Dissertations / Theses on the topic 'Regularity'

Graphs with high regularity and transitivity conditions are studied. The first graphs considered are graphs where each vertex has an intersection array (possibly differing from that of other vertices). These graphs are called distance-regularised and are shown to be distance-regular or bipartite with each bipartition having the same intersection array. The latter graphs are called distance-biregular. This leads to the study of distance-biregular graphs. The derived graphs of a distance-biregular graph are shown to be distance-regular and the notion of feasibility for a distance-regular graph is extended to the biregular case. The study of the intersection arrays of distance-biregular graphs is concluded with a bound on the diameter in terms of the girth and valencies. Special classes of distance-biregular graphs are also studied. Distance-biregular graphs with 2-valent vertices are shown to be the subdivision graphs of cages. Distance-biregular graphs with one derived graph complete and the other strongly-regular are characterised according to the minimum eigenvalue of the strongly-regular graph. Distance-biregular graphs with prescribed derived graph are classified in cases where the derived graph is from some classes of classical distance-regular graphs. A graph theoretic proof of part of the Praeger, Saxl and Yokoyama theorem is given. Finally imprimitivity in distance-biregular graphs is studied and the Praeger, Saxl and Yokoyama theorem is used to show that primitive non-regular distance-bitransitive graphs have almost simple automorphism groups. Many examples of distance-biregular and distance-bitransitive graphs are given.

Cette thèse s’intéresse principalement à l’existence et à la régularité desensembles minimaux. On commence par montrer, dans le chapitre 3, que le problème de Plateau étudié par Reifenberg admet au moins une solution. C’est-à-dire que, si l’onse donne un ensemble compact B⊂R^n et un sous-groupe L du groupe d’homologie de Čech H_(d-1) (B;G) de dimension (d-1) sur un groupe abelien G, on montre qu’il existe un ensemble compact E⊃B tel que L est contenu dans le noyau de l’homomorphisme H_(d-1) (B;G)→H_(d-1) (E;G) induit par l’application d’inclusion B→E, et pour lequel la mesure de Hausdorff H^d (E∖B) est minimale (sous ces contraintes). Ensuite, on montre au chapitre 4, que pour tout ensemble presque minimal glissant E de dimension 2, dans un domaine régulier Σ ressemblant localement à un demi espace, associé à la frontière glissante ∂Σ, et tel que E⊃∂Σ, il se trouve qu’à la frontière E est localement équivalent, par un homéomorphisme biHöldérien qui préserve la frontière, à un cône minimal glissant contenu dans un demi plan Ω, avec frontière glissante ∂Ω. De plus les seuls cônes minimaux possibles dans ce cas sont ∂Ω seul, ou son union avec un cône de type P_+ ou Y_+
This thesis focuses on the existence and regularity of minimal sets. First we show, in Chapter 3, that there exists (at least) a minimizerfor Reifenberg Plateau problems. That is, Given a compact set B⊂R^n, and a subgroup L of the Čech homology group H_(d-1) (B;G) of dimension (d-1)over an abelian group G, we will show that there exists a compact set E⊃B such that L is contained in the kernel of the homomorphism H_(d-1) (B;G)→H_(d-1) (E;G) induced by the natural inclusion map B→E, and such that the Hausdorff measure H^d (E∖B) is minimal under these constraints. Next we will show, in Chapter 4, that if E is a sliding almost minimal set of dimension 2, in a smooth domain Σ that looks locally like a half space, and with sliding boundary , and if in addition E⊃∂Σ, then, near every point of the boundary ∂Σ, E is locally biHölder equivalent to a sliding minimal cone (in a half space Ω, and with sliding boundary ∂Ω). In addition the only possible sliding minimal cones in this case are ∂Ω or the union of ∂Ω with a cone of type P_+ or Y_+

In nowadays nanometer technology nodes, the semiconductor industry has to deal with the new challenges associated to technology scaling. On one hand, process developers face increasing manufacturing cost and variability, but also decreasing manufacturing yield. On the other hand, circuit designers and electronic design automation (EDA) developers have to reduce design turnaround time and provide the tools to cope with increasing design complexity and reduce the time-to-market. In this scenario, closer collaboration between all the actors involved is required. New approaches considering both design and manufacturing need to be explored. These are the so called design for manufacturability (DFM) techniques. A DFM trend that is becoming dominant is to make circuit layouts more regular and repetitive. The regular layout fabrics are based on the configuration of a simplied mask set, therefore reducing the manufacturing cost. Moreover, a reduced number of layout patterns is used, allowing better process variability control and optimization. Hence, regularity reduces layout complexity and therefore design complexity, allowing faster time-to-market. In this thesis, we explore forcing maximum layout regularity focusing on future technology nodes, with increasing design and manufacturability issues, where we expect layout regularity to be mandatory. With this objective, we have developed a new regular layout fabric called Via-Configurable Transistor Array (VCTA). The physical design is fully explained involving layout and geometrical considerations for transistors and interconnects. Initially, VCTA layouts developed manually have been evaluated in terms of manufacturability, but also in terms of area, energy and delay. For digital design, 32-bit binary adders designed with VCTA have been compared to standard cell layouts. For analog design, a delay-locked loop design using VCTA has been compared to its full custom version. We have also developed a physical synthesis tool that allows us to obtain VCTA circuit layouts in an automated way. Developing our own automation tool lets us controlling all the decisions made during the physical design flow to ensure that maximum layout regularity is respected. In this case the work is based on several algorithms, for instance for routing, that we have oriented to the area optimization of the layouts. Finally, in order to demonstrate the benefits of layout regularity, we have proposed a new layout regularity metric called Fixed Origin Corner Square Inspection (FOCSI). It is based on the geometrical inspection of the patterns in the layouts and it allows designers to compare regularity of designs but also how their regularity will impact their manufacturability. The FOCSI layout analysis tool can be used to optimize manufacturability.

Aim: Regular repeating patterns are prominent features in a visual scene. Here I consider whether regularity is an adaptable feature that produces a subsequent after-effect and whether a first- or second-order process mediates that after-effect. Method: Stimuli consisted of a 7 by 7 arrangement of elements on a baseline grid. The position of each element was randomly jittered from its baseline position by an amount that determined its degree of pattern irregularity. The elements of the pattern consisted of dark Gaussian blobs (GB), difference of Gaussians (DOG) or random binary patterns (RBP). Observers adapted for 60 seconds to a pair of patterns above and below fixation with a different degree of regularity, then adjusted the relative degree of regularity of two subsequently presented test patterns. The size of the after-effect at the point of subjective equality (PSE) was given by the baseline removed difference in regularity at the PSE or log ratio of the physical element jitter of the two test patterns at the PSE. Results: PSEs revealed that regularity is an adaptable feature that produces a unidirectional after-effect; specifically that adaptation only causes test patterns to appear less regular. The after-effect displayed transfer from GB adaptors to both DOG and RB test patterns and from DOG and (RBP) adaptors to GB patterns. Conclusion: Pattern regularity is an adaptable feature in vision, which produces a novel unidirectional after-effect I have termed Regularity After-Effect, or RAE. I propose second-order spatial-frequency channels as candidate mechanisms of regularity processing.
Objectif: Les motifs réguliers répétitifs sont des caractéristiques de premier plan dans la scène visuelle. Cette communication a comme objectif de découvrir si la régularité est une caractéristique adaptable du système visuel produisant un effet consécutif et si cet effet-consécutif est lié à un processus de premier- ou de second-ordre. Méthode: Les stimuli étaient constitués en un arrangement 7 par 7 éléments sur une grille. La position de chaque élément a été giguer au hasard à partir de sa position d'origine avec une valeur qui détermine son degré d'irrégularité. Les éléments qui constituent chaque grille pouvaient être des blobs de Gaussiennes (GB), des différence de Gaussiennes (DOG) ou de motif binaire aléatoire (RBP). Les participants ont été adaptés pour 60 secondes à une paire de motifs placée de part et d'autre d'un point de fixation ou chaque motif avait un degré différent de régularité. Les participants devaient ajuster le degré relatif de régularité de deux motifs présentés après. La taille de l'effet-consécutif est obtenue par la différence de régularité au point subjectif d'égalité soustrait à la régularité mesurée entre les deux motifs test ou par le logarithme du ratio de la différence de régularité entre les deux motifs test au point subjectif d'égalité. Résultats: Les point-subjectif d'égalité mesurée ont montrées que la régularité est une caractéristique adaptable qui produit un effet-consécutif unidirectionnel, plus précisément que les motifs sont perçus comme plus irréguliers après adaptation. On a observe un transfert à partir des stimuli d'élément GB a des motifs de test RBP et DOG et un transfert a partir des stimuli DOG et RBP vers des test de GB. Conclusion: La régularité est un élément du système visuel adaptable, produisant effet-consécutif unidirectionnel nouveau que appelé l'effet consécutif de la régularité. Je propose les canaux de fréquence-spatial de second-ordre comme mécanisme candidat au traitement de la régularité.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 171-179).
We extend various fundamental combinatorial theorems and techniques from the dense setting to the sparse setting. First, we consider Szemerédi regularity lemma, a fundamental tool in extremal combinatorics. The regularity method, in its original form, is effective only for dense graphs. It has been a long standing problem to extend the regularity method to sparse graphs. We solve this problem by proving a so-called "counting lemma," thereby allowing us to apply the regularity method to relatively dense subgraphs of sparse pseudorandom graphs. Next, by extending these ideas to hypergraphs, we obtain a simplification and extension of the key technical ingredient in the proof of the celebrated Green-Tao theorem, which states that there are arbitrarily long arithmetic progressions in the primes. The key step, known as a relative Szemerédi theorem, says that any positive proportion subset of a pseudorandom set of integers contains long arithmetic progressions. We give a simple proof of a strengthening of the relative Szemerédi theorem, showing that a much weaker pseudorandomness condition is sufficient. Finally, we give a short simple proof of a multidimensional Szemerédi theorem in the primes, which states that any positive proportion subset of Pd (where P denotes the primes) contains constellations of any given shape. This has been conjectured by Tao and recently proved by Cook, Magyar, and Titichetrakun and independently by Tao and Ziegler.
by Yufei Zhao.
Ph. D.

Thesis (MSc (Mathematical Sciences. Physical and Mathematical Analysis))--University of Stellenbosch, 2006.
This work studies the regularity (or smoothness) of continuous finitely supported refinable functions which are mainly encountered in multiresolution analysis, iterative interpolation processes, signal analysis, etc. Here, we present various kinds of sufficient conditions on a given mask to guarantee the regularity class of the corresponding refinable function. First, we introduce and analyze the cardinal B-splines Nm, m ∈ N. In particular, we show that these functions are refinable and belong to the smoothness class Cm−2(R). As a generalization of the cardinal B-splines, we proceed to discuss refinable functions with positive mask coefficients. A standard result on the existence of a refinable function in the case of positive masks is quoted. Following [13], we extend the regularity result in [25], and we provide an example which illustrates the fact that the associated symbol to a given positive mask need not be a Hurwitz polynomial for its corresponding refinable function to be in a specified smoothness class. Furthermore, we apply our regularity result to an integral equation. An important tool for our work is Fourier analysis, from which we state some standard results and give the proof of a non-standard result. Next, we study the H¨older regularity of refinable functions, whose associated mask coefficients are not necessarily positive, by estimating the rate of decay of their Fourier transforms. After showing the embedding of certain Sobolev spaces into a H¨older regularity space, we proceed to discuss sufficient conditions for a given refinable function to be in such a H¨older space. We specifically express the minimum H¨older regularity of refinable functions as a function of the spectral radius of an associated transfer operator acting on a finite dimensional space of trigonometric polynomials. We apply our Fourier-based regularity results to the Daubechies and Dubuc-Deslauriers refinable functions, as well as to a one-parameter family of refinable functions, and then compare our regularity estimates with those obtained by means of a subdivision-based result from [28]. Moreover, we provide graphical examples to illustrate the theory developed.

This thesis consists of three scientific papers, devoted to the regu-larity theory of free boundary problems. We use iteration arguments to derive the optimal regularity in the optimal switching problem, and to analyse the regularity of the free boundary in the biharmonic obstacle problem and in the double obstacle problem.In Paper A, we study the interior regularity of the solution to the optimal switching problem. We derive the optimal C1,1-regularity of the minimal solution under the assumption that the zero loop set is the closure of its interior.In Paper B, assuming that the solution to the biharmonic obstacle problem with a zero obstacle is suÿciently close-to the one-dimensional solution (xn)3+, we derive the C1,-regularity of the free boundary, under an additional assumption that the noncoincidence set is an NTA-domain.In Paper C we study the two-dimensional double obstacle problem with polynomial obstacles p1 p2, and observe that there is a new type of blow-ups that we call double-cone solutions. We investigate the existence of double-cone solutions depending on the coeÿcients of p1, p2, and show that if the solution to the double obstacle problem with obstacles p1 = −|x|2 and p2 = |x|2 is close to a double-cone solution, then the free boundary is a union of four C1,-graphs, pairwise crossing at the origin.

QC 20161103

Mesh quality plays an important role in improving the accuracy of the numerical simulation. There are different quality metrics for specific numerical cases. A regular mesh consisting of the equilateral triangles is one of them and is expected to improve the error performance. In this study, Engwirda’s frontal-Delaunay scheme and Marcum’s advancing front local reconnection scheme are described along with the conventional Delaunay triangulation. They are shown to improve the mesh regularity effectively. Even though several numerical test cases show that more regular meshes barely improve the error performance, the time cost in the solver of regular meshes is smaller than the Delaunay mesh. The time cost decrease in the solver pays off the additional cost in the mesh generation stage. For simple test cases, more regular meshes obtain lower errors than conventional Delaunay meshes with similar time costs. For more complicated cases, the improvement in errors is small but regular meshes can save time, especially for a high order solver. Generally speaking, a regular mesh does not improve the error performance as much as we expect, but it is worth generating.
Applied Science, Faculty of
Mechanical Engineering, Department of
Graduate

In this work local behavior for solutions to the inhomogeneous p-Laplace in divergence form and its parabolic version are studied. It is parabolic and non-linear generalization of the Calderon-Zygmund theory for the Laplace operator. I.e. the borderline case BMO is studied. The two main results are local BMO and Hoelder estimates for the inhomogenious p-Laplace and the parabolic p-Laplace system. An adaption of some estimates to fluid mechanics, namely on the p-Stokes equation are also proven. The p-Stokes system is a very important physical model for so-called non Newtonian fluids (e.g. blood). For this system BMO and Hoelder estimates are proven in the stationary 2-dimensional case.

Our purpose is to define, and develop a regularity theory for, Intrinsic Minimising Fractional Harmonic Maps from Euclidean space into smooth compact Riemannian manifolds for fractional powers strictly between zero and one. Our aims are motivated by the theory for Intrinsic Semi-Harmonic Maps, corresponding to the power one-half, developed by Moser. Our definition and methodology are based on an extension method used for the analysis of real valued fractional harmonic functions. We define and derive regularity properties of Fractional Harmonic Maps by regarding their domain as part of the boundary of a half-space, equipped with a Riemannian metric which degenerates or becomes singular on the boundary, and considering the regularity of their extensions to this half-space. We show that Fractional Harmonic Maps, and their first order derivatives, are locally Hölder continuous away from a set with Hausdorff dimension depending on the dimension of the domain and the fractional power in question. We achieve this by establishing the corresponding partial regularity of extensions of Fractional Harmonic Maps which minimise the Dirichlet energy on the half-space. To prove local Hölder continuity, we develop several results in the spirit of the regularity theory for harmonic maps. We combine a monotonicity formula with the construction of comparison maps, scaling in the Poincaré inequality and results from the theory of harmonic maps, to prove energy decay sufficient for the application of a modified decay lemma of Morrey. Using the Hölder continuity of minimisers, we prove a bound for the essential supremum of their gradient. Then we consider the derivatives in directions tangential to the boundary of the half-space; we establish the existence of their gradients using difference quotients. A Caccioppoli-type inequality and scaling in the Poincaré inequality then imply decay estimates sufficient for the application of the modified decay lemma to these derivatives.

Cylindrical algebraic decomposition is a powerful algorithmic technique in semi-algebraic geometry. Nevertheless, there is a disparity between what algorithms output and what the abstract definition of a cylindrical algebraic decomposition allows. Some work has been done in trying to understand what the ideal class of cylindrical algebraic decom- positions should be — especially from a topological point of view. We prove a special case of a conjecture proposed by Lazard in [22]; the conjecture relates a special class of cylindrical algebraic decompositions to regular cell complexes. Moreover, we study the properties that define this special class of cell decompositions, as well as their implications for the actual topology of the cells that make up the cell decompositions.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2017.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 123-127).
In this thesis, we analyze the regularity method pioneered by Szemerédi, and also discuss one of its prevalent applications, the removal lemma. First, we prove a new lower bound on the number of parts required in a version of Szemerédi's regularity lemma, determining the order of the tower height in that version up to a constant factor. This addresses a question of Gowers. Next, we turn to algorithms. We give a fast algorithmic Frieze-Kannan (weak) regularity lemma that improves on previous running times. We use this to give a substantially faster deterministic approximation algorithm for counting subgraphs. Previously, only an exponential dependence of the running time on the error parameter was known; we improve it to a polynomial dependence. We also revisit the problem of finding an algorithmic regularity lemma, giving approximation algorithms for some co-NP-complete problems. We show how to use the Frieze-Kannan regularity lemma to approximate the regularity of a pair of vertex sets. We also show how to quickly find, for each [epsilon]' > [epsilon], an [epsilon]'-regular partition with k parts if there exists an [epsilon]-regular partition with k parts. After studying algorithms, we turn to the arithmetic setting. Green proved an arithmetic regularity lemma, and used it to prove an arithmetic removal lemma. The bounds obtained, however, were tower-type, and Green posed the problem of improving the quantitative bounds on the arithmetic triangle removal lemma, and, in particular, asked whether a polynomial bound holds. The previous best known bound was tower-type with a logarithmic tower height. We solve Green's problem, proving an essentially tight bound for Green's arithmetic triangle removal lemma in Fn/p. Finally, we give a new proof of a regularity lemma for permutations, improving the previous tower-type bound on the number of parts to an exponential bound.
by László Miklós Lovász.
Ph. D.

We investigate solutions of the coupled Diral Klein-Gordon Equations in one and three space dimensions. Through analysis of the Fourier representations of the solutions to these equations, we introduce the ‘Null Structure’ as developed by Klainerman and Machedon. This structure allows us to prove the necessary estimates, both fixed time and bilinear space-time, that allow us to show existence of solutions of these equations with initial data of lower regularity than previously required. We also study global existence for a two dimensional wave equation with a critical non-linearity.

CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior
Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico
In this work we establish local regularity estimates for at solutions to non-convex fully nonlinear elliptic equations and we study cavitation type equations modeled within coef- icients bounded and measurable.
Neste trabalho estabelecemos estimativas de regularidade local para soluÃÃes "flat" de equaÃÃes elÃpticas totalmente nÃo-lineares nÃo-convexas e estudamos equations do tipo cavidade com coeficientes meramente mensurÃveis.

In this thesis we provide regularity results for convex and semiconvex variational problems which are of linear growth and depend on the symmetric rather than the full gradient. By the non-availability of Korn's Inequality (known as Ornstein's Non-Inequality), usual approaches need to be modified in order to obtain higher regularity of generalised minima.

Doutoramento em Matemática
In this thesis we consider Wiener-Hopf-Hankel operators with Fourier symbols in the class of almost periodic, semi-almost periodic and piecewise almost periodic functions. In the first place, we consider Wiener-Hopf-Hankel operators acting between L2 Lebesgue spaces with possibly different Fourier matrix symbols in the Wiener-Hopf and in the Hankel operators. In the second place, we consider these operators with equal Fourier symbols and acting between weighted Lebesgue spaces Lp(R;w), where 1 < p < 1 and w belongs to a subclass of Muckenhoupt weights. In addition, singular integral operators with Carleman shift and almost periodic coefficients are also object of study. The main purpose of this thesis is to obtain regularity properties characterizations of those classes of operators. By regularity properties we mean those that depend on the kernel and cokernel of the operator. The main techniques used are the equivalence relations between operators and the factorization theory. An invertibility characterization for the Wiener-Hopf-Hankel operators with symbols belonging to the Wiener subclass of almost periodic functions APW is obtained, assuming that a particular matrix function admits a numerical range bounded away from zero and based on the values of a certain mean motion. For Wiener-Hopf-Hankel operators acting between L2-spaces and with possibly different AP symbols, criteria for the semi-Fredholm property and for one-sided and both-sided invertibility are obtained and the inverses for all possible cases are exhibited. For such results, a new type of AP factorization is introduced. Singular integral operators with Carleman shift and scalar almost periodic coefficients are also studied. Considering an auxiliar and simpler operator, and using appropriate factorizations, the dimensions of the kernels and cokernels of those operators are obtained. For Wiener-Hopf-Hankel operators with (possibly different) SAP and PAP matrix symbols and acting between L2-spaces, criteria for the Fredholm property are presented as well as the sum of the Fredholm indices of the Wiener-Hopf plus Hankel and Wiener-Hopf minus Hankel operators. By studying dependencies between different matrix Fourier symbols of Wiener-Hopf plus Hankel operators acting between L2-spaces, results about the kernel and cokernel of those operators are derived. For Wiener-Hopf-Hankel operators acting between weighted Lebesgue spaces, Lp(R;w), a study is made considering equal scalar Fourier symbols in the Wiener-Hopf and in the Hankel operators and belonging to the classes of APp;w, SAPp;w and PAPp;w. It is obtained an invertibility characterization for Wiener-Hopf plus Hankel operators with APp;w symbols. In the cases for which the Fourier symbols of the operators belong to SAPp;w and PAPp;w, it is obtained semi-Fredholm criteria for Wiener-Hopf-Hankel operators as well as formulas for the Fredholm indices of those operators.
Nesta tese consideramos operadores de Wiener-Hopf-Hankel com símbolos de Fourier nas classes das funções quase-periódicas, semi-quase periódicas e quase periódicas por troços. Começamos por considerar operadores de Wiener-Hopf-Hankel actuando entre espaços de Lebesgue L2 com símbolos matriciais de Fourier possivelmente diferentes nos operadores de Wiener- Hopf e de Hankel. Seguidamente, consideramos estes operadores com símbolos de Fourier iguais actuando entre espaços de Lebesgue com pesos Lp(R;w), onde 1 < p < 1 e w pertence a uma subclasse de pesos de Muckenhoupt. Adicionalmente, são também objecto de estudo operadores singulares integrais com deslocamento de Carleman e coeficientes quaseperiódicos. O objectivo principal desta tese é obter caracterizações para tais classes de operadores no que refere às suas propriedades de regularidade. Por propriedades de regularidade nós designamos aquelas propriedades que dependem do núcleo e do co-núcleo do operador. As principais técnicas usadas são as relações de equivalência entre operadores e a teoria da factorização. Uma caracterização da invertibilidade de operadores de Wiener-Hopf-Hankel com símbolos pertencentes à subclasse de Wiener de funções quaseperiódicas APW é obtida, assumindo que uma particular função matricial admite um contradomínio numérico limitado fora de zero e baseando-nos nos valores uma certa média de deslocamento. Para os operadores de Wiener-Hopf-Hankel actuando entre espaços de Lebesgue L2 e com símbolos AP possivelmente diferentes, critérios para a propriedade de semi-Fredholm e para a invertibilidade lateral e bi-lateral são obtidos e inversos para todos os casos possíveis são apresentados. Com vista a tais resultados, um novo tipo de factorização AP é introduzido. Operadores singulares integrais com deslocamento de Carleman e com coeficientes escalares quase-periódicos são também estudados. Considerando um operador auxiliar mais simples e usando factorizações apropriadas, as dimensões dos núcleos e dos co-núcleos destes operadores são obtidas. Para operadores de Wiener-Hopf-Hankel com símbolos matriciais SAP e PAP (possivelmente diferentes) actuando entre espaços de Lebesgue L2, critérios para a propriedade de Fredholm são apresentados tal como a soma dos índices de Fredholm dos operadores de Wiener-Hopf mais Hankel e Wiener-Hopf menos Hankel. Estudando dependências entre diferentes símbolos matriciais de Fourier dos operadores de Wiener-Hopf mais Hankel actuando entre espaços de Lebesgue L2, conclusões são obtidas acerca do núcleo e do co-núcleo destes operadores. Para operadores de Wiener-Hopf-Hankel actuando entre espaços de Lebesgue com pesos, Lp(R;w), é feito um estudo considerando símbolos de Fourier escalares e iguais nos operadores de Wiener-Hopf e de Hankel e pertencentes às classes APp;w, SAPp;w e PAPp;w. É obtida uma caracterização da invertibilidade para operadores de Wiener-Hopf mais Hankel com símbolos APp;w. No caso em que os símbolos de Fourier dos operadores pertencem a SAPp;w e PAPp;w, são obtidos critérios de semi-Fredholm para os operadores de Wiener-Hopf-Hankel assim como fórmulas para os índices de Fredholm de tais operadores.

Cette thèse est dédiée a l''etude de certaines propriétés des équations d' évolutions non-autonomes $u'(t)+A(t)u(t)=f(t), u(0)=x.$ Il s'agit précisément de la propriété de la régularité maximale $L^p$: étant donnée $fin L^{p}(0,tau;H)$, montrer l'existence et unicité de la solution $u in W^{1,p}(0,tau;H)$. Ce problème a 'et'e intensivement étudie dans le cas autonome, i.e., $A(t)=A$ pour tout $t$. Dans le cas non-autonome, le problème a été considéré par J.L.Lions en 1960. Nous montrons divers résultats qui étendent tout ce qui est connu sur ce problème. On suppose ici que la famille des opérateurs $(mathcal{A}(t))_{tin [0,tau]}$ est associée à des formes quasi-coercives, non autonomes $(fra(t))_{t in [0,tau]}.$ Nous considérons également le problème de régularité maximale pour les d'ordre 2 (équations des ondes). Plusieurs exemples et applications sont considérés
This Thesis is devoted to certain properties of non-autonomous evolution equations $u'(t)+A(t)u(t)=f(t), u(0)=x.$ More precisely, we are interested in the maximal $L^p$-regularity: given $fin L^{p}(0,tau;H),$ prove existence and uniqueness of the solution $u in W^{1,p}(0,tau;H)$. This problem was intensively studied in the autonomous cas, i.e., $A(t)=A$ for all $t.$ In the non-autonomous cas, the problem was considered by J.L.Lions in 1960. We prove serval results which extend all previously known ones on this problem. Here we assume that the familly of the operators $(mathcal{A}(t))_{tin [0,tau]}$ is associated with quasi-coercive, non-autonomous forms $(fra(t))_{t in [0,tau]}.$ We also consider the problem of maximal regularity for second order equations (the wave equation). Serval examples and applications are given in this Thesis

In this thesis we investigate the properties of various Banach function algebras and uniform algebras. We are particularly interested in regularity of Banach function algebras and extensions of uniform algebras. The first chapter contains the background in normed algebras, Banach function algebras, and uniform algebras which will be required throughout the thesis. In the second chapter we investigate the classicalisation of certain compact subsets of the complex plane obtained by deleting a sequence of open disks from a closed disk. Sets obtained in this manner are called Swiss cheese sets. We give a new topological proof of the Feinstein-Heath classicalisation theorem along with similar results. We conclude the chapter with an application of the classicalisation results. The results in this chapter are joint with H. Yang. In the third chapter we study Banach function algebras of functions satisfying a generalised notion of differentiability. These algebras were first investigated by Bland and Feinstein as a way to describe the completion of certain normed algebras of complex-differentiable functions. We prove a new version of chain rule in this setting, generalising a result of Chaobankoh, and use this chain rule to give a new proof of the quotient rule. We also investigate naturality and homomorphisms between these algebras. In the fourth chapter we continue the study of the notion of differentiability from the third chapter. We investigate a new notion of quasianalyticity in this setting and prove an analogue of the classical Denjoy-Carleman theorem. We describe those functions which satisfy a notion of analyticity, and give an application of these results. In the fifth chapter we investigate various methods for constructing extensions of uniform algebras. We study the structure of Cole extensions, introduced by Cole and later investigated by Dawson, relative to certain projections. We also discuss a larger class of extensions, which we call generalised Cole extensions, originally introduced by Cole and Feinstein. In the final chapter we investigate extensions of derivations from uniform algebras. We prove that there exists a non-trivial uniform algebra such that every derivation extends with the same norm to every generalised Cole extension of that algebra. A non-trivial, weakly amenable uniform algebra satisfies this property. We also investigate a sequence of extensions of a derivation from the disk algebra.

PRAZERES, Disson Soares dos. Improved regularity estimates in nonlinear elliptic equations. 2015. 48 f. Tese (doutorado) - Universidade Federal do Ceará, Centro de Ciências, Programa de Pós-Graduação em Matemática, Fortaleza-Ce, 2015
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In this work we establish local regularity estimates for at solutions to non-convex fully nonlinear elliptic equations and we study cavitation type equations modeled within coef- cients bounded and measurable.
Neste trabalho estabelecemos estimativas de regularidade local para soluções "flat" de equações elípticas totalmente não-lineares não-convexas e estudamos equations do tipo cavidade com coeficientes meramente mensuráveis.

The aim of this thesis is to study the Castelnuovo-Mumford regularity of powers and products of ideals of polynomial rings. In general, it is known that the regularity of large enough powers of ideals are given by a linear function. Moreover, similar result holds for products of large enough powers of ideals which are generated in the same degrees. These are fascinating results, however, there are still missing mysteries on specics of the asymptotic linear functions. For example, the stabilizing index (the smallest power for which the regularity is a linear function) and the so called constant (the constant term in the linear function) is unknown in general. We compute these two missing parts for two deferent classes of ideals using two deferent methods.

In this work, we deal with the study of some free boundary problems governed by non-homogeneous equations. In particular, we are interested in the regularity of the free boundaries for solutions of one-phase problems associated with non-divergence elliptic operators with variable coefficients.

The main topic of the thesis is the study of Elliptic PDEs. It is divided into three parts: (I) integro-differential equations, (II) stable solutions to reaction-diffusion problems, and (III) weighted isoperimetric and Sobolev inequalities. Integro-differential equations arise naturally in the study of stochastic processes with jumps, and are used in Finance, Physics, or Ecology. The most canonical example of integro-differential operator is the fractional Laplacian (the infinitesimal generator of the radially symmetric stable process). In the first Part of the thesis we find and prove the Pohozaev identity for such operator. We also obtain boundary regularity results for general integro-differential operators, as explained next. In the classical case of the Laplacian, the Pohozaev identity applies to any solution of linear or semilinear problems in bounded domains, and is a very important tool in the study of elliptic PDEs. Before our work, a Pohozaev identity for the fractional Laplacian was not known. It was not even known which form should it have, if any. In this thesis we find and establish such identity. Quite surprisingly, it involves a local boundary term, even though the operator is nonlocal. The proof of the identity requires fine boundary regularity properties of solutions, that we also establish here. Our boundary regularity results apply to fully nonlinear integro-differential equations, but they improve the best known ones even for linear ones. Our work in Part II concerns the regularity of local minimizers to some elliptic equations, a classical problem in the Calculus of Variations. More precisely, we study the regularity of stable solutions to reaction-diffusion problems in bounded domains. It is a long standing open problem to prove that all stable solutions are bounded, and thus regular, in dimensions n<10. In dimensions n>=10 there are examples of singular stable solutions. The question is still open in dimensions 4El tema principal de la tesi és l'estudi d'EDPs el·líptiques. La tesi està dividida en tres parts: (I) equacions integro-diferencials, (II) solucions estables de problemes de reacció-difusió, i (III) desigualtats isoperimètriques i de Sobolev amb pesos. Les equacions integro-differencials apareixen de manera natural en l'estudi de processos estocàstics amb salts (processos de Lévy), i s'utilitzen per modelitzar problemes en Finances, Física, o Ecologia. L'exemple més canònic d'operador integro-diferencial és el Laplacià fraccionari (el generador infinitesimal d'un procés estable i radialment simètric). A la Part I de la tesi trobem i demostrem la identitat de Pohozaev per aquest operador. També obtenim resultats de regularitat a la vora per operadors integro-diferencials més generals, tal com expliquem a continuació. En el cas clàssic del Laplacià, la identitat de Pohozaev s'aplica a qualsevol solució de problemes lineals o semilineals en dominis acotats, i és una eina molt important en l'estudi d'EDPs el·líptiques. Abans del nostre treball, no es coneixia cap identitat de Pohozaev pel Laplacià fraccionari. Ni tan sols es sabia quina forma hauria de tenir, en cas que existís. En aquesta tesi trobem i demostrem aquesta identitat. Sorprenentment, la identitat involucra un terma de vora local, tot i que l'operador és no-local. La demostració de la identitat requereix conèixer el comportament precís de les solucions a la vora, cosa que també obtenim aquí. Els nostres resultats de regularitat a la vora s'apliquen a equacions integro-diferencials completament no-lineals, però milloren els resultats anteriors fins i tot per a equacions lineals. A la Part II estudiem la regularitat dels minimitzants locals d'algunes equacions el·líptiques, un problema clàssic del Càlcul de Variacions. En concret, estudiem la regularitat de les solucions estables a problemes de reacció-difusió en dominis acotats. És un problema obert des de fa molts anys demostrar que totes les solucions estables són acotades (i per tant regulars) en dimensions n<10. En dimensions n>=10 hi ha exemples de solucions estables singulars. La questió encara està oberta en dimensions 4