Η μετάλλαξη των συναρτήσεων είναι το κυρίως θέμα της παρούσας εργασίας που συνδυάζει τη γεωμετρία με τη θεωρία μέτρου και την ανάλυση με έναν ουσιώδη τρόπο. Δεδομένης μιας πραγματικής συνάρτησης f που ορίζεται σε ένα υποσύνολο του, κατασκευάζεται μία καινούρια συνάρτηση Ω n \ f ∗, η οποία έχει σημαντικές ιδιότητες. Οι εφαρμογές των θεωρημάτων που προκύπτουν είναι πολλές και ιδιαίτερα σημαντικές. Ένα από τα βασικότερα θεωρήματα είναι η ανισότητα HardyLittlewood-Sobolev που αποδεικνύεται με τη βοήθεια της συμμετρικο- ποίησης. Στο θεώρημα αυτό έχουμε ότι οι συναρτήσεις που ελαχιστοποιούν αυτήν την ανισότητα και την καθιστούν ισότητα (ονομάζονται ακραίες συναρτήσεις) είναι σφαιρικά συμμετρικές συναρτήσεις. Επίσης, μία πολύ ενδιαφέρουσα και σημαντική εφαρμογή της συμμετρικοποίησης είναι η γνωστή ισοπεριμετρική ανισότητα (δηλαδή η μπάλα έχει την ελάχιστη επιφάνεια μεταξύ όλων των σωμάτων δοσμένου όγκου). Στο κεφάλαιο 1 παρουσιάζονται κάποιες απαραίτητες έννοιες, ορισμοί και θεωρήματα από τη θεωρία μέτρου και ολοκλήρωσης, καθώς χρησιμοποιούνται συχνά στους ορισμούς και τις αποδείξεις των θεωρημάτων που αναπτύσσονται. Στο κεφάλαιο 2 παρουσιάζεται η φθίνουσα μετάλλαξη συναρτήσεων που ορίζονται σε υποσύνολα Ω του . Θεωρώντας μια πραγματική συνάρτηση σε ένα τέτοιο σύνολο, κατασκευάζουμε μια νέα συνάρτηση, η οποία έχει πεδίο ορισμού τη μπάλα με κέντρο την αρχή των αξόνων, η οποία έχει το ίδιο μέτρο (όγκο) με το και η νέα συνάρτηση έχει σημαντικές ιδιότητες. Γενικά, επιθυμούμε η νέα συνάρτηση να είναι ακτινική και ακτινικά φθίνουσα. Για να δοθεί ο ορισμός αυτός, πρώτα κατασκευάζουμε τη μονοδιάστατη φθίνουσα μετάλλαξη της δοσμένης συνάρτησης. n \ Ω iΣτο κεφάλαιο 3 παρουσιάζονται ανισότητες για μεταλλαγμένες συναρτήσεις μεταξύ των οποίων η γνωστή ανισότητα του Riesz και η αναφερθείσα σημαντική ανισότητα Hardy- Littlewood-Sobolev. Στο κεφάλαιο 4 δίνεται η συμμετρικοποίηση Steiner με μία πιο γεωμετρική σκοπιά καθώς ένα από τα πιο σημαντικά αποτελέσματά της είναι η γνωστή ισοπεριμετρική ανισότητα και άλλες πολύ ενδιαφέρουσες εφαρμογές. / -

The first part of this thesis is devoted to symmetrizations. Symmetrizations are tranformations of functions that preserve many properties of functions and enhance their symmetry. In the calculus of variation they are a simple and powerful tool to prove that minimizers of functionals are symmetric functions. In this work, the approximation of symmetrizations by simpler symmetrizations is investigated: The existence of a universal approximating sequence is proved, sufficient conditions for deterministic and random sequences to be approximating are given. These approximation methods are then used to prove some symmetry properties of critical points obtained by minimax methods: For example if there is a solution obtained by the mountain pass theorem, then there is a symmetric solution with the same energy. This part ends with a study of the properties of anisotropic symmetrizations i.e. symmetrizations performed with respect to noneuclidean norms. The second part is devoted to L^1 estimates.  In general, the second derivative of the solution of the Poisson equation with L^1 data fails to be in L^1. Recently it was proved that if the data is a L^1 divergence-free vector-field, then even if in general it is false that the second derivative of the solution is in L^1, all the consequences thereof by Sobolev embeddings hold. Elementary proofs of such results, as well as a generalization with a second order operator replacing the divergence, are given. / La première partie de cette thèse est consacrée aux symétrisations. Les symétrisations sont des transformations de fonctions qui préservent de nombreuses propriétés des fonctions et qui améliorent leur symétrie. Elles sont un outil simple et puissant pour montrer dans le calcul des variations que les minimiseurs de certaines fonctionnelles sont des fonctions symétriques. Dans ce travail, nous étudions l'approximation des symétrisations par des symétrisations plus simples. Nous prouvons l'existence d'une suite approximante universelle et nous donnons des conditions suffisantes pour que des suites déterministes et aléatoires soient approximantes. Nous utilisons ensuite ces méthodes d'approximation pour prouver des propriétés de symétrie de points critiques obtenus par des méthodes de minimax. Par exemple, s'il y a une solution obtenue par le théorème du col, alors il y a une solution symétrique de même énergie. Nous achevons cette partie par une étude des symétrisations anisotropes (symétrisations par rapport à des normes non euclidiennes). La seconde partie est consacrée aux estimations L^1. En général, les dérivées secondes de la solution de l'équation de Poisson avec des données L^1 ne sont pas dans L^1. Recemment, on a prouvé que si les données sont un champ de vecteurs L^1 à divergence nulle, même si en général les dérivées secondes ne sont toujours pas dans L^1, toutes les conséquences qui en suivraient par les injections de Sobolev sont vraies. Nous donnons des preuves élémentaires de ces résultats, avec une extension où la divergence est remplacée par un opérateur différentiel du second ordre.

In this thesis we collect the study of two problems in the Calculus of Variations and Partial Differential Equations. Our first group of results concern the analysis of minimizers in a variational model describing the shape of liquid drops and crystals under the influence of gravity, resting on a horizontal surface. Making use of anisotropic symmetrization techniques and an analysis of fine properties of minimizers within the class of sets of finite perimeter, we establish existence, convexity and symmetry of minimizers. In the case of smooth surface tensions, we obtain uniqueness of minimizers via an ODE characterization. In the second group of results discussed in this thesis, which is joint work with A. Vasseur, we treat a problem in compressible fluid dynamics, establishing a uniform bound from below on the temperature for a variant of the compressible Navier-Stokes-Fourier system under suitable hypotheses. This system of equations forms a mathematical model of the motion of a compressible fluid subject to heat conduction. Building upon the work of (Mellet, Vasseur 2009), we identify a class of weak solutions satisfying a localized form of the entropy inequality (adapted to measure the set where the temperature becomes small) and use a form of the De Giorgi argument for L[superscript infinity] bounds of solutions to elliptic equations with bounded measurable coefficients. / text

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A partir da problematização da instituição de privação de liberdade para adolescentes em São Paulo, a Fundação CASA, busca-se, como objetivo central, identificar os processos de simetrização entre aquela entidade e as instituições de internação voltadas para adultos, as "prisões”. A partir disso, destaca-se, como objetivo específico, a ocorrência de rebeliões enquanto uma simetria-chave para a presente análise: esses acontecimentos se dão como situações-limite, a partir das quais se pode apreender o que é acionado nas relações e instituições que envolvem esses processos. Constata-se como as rebeliões, vistas como ruptura do cotidiano, fazem parte da sociabilidade institucional, estando presente no próprio dia-a-dia dos atores que circulam pela instituição. / This study is part of the debate on adolescents in conflict with the law. 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In this Ph. D. thesis we discuss four different problems in analysis: (a) sharp inequalities related to the restriction phenomena for the Fourier transform, with emphasis on some Strichartz-type estimates; (b) extremal approximations of exponential type for the Gaussian and for a class of even functions, with applications to analytic number theory; (c) radial symmetrization approach to convolution-like inequalities for the Boltzmann collision operator; (d) regularity of maximal operators with respect to weak derivatives and weak continuity. / text

The first part of the thesis is devoted to the analysis on a capacity space, with capacities as substitutes of measures in the study of function spaces. The goal is to extend to the associated function lattices some aspects of the theory of Banach function spaces, to show how the general theory can be applied to classical function spaces such as Lorentz spaces, and to complete the real interpolation theory for these spaces included in [CeClM] and [Ce]. In the second part of the thesis, we present an integral inequality connecting a function space norm of the gradient of a function to an integral of the corresponding capacity of the conductor between two level surfaces of the function, which extends the estimates obtained by V. Maz’ya and S. Costea, and sharp capacitary inequalities due to V. Maz’ya in the case of the Sobolev norm. The inequality, obtained under appropriate convexity conditions on the function space, gives a characterization of Sobolev type inequalities involving two measures, necessary and sufficient conditions for Sobolev isocapacitary type inequalities, and self-improvements for integrability of Lipschitz functions. / La primera part està dedicada a l’anàlisi d’un espai de capacitat, amb capacitats com a substituts de les mesures en l’estudi d’espais de funcions. L’objectiu és estendre als recicles de funcions associats alguns aspectes de la la teoria d’espais de funcions de Banach, mostrar com la teoria general pot ser aplicada a espais funcionals clàssics com els espais de Lorentz, i completar la teoria d’interpolació real d’aquests espais inclosos en [CeClM] i [Ce]. A la segona part de la tesi es presenta una desigualtat integral que connecta la norma del gradient d’una funció en un espai de funcions amb la integral de la corresponent capacitat del conductor entre dues superfícies de nivell de la funció, que estén les estimacions obtingudes per V. Maz’ya i S. Costea, i desigualtats capacitàries fortes de V. Maz’ya en el cas de la norma de Sobolev. La desigualtat, obtinguda sota condicions de convexitat pel espai funcional, permet una caracterització de les desigualtats de tipus Sobolev per dues mesures, condicions necessàries i suficients per desigualtats isocapacitàries de tipus Sobolev, i la millora de l’autointegrabilitat de les funcions de Lipschitz.

La détermination de la structure des protéines à une résolution atomique est cruciale pour la compréhension de leur fonction cellulaire. Actuellement, la cristallographie aux rayons X est la méthode la plus efficace pour la détermination à haute résolution de la structure de protéines monomériques allant 40 et 100 kDa. Par contre, elle est limitée par la croissance de cristaux de bonne qualité, qui est problématique pour nombreuses cibles. La cryo-microscopie électronique (cryoME) permet la détermination structurale à résolution quasi-atomique de larges structures protéiques, de préférence symétrique et en solution. Cependant, les images de cryoME sont très bruitées, car une faible dose d'électrons est appliquée de manière à limiter les dommages d'irradiation. En moyennant des dizaines d'images correspondant à la même orientation moléculaire, le rapport signal sur bruit est amélioré. La combinaison des images moyennées de plusieurs orientations permet l'obtention d'une carte de densité électronique 3D de la molécule d'intérêt. Si la taille et la symétrie de la molécule diminuent, l'analyse cryoME devient de moins en moins précise, il est alors impossible d'analyser des protéines monomériques de taille inférieure à 100 kDa. Le but de ce travail a été de développer une nouvelle approche pour réduire cette limite de poids moléculaire. Elle consiste à fusionner la protéine d'intérêt (cible) à une matrice homo-oligomérique, générant une particule symétrique et de taille importante adaptée à l'analyse par cryoME. Dans cette thèse, nous avons cherché à tester et démontrer la faisabilité de cette approche de symétrisation en utilisant des protéines cibles de structure connue.Pour mettre en place notre étude pilote, nous avons choisi différentes combinaisons de cibles et de matrices connectées par des peptides de liaison (linker) de longueur différentes. Nous avons caractérisé les fusions exprimées en bactéries par microscopie électronique après coloration négative et par plusieurs techniques biophysiques. Grace à ces techniques, nous avons trouvé que la meilleure combinaison est la fusion entre la protéine matrice glutamine synthétase (GS), un 12-mer de symétrie D6 et la cible maltose binding protein (Mbp), connectées par un linker contenant trois alanines, que nous avons appelée « Mag ». En jouant sur la longueur du linker nous avons ensuite sélectionné la fusion la plus compacte pour l'analyse cryoME: MagΔ5. Nous avons obtenu la carte cryoME à 10 Å de MagΔ5, qui présente une bonne corrélation avec les modèles atomiques de Mbp et GS. Plus particulièrement, le site catalytique et quelques hélices α sont identifiables. Ces résultats sont confirmés par l'étude cristallographique que nous avons conduite sur MagΔ5. L'ensemble de ce travail souligne que la présence d'une grande interface d'interactions cible-matrice stabilise la fusion et améliore la résolution en cryoME. Pour la symétrisation d'une cible inconnue, nous envisageons la même procédure expérimentale que celle développée pour MagΔ5. La matrice et le linker les plus adaptés devront être identifiés en utilisant les mêmes méthodes biophysiques.En conclusion, ce travail établit la preuve de concept que la méthode de symétrisation des protéines permet la détermination de la structure de protéines de poids moléculaire inférieur à 100 kDa par cryoME. Cette méthode a le potentiel d'être un nouvel outil prometteur, qui faciliterait l'analyse de cibles résistantes à l'analyse structurale conventionnelle. / Structural determination of proteins at atomic level resolution is crucial for unravelling their function. X-ray crystallography has successfully been used to determine macromolecular structures with sizes ranging from kDa to MDa, and currently remains the most efficient method for the high-resolution structure determination of monomeric proteins within the 40-100 kDa range. However, this method is limited by the ability to grow well diffracting crystals, which is problematic for several targets, such as membrane proteins. Single particle cryo electron microscopy (cryoEM) allows near atomic (3-4Å) resolution structural determination of large, preferably symmetric, assemblies in solution. Biological molecules scatter electrons weakly and, to avoid radiation damage, only low electron doses can be used during imaging. Consequently, raw cryoEM images are extremely noisy. However, averaging many molecular images aligned in the same orientation permits one to increase the signal-to-noise ratio, ultimately allowing the achievement of a 3D density map of the molecule of interest. Nevertheless, as the molecular size and degree of symmetry decrease, the individual images loose adequate features for accurate alignment. Currently, cryoEM analysis is practically impossible for monomeric proteins below ~100 kDa in mass. We propose to circumvent this obstacle by fusing such monomeric target proteins to a homo-oligomeric protein (template), thereby generating a self-assembling particle whose large size and symmetry should facilitate cryoEM analysis. In the present thesis we seek to test and demonstrate the feasibility of this ‘protein symmetrization' approach and to evaluate its usefulness for protein structure determination. To set up the pilot study we combined selected targets of known structure with two templates: Glutamine Synthetase (GS), a 12-mer with D6 symmetry and a helical N-terminus, and the E2 subunit of the pyruvate dehydrogenase complex, a 60-mer with icosahedral symmetry and an unstructured N-terminus. After recombinant production in E.coli we identified by negative stain EM a promising dodecameric chimera for structural analysis, comprising maltose binding protein (Mbp) connected to GS by a tri-alanine linker (denoted “Mag”). In order to optimize sample homogeneity we produced a panel of Mag deletion constructs by sequentially truncating the 17 residues between the Mbp and GS domains. A combination of biophysical techniques (thermal shift assay, dynamic light scattering, size exclusion chromatography) and negative stain EM allowed us to select the best candidate for cryoEM analysis, MagΔ5. By enforcing D6 symmetry we obtained a cryoEM map with a resolution of 10Å (FSC 0.5 criterion). The density of the symmetrized 40 kDa Mbp presents shape and features corresponding to the known atomic structure. In particular, the catalytic pocket and specific α-helical elements are distinguishable. The cryoEM map is additionally validated by a 7Å crystal structure of the MagΔ5 oligomer. The presence of a continuous helical connection between target (Mbp) and template (GS) likely contributed to the conformational homogeneity of MagΔ5. Moreover, comparing MagΔ5 with other chimeras studied in this work suggests that a large buried surface area and favorable interactions between the target and template limit the flexibility of the chimera and improve its resolution by cryoEM. For the symmetrization of a target of unknown structure, we envisage proceeding by a trial and error approach by fusing it to a panel of templates with helical termini and different surface properties, and subsequently selecting the best ones using biophysical assays. In conclusion, the present work establishes the proof-of-concept that protein symmetrization can be used for the structure determination of monomeric proteins below 100 kDa by cryoEM, thereby providing a promising new tool for analyzing targets resistant to conventional structural analysis.

<p>The goal of this thesis is to establish generalized isoperimetric inequalities on first exit time distributions as well as expectations of L\'evy processes.</p> <p>Firstly, we prove inequalities on first exit time distributions in the case that the L\'evy process is an $\alpha$-stable symmetric process $A_t$ on $\R^d$, $\alpha\in(0,2]$. Given $A_t$ and a bounded domain $D\subset\R^d$, we present a proof, based on the classical Brascamp-Lieb-Luttinger inequalities for multiple integrals, that the distribution of the first exit time of $A_t$ from $D$ increases under Steiner symmetrization. Further, it is shown that when a sequence of domains $\{D_m\}$ each contained in a ball $B\subset\R^d$ and satisfying the $\varepsilon$-cone property converges to a domain $D'$ with respect to the Hausdorff metric, the sequence of distributions of first exit times for Brownian motion from  $D_m$  converges to the distribution of the exit time of Brownian motion from $D'$. The second set of results in this thesis extends the theorems from \cite{BanMen} by proving generalized isoperimetric inequalities on expectations of L\'evy processes in the case of Steiner symmetrization.% using the Brascamp-Lieb-Luttinger inequalities used above. </p> <p>These results will then be used to establish inequalities involving distributions of first exit times of $\alpha$-stable symmetric processes $A_t$ from triangles and quadrilaterals. The primary application of these inequalities is verifying a conjecture from Ba\~nuelos for these planar domains. This extends a classical result of P\'olya and Szeg\"o to the fractional Laplacian with Dirichlet boundary conditions.</p>

In this thesis a set of recently proposed multigravity theories is analysed. In the special case of bimetric gravity, the theory has been conclusively shown to be ghost-free. On the other hand, for multigravity theories in general, the ghost-issue has not been settled conclusively. Motivated by this fact, the main object of this thesis is to clarify what has been proven so far and what issues that still needs to be addressed. We also provide new calculations and results pointing in the direction that the multigravity theories must be restricted to a set of bimetric Hassan-Rosen couplings in a tree-type structure in order to be consistent. In particular, we prove that for a multivielbein theory of <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathscr%7BN%7D" /> interacting vielbeins, the Lorentz equations of motion is a set of <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?(%5Cmathscr%7BN%7D-1)" /> Deser-van Nieuwenhuizen conditions if and only if the theory consists of bimetric Hassan-Rosen couplings in a tree-type structure.

In the modern era of high and infinite dimensional data, classical statistical methodology is often rendered inefficient and ineffective when confronted with such big data problems as arise in genomics, medical imaging, speech analysis, and many other areas of research. Many problems manifest when the practitioner is required to take into account the covariance structure of the data during his or her analysis, which takes on the form of either a high dimensional low rank matrix or a finite dimensional representation of an infinite dimensional operator acting on some underlying function space. Thus, novel methodology is required to estimate, analyze, and make inferences concerning such covariances. In this manuscript, we propose using tools from the concentration of measure literature–a theory that arose in the latter half of the 20th century from connections between geometry, probability, and functional analysis–to construct rigorous descriptive and inferential statistical methodology for covariance matrices and operators. A variety of concentration inequalities are considered, which allow for the construction of nonasymptotic dimension-free confidence sets for the unknown matrices and operators. Given such confidence sets a wide range of estimation and inferential procedures can be and are subsequently developed. For high dimensional data, we propose a method to search a concentration in- equality based confidence set using a binary search algorithm for the estimation of large sparse covariance matrices. Both sub-Gaussian and sub-exponential concentration inequalities are considered and applied to both simulated data and to a set of gene expression data from a study of small round blue-cell tumours. For infinite dimensional data, which is also referred to as functional data, we use a celebrated result, Talagrand’s concentration inequality, in the Banach space setting to construct confidence sets for covariance operators. From these confidence sets, three different inferential techniques emerge: the first is a k-sample test for equality of covariance operator; the second is a functional data classifier, which makes its decisions based on the covariance structure of the data; the third is a functional data clustering algorithm, which incorporates the concentration inequality based confidence sets into the framework of an expectation-maximization algorithm. These techniques are applied to simulated data and to speech samples from a set of spoken phoneme data. Lastly, we take a closer look at a key tool used in the construction of concentration based confidence sets: Rademacher symmetrization. The symmetrization inequality, which arises in the probability in Banach spaces literature, is shown to be connected with optimal transport theory and specifically the Wasserstein distance. This insight is used to improve the symmetrization inequality resulting in tighter concentration bounds to be used in the construction of nonasymptotic confidence sets. A variety of other applications are considered including tests for data symmetry and tightening inequalities in Banach spaces. An R package for inference on covariance operators is briefly discussed in an appendix chapter.