The thesis is made up of a number of studies involving long-range dependence (LRD), that is, a slow power-law decay in the temporal correlation of stochastic models. Such a phenomenon has been frequently observed in practice. The models with LRD often yield non-standard probabilistic and statistical results. The thesis includes in particular the following topics: Multivariate limit theorems. We consider a vector made of stationary sequences, some components of which have LRD, while the others do not. We show that the joint scaling limits of the vector exhibit an asymptotic independence property. Non-central limit theorems. We introduce new classes of stationary models with LRD through Volterra-type nonlinear filters of white noise. The scaling limits of the sum lead to a rich class of non-Gaussian stochastic processes defined by multiple stochastic integrals. Limit theorems for quadratic forms. We consider continuous-time quadratic forms involving continuous-time linear processes with LRD. We show that the scaling limit of such quadratic forms depends on both the strength of LRD and the decaying rate of the quadratic coefficient. Behavior of the generalized Rosenblatt process. The generalized Rosenblatt process arises from scaling limits under LRD. We study the behavior of this process as its two critical parameters approach the boundaries of the defining region. Inference using self-normalization and resampling. We introduce a procedure called "self-normalized block sampling" for the inference of the mean of stationary time series. It provides a unified approach to time series with or without LRD, as well as with or without heavy tails. The asymptotic validity of the procedure is established.

Le but principal de cette thèse est de développer de nouveaux modèles non linéaires à longue mémoire pour modéliser des rendements financiers et leur estimation statistique. En plus de la longue mémoire, ces modèles sont capables de mettre en lumière d’autres faits stylisés comme l’asymétrie ou l’effet de levier. Les processus étudiés dans la thèse sont des solutions stationnaires de certaines équations aux différences stochastiques non linéaires impliquant un “bruit” i.i.d. Outre le fait de résoudre ces équations, qui est non trivial en lui-même, nous prouvons que leur solutions sont dépendantes à longue portée. Enfin pour un modèle non linéaire particulier à longue portée (GQARCH) nous prouvon la consistence et la normalité asymptotique de l’estimateur du quasi-maximum de vraisemblance (QMLE). / The thesis introduces new nonlinear models with long memory which can be used for modelling of financial returns and statistical inference. Apart from long memory, these models are capable to exhibit other stylized facts such as asymmetry and leverage. The processes studied in the thesis are defined as stationary solutions of certain nonlinear stochastic difference equations involving a given i.i.d. “noise”. Apart from solvability issues of these equations which are not trivial by itself, it is proved that their solutions exhibit long memory properties. Finally, for a particularly tractable nonlinear parametric model with long memory (GQARCH) we prove consistency and asymptotic normality of quasi-ML estimators.

In this thesis we study a mathematical model that describes task partitioning in a colony of ants. This process of self-organization is modeled by a nonlinear coupled system of rst order autonomous ordinary dierential equations. We discuss how this system of equations can be derived based on the behavior of ants in a colony. We use GNU Octave (a high-level programming language) to solve the system of equations numerically for dierent sets of parameters and show how the solutions respond to changes in the parameter values. Finally, we prove that the model is well-posed locally in time. We rewrite the system of ordinary dierential equations in terms of a system of coupled Volterra integral equations and look at the right-hand side of the system as a nonlinear operator on a Banach space. By doing so, we have transformed the problem of showing existence and uniqueness of solutions to a system of ordinary dierential equations into a problem of showing existence and uniqueness of a xed point to the corresponding integral operator. Additionally, we use Gronwall's inequality to prove the stability of solutions with respect to data and parameters.

Este trabalho trata de estimativas inferiores para o primeiro autovalor do problema de Dirichlet para o Laplaciano para domínios relativamente compactos contidos em variedades riemannianas. Essas estimativas são obtidas com hipóteses sobre a curvatura seccional ou a curvatura de Ricci radial e a curvatura do bordo do domínio. / This paper deals of lower estimates for the first eigenvalue of the Dirichlet problem for the Laplacian for relatively compact domains contained in Riemannian manifolds. These estimates are obtained with assumptions on the sectional or Ricci radial curvature and the curvature of the boundary of the domain.

Made available in DSpace on 2014-06-11T19:35:38Z (GMT). No. of bitstreams: 0 Previous issue date: 2010-04-30Bitstream added on 2014-06-13T20:07:08Z : No. of bitstreams: 1 bertin_mcfg_dr_ift.pdf: 756466 bytes, checksum: ce1f33918fe3aabd6f7ec3c8bae37297 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Neste trabalho apresentaremos o formalismo de Hamilton-Jacobi para sistemas singulares em teorias de campos, com foco em teorias com derivadas de ordem superior. Iniciaremos com uma análise preliminar do cálculo variacional para esses sistemas, que envolve as condições para a extremização de uma integral fundamental múltipla e a análise dos teoremas de Noether. Buscaremos seguir este caminho na construção do formalismo de Hamilton- Jacobi em forma covariante, em que nos utilizaremos da clássica abordagem de Carathéodory adaptada a teorias de campos. No terceiro capítulo, mostraremos como o formalismo pode ser construído dada a escolha de uma dinâmica relativística específica e como esta escolha nos permite tratar de sistemas singulares de forma natural. No quarto capítulo abordaremos o problema das condições de integrabilidade, análise que garantirá um método autoconsistente de análise de vínculos. Nesta análise, seremos capazes de relacionar um conjunto de geradores a simetrias da integral fundamental e um segundo tipo a uma modificação da dinâmica com a introdução de parênteses generalizados. Nos dois últimos capítulos apresentaremos aplicações deste método / In this work we will develop the Hamilton-Jacobi formalism to singular and higher-order derivative field theories. We will begin with a preliminary approach to the variational problem concerning the search for extrema of a given fundamental integral, and the analysis of the Noether’s theorems. Next, we will present a covariant Hamilton-Jacobi theory using the classical approach of Carathéodory applied to field theories. In the third chapter we will show how this formalism can be derived given a choice of relativistic dynamics, and how this choice allows us to deal with singular systems. In the fourth chapter we will address the problem of integrability conditions. This analysis will be the basic tool for a self consistent constraint analysis. We will see that we can relate a certain set of generators to symmetries of the action, as well as a second type of generators to a modification of the dynamics by means of generalized brackets. The two last chapters will be used for applications

abstract: Since Duffin and Schaeffer's introduction of frames in 1952, the concept of a frame has received much attention in the mathematical community and has inspired several generalizations. The focus of this thesis is on the concept of an operator-valued frame (OVF) and a more general concept called herein an operator-valued frame associated with a measure space (MS-OVF), which is sometimes called a continuous g-frame. The first of two main topics explored in this thesis is the relationship between MS-OVFs and objects prominent in quantum information theory called positive operator-valued measures (POVMs). It has been observed that every MS-OVF gives rise to a POVM with invertible total variation in a natural way. The first main result of this thesis is a characterization of which POVMs arise in this way, a result obtained by extending certain existing Radon-Nikodym theorems for POVMs. The second main topic investigated in this thesis is the role of the theory of unitary representations of a Lie group G in the construction of OVFs for the L^2-space of a relatively compact subset of G. For G=R, Duffin and Schaeffer have given general conditions that ensure a sequence of (one-dimensional) representations of G, restricted to (-1/2,1/2), forms a frame for L^{2}(-1/2,1/2), and similar conditions exist for G=R^n. The second main result of this thesis expresses conditions related to Duffin and Schaeffer's for two more particular Lie groups: the Euclidean motion group on R^2 and the (2n+1)-dimensional Heisenberg group. This proceeds in two steps. First, for a Lie group admitting a uniform lattice and an appropriate relatively compact subset E of G, the Selberg Trace Formula is used to obtain a Parseval OVF for L^{2}(E) that is expressed in terms of irreducible representations of G. Second, for the two particular Lie groups an appropriate set E is found, and it is shown that for each of these groups, with suitably parametrized unitary duals, the Parseval OVF remains an OVF when perturbations are made to the parameters of the included representations. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2014

L'objet principal de cette thèse est de généraliser un certain nombre de résultats bien connus de la géométrie algébrique au cas k"{a}hlerien non nécessairement projectif. On généralise d'abord le théorème d'annulation de Nadel au cas k"{a}hlerien arbitraire. On obtient aussi un cas particulier du théorème d'annulation de Kawamata-Viehweg pour les variétés qui admettent une fibration vers un tore dont la fibre générique est projective. En utilisant ce résultat, on étudie le problème de déformation pour les variétés k"{a}hlériennes compactes sous une hypothèse portant sur les fibrés canoniques. On étudie enfin les variétés à fibré anticonique nef. On montre que si le fibré anticanonique est nef, alors le fibré tangent est à pentes semi-positif relative à la filtration de Harder-Narasimhan pour la polarization $omega_X ^{n-1}$. Comme application, on donne une preuve simple de la surjectivité de l'application d'Albanese, et on étudie aussi la trivialité locale de l'application d'Albanese. / The aim of this thesis is to generalize a certain number of results of algebraic geometry to K"{a}hler geometry. We first generalize the Nadel vanishing theorem to arbitrary compact K"{a}hler manifolds. We prove also a particular version of the Kawamata-Viehweg vanishing theorem for manifolds admitting a fibration to a torus such that the generic fiber is projective. Using this result, we study the theory of deformations of compact Kähler manifolds under certain assumptions on their canonical bundles. Finally, we study varieties with nef anticanonical bundles. We prove that the slopes of the Harder-Narasimhan filtration of the tangent bundles with respect to a polarization of the form $omega_X^{n-1}$ are semi-positive. As an application, we give a simple proof of the surjectivity of the Albanese map, and we investigate also the local triviality of the Albanese map.

One of the basic questions in harmonic analysis is to study the decay properties of the Fourier transform of measures or distributions supported on thin sets in Rn. When the support is a smooth enough manifold, an almost complete picture is available. One of the early results in this direction is the following: Let f in Cc∞(dσ), where dσ is the surface measure on the sphere Sn-1 Rn.Then the modulus of the Fourier transform of fdσ is bounded above by (1+|x|)(n-1)/2. Also fdσ in Lp(Rn) for all p > 2n/(n-1) . This result can be extended to compactly supported measure on (n-1)-dimensional manifolds with appropriate assumptions on the curvature. Similar results are known for measures supported in lower dimensional manifolds in Rn under appropriate curvature conditions. However, the picture for fractal measures is far from complete. This thesis is a contribution to the study of asymptotic properties of the Fourier transform of measures supported in sets of fractal dimension 0 < α < n for p ≤ 2n/α. In 2004, Agranovsky and Narayanan proved that if μ is a measure supported in a C1-manifold of dimension d < n, then the Fourier transform of μ is not in Lp(Rn) for 1 ≤ p ≤ 2n/d. We prove that the Fourier transform of a measure μ supported in a set E of fractal dimension α does not belong to Lp(Rn) for p≤ 2n/α. As an application we obtain Wiener-Tauberian type theorems on Rn and M(2). We also study Lp-asymptotics of the Fourier transform of fractal measures μ under appropriate conditions and give quantitative versions of the above statement by obtaining lower and upper bounds for the following limsup L∞ L-k∫|x|≤L|(fdµ)^(x)|pdx

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Apresentamos vÃrios resultados de existÃncia, unicidade, rigidez e bifurcaÃÃo para o problema da prescriÃÃo de diversas estruturas geomÃtricas em variedades Riemannianas, entre os quais incluem-se: i) deformaÃÃo e rigidez para estruturas 2k-Einstein em variedades com (2k &#8722; 2)-curvatura seccional constante; ii) deformaÃÃo conforme de mÃtricas no contexto do problema de Yamabe para curvaturas de Gauss-Bonnet; iii) unicidade, bifurcaÃÃo e rigidez local no Ãmbito do problema de Yamabe para as funÃÃes simÃtricas dos autovalores do tensor de Schouten. / We present several results of existence, uniqueness, rigidity and bifurcation for the problem of prescribing various geometric structures on Riemannian manifolds, among which include: i) deformation and rigidity for 2k-Einstein structures on manifolds with constant (2k &#8722; 2)-sectional curvature; ii) conformal deformation of metrics in the context of the Yamabe Problem for Gauss-Bonnet curvatures; iii) uniqueness, bifurcation and local rigidity in scope of the Yamabe Problem for symmetric functions of eigenvalues of the Schouten tensor.

Orientador: Plamen Emilov Kochloukov / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-10T10:59:05Z (GMT). No. of bitstreams: 1 SantuloJunior_EdneiAparecido_D.pdf: 675335 bytes, checksum: ff19aaa47432de596122e88eeede9a05 (MD5) Previous issue date: 2007 / Resumo: Kemer classificou, a menos de PI-equivalência, todas as álgebras T-primas no caso de caracterísitica zero e, em seu importante Teorema sobre o Produto Tensorial (TPT), demonstrou que o produto tensorial entre duas álgebras T-primas (ainda sobre corpos de característica zero) resulta igualmente numa álgebra T-prima. Neste trabalho é fornecida uma generalização para o último caso do TPT utilizando identidades graduadas. Além disso, é estudada a existência de mergulhos nas álgebras que aparecem no TPT. Mais especificamente, são encontradas condições necessárias e suficientes para a existência de um mergulho graduado de uma álgebra que satisfaz todas as identidades graduadas da álgebra de matrizes cujas entradas pertencem à álgebra de Grassmann em uma álgebra de matrizes cujas entradas se encontram numa álgebra supercomutativa com unidade, quando todas essas álgebras são consideradas sobre corpos infinitos de característica diferente de dois. Por fim, são fornecidas bases de identidades graduadas para os T-ideais graduados da nésima potência tensorial da' álgebra de Grassmann, das álgebras de matrizes cuja ordem é uma potência de dois, e do produto tensorial de quaisquer duas dentre as álgebras previamente citadas. A partir destas deduz-se o TPT no caso em que a ordem das álgebras de matrizes é uma potência de dois / Abstract: Kemer classified, up to PI-equivalence, the T-prime algebras in the case of characteristic zero, and in his celebrated Tenso r Product Theorem (TPT) he showed that the tensor product of two T-prime algebras considered over a field of characteristic zero, is another T-prime algebra. In this work, a generalization for the last case of the TPT is given using graded identities. The existence of embeddings into the algebras cited on the TPT is also studied. More specifically, necessary and sufficient conditions for the existence of a graded embedding of an algebra satisfying all graded polynomial identities for the matrix algebra with entries in the Grassmann algebra, into a matrix algebra with entries in a supercommutative algebra with unity are found when these algebras are taken over fields of characteristic different from two. Graded identities that generate the graded T-ideals of the n-th tensor power of the Grassmann algebra, of the matrix algebras cited in Kemer's TPT (whose order is a power of two) and of the tensor product between any two of those algebras are provided. As a consequence, Kemer's TPT is derived from those results in the special case when the order of the matrices in the matrix algebras under consideration, is a powers of two / Doutorado / Algebra / Doutor em Matemática