This thesis proposes new analysis tools for simulation models in the presence of data. To achieve a representation close to reality, simulation models are typically endowed with a set of inputs, termed parameters, that represent several controllable, stochastic or unknown components of the system. Because these models often utilize computationally expensive procedures, even modern supercomputers require a nontrivial amount of time, money, and energy to run for complex systems. Existing statistical frameworks avoid repeated evaluations of deterministic models through an emulator, constructed by conducting an experiment on the code. In high dimensional scenarios, the traditional framework for emulator-based analysis can fail due to the computational burden of inference. This thesis proposes a new class of experiments where inference from half a million observations is possible in seconds versus the days required for the traditional technique. In a case study presented in this thesis, the parameter of interest is a function as opposed to a scalar or a set of scalars, meaning the problem exists in the high dimensional regime. This work develops a new modeling strategy to nonparametrically study the functional parameter using Bayesian inference. Stochastic simulations are also investigated in the thesis. I describe the development of emulators through a framework termed quantile kriging, which allows for non-parametric representations of the stochastic behavior of the output whereas previous work has focused on normally distributed outputs. Furthermore, this work studied asymptotic properties of this methodology that yielded practical insights. Under certain regulatory conditions, there is the following result: By using an experiment that has the appropriate ratio of replications to sets of different inputs, we can achieve an optimal rate of convergence. Additionally, this method provided the basic tool for the study of defect patterns and a case study is explored.

[pt] O problema de decomposição de contrações em espaços de Hilbert é motivado pelo problema do subespaço invariante, o qual é um famoso problema em aberto em Teoria de Operadores. Se T (pertence) B [H] é uma contração, define- se o operador A como o limite forte da seqüência { T* n Tn (pertence) B [H]; n > ou = 1}. Este operador caracteriza as isometrias, uma vez que T é uma isometria se e somente se A = I. A decomposição de Von Neumann-Wold para isometrias estabelece que toda isometria é a soma direta ortogonal de um Shift unilateral com um operador unitário. O presente trabalho estende a decomposição de Von Neumann-Wold para contrações tais que o operador A é uma projeção ortogonal arbitrária. Através desta decomposição, conclui-se que se uma contração não possui subespaço invariante próprio, então T (pertence) C00 U C01 U C10. uma análise abrangente do efeito dessa nova decomposição é desenvolvida, interceptando a classe de contrações em questão com as classes dos operadores compactos, normais, quasinormais, subnormais, hiponormais e normalóides. Como se conclui que o operador A é uma projeção ortogonal apenas até a classe das contrações quasinormais, também é analisado o quanto o operador A referente a uma contração subnormal não-quasinormal pode se afastar de uma projeção ortogonal. Além disso, estabelece-se para contrações hipornormais o subespaço onde A é uma projeção ortogonal. / [en] Decomposition of Hilbert-space contractions is motivated the invariant subspace problem, which is a famous open problem in Operator Theory. If T (pertenc) B [H] is a contraction, {T*n Tn (pertenc) B [H]; n > = 1} converger strongly. Let the operator A be its (strongly) limit. T is a isometry if and only if A = I. The von Neumann-Wold decomposition for isometries says that a isometry is the direct orthogonal sum of a unilateral shift and a unitary operator. The present work extends the von Neumann-Wold decomposition to a contrataction for wich A is an orthogonal projection. According to such a decomposition it is established that a contractin with no nontrivial invariant subspace is such that T (pertenc) C00 U C01 U C10. it follows a detailed investigation n the impact of such a new decomposition on several classes of operators; viz. compact, normal, quasinormal, subnormal, hyponormal and normaloid. It is verified that the operator A is an orthogonal projection up to the class of all quasinormal contraction T, but not for every subnormal contraction. Thus it is investigated how the operator A, for a susbnormal contraction T, can distanciate from an orthogonal projection, for hyponormal contraction T, is exhibited as well

[pt] As transformações F e F(diferente) surgiram associados ao problema de estabilidade em média-quadrática de sistemas bilineares discretos de dimensão infinita evoluindo em espaços de Hilbert separáveis, tendo sido originariamente definidas através de séries infinitas na álgebra de Banach dos operadores lineares e limitados no espaço de Hilbert em que o sistema evolui. O presente trabalho parte de uma condição suficiente para a estabilidade, condição esta anteriormente determinada, que se traduz imposições sobre os raios espectrais das transformações mencionadas ambos estritamente menores que um- e do fato já conhecido de que, a condição sendo parcialmente satisfeita, isto é, um dos raios espectrais menor que um, não implica que ela o seja por completo. Deste modo, coloca-se uma primeira questão: em que casos tal implicação existe? O estudo é então desenvolvido sobre a simplificação das condições que originaram: as transformações F e F (diferente) são tomadas simplificadamente como somas de apenas dois termos, e a questão inicial se converte na pesquisa de casos em que a igualdade entre raios espectrais de F e F(diferente) ocorre. Mais precisamente, os termos que compõem F e F(diferente) se constitui em produtos de operadores pertencentes à álgebra de Banach inicialmente referida, de modo que é feita uma análise do comportamento dos raios espectrais de F e F(diferente) situando-se esses operadores em classes específicas nessa álgebra. Sob estas condições são apresentados resultados relativos às classes dos operadores auto-adjuntos, unitários, normais, isometrias e subnormais, assim como um resultado referente aos shifts ponderados. Além disto, é apresentado um resultado geral para o caso de espaços de dimensão finita. / [en] The transformations f and F(different) appeared associated to the mean-square stability problem for infinite dimensional discrete bilinear systems evolving in a separable Hilbert space, being originally defined as infinite series in the Banach algebra of bounded linear operators on the Hilbert space where the system evolves. The present work starts with a previously defined sufficient stability condition, expressed by assumptions on the spectral radiuses of the mentioned transformations - both strictly less than one- and from the already known fact that the condition being partially fulfilled, that is, one of the spectral radiuses less than one, does not imply that it be so completely. Thus one poses a first question: in which cases does one have such an implication? The study is then developed on a simplification of the conditions from which it arose: both F and F(different) are taken as sums of only terms, and the initial question becomes the search for cases in which the equality betweem the spectral radiuses of F and F(different) occurs. More precisely, the terms that compose F and F(different) are products of operators in the above mentioned algebra, so that the behaviour of the spectral radiuses of F and F (different) is analysed by placing those operators in specific classes in that algebra. Under these assumptions, results related to the classes of self-adjoint, unitary, normal, isometries and subnormal operators are presented, as well as result referring to weighted shifts. Besides, a general result related to finite-dimensional spaces is also presented.

In this thesis we consider problems related to rational inner functionsand several different Hilbert spaces on the unit polydisc. In the general introduction the functions and the function spaces we will be interested in are introduced, and in particular we point outproblems and phenomena that occur in higher dimensions and are notpresent for one variable functions. For example, we provide a detailed construction of a non-trivial shift-invariant subspace of Dirichlet-type spaces on the bidisc which is not fnitely generated. Furthermore, Clark-Aleksandrov measures are generalized to higher dimensions, and certain results about such measures are proved. Paper I concerns containment of rational inner functions in Dirichlet-type spaces on polydiscs. In particular a theorem relating H^p integrability of the partial derivatives of a rational inner function to containment of the function in certain Dirichlet-type spaces is proved. As a corollary, we see that every rational inner function on D^n belongs to the isotropic Dirichlet-type space with weight 1/n. In Paper II, Zhu's sub-Bergman spaces of one variable functions on the unit disc are generalized to weighted Bergman spaces on D^n. Unlike in one variable,we show that sub-Bergman spaces associated to a rational inner function are generally not contained in a weighted Bergman space of higher regularity. We also show how Clark measures on the n-torus can be used to study model spaces on D^n associated to rational inner functions.

Yes / A total set of states for which we have no resolution of the identity (a `pre-basis'), is considered in a finite dimensional Hilbert space. A dressing formalism renormalizes them into density matrices which resolve the identity, and makes them a `generalized basis', which is practically useful. The dresssing mechanism is inspired by Shapley's methodology in cooperative game theory, and it uses Mobius transforms. There is non-independence and redundancy in these generalized bases, which is quantifi ed with a Shannon type of entropy. Due to this redundancy, calculations based on generalized bases, are sensitive to physical changes and robust in the presence of noise. For example, the representation of an arbitrary vector in such generalized bases, is robust when noise is inserted in the coeffcients. Also in a physical system with ground state which changes abruptly at some value of the coupling constant, the proposed methodology detects such changes, even when noise is added to the parameters in the Hamiltonian of the system.

In this paper, there are three chapters. The first chapter discusses interpolation. Here a theorem about the uniqueness of the solution to the general interpolation problem is proven. Then the problem of how to represent this unique solution is discussed. Finally, the error involved in the interpolation and the convergence of the interpolation process is developed. In the second chapter a theorem about the uniform approximation to continuous functions is proven. Then the best approximation and the least squares approximation (a special case of best approximation) is discussed. In the third chapter orthogonal polynomials as discussed as well as bounded linear functionals in Hilbert spaces, interpolation and approximation and approximation in Hilbert space.

Let $\Omega\subset\mathbb{R}^n$ be an open and connected subset of $\mathbb{R}^n$. We say that a function $F\colon \Omega-\Omega\to\mathbb{C}$, where $\Omega-\Omega=\{x-y\colon x,y\in\Omega\}$, is positive definite if for any $x_1,\ldots,x_m\in\Omega$ and any $c_1,\ldots,c_m\in \mathbb{C}$ we have that $\sum_{j,k=1}^m F(x_j-x_k)c_j\overline{c_k}\geq 0$. Let $F\colon\Omega-\Omega\to\mathbb{C}$ be a continuous positive definite function. We give necessary and sufficient conditions for $F$ to have an extension to a continuous and positive definite function defined on the entire Euclidean space $\mathbb{R}^n$. The conditions are formulated in terms of strong commutativity of some certain selfadjoint operators defined on a Hilbert space associated to our positive definite function.

In the first part of this paper we show a similarity between the principle of Structural Risk Minimization Principle (SRM) (Vapnik, 1982) and the idea of Sparse Approximation, as defined in (Chen, Donoho and Saunders, 1995) and Olshausen and Field (1996). Then we focus on two specific (approximate) implementations of SRM and Sparse Approximation, which have been used to solve the problem of function approximation. For SRM we consider the Support Vector Machine technique proposed by V. Vapnik and his team at AT&T Bell Labs, and for Sparse Approximation we consider a modification of the Basis Pursuit De-Noising algorithm proposed by Chen, Donoho and Saunders (1995). We show that, under certain conditions, these two techniques are equivalent: they give the same solution and they require the solution of the same quadratic programming problem.

Mutually unbiased bases is an important topic in the recent quantum system researches. Although there is much work in this area, many problems related to mutually unbiased bases are still open. For example, constructing a complete set of mutually unbiased bases in the Hilbert spaces with composite dimensions has not been achieved yet. This thesis defines a weaker concept than mutually unbiased bases in the Hilbert spaces with composite dimensions. We call this concept, weak mutually unbiased bases. There is a duality between such bases and the geometry of the phase space Zd × Zd, where d is the phase space dimension. To show this duality we study the properties of lines through the origin in Zd × Zd, then we explain the correspondence between the properties of these lines and the properties of the weak mutually unbiased bases. We give an explicit construction of a complete set of weak mutually unbiased bases in the Hilbert space Hd, where d is odd and d = p1p2; p1, p2 are prime numbers. We apply the concept of weak mutually unbiased bases in the context of quantum tomography and quantum cryptography.

Given a C*-algebra $A$ endowed with an action $\alpha$ of $\R$ and an $\alpha$-invariant trace $\tau$, there is a canonical dual trace $\widehat \tau$ on the crossed product $A \rtimes_\alpha \R$. This dual trace induces (as would any suitable trace) a real-valued homomorphism $\widehat \tau_* : K_0(A \rtimes_\alpha \R) \to \R$ on the even $K$-theory group. Recall there is a natural isomorphism $\phi_\alpha^i : K_i(A) \to K_{i+1}(A \rtimes_\alpha \R)$, the Connes-Thom isomorphism. The attraction of describing $\widehat \tau_* \circ \phi_\alpha^1$ directly in terms of the generators of $K_1(A)$ is clear. Indeed, the paper where the isomorphisms $\{\phi_\alpha^0,\phi_\alpha^1\}$ first appear sees Connes show that $\widehat \tau_* \phi_\alpha^1[u] = \frac{1}{2 \pi i} \tau(\delta(u) u^*)$, where $\delta = \frac{d}{dt} \big|_{t=0} \alpha_t(\cdot)$ and $u$ is any appropriate unitary. A careful proof of the aforementioned result occupies a central place in this thesis. To place the result in its proper context, the right-hand side is first considered in its own right, i.e., in isolation from mention of the crossed-product. A study of 1-parameter dynamical systems and exterior equivalence is undertaken, with several useful technical results being proven. A connection is drawn between a lemma of Connes on exterior equivalence and projections, and a quantum-mechanical theorem of Bargmann-Wigner. An introduction to the Connes-Thom isomorphism is supplied and, in the course of this introduction, a refined version of suspension isomorphism $K_1(A) \to K_0(\susp A)$ is formulated and proven. Finally, we embark on a survey of unbounded traces on C*-algebras; when traces are allowed to be unbounded, there is inevitably a certain amount of hard, technical work needed to resolve various domain issues and justify various manipulations. / Graduate / 0280