the Horizon Quantum Mechanics to describe the gravitational radius of compact sources by means of an operator and derive a Horizon Wave Function which will allow us to estimate the probability of formation of black holes in scattering processes. If the Planck scale is kept at its standard value, however, it will be impossible to test that regime with any foreseeable technology. We then review how the introduction of extra dimensions can potentially lower the Planck scale down to the TeV range in an attempt to solve the hierarchy problem. In this context, we proceed by studying black holes described by a generalisation of the Kerr metric for higher dimensional spacetime known as the Myers-Perry metric. Our computation of the probability that a rotating source in higher dimensions is a black hole suggests that, even if the fundamental Planck scale is as low as a few TeV's, we should not be able to detect any black hole in colliders as is indeed the case.

Como professora de matemática, com experiência no Ensino Secundário, tenho constatado que os alunos revelam bastantes dificuldades de compreensão, aquisição e aplicação dos conteúdos trigonométricos. Também os conteúdos de trigonometria dos actuais manuais escolares do ensino secundário não são suficientes para aqueles alunos que pretendem seguir estudos superiores nas áreas da Física, da Matemática, da Engenharia, Este trabalho destina-se essencialmente a esse grupo de alunos, mas também pode servir de material de apoio a professores do Ensino Básico/Secundário que queiram enriquecer e aprofundar os seus conhecimentos nesta área. Ao longo deste trabalho, propomos uma forma alternativa de apresentação deste tema, pensamos que deste modo os alunos possam compreender "como, donde e porquê" aparecem as relações trigonométricas. Classificamos e analisamos em profundidade equações trigonométricas, com a finalidade de propor abordagens diferentes de resolução, umas típicas outras originais. Analisamos desigualdades trigonométricas e sistemas de equações trigonométricas utilizando métodos de resolução distintos. Apresentamos versões distintas de demonstrações para as mesmas afirmações, e apresentamos problemas de Geometria, de Topografia e de Física, mostrando desta forma a aplicação prática da Trigonometria. Finalizamos este trabalho com a apresentação de algumas relações trigonométricas na teoria dos Números Complexos. – ABSTRACT: School students, I have noticed that students have many difficulties in understanding trigonometric contents. The contents related to trigonometry that are presented, in modem textbooks adopted by Secondary Schools are not enough for those students who intend to take a degree in areas in field of physics, mathematics and engineering. This study is essentially intended for the above-mentioned students, but it can also be used as a support material for a secondary school teachers who want to enrich and increase their scientific knowledge in the field of trigonometry. This project suggests an alternative way to introduce the topic so that student can understand "how, from, where and why" trigonometric relationships appear. Trigonometric equations are classified and examined deeply in order to suggest different approaches for their solution, typical ones and original others. Trigonometric inequalities and systems of trigonometric equations of special types are also analysed using different solution methods. Different versions of demonstrations for the same statements are presented as well as problems of Geometry, Topography and Physics, with the purpose of showing the practical use of trigonometry. This project ends with the presentation of some trigonometric relationships in the theory of Complex Numbers.

In this thesis two studies were done using MRI. In chapter two, in vivo 3.0 T MRI data from white matter and grey matter in brain from 4 healthy volunteers was studied using a multi component T₂ relaxation analysis. The goal of this study was to find the dependence of myelin water fraction (MWF), the ratio of water in myelin bilayers to the total water component, with the repetition time of the MR sequence TR. Results showed that MWF increased with decreasing TR time. This behavior is believed to be influenced by the exchange of water between the myelin water and the intra/extracellular water pools. Several models were explored to explain this result, including a fast exchange model, a slow exchange model and a hybrid model in which myelin was proposed to contain regions of fast exchange and regions of slow exchange. In chapter three, we addressed the questions: Does the intra/extra cellular (IE) water geometric mean T₂ (gmT₂) of white and grey matter depend upon the refocusing interval? To answer these questions IE water gmT₂ times for different white and grey matter regions of interest were obtained from 5 healthy subjects. It was found that IE water gmT₂ times from both white matter and grey matter tissue decreased by approximately the same amount with refocusing interval prolongation from 10ms to 40 ms. Several mechanisms for this dependence were considered, including water exchange, existence of myelin, non-heme iron accumulation, or the effect of blood oxygenation. In this case, based on our simulations, exchange did not appear to play a role. Non-heme iron accumulation was related to T₂ time but not to the change in T₂ with echo spacing. Deoxygenation of blood results in the presence of paramagnetic deoxyhemoglobin (dHb), which increases the magnetic susceptibility, and thus the local magnetic field, which can shorten the transverse relaxation time; however, this mechanism would be expected to affect grey matter T₂’s more than white matter T₂s.
Science, Faculty of
Physics and Astronomy, Department of
Graduate

The continuously increasing complexity of embedded systems is a major issue for their development, which, in addition, must also consider specific extra-functional requirements and constraints, such as limited and shared resources, distribution, timing, and dependability. Thus, embedded systems call for development solutions that can efficiently and predictably cope with these issues. Component-based software engineering is a proven paradigm to handle complexity. Yet, for efficiently managing extra-functional properties, a component model needs to have dedicated mechanisms that provide a suitable support for their management. The objective of this thesis is to build this support. We have performed a systematic analysis of existing component models and identified challenges of applying a component-based approach to embedded system development. Based on these challenges we have advanced the current state-of-the-art by developing a new component model, called ProCom, that accommodates the specifics of embedded systems through its well-defined execution semantics and layered structure. Centered around ProCom, we have also developed PRIDE, the ProCom Integrated Development Environment. PRIDE supports the development from early specification to synthesis and deployment, providing the means to aggregate various analysis and verification tools. The main contribution of the thesis is in the design and implementation of an extra-functional property management framework that enables to seamlessly specify, manage and integrate multi-valued context-aware extra-functional properties of component-based embedded systems. Properties can be attached to architectural elements of component models and their values can be compared and refined during the development process. In particular, having multiple context-aware values allows values from different sources to be compared. The proposed concepts have been demonstrated on several representative example systems.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
Optimization problems seek to maximize or minimize a certain function, they play an important role in the real world. Many practical applications of sciences like engineering and economics can be formulated through optimization problems, for example, minimizing the amount of energy used in a factory or maximizing the investment of a person. In this work we will deal with issues of functions of one, two or three variables in the search for the point whose value is maximum or minimum, will be treated contents that can be worked in high school, as the Inequality of Means, Quadratic Function and also themes that can be explored in graduate course, in the case of Derivative and Method of Lagrange, all threads have interesting examples of applications of the concepts studied.
Problemas de otimização procuram maximizar ou minimizar uma determinada função, eles desempenham um papel importante no mundo real. Muitas aplicações práticas de ciências como engenharia e economia, podem ser formuladas através de problemas de otimização, como, por exemplo, a minimização da quantidade de energia usada em uma fábrica ou a maximização dos investimentos de uma pessoa. Nesse trabalho trataremos com questões de funções de uma, duas ou três variáveis na busca pelo ponto cujo o valor e máximo ou mínimo, ser a tratado conte udos que podem ser trabalhados no ensino médio, como a Desigualdade das Médias, Função Quadrática e também, temas que podem ser explorados em cursos de graduações, no caso das Derivadas e o Método de Lagrange, todos os tópicos possuem exemplos interessantes sobre aplicações dos conceitos estudados.

Data networks are an important part in an increasing number of applications with real-time and reliability requirements. To meet these demands a variety of approaches have been proposed. Forward error correction, which adds redundancy to the communicated data, is one of them. However, the redundancy occupies communication bandwidth, so it is desirable to control the amount of redundancy in order to achieve high reliability without adding excessive communication delay. The main contribution of the thesis is to formulate the problem of adjusting the redundancy in a control framework, which enables the dynamic properties of error correction control to be analyzed using control theory. The trade-off between application quality and resource usage is captured by introducing an optimal control problem. Its dependence on the knowledge of the network state at the transmission side is discussed. An error correction controller that optimizes the amount of redundancy without relying on network state information is presented. This is achieved by utilizing an extremum seeking control algorithm to optimize the cost function. Models with varying complexity of the resulting feedback system are presented and analyzed. Conditions for convergence are given. Multiple-input describing function analysis is used to examine periodic solutions. The results are illustrated through computer simulations and experiments on a wireless sensor network.

Cette thèse se place dans le cadre de la Statistique des valeurs extrêmes. Elle y apporte trois contributions principales. L'estimation des quantiles extrêmes se fait dans la littérature en deux étapes. La première étape consiste à utiliser une approximation des quantiles basée sur la théorie des valeurs extrêmes. La deuxième étape consiste à estimer les paramètres inconnus de l'approximation en question, et ce en utilisant les valeurs les plus grandes du jeu de données. Cette décomposition mène à deux erreurs de nature différente, la première étant une erreur systémique de modèle, dite d'approximation ou encore d'extrapolation, la seconde consituant une erreur d'estimation aléatoire. La première contribution de cette thèse est l'étude théorique de cette erreur d'extrapolation mal connue.Cette étude est menée pour deux types d'estimateur différents, tous deux cas particuliers de l'approximation dite de la "loi de Pareto généralisée" : l'estimateur Exponential Tail dédié au domaine d'attraction de Gumbel et l'estimateur de Weissman dédié à celui de Fréchet.Nous montrons alors que l'erreur en question peut s'interpréter comme un reste d'ordre un d'un développement de Taylor. Des conditions nécessaires et suffisantes sont alors établies de telle sorte que l'erreur tende vers zéro quand la taille de l'échantillon augmente. De manière originale, ces conditions mènent à une division du domaine d'attraction de Gumbel en trois parties distinctes. En comparaison, l'erreur d'extrapolation associée à l'estimateur de Weissman présente un comportement unifié sur tout le domaine d'attraction de Fréchet. Des équivalents de l'erreur sont fournis et leur comportement est illustré numériquement. La deuxième contribution est la proposition d'un nouvel estimateur des quantiles extrêmes. Le problème est abordé dans le cadre du modèle ``log Weibull-tail'' généralisé, où le logarithme de l'inverse du taux de hasard cumulé est supposé à variation régulière étendue. Après une discussion sur les conséquences de cette hypothèse, nous proposons un nouvel estimateur des quantiles extrêmes basé sur ce modèle. La normalité asymptotique dudit estimateur est alors établie et son comportement en pratique est évalué sur données réelles et simulées.La troisième contribution de cette thèse est la proposition d'outils permettant en pratique de quantifier les limites d'extrapolation d'un jeu de données. Dans cette optique, nous commençons par proposer des estimateurs des erreurs d'extrapolation associées aux approximations Exponential Tail et Weissman. Après avoir évalué les performances de ces estimateurs sur données simulées, nous estimons les limites d'extrapolation associées à deux jeux de données réelles constitués de mesures journalières de variables environnementales. Dépendant de l'aléa climatique considéré, nous montrons que ces limites sont plus ou moins contraignantes
This thesis takes place in the extreme value statistics framework. It provides three main contributions to this area. The extreme quantile estimation is a two step approach. First, it consists in proposing an extreme value based quantile approximation. Then, estimators of the unknown quantities are plugged in the previous approximation leading to an extreme quantile estimator.The first contribution of this thesis is the study of this previous approximation error. These investigations are carried out using two different kind of estimators, both based on the well-known Generalized Pareto approximation: the Exponential Tail estimator dedicated to the Gumbel maximum domain of attraction and the Weissman estimator dedicated to the Fréchet one.It is shown that the extrapolation error can be interpreted as the remainder of a first order Taylor expansion. Necessary and sufficient conditions are then provided such that this error tends to zero as the sample size increases. Interestingly, in case of the so-called Exponential Tail estimator, these conditions lead to a subdivision of Gumbel maximum domain of attraction into three subsets. In constrast, the extrapolation error associated with Weissmanestimator has a common behavior over the whole Fréchet maximum domain of attraction. First order equivalents of the extrapolation error are thenderived and their accuracy is illustrated numerically.The second contribution is the proposition of a new extreme quantile estimator.The problem is addressed in the framework of the so-called ``log-Generalized Weibull tail limit'', where the logarithm of the inverse cumulative hazard rate function is supposed to be of extended regular variation. Based on this model, a new estimator of extreme quantiles is proposed. Its asymptotic normality is established and its behavior in practice is illustrated on both real and simulated data.The third contribution of this thesis is the proposition of new mathematical tools allowing the quantification of extrapolation limits associated with a real dataset. To this end, we propose estimators of extrapolation errors associated with the Exponentail Tail and the Weissman approximations. We then study on simulated data how these two estimators perform. We finally use these estimators on real datasets to show that, depending on the climatic phenomena,the extrapolation limits can be more or less stringent

In this thesis several problems are addressed. The problems considered vary from second order problems up to high order problems where generaliza- tions to nth order are studied. Such problems range from problems without functional dependence up to problems where the functional dependence is featured both in the equation and on the boundary conditions. Functional boundary conditions include most of the classical conditions as multipoint cases, conditions with delay and/or advances, nonlocal or in- tegral, with maximum or minimum arguments,... Existence, nonexistence, multiplicity and localization results are then discussed in accordance with these conditions. The method used is the lower and upper solutions combined with di¤erent techniques (degree theory, Nagumo condition, iterative technique, Green s function) to obtain such results. Several applications are studied such as the periodic oscillations of the axis of a satellite and conjugate boundary value problems, to emphasize the applicability of the method used; RESUMO:Nesta tese, intitulada em português, Resultados de existência, localiza- ção e multiplicidade para problemas não lineares e funcionais de ordem su- perior com valores na fronteira , diferentes problemas são abordados. Estes problemas variam desde problemas de segunda ordem até problemas de or- dem superior, onde generalizações de ordem n são feitas e onde os problemas apresentados variam desde o caso em que não existe dependência funcional até aos em que esta dependência funcional está presente tanto na equação como nas condições de fronteira. Sobre estas condições, que incluem a maioria das condições clássicas, re- sultados de existência, não existência, multiplicidade e localização de solução são discutidos de acordo com estas condições. O método utilizado é o método da sub e sobre-solução combinado com diferentes técnicas. Várias aplicações são estudadas, nomeadamente as oscilações periódicas do eixo de um satélite e problemas conjugados, de forma a dar ênfase à aplicabilidade do método utilizado.