This paper deals with Lebesgue points and studies properties of the set of Lebesgue points for various classes of functions. We consider continuous functions, L1 functions and Sobolev functions. In the case of uniformly continuous functions and Hölder continuous functions we develop a characterization in terms of Lebesgue points. For Sobolev functions we study the dimension of the set of non-Lebesgue points.

Made available in DSpace on 2014-06-11T19:27:09Z (GMT). No. of bitstreams: 0 Previous issue date: 2013-07-26Bitstream added on 2014-06-13T19:34:55Z : No. of bitstreams: 1 santos_ln_me_rcla_parcial.pdf: 254454 bytes, checksum: e32c8bbf83212b9a080a05b6df96f529 (MD5) Bitstreams deleted on 2015-06-25T13:00:45Z: santos_ln_me_rcla_parcial.pdf,. Added 1 bitstream(s) on 2015-06-25T13:03:12Z : No. of bitstreams: 1 000719346_20160726.pdf: 224220 bytes, checksum: 33187ebbdf2a29afe4f365dc6ce932e7 (MD5) Bitstreams deleted on 2016-07-29T12:53:55Z: 000719346_20160726.pdf,. Added 1 bitstream(s) on 2016-07-29T12:54:49Z : No. of bitstreams: 1 000719346.pdf: 949531 bytes, checksum: f249fa8a2707372138d0d3be07ff83fd (MD5) / Este trabalho apresenta resultados importantes sobre a Teoria de Integração. Inicialmente é desenvolvida uma parte sobre Teoria da Medida, necessária para introduzir a integral de Lebesgue e suas propriedades. Também é apresentada a integral de Riemann-Stieltjes. Em seguida, são demonstrados resultados importantes sobre converg ência envolvendo as integrais de Lebesgue, resultados estes que não são válidos para integrais de Riemann. Para apresentar tais temas, usa-se mais fortemente as referências [1], [2], [3] e [4] / This study presents important results on Integration of Theory. The rst of all part is developed on Measure Theory which is necessary to introduce the Lebesgue integral and its properties and we introduce. It also shows the Riemann-Stieltjes integral. Important results are proved on convergence involving the integrals of Lebesgue, which are not valid for the Riemann integral. Im order to present these themes we strongly use the references [1], [2], [3] and [4]

Hattendorff's theorem on the zero means and uncorrelatedness of losses in disjoint time periods on a life insurance policy is derived for payment streams, discount functions and time periods that are all stochastic. Thiele's differential equation, describing the development of life insurance policy reserves over the contract period, is derived for stochastic payment streams generated by point processes with intensities. The development follows that by Norberg. In pursuit of these aims, the basic properties of Lebesgue-Stieltjes integration are spelled out in detail. An axiomatic approach to the discounting of payment streams is presented, and a characterization in terms of the integral of a discount function is derived, again following the development by Norberg. The required concepts and tools from the theory of continuous time stochastic processes, in particular point processes, are surveyed. / Dissertation (MSc (Actuarial Science))--University of Pretoria, 2007. / Insurance and Actuarial Science / unrestricted

This dissertation explores geometric measure theory; the first part explores a question posed by Paul Erdös -- Is there a number c > 0 such that if E is a Lebesgue measurable subset of the plane with λ²(E) (planar measure)> c, then E contains the vertices of a triangle with area equal to one? -- other related geometric questions that arise from the topic. In the second part, "we parametrize the theorems from general topology characterizing the continuous images and the homeomorphic images of the Cantor set, C" (abstract, para. 5).

abstract: I focus on algorithms that generate good sampling points for function approximation. In 1D, it is well known that polynomial interpolation using equispaced points is unstable. On the other hand, using Chebyshev nodes provides both stable and highly accurate points for polynomial interpolation. In higher dimensional complex regions, optimal sampling points are not known explicitly. This work presents robust algorithms that find good sampling points in complex regions for polynomial interpolation, least-squares, and radial basis function (RBF) methods. The quality of these nodes is measured using the Lebesgue constant. I will also consider optimal sampling for constrained optimization, used to solve PDEs, where boundary conditions must be imposed. Furthermore, I extend the scope of the problem to include finding near-optimal sampling points for high-order finite difference methods. These high-order finite difference methods can be implemented using either piecewise polynomials or RBFs. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2019

We consider the problem of optimizing the choice of interpolation nodes such that the interpolation error is minimized, given the constraint that none of the nodes may be placed inside a forbidden region. Restricting the problem to using one-dimensional polynomial interpolants, we explore different ways of quantifying the interpolation error; such as the integral of the absolute/squared difference between the interpolated function and the interpolant, or the Lebesgue constant, which compares the interpolant with the best possible approximating polynomial of a given degree. The interpolation error then serves as a cost function that we intend to minimize using gradient-based optimization algorithms. The results are compared with existing theory about the optimal choice of interpolation nodes in the absence of a forbidden region (mainly due to Chebyshev) and indicate that the Chebyshev points of the second kind are near-optimal as interpolation nodes for optimizing the Lebesgue constant, whereas placing the points as close as possible to the forbidden region seems optimal for minimizing the integral of the difference between the interpolated function and the interpolant. We conclude that the Chebyshev points of the second kind serve as a great choice of interpolation nodes, even with the constraint on the placement of the nodes explored in this paper, and that the interpolation nodes should be placed as close as possible to the forbidden region in order to minimize the interpolation error.

Cette thèse, comporte deux parties distinctes.Dans la première partie, on étudie l'existence et l'inexistence d'une inégalité qu'on a appelée l'inégalité de Hopf Uniforme (IHU), pour une équation linéaire de la forme Lv = f à coefficients bornés mesurables et sous les conditions de Dirichlet homogènes. L'IHU est une variante du principe de maximum, on l'a appliquée dans la preuve de la régularité W1;p 0 pour un problème semi-linéaire singulier : Lu = F(u) où les coefficients de L sont dans l'espace vmor (fonctions à oscillation moyenne évanescente) et F(u) est singulier en u = 0 F(0) = +∞. De plus, si les coefficients sont lipschitziens, on prouve que la régularité optimale du gradient de la solution u est bmor (fonctions à oscillation moyenne bornée i.e Grad u dans bmor).Dans la seconde partie, on s'intéresse à la régularité du système d'élasticité (équations stationnaires des ondes élastiques) avec une fonction source singulière au sens qu'elle n’est qu'intégrable par rapport à la fonction distance au bord du domaine. Via la dualité, nous montrons, selon ~f , que le problème admet une solution dite très faible dont le gradient n'est pas nécessairement intégrable sur tout le domaine mais uniquement localement. Nous déterminons aussi les fonctions vectorielles ~f pour lesquelles, ~u a son gradient intégrable sur tout l'espace de travail. / We discuss the existence and non existence of the so called Hopf uniform Inequality (variant of a maximum principle) for the linear equation Lv = f with measurable coefficients and under the homogeneous Dirichlet Boundary condition. Then we apply such inequality to prove the W1;p 0 -regularity of a semi linear problem Lu = F(u), singular at u = 0, with the coefficients of the main operator of L in the space of vanishing mean oscillation. Moreover, when those coefficients are Lipschitz, we show that the gradient of the solution is at most in the space of bounded mean oscillation : bmor. In the last part of this thesis, we are concerned with the linear easticity system (Stationnary equation of the waves elasticity). But, here the second terms varies with respect to the distance function until the boundary.Using the duality method, we study the regularity of the solution of the elasticity system for the data belonging to various weighted spaces.

Dataloggern DL141E möjliggör kontinuerlig loggnig av mätdata från sensorer på upp till 30k sampel/s, som vidare kan överföras till mobiltelefoner via trådlös kommunikation. Detta är dock för stora datamängder per tidsenhet för mobiltelefoner som är tekniskt begränsade. Därför önskas bara relevant mätdata för att reducerar denna onödiga datamängd. I denna studie föreslås ett tillvägagångssätt där enspecifik mindre samling av diskreta försampel loggas i tur och ordning. Varje samling signalbehandlas genom att ställas mot fördefinierade triggningskriterier för att trigga loggning av en stor uppsättning av sampel på bara intressanta signalavvikelser. Dessa triggningskriterier är en särskild nivåöverskridning och signalriktning i kombination med ett antal sampel i följd. Studien förser en granskning av hur signalberäkningsmetoden ”Lebesgue sampling” kan tillämpas med kriterierna för god träffsäkerhet och en skälig beräkningstid i mobiltelefoner. Detta beaktas med dataloggerns vanligaste signaltyper puls och ramp i en miljö där småbrus och transienter förekommer. Träffsäkerheten och beräkningsbördan beaktasför att bedöma Lebesgue metodens effektivitet och antal nödvändiga försampel per uppsättning. Implementeringen görs i Java Android plattform och integreras därefter i en digital triggningsmodul med Graphical User Interface (GUI). / With the data logger DL141E it’s possible to continuously log measurement data from sensors up to 30k samples per second, and then transferring them to a mobile phone with Bluetooth technology. But this is by far too much sample data in a small time for a mobile phone with technical limitations to receive. That’s why only relevant measurement data should be mass logged to reduce the unnecessary data amount. Int his study a new approach is proposed where a specific and smaller amount of discrete pre-samples are logged in sequence. Every set of pre-samples is processed by comparing them to the user pre-defined trigger criterias. Met criterias will trigger logging of a massive set of samples on basis of only interesting signal deviations. The following trigger criterias are used: a specific signal level to cross, a specific signal direction, and both of these in combination with an amount of consecutive samples. The study provides an examination on how the signal processing method ”Lebesgue sampling” can be applied with the above criterias to achieve a god accuracy with reasonable processing time on mobile phones. This is observed using sensors with the most common signal types ramp and pulse in an environment where small noises and transients occur. The accuracy and the processing load are taken into account when estimating the efficiency of Lebesgue method and when estimating how many pre-samples per set might be sufficient. The implementation is written in Java Android platform and then integrated into a digital triggering module with Graphical User Interface (GUI).

La méthode des régions-solutions a été développée par Frigon [7] en 2018 pour montrer l'existence de solutions à des équations différentielles ordinaires de premier ordre, dont le graphe d'une solution se trouve à l'intérieur d'une région-solution \(R \subset [0, T] \times \mathbb{R}^{N}\). Cette méthode est en particulier une généralisation des sous et sur-solutions et des tubes-solutions. On présente cette méthode et certains résultats d'existence qui en découlent. D'autre part, la dérivée de Stieltjes, communément appelée \(g\)-dérivée, est le fruit du travail de Pouso et Rodríguez [20] en 2014, permettant l'unification des équations différentielles classiques, des équations aux échelles de temps et des équations différentielles avec impulsions. Elle est en particulier liée au théorème fondamental du calcul pour l'intégrale de Lebesgue-Stieltjes. On présente la base de cette théorie dans un premier temps, puis la façon dont cette \(g\)-dérivée généralise d'autres types d'équations différentielles ou aux échelles de temps. On introduit en particulier la notion de \((g \times I_{\mathbb{R}^{N}})\)-différentiabilité et des résultats qui découlent de cette définition. On présente de plus une fonction exponentielle qui permet de résoudre les équations différentielles de Stieltjes linéaires, introduite par Frigon et Pouso [8]. Le but de ce mémoire est de généraliser la méthode des régions-solutions, dont la généralisation s'appellera \(g\)-région-solution, afin de montrer l'existence de solutions aux équations différentielles de Stieltjes. On présente plusieurs exemples de \(g\)-régions admissibles et de \(g\)-régions-solutions, puis des théorèmes d'existence se basant sur cette méthode. On donne de plus des exemples où on applique ces théorèmes. On termine ce mémoire en présentant deux applications des théorèmes d'existence à l'évolution d'une population de cerfs de Virginie ainsi qu'à l'évolution de la tension générée par une diode à effet tunnel résonnant (DTR) dirigée vers une diode laser (DL). / The method of solution-regions has been developed by Frigon [7] in 2018 to show the existence of solutions for first-order ordinary differential equations, where the graph of a solution is inside a solution-region \(R \subset [0, T] \times \mathbb{R}^{N}\). This method is in particular a generalization of the lower and upper solutions and of the solution-tubes. We show this method and some existence results which follow. On the other hand, the Stieltjes derivative, more commonly called \(g\)-derivative, is the fruit of the work of Pouso and Rodríguez [20] in 2014, which unifies classic differential equations, equations on time scales and differential equations with impulses. In particular, it leads to the fundamental theorem of calculus for the Lebesgue-Stieltjes integral. We start by showing the basis of this theory, and then the way this \(g\)-derivative generalizes other types of differential or time scale equations. We introduce in particular the \((g \times I_{\mathbb{R}^{N}})\)-differentiability and results that follow from this definition. Furthermore, we present an exponential function which solves linear Stieltjes differential equations. The goal of this thesis is to generalize the method of solution-regions, where the generalization will be called \(g\)-solution-region, to show the existence of solutions for Stieltjes differential equations. We present multiple examples of \(g\)-admissible regions and \(g\)-solution-regions, then we establish existence theorems based on this method. We also show examples where we apply these theorems. Finally, we end this thesis by showing two applications of the existence theorems to the evolution of a population of white-tailed deer and to the evolution of the voltage generated by a resonant tunneling diode (RTD) to a laser diode (LD).

Nós estudaremos o comportamento de difeomorfismos do círculo, tanto do ponto de vista combinatório quanto do ponto de vista topológico e da teoria da medida, seguindo os trabalhos de Michael Herman. A cada homeomorfismo do círculo podemos associar um número real positivo, denominado número de rotação. Mostraremos que existe um conjunto de números irracionais de medida de Lebesgue total na reta tal que, se f é um difeomorfismo do círculo de classe \'C POT. r \' que preserva a orientação, com r maior ou igual a 3 e com número de rotação nesse conjunto, então f é pelo menos \'C POT. r - 2\' -conjugada a uma translação irracional. Além disso, mostraremos que dado um caminho \'f IND. t\' de classe \'C POT. 1\' definido em um intervalo [a;b] no conjunto dos difeomorfismos do círculo de classe \'C POT. r\' que preservam a orientação, com r maior ou igual a 3, o conjunto dos parâmetros em que \'f IND. t\' é \'C POT. r - 2\' -conjugada a uma translação irracional tem medida de Lebesgue positiva, desde que os números de rotação em \'f IND. a\' e \'f IND. b\' sejam distintos / We will study the generic behavior of circle diffeomorphisms, in the combinatorial, topological and measure-theoretical sense, following the work of Michael Herman. To each order preserving homeomorphism of the circle we can associate a positive real number, called rotation number, which is invariant under conjugacy. We will show that there is a set of irrational numbers with full Lebesgue measure on R such that, if f is a circle diffeomorphism of class \'C POT. r\', with r greater or equal 3 and with rotation number in that set, then f is at least \'C POT. r - 2\' -conjugated to an irrational translation. Moreover, we will show that if ft is a \'C POT. 1\' -path defined on a interval [a;b] over the set of the circle diffeomorphisms orientation preserving, with r \'> or =\' 3, then the set of parameters where \'f IND. t\' is \'C POT. r - 2\' -conjugated to a irrational translation has positive Lebesgue measure, since the rotation numbers of \'f IND. a\' and \'f IND. b\' are distinct