In this thesis we are studying some properties of weakly sequential homeomorphisms between Banach spaces. First, we show some results that summarize how are some clas- ses of Banach spaces (specifically separable spaces, spaces with separable dual, Asplund spaces, reflexive spaces, weakly compactly generated spaces and spaces not containing isomorphic copy of ℓ1) determined by weak topology of the space. Then we show that to preserve some properties (separability, reflexivity and being weakly compactly gene- rated) it is enough for the spaces to be weakly sequentially homeomorphic. Furthermore we show that if two spaces are weakly sequentially uniformly homeomorphic then one contains isomorphic copy of ℓ1 if and only if the other spaces has this property. Finally we construct weakly sequential homeomorphisms between some class of Banach spaces.

The thesis consists of two papers and one preprint. The two papers are de- voted to the approximation properties of Lipschitz-free spaces. In the first pa- per we prove that the Lipschitz-free space over a doubling metric space has the bounded approximation property. In particular, the Lipschitz-free space over a closed subset of Rn has the bounded approximation property. We also show that the Lipschitz-free spaces over ℓ1 and over ℓn 1 admit a monotone finite-dimensional Schauder decomposition. In the second paper we improve this work and obtain even a Schauder basis in the Lipschitz-free spaces over ℓ1 and ℓn 1 . The topic of the preprint is rigidity of ℓ∞ and ℓn ∞ with respect to uniformly differentiable map- pings. Our main result is a non-linear analogy of the classical result on rigidity of ℓ∞ with respect to non-weakly compact linear operators by Rosenthal, and it generalises the theorem on non-complementability of c0 in ℓ∞ due to Phillips. 1

The Ph.D. Thesis ¿Operators on weighted spaces of holomorphic functions¿ presented here treats different areas of functional analysis such as spaces of holomorphic functions, infinite dimensional holomorphy and dynamics of operators. After a first chapter that introduces the notation, definitions and the basic results we will use throughout the thesis, the text is divided into two parts. A first one, consisting of Chapters 1 and 2, focused on a study of weighted (LB)-spaces of entire functions on Banach spaces, and a second one, corresponding to Chapters 3 and 4, where we consider differentiation and integration operators acting on different classes of weighted spaces of entire functions to study its dynamical behaviour. In what follows, we give a brief description of the different chapters: In Chapter 1, given a decreasing sequence of continuous radial weights on a Banach space X, we consider the weighted inductive limits of spaces of entire functions VH(X) and VH0(X). Weighted spaces of holomorphic functions appear naturally in the study of growth conditions of holomorphic functions and have been investigated by many authors since the work of Williams in 1967, Rubel and Shields in 1970 and Shields and Williams in 1971. We determine conditions on the family of weights to ensure that the corresponding weighted space is an algebra or has polynomial Schauder decompositions. We study Hörmander algebras of entire functions defined on a Banach space and we give a description of them in terms of sequence spaces. We also focus on algebra homomorphisms between these spaces and obtain a Banach-Stone type theorem for a particular decreasing family of weights. Finally, we study the spectra of these weighted algebras, endowing them with an analytic structure, and we prove that each function f ¿ VH(X) extends naturally to an analytic function defined on the spectrum. Given an algebra homomorphism, we also investigate how the mapping induced between the spectra acts on the corresponding analytic structures and we show how in this setting composition operators have a different behavior from that for holomorphic functions of bounded type. This research is related to recent work by Carando, García, Maestre and Sevilla-Peris. The results included in this chapter are published by Beltrán in [14]. Chapter 2 is devoted to study the predual of VH(X) in order to linearize this space of entire functions. We apply Mujica¿s completeness theorem for (LB)-spaces to find a predual and to prove that VH(X) is regular and complete. We also study conditions to ensure that the equality VH0(X) = VH(X) holds. At this point, we will see some differences between the finite and the infinite dimensional cases. Finally, we give conditions which ensure that a function f defined in a subset A of X, with values in another Banach space E, and admitting certain weak extensions in a space of holomorphic functions can be holomorphically extended in the corresponding space of vector-valued functions. Most of the results obtained have been published by the author in [13]. The rest of the thesis is devoted to study the dynamical behaviour of the following three operators on weighted spaces of entire functions: the differentiation operator Df(z) = f (z), the integration operator Jf(z) = z 0 f(¿)d¿ and the Hardy operator Hf(z) = 1 z z 0 f(¿)d¿, z ¿ C. In Chapter 3 we focus on the dynamics of these operators on a wide class of weighted Banach spaces of entire functions defined by means of integrals and supremum norms: the weighted spaces of entire functions Bp,q(v), 1 ¿ p ¿ ¿, and 1 ¿ q ¿ ¿. For q = ¿ they are known as generalized weighted Bergman spaces of entire functions, denoted by Hv(C) and H0 v (C) if, in addition, p = ¿. We analyze when they are hypercyclic, chaotic, power bounded, mean ergodic or uniformly mean ergodic; thus complementing also work by Bonet and Ricker about mean ergodic multiplication operators. Moreover, for weights satisfying some conditions, we estimate the norm of the operators and study their spectrum. Special emphasis is made on exponential weights. The content of this chapter is published in [17] and [15]. For differential operators ¿(D) : Bp,q(v) ¿ Bp,q(v), whenever D : Bp,q(v) ¿ Bp,q(v) is continuous and ¿ is an entire function, we study hypercyclicity and chaos. The chapter ends with an example provided by A. Peris of a hypercyclic and uniformly mean ergodic operator. To our knowledge, this is the first example of an operator with these two properties. We thank him for giving us permission to include it in our thesis. The last chapter is devoted to the study of the dynamics of the differentiation and the integration operators on weighted inductive and projective limits of spaces of entire functions. We give sufficient conditions so that D and J are continuous on these spaces and we characterize when the differentiation operator is hypercyclic, topologically mixing or chaotic on projective limits. Finally, the dynamics of these operators is investigated in the Hörmander algebras Ap(C) and A0 p(C). The results concerning this topic are included by Bonet, Fernández and the author in [16]. / Beltrán Meneu, MJ. (2014). Operators on wighted spaces of holomorphic functions [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/36578 / Premios Extraordinarios de tesis doctorales

Positive kernels and their associated reproducing kernel Hilbert spaces have played a key role in the development of complex analysis and Hilbert-space operator theory, and they have recently been extended to the setting of free noncommutative function theory. In this paper, we develop the subject further in a number of directions. We give a characterization of completely positive noncommutative kernels in the setting of Hilbert C*-modules and Hilbert W*-modules. We prove an Arveson-type extension theorem for completely positive noncommutative kernels, and we show that a uniformly bounded noncommutative kernel can be decomposed into a linear combination of completely positive noncommutative kernels. / Ph. D. / Over the last several decades, positive kernels and their associated reproducing kernel Hilbert spaces have played a key role in the development of complex analysis and Hilbert-space operator theory. Recently, they have been extended to the setting of free noncommutative function theory which is an active area of research with motivation from several different sources including free probability and noncommutative real semialgebraic geometry. In this paper, we develop further the theory of positive kernels in the noncommutative setting.

This thesis concerns the boundary behavior of solutions to parabolic equations. It consists of a comprehensive summary and four scientific papers. The equations concerned are different generalizations of the heat equation. Paper I concerns the solutions to non-linear parabolic equations with linear growth. For non-negative solutions that vanish continuously on the lateral boundary of an NTA cylinder the following main results are established: a backward Harnack inequality, the doubling property for the Riesz measure associated with such solutions, and the Hölder continuityof the quotient of two such solutions up to the boundary. Paper 2 concerns the solutions to linear degenerate parabolic equations, where the degeneracy is controlled by a Muckenhoupt weight of class 1+2/n. For non-negative solutions that vanish continuously on the lateral boundary of an NTA cylinder the following main results are established: a backward Harnack inequality, the doubling property for the parabolic measure, and the Hölder continuity of the quotient of two such solutions up to the boundary. Paper 3 concerns a fractional heat equation. The first main result is that a solution to the fractional heat equation in Euclidean space of dimension n can be extended as a solution to a certain linear degenerate parabolic equation in the upper half space of dimension n+1. The second main result is the Hölder continuity of quotients of two non-negative solutions that vanish continuously on the latteral boundary of a Lipschitz domain. Paper 4 concerns the solutions to uniformly parabolic linear equations with complex coefficients. The first main result is that under certain assumptions on the opperator the bounds for the single layer potentials associated to the opperator are bounded. The second main result is that these bounds always hold if the opperator is realvalued and symmetric.

違約機率校準檢定 - global test 由兩部分組成：第一部分為 level，探討真實的平均違約機率是否被高估；第二部分 shape，探討高低違約機率的表現情形。但 global test 與相關違約事件下的 level test 檢定尺度皆遠高於顯著水準 $\alpha$。本文先是針對相關違約事件，利用截斷分配使 level test 犯型一誤差機率更接近顯著水準，並提出虛無假設及對立假設為 $H_0: \theta \in \cup_{i=1}^2 \Theta_{i0}$ vs. $H_1: \theta \in \cap_{i=1}^2 \Theta_{i1}$ 的形式，引用交聯集檢定。更進一步透過 Liu \& Berger (1995, \textit{The Annals of Statistics}, 23, 1, 55-72) 建構齊一較強檢力檢定，改善檢定力。模擬結果顯示交聯集檢定與齊一較強檢力檢定的檢定尺度皆為 $\alpha$，且齊一較強檢力檢定的檢定力皆高於交聯集檢定。 / The calibration test of the PDs (probabilities of default) --- global test is twofold, the first part is the level test, which is about the mean of calibrated PDs. Second, the shape test is about whether a calibrated PD model differentiates correctly between low and high default probability events. In simulation results, we found that the type I error of global test is much greater than significant level $\alpha$, so is level test in correlation default events. In this study, firstly, we use the truncated level test to control previous error and suggest the hypothesis $H_0: \theta \in \cup_{i=1}^2 \Theta_{i0}$ vs. $H_1: \theta \in \cap_{i=1}^2 \Theta_{i1}$. Secondly, we introduce the intersection union test (IUT). Moreover, we construct an uniformly more powerful test (UMP test) by Liu \& Berger (1995, \textit{The Annals of Statistics}, 23, 1, 55-72). Simulation results show that the IUT and UMP test are size $\alpha$ tests, and the power of UMP test is greater than IUT.

Submitted by Alessandro Ribeiro de Carvalho Casalecchi (alercc@gmail.com) on 2017-07-03T21:17:55Z No. of bitstreams: 1 Tese_Alessandro_Casalecchi.pdf: 2174297 bytes, checksum: 27298549cf220c58b7eb52f7323446d7 (MD5) / Approved for entry into archive by Suzinei Teles Garcia Garcia (suzinei.garcia@fgv.br) on 2017-07-04T11:10:20Z (GMT) No. of bitstreams: 1 Tese_Alessandro_Casalecchi.pdf: 2174297 bytes, checksum: 27298549cf220c58b7eb52f7323446d7 (MD5) / Made available in DSpace on 2017-07-05T13:46:08Z (GMT). No. of bitstreams: 1 Tese_Alessandro_Casalecchi.pdf: 2174297 bytes, checksum: 27298549cf220c58b7eb52f7323446d7 (MD5) Previous issue date: 2017-05-25 / Os dois artigos desta tese, os capítulos 2 e 3, referem-se a testes de hipótese mas têm focos diferentes. O capítulo 2, intitulado "Improvements for external validity tests in fuzzy regression discontinuity designs," apresenta condições --- hipóteses de continuidade, monotonicidade estrita e convergência pontual --- sob as quais testes de qualidade de ajuste para duas amostras podem ser usados para testes de validade externa em modelos de tratamento-controle que sofrem de "compliance" imperfeito. Modelos com "compliance" imperfeito permitem a estimação de efeitos de tratamento apenas para a subpopulação de "compliers", sendo que tais estimativas não são necessariamente válidas para outras subpopulações ("always-takers" e "never-takers"). Sob as condições do capítulo 2, o uso do teste de qualidade de ajuste no lugar do teste de diferença de médias representa um avanço para testes de validade externa, uma vez que mais hipóteses alternativas são detectáveis pelo primeiro teste. Sugerimos combinar duas estatísticas de teste de qualidade de ajuste (uma para tratados e outra para não tratados) na forma de um teste múltiplo ao invés de um teste conjunto. O capítulo 3, intitulado "Higher-order UMP tests", sugere uma estratégia para se escolher, dentro de um conjunto de estatísticas de teste disponíveis, aquela que fornece o teste mais poderoso quando as funções de poder dos testes em questão não podem ser diferenciadas através de métodos assintóticos usuais, como análise de poder local ("local power analysis"). Propomos o uso de aproximações assintóticas de ordem mais alta, como expansões de Edgeworth, para se aproximar as densidades amostrais das estatísticas disponíveis e, com isso, verificar-se quais delas possuem a propriedade da razão monotônica de verossimilhança. Tal propriedade implica, pelo Teorema de Karlin-Rubin, que o teste é uniformemente mais poderoso (UMP) --- ao menos até certa ordem de aproximação --- se a estatística for suficiente para o parâmetro relevante. Para o caso em que as estatísticas sendo comparadas não são suficientes, argumentamos que frequentemente elas podem se tornar suficientes para uma família paramétrica de interesse após reparametrizações apropriadas. Para fins de ilustração, nós aplicamos o método proposto para determinar o valor ótimo, em termos de poder, do parâmetro de suavização do estimador de densidade por kernel em bases de dados simuladas e concluímos que a ordem de aproximação usada nesta aplicação (segunda ordem) não é alta o suficiente para permitir a diferenciação das funções de poder associadas aos diferentes valores do parâmetro de suavização. / The two papers in this work, chapters 2 and 3, regard hypothesis testing but address different issues. Chapter 2, entitled "Improvements for external validity tests in fuzzy regression discontinuity designs", shows conditions --- assumptions of continuity, strict monotonicity and pointwise convergence --- under which two-sample goodness-of-fit (GOF) tests can be used to test for external validity in treatment-control models that suffer from imperfect compliance of units with respect to the assigned treatment. Imperfect compliance allows researchers to estimate only treatment effects for the subpopulation of compliers, and the validity of these estimates for other subpopulations (always-takers and never-takers) remains an open problem. Under the conditions in Chapter 2, the use of GOF tests in place of mean difference tests represents an improvement over other external validity tests in the literature, since more alternative hypotheses are detectable by the test statistic. We suggested to combine two GOF test statistics (one for the treated and one for the untreated) in a multiple test instead of a joint test. Chapter 3, entitled "Higher-order UMP tests", suggests a strategy to choose among candidate test statistics, according to a power criterion, when their power performances are not distinguishable by usual methods of asymptotic comparison like local power analysis. We propose the use of higher-order asymptotic expansions, like Edgeworth expansions, to approximate the sample densities of the candidate test statistics and verify which of them has the monotone likelihood ratio property. This property implies, by the Karlin-Rubin Theorem, that the test is uniformly most powerful (UMP) --- at least to an order of approximation --- if the statistic is sufficient for the relevant parameter. When the statistics under study are not sufficient, we argue that they can often be made sufficient for a desired parametric family after appropriate reparameterization. We applied the method to search for the power-optimal bandwidth for the kernel density estimator in simulated data sets, and concluded that the order of approximation that we used (second order) is still too low to allow us to distinguish among bandwidths.

Baarda’s outlier test is one of the best established theories in geodetic practice. The optimal test statistic of the local model test for a single outlier is known as the normalized residual. Also other model disturbances can be detected and identified with this test. It enjoys the property of being a uniformly most powerful invariant (UMPI) test, but is not a uniformly most powerful (UMP) test. In this contribution we will prove that in the class of test statistics following a common central or non-central chi2 distribution, Baarda’s solution is also uniformly most powerful, i.e. UMPchi2 for short. It turns out that UMPchi2 is identical to UMPI, such that this proof can be seen as another proof of the UMPI property of the test. We demonstrate by an example that by means of the Monte Carlo method it is even possible to construct test statistics, which are regionally more powerful than Baarda’s solution. They follow a socalled generalized chi2 distribution. Due to high computational costs we do not yet propose this as a ”new outlier detection method”, but only as a proof that it is in principle possible to outperform the statistical power of Baarda’s test. / Der Ausreißertest nach Baarda ist eine der am besten etablierten Theorien in der geodätischen Praxis. Die optimale Teststatistik des lokalen Modelltests für einen einzelnen Ausreißer wird als normierte Verbesserung bezeichnet. Auch andere Modellabweichungen können mit diesem Test aufgedeckt und identifiziert werden. Er hat die Eigenschaft, der gleichmäßig stärkste invariante (uniformly most powerful invariant, UMPI) Test zu sein, aber er ist kein gleichmäßig stärkster (uniformly most powerful, UMP) Test. In diesem Beitrag werden wir beweisen, dass in der Klasse der Teststatistiken mit einer gewöhnlichen zentralen oder nichtzentralen chi2 Verteilung die Baardaschen Lösung auch gleichmäßig stärkster Test ist, abgekürzt UMPchi2. Es stellt sich heraus, dass UMPchi2 gleichwertig mit UMPI ist, so dass dieser Beweis als ein weiterer Beweis der UMPI-Eigenschaft des Testes angesehen werden kann. Wir zeigen an einem Beispiel, dass es mittels der Monte Carlo Methode sogar möglich ist, Teststatistiken zu konstruieren, die regional stärker ist, als die Baardasche Lösung. Diese weisen eine sogenannte verallgemeinerte chi2 Verteilung auf. Wegen der hohen Rechenkosten schlagen wir das noch nicht als neue Ausreißererkennungsmethode vor, sondern nur als Beweis, dass es im Prinzip möglich ist, die Teststärke des Ausreißertests nach Baarda zu übertreffen.

The notion of a fuzzy metric space due to George and Veeramani has many advantages in analysis since many notions and results from classical metric space theory can be extended and generalized to the setting of fuzzy metric spaces, for instance: the notion of completeness, completion of spaces as well as extension of maps. The layout of the dissertation is as follows: Chapter 1 provide the necessary background in the context of metric spaces, while chapter 2 presents some concepts and results from classical metric spaces in the setting of fuzzy metric spaces. In chapter 3 we continue with the study of fuzzy metric spaces, among others we show that: the product of two complete fuzzy metric spaces is also a complete fuzzy metric space. Our main contribution is in chapter 4. We introduce the concept of a standard fuzzy pseudo metric space and present some results on fuzzy metric identification. Furthermore, we discuss some properties of t-nonexpansive maps. / Mathematical Sciences / M. Sc. (Mathematics)

The main body of this thesis consists of six appended papers. In the  first two, different  cooperative surveillance problems are considered. The second two consider different aspects of the trajectory planning problem, while the last two deal with observer design for mobile robotic and Euler-Lagrange systems respectively.In Papers A and B,  a combinatorial optimization based framework to cooperative surveillance missions using multiple Unmanned Ground Vehicles (UGVs) is proposed. In particular, Paper A  considers the the Minimum Time UGV Surveillance Problem (MTUSP) while Paper B treats the Connectivity Constrained UGV Surveillance Problem (CUSP). The minimum time formulation is the following. Given a set of surveillance UGVs and a polyhedral area, find waypoint-paths for all UGVs such that every point of the area is visible from  a point on a waypoint-path and such that the time for executing the search in parallel is minimized.  The connectivity constrained formulation  extends the MTUSP by additionally requiring the induced information graph to be  kept recurrently connected  at the time instants when the UGVs  perform the surveillance mission.  In these two papers, the NP-hardness of  both these problems are shown and decomposition techniques are proposed that allow us to find an approximative solution efficiently in an algorithmic manner.Paper C addresses the problem of designing a real time, high performance trajectory planner for an aerial vehicle that uses information about terrain and enemy threats, to fly low and avoid radar exposure on the way to a given target. The high-level framework augments Receding Horizon Control (RHC) with a graph based terminal cost that captures the global characteristics of the environment.  An important issue with RHC is to make sure that the greedy, short term optimization does not lead to long term problems, which in our case boils down to two things: not getting into situations where a collision is unavoidable, and making sure that the destination is actually reached. Hence, the main contribution of this paper is to present a trajectory planner with provable safety and task completion properties. Direct methods for trajectory optimization are traditionally based on a priori temporal discretization and collocation methods. In Paper D, the problem of adaptive node distribution is formulated as a constrained optimization problem, which is to be included in the underlying nonlinear mathematical programming problem. The benefits of utilizing the suggested method for  online  trajectory optimization are illustrated by a missile guidance example.In Paper E, the problem of active observer design for an important class of non-uniformly observable systems, namely mobile robotic systems, is considered. The set of feasible configurations and the set of output flow equivalent states are defined. It is shown that the inter-relation between these two sets may serve as the basis for design of active observers. The proposed observer design methodology is illustrated by considering a  unicycle robot model, equipped with a set of range-measuring sensors. Finally, in Paper F, a geometrically intrinsic observer for Euler-Lagrange systems is defined and analyzed. This observer is a generalization of the observer proposed by Aghannan and Rouchon. Their contractivity result is reproduced and complemented  by  a proof  that the region of contraction is infinitely thin. Moreover, assuming a priori bounds on the velocities, convergence of the observer is shown by means of Lyapunov's direct method in the case of configuration manifolds with constant curvature. / QC 20100622 / TAIS, AURES