Neste trabalho estudamos diferentes noções de diferenciabilidade para funções definidas na esfera unitária S^n-1 de R^n, n>=2. Em relação à derivada usual, encontramos condições necessárias e/ou suficientes para que uma função seja diferenciável até uma ordem fixada. Para as outras duas, a derivada forte de Laplace-Beltrami e a derivada fraca, apresentamos algumas propriedades básicas e procuramos condições que garantam a equivalência destas com a diferenciabilidade usual. / In this work we study different notions of differentiability for functions defined on the unit sphere S^n-1 of R^n, n>=2. With respect to the usual derivative, we find necessary and/or sufficient conditions in order that a function be differentiable up to a fixed order. As for the other two, the strong Laplace-Beltrami derivative and the weak derivative, we present some basic properties about them and search for conditions that guarantee the equivalence of them with the previous one.

Neste trabalho obtemos taxas de decaimento para autovalores e valores singulares de operadores integrais gerados por núcleos de quadrado integrável sobre a esfera unitária em \'R POT. m+1\', m 2, sob hipóteses sobre ambos, certas derivadas do núcleo e o operador integral gerado por tais derivadas. Este tipo de problema é comum na literatura, mas as hipóteses geralmente são definidas via diferenciação usual em \'R POT m+1\'. Aqui, as hipóteses são todas definidas via derivada de Laplace-Beltrami, um conceito genuinamente esférico investigado primeiramente por W. Rudin no começo dos anos 50. As taxas de decaimento apresentadas são ótimas e dependem da dimensão m e da ordem de diferenciabilidade usada para definir as condições de suavidade / In this work we obtain decay rates for singular values and eigenvalues of integral operators generated by square integrable kernels on the unit sphere in \'R m+1\', m 2, under assumptions on both, certain derivatives of the kernel and the integral operators generated by such derivatives. This type of problem is common in the literature but the assumptions are usually defined via standard differentiation in \'R POT. m+1\'. Here, the assumptions are all defined via the Laplace-Beltrami derivative, a concept first investigated by W. Rudin in the early fifties and genuinely spherical in nature. The rates we present are optimal and depend on both, the differentiability order used to define the smoothness conditions and the dimension m

Orientador: Edmundo Capelas de Oliveira / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-26T00:53:38Z (GMT). No. of bitstreams: 1 Oliveira_DanieladosSantosde_M.pdf: 3702602 bytes, checksum: c0b05792ff3ac3c5bdd5fad1b7586dd5 (MD5) Previous issue date: 2014 / Resumo: Neste trabalho apresentamos um estudo sobre as funções de Mittag-Leffler de um, dois e três parâmetros. Apresentamos a função de Mittag-Leffler como uma generalização da função exponencial bem como a relação que esta possui com outras funções especiais, tais como as funções beta, gama, gama incompleta e erro. Abordamos, também, a integração fracionária que se faz necessária para introduzir o conceito de derivação fracionária. Duas formulações para a derivada fracionária são estudadas, as formulações proposta por Riemann-Liouville e por Caputo. Investigamos quais regras clássicas de derivação são estendidas para estas formulações. Por fim, como uma aplicação, utilizamos a metodologia da transformada de Laplace para resolver a equação diferencial fracionária associada ao problema do oscilador harmônico fracionário / Abstract: This work presents a study about the one- two- and three-parameters Mittag-Leffler functions. We show that the Mittag-Leffler function is a generalization of the exponential function and present its relations to other special functions beta, gamma, incomplete gamma and error functions. We also approach fractional integration, which is necessary to introduce the concept of fractional derivatives. Two formulations for the fractional derivative are studied, the formulations proposed by Riemann-Liouville and by Caputo. We investigate which classical derivatives rules can be extended to these formulations. Finally, as an application, using the Laplace transform methodology, we discuss the fractional differential equation associated with the harmonic oscillator problem / Mestrado / Matematica Aplicada / Mestra em Matemática Aplicada

In many applications nonsmooth nonconvex energy functions, which are Lipschitz continuous, appear quite naturally. Contact mechanics with friction is a classic example. A second example is the 1-Laplace operator and its eigenfunctions. In this work we will give an algorithm such that for every locally Lipschitz continuous function f and every sequence produced by this algorithm it holds that every accumulation point of the sequence is a critical point of f in the sense of Clarke. Here f is defined on a reflexive Banach space X, such that X and its dual space X' are strictly convex and Clarkson's inequalities hold. (E.g. Sobolev spaces and every closed subspace equipped with the Sobolev norm satisfy these assumptions for p>1.) This algorithm is designed primarily to solve variational problems or their high dimensional discretizations, but can be applied to a variety of locally Lipschitz functions. In elastic contact mechanics the strain energy is often smooth and nonconvex on a suitable domain, while the contact and the friction energy are nonsmooth and have a support on a subspace which has a substantially smaller dimension than the strain energy, since all points in the interior of the bodies only have effect on the strain energy. For such elastic contact problems we suggest a specialization of our algorithm, which treats the smooth part with Newton like methods. In the case that the gradient of the entire energy function is semismooth close to the minimizer, we can even prove superlinear convergence of this specialization of our algorithm. We test the algorithm and its specialization with a couple of benchmark problems. Moreover, we apply the algorithm to the 1-Laplace minimization problem restricted to finitely dimensional subspaces of piecewise affine, continuous functions. The algorithm developed here uses ideas of the bundle trust region method by Schramm, and a new generalization of the concept of gradients on a set. The basic idea behind this gradients on sets is that we want to find a stable descent direction, which is a descent direction on an entire neighborhood of an iteration point. This way we avoid oscillations of the gradients and very small descent steps (in the smooth and in the nonsmooth case). It turns out, that the norm smallest element of the gradient on a set provides a stable descent direction. The algorithm we present here is the first algorithm which can treat locally Lipschitz continuous functions in this generality, up to our knowledge. In particular, large finitely dimensional Banach spaces haven't been studied for nonsmooth nonconvex functions so far. We will show that the algorithm is very robust and often faster than common algorithms. Furthermore, we will see that with this algorithm it is possible to compute reliably the first eigenfunctions of the 1-Laplace operator up to disretization errors, for the first time. / In vielen Anwendungen tauchen nichtglatte, nichtkonvexe, Lipschitz-stetige Energie Funktionen in natuerlicher Weise auf. Ein klassische Beispiel bildet die Kontaktmechanik mit Reibung. Ein weiteres Beispiel ist der $1$-Laplace Operator und seine Eigenfunktionen. In dieser Dissertation werden wir ein Abstiegsverfahren angeben, so dass fuer jede lokal Lipschitz-stetige Funktion f jeder Haeufungspunkt einer durch dieses Verfahren erzeugten Folge ein kritischer Punkt von f im Sinne von Clarke ist. Hier ist f auf einem einem reflexiver, strikt konvexem Banachraum definierert, fuer den der Dualraum ebenfalls strikt konvex ist und die Clarkeson Ungleichungen gelten. (Z.B. Sobolevraeume und jeder abgeschlossene Unterraum mit der Sobolevnorm versehen, erfuellt diese Bedingung fuer p>1.) Dieser Algorithmus ist primaer entwickelt worden um Variationsprobleme, bzw. deren hochdimensionalen Diskretisierungen zu loesen. Er kann aber auch fuer eine Vielzahl anderer lokal Lipschitz stetige Funktionen eingesetzt werden. In der elastischen Kontaktmechanik ist die Spannungsenergie oft glatt und nichtkonvex auf einem geeignetem Definitionsbereich, waehrend der Kontakt und die Reibung durch nicht glatte Funktionen modelliert werden, deren Traeger ein Unterraum mit wesentlich kleineren Dimension ist, da alle Punkte im Inneren des Koerpers nur die Spannungsenergie beeinflussen. Fuer solche elastischen Kontaktprobleme schlagen wir eine Spezialisierung unseres Algorithmuses vor, der den glatten Teil mit Newton aehnlichen Methoden behandelt. Falls der Gradient der gesamten Energiefunktion semiglatt in der Naehe der Minimalstelle ist, koennen wir sogar beweisen, dass der Algorithmus superlinear konvergiert. Wir testen den Algorithmus und seine Spezialisierung an mehreren Benchmark Problemen. Ausserdem wenden wir den Algorithmus auf 1-Laplace Minimierungsproblem eingeschraenkt auf eine endlich dimensionalen Unterraum der stueckweise affinen, stetigen Funktionen an. Der hier entwickelte Algorithmus verwendet Ideen des Bundle-Trust-Region-Verfahrens von Schramm, und einen neu entwickelten Verallgemeinerung von Gradienten auf Mengen. Die zentrale Idee hinter den Gradienten auf Mengen ist die, dass wir stabile Abstiegsrichtungen auf einer ganzen Umgebung der Iterationspunkte finden wollen. Auf diese Weise vermeiden wir das Oszillieren der Gradienten und sehr kleine Abstiegsschritte (im glatten, wie im nichtglatten Fall.) Es stellt sich heraus, dass das normkleinste Element dieses Gradienten auf der Umgebung eine stabil Abstiegsrichtung bestimmt. So weit es uns bekannt ist, koennen die hier entwickelten Algorithmen zum ersten Mal lokal Lipschitz-stetige Funktionen in dieser Allgemeinheit behandeln. Insbesondere wurden nichtglatte, nichtkonvexe Funktionen auf derart hochdimensionale Banachraeume bis jetzt nicht behandelt. Wir werden zeigen, dass unser Algorithmus sehr robust und oft schneller als uebliche Algorithmen ist. Des Weiteren, werden wir sehen, dass es mit diesem Algorithmus das erste mal moeglich ist, zuverlaessig die erste Eigenfunktion des 1-Laplace Operators bis auf Diskretisierungsfehler zu bestimmen.

Cette thèse sur travaux est la synthèse de publications réalisées entre 2005 et 2012 ainsi que de papiers de travail. Elle est organisée autour de trois axes : des questions relatives au Club de Paris, des articles au sujet de la loi de Gibrat et des travaux autour de l’Histoire de la Banque de France. Le premier axe comprend deux papiers publiés dans le bulletin de la Banque de France : l’un sur l’évaluation de l’initiative PPTE (Pays pauvres très endettés, mécanismes et éléments d’évaluation, Bulletin N°140, août 2005) et le second sur la modélisation des buybacks de créance au sein du club de Paris. Ce dernier papier a été sous deux formes (grand public : Modélisation et analyse des mécanismes du Club de Paris de rachat de créances par prépaiement, avec Laurent Daniel, Bulletin N° 152, août 2006, et recherche : Pricing the implicit contracts in the Paris Club debt buybacks avec Laurent Daniel, working paper, December 2007). Le second axe concerne la validation de la loi de Gibrat, avec la publication de trois articles (French butchers don't do Quantum Physics in Economics Letters, Vol. 103, May 2009, Pp. 101-106 ; The Paretian Ratio Distribution - An application to the volatility of GDP in Economics Letters, Vol. 111, May 2011, pp. 180-183 ; The Laplace Illusion in Physica A, Vol. 391, August 2012, pp. 3963–3970). Le dernier axe regroupe des travaux sur l’Histoire de la Banque de France. Certains sont publiés comme La Caisse de Réserve des Employés de la Banque de France 1800-1950, (Économies et Sociétés, série « Histoire Économique Quantitative », août 2007, n°37, pp. 1365-1383 ou en cours. / This dissertation is made of several papers published between 2005 and 2012 and somme working papers. The first part deals with the Paris Club. Two papers published in the Bulletin of the Banque de France deal with the very indebted countries and debt buybacks ( Pricing the implicit contracts in the Paris Club debt buybacks). The second axis is oriented on the Gibrat's law (French butchers don't do Quantum Physics in Economics Letters, Vol. 103, May 2009, Pp. 101-106 ; The Paretian Ratio Distribution - An application to the volatility of GDP in Economics Letters, Vol. 111, May 2011, pp. 180-183 ; The Laplace Illusion in Physica A, Vol. 391, August 2012, pp. 3963–3970). The third axis deals with the history of the Banque de France.

L'application exponentielle duale est une façon d'encoder les matrices de Stokes d'une connexion sur un fibré trivial sur la sphère de Riemann avec deux pôles : un pôle double en 0 et un pôle simple en l'infini.On donne ici une formule pour l'application exponentielle duale comme une série formelle non commutative. D'autres généralisations de cette formule sont données. / The exponential dual map is a way to encode Stokes data of a connection on a trivial vector bundle on the Riemann sphere with two poles: one double pole at 0 and one simple pole at infinity.We give here a formula for the exponential dual map expressed as a non commutative serie. Others generalizations of this formula are given.

In many applications nonsmooth nonconvex energy functions, which are Lipschitz continuous, appear quite naturally. Contact mechanics with friction is a classic example. A second example is the 1-Laplace operator and its eigenfunctions. In this work we will give an algorithm such that for every locally Lipschitz continuous function f and every sequence produced by this algorithm it holds that every accumulation point of the sequence is a critical point of f in the sense of Clarke. Here f is defined on a reflexive Banach space X, such that X and its dual space X' are strictly convex and Clarkson's inequalities hold. (E.g. Sobolev spaces and every closed subspace equipped with the Sobolev norm satisfy these assumptions for p>1.) This algorithm is designed primarily to solve variational problems or their high dimensional discretizations, but can be applied to a variety of locally Lipschitz functions. In elastic contact mechanics the strain energy is often smooth and nonconvex on a suitable domain, while the contact and the friction energy are nonsmooth and have a support on a subspace which has a substantially smaller dimension than the strain energy, since all points in the interior of the bodies only have effect on the strain energy. For such elastic contact problems we suggest a specialization of our algorithm, which treats the smooth part with Newton like methods. In the case that the gradient of the entire energy function is semismooth close to the minimizer, we can even prove superlinear convergence of this specialization of our algorithm. We test the algorithm and its specialization with a couple of benchmark problems. Moreover, we apply the algorithm to the 1-Laplace minimization problem restricted to finitely dimensional subspaces of piecewise affine, continuous functions. The algorithm developed here uses ideas of the bundle trust region method by Schramm, and a new generalization of the concept of gradients on a set. The basic idea behind this gradients on sets is that we want to find a stable descent direction, which is a descent direction on an entire neighborhood of an iteration point. This way we avoid oscillations of the gradients and very small descent steps (in the smooth and in the nonsmooth case). It turns out, that the norm smallest element of the gradient on a set provides a stable descent direction. The algorithm we present here is the first algorithm which can treat locally Lipschitz continuous functions in this generality, up to our knowledge. In particular, large finitely dimensional Banach spaces haven't been studied for nonsmooth nonconvex functions so far. We will show that the algorithm is very robust and often faster than common algorithms. Furthermore, we will see that with this algorithm it is possible to compute reliably the first eigenfunctions of the 1-Laplace operator up to disretization errors, for the first time. / In vielen Anwendungen tauchen nichtglatte, nichtkonvexe, Lipschitz-stetige Energie Funktionen in natuerlicher Weise auf. Ein klassische Beispiel bildet die Kontaktmechanik mit Reibung. Ein weiteres Beispiel ist der $1$-Laplace Operator und seine Eigenfunktionen. In dieser Dissertation werden wir ein Abstiegsverfahren angeben, so dass fuer jede lokal Lipschitz-stetige Funktion f jeder Haeufungspunkt einer durch dieses Verfahren erzeugten Folge ein kritischer Punkt von f im Sinne von Clarke ist. Hier ist f auf einem einem reflexiver, strikt konvexem Banachraum definierert, fuer den der Dualraum ebenfalls strikt konvex ist und die Clarkeson Ungleichungen gelten. (Z.B. Sobolevraeume und jeder abgeschlossene Unterraum mit der Sobolevnorm versehen, erfuellt diese Bedingung fuer p>1.) Dieser Algorithmus ist primaer entwickelt worden um Variationsprobleme, bzw. deren hochdimensionalen Diskretisierungen zu loesen. Er kann aber auch fuer eine Vielzahl anderer lokal Lipschitz stetige Funktionen eingesetzt werden. In der elastischen Kontaktmechanik ist die Spannungsenergie oft glatt und nichtkonvex auf einem geeignetem Definitionsbereich, waehrend der Kontakt und die Reibung durch nicht glatte Funktionen modelliert werden, deren Traeger ein Unterraum mit wesentlich kleineren Dimension ist, da alle Punkte im Inneren des Koerpers nur die Spannungsenergie beeinflussen. Fuer solche elastischen Kontaktprobleme schlagen wir eine Spezialisierung unseres Algorithmuses vor, der den glatten Teil mit Newton aehnlichen Methoden behandelt. Falls der Gradient der gesamten Energiefunktion semiglatt in der Naehe der Minimalstelle ist, koennen wir sogar beweisen, dass der Algorithmus superlinear konvergiert. Wir testen den Algorithmus und seine Spezialisierung an mehreren Benchmark Problemen. Ausserdem wenden wir den Algorithmus auf 1-Laplace Minimierungsproblem eingeschraenkt auf eine endlich dimensionalen Unterraum der stueckweise affinen, stetigen Funktionen an. Der hier entwickelte Algorithmus verwendet Ideen des Bundle-Trust-Region-Verfahrens von Schramm, und einen neu entwickelten Verallgemeinerung von Gradienten auf Mengen. Die zentrale Idee hinter den Gradienten auf Mengen ist die, dass wir stabile Abstiegsrichtungen auf einer ganzen Umgebung der Iterationspunkte finden wollen. Auf diese Weise vermeiden wir das Oszillieren der Gradienten und sehr kleine Abstiegsschritte (im glatten, wie im nichtglatten Fall.) Es stellt sich heraus, dass das normkleinste Element dieses Gradienten auf der Umgebung eine stabil Abstiegsrichtung bestimmt. So weit es uns bekannt ist, koennen die hier entwickelten Algorithmen zum ersten Mal lokal Lipschitz-stetige Funktionen in dieser Allgemeinheit behandeln. Insbesondere wurden nichtglatte, nichtkonvexe Funktionen auf derart hochdimensionale Banachraeume bis jetzt nicht behandelt. Wir werden zeigen, dass unser Algorithmus sehr robust und oft schneller als uebliche Algorithmen ist. Des Weiteren, werden wir sehen, dass es mit diesem Algorithmus das erste mal moeglich ist, zuverlaessig die erste Eigenfunktion des 1-Laplace Operators bis auf Diskretisierungsfehler zu bestimmen.

We consider the eigenvalue problem associated to the 1-Laplace operator. We define higher eigensolutions by means of weak slope and establish existence of a sequence of eigensolutions by using nonsmooth critical point theory. In addition, we deduce a new necessary condition for the first eigenvalue of the 1-Laplace operator by means of inner variations. / Wir betrachten das zum 1-Laplace-Operator gehörige Eigenwertproblem. Wir definieren höhere Eigenlösungen mittels weak slope und weisen die Existenz einer Folge von Eigenlösungen nach, indem wir die nichtglatte Theorie kritischer Punkte anwenden. Zusätzlich leiten wir eine neue notwendige Bedingung für den ersten Eigenwert des 1-Laplace-Operators mittels innerer Variationen her.

In this thesis we study the asymptotic behavior of bi-Lipschitz diffeomorphic weighted Riemannian manifolds with techniques from the theory of homogenization. To do so we re-interpret the problem as different induced metrics on one reference manifold. Our analysis is twofold. On the one hand we consider second-order uniformly elliptic operators on weighted Riemannian manifolds. They naturally emerge when studying spectral properties of the Laplace-Beltrami operator on families of manifolds with rapidly oscillating metrics. We appeal to the notion of H-convergence introduced by Murat and Tartar. In our first main result we establish an H-compactness result that applies to elliptic operators with measurable, uniformly elliptic coefficients on weighted Riemannian manifolds. We further discuss the special case of locally periodic coefficients and study the asymptotic spectral behavior of Euclidean submanifolds with rapidly oscillating geometry. On the other hand we study integral functionals featuring non-convex integrands with non-standard growth on the Euclidean space in a stochastic framework. Our second main result is a Γ-convergence statement under certain assumptions on the statistics of their integrands. Such functionals provide a tool to study the Dirichlet energy on non-uniformly bi-Lipschitz diffeomorphic manifolds. We show Mosco-convergence of the Dirichlet energy and deduce conditions for the spectral behavior of weighted Riemannian manifolds with locally oscillating random structure, especially in the case of Euclidean submanifolds.:Introduction Outline Notation I. Preliminaries 1. Convergence of Riemannian Manifolds 1.1. Hausdorff-Convergence 1.2. Gromov-Hausdorff-Convergence 1.3. Spectral Convergence 1.4. Mosco-Convergence 2. Homogenization 2.1. Periodic Homogenization 2.2. Stochastic Homogenization II. Uniformly bi-Lipschitz Diffeomorphic Manifolds 3. Uniformly Elliptic Operators on a Riemannian Manifold 3.1. Setting 3.2. Main Results 3.3. Strategy of the Proof and Auxiliary Results 3.4. Identi cation of the Limit via Local Coordinate Charts 3.5. Examples 3.6. Proofs 4. Application to Uniformly bi-Lipschitz Diffeomorphic Manifolds 4.1. Setting and Results 4.2. Examples 4.3. Proofs III. Rapidly Oscillating Random Manifolds 5. Integral Functionals with Non-Uniformal Growth 5.1. Setting 5.2. Main Results 5.3. Strategy of the Proof and Auxiliary Results 5.4. Proofs 6. Application to Rapidly Oscillating Riemannian Manifolds 6.1. Setting and Results 6.2. Examples 6.3. Proofs Summary and Discussion Bibliography List of Figures

This thesis investigates Parameter-Efficient Fine-Tuning (PEFT) methods, specifically Low-Rank Adaptation (LoRA) (Hu et al. 2021) and Adapters (Houlsby et al. 2019), using the General Language Understanding Evaluation (GLUE) dataset (Wang et al. 2019). The primary focus is to evaluate the effectiveness and efficiency of these methods in fine-tuning pre-trained language models. Additionally, we introduce a novel application by applying the methodology from Yang et al. 2024 to the adapter module weights. We utilize Laplace approximations over both the LoRA (Yang et al. 2024, Daxberger et al. 2022a) and the newly adapted Adapter weights, assessing the Expected Calibration Error (ECE) and Negative Log-Likelihood (NLL). Furthermore, we discuss practical considerations such as training time, memory usage, and storage space implications of these PEFT techniques. The findings provide valuable insights into the trade-offs and benefits of using LoRA and Adapters for fine-tuning in resource-constrained environments.