Dissertations / Theses on the topic 'Functions of bounded variation'

The paper begins with a short survey of monotone functions. The functions of bounded variation are introduced and some basic properties of these functions are given. Finally the jump function of a function of bounded variation is defined.

This thesis is devoted to the study of different generalizations of the classical conception of a function of bounded variation. First, we study the functions of bounded p-variation introduced by Wiener in 1924. We obtain estimates of the total p-variation (1<p<∞) and other related functionals for a periodic function f in Lp([0,1]) in terms of its Lp-modulus of continuity ω(f;δ)p. These estimates are sharp for any rate of decay of ω(f;δ)p. Moreover, the constant coefficients in them depend on parameters in an optimal way. Inspired by these results, we consider the relationship between the Riesz type generalized variation vp,α(f) (1<p<∞, 0≤α≤1-1/p) and the modulus of p-continuity  ω1-1/p(f;δ). These functionals generate scales of spaces that connect the space of functions of bounded p-variation and the Sobolev space Wp1. We prove sharp estimates of vp,α(f) in terms of ω1-1/p(f;δ). In the same direction, we study relations between moduli of p-continuity and q-continuity for 1<p<q<∞. We prove an inequality that estimates ω1-1/p(f;δ) in terms of ω1-1/q(f;δ). The inequality is sharp for any order of decay of ω1-1/q(f;δ). Next, we study another generalization of bounded variation: the so-called bounded Λ-variation, introduced by Waterman in 1972. We investigate relations between the space ΛBV of functions of bounded Λ-variation, and classes of functions defined via integral smoothness properties. In particular, we obtain the necessary and sufficient condition for the embedding of the class Lip(α;p) into ΛBV. This solves a problem of Wang (2009). We consider also functions of two variables. Applying our one-dimensional result, we obtain sharp estimates of the Hardy-Vitali type p-variation of a bivariate function in terms of its mixed modulus of continuity in Lp([0,1]2). Further, we investigate Fubini-type properties of the space Hp(2) of functions of bounded Hardy-Vitali p-variation. This leads us to consider the symmetric mixed norm space Vp[Vp]sym of functions of bounded iterated p-variation. For p>1, we prove that Hp(2) is not embedded into Vp[Vp]sym, and that Vp[Vp]sym is not embedded into Hp(2). In other words, Fubini-type properties completely fail in the class of functions of bounded Hardy-Vitali type p-variation for p>1.
Baksidestext The classical concept of the total variation of a function has been extended in several directions. Such extensions find many applications in different areas of mathematics. Consequently, the study of notions of generalized bounded variation forms an important direction in the field of mathematical analysis. This thesis is devoted to the investigation of various properties of functions of generalized bounded variation.  In particular, we obtain the following results: sharp relations between spaces of generalized bounded variation and spaces of functions  defined by integral smoothness conditions  (e.g., Sobolev and Besov spaces); optimal properties of certain scales of function spaces of frac- tional smoothness generated by functionals of variational type; sharp embeddings within  the scale of spaces of functions of bounded p-variation; results concerning bivariate functions of bounded p-variation, in particular sharp estimates of total variation in terms of the mixed Lp-modulus of continuity, and Fubini-type properties.

We study properties of functions with bounded variation in Carnot-Carathéodory spaces. In Chapter 2 we prove their almost everywhere approximate differentiability and we examine their approximate discontinuity set and the decomposition of their distributional derivatives. Under an additional assumption on the space, called property R, we show that almost all approximate discontinuities are of jump type and we study a representation formula for the jump part of the derivative. In Chapter 3 we prove a rank-one theorem à la G. Alberti for the derivatives of vector-valued maps with bounded variation in a class of Carnot groups that includes all Heisenberg groups H^n with n ≥ 2. Some important tools for the proof are properties linking the horizontal derivatives of a real-valued function with bounded variation to its subgraph. In Chapter 4 we prove a compactness result for bounded sequences (u_j) of functions with bounded variation in metric spaces (X, d_j) where the space X is fixed, but the metric may vary with j. We also provide an application to Carnot-Carathéodory spaces. The results of Chapter 4 are fundamental for the proofs of some facts of Chapter 2.
Analizziamo alcune proprietà di funzioni a variazione limitata in spazi di Carnot-Carathéodory. Nel Capitolo 2 dimostriamo che esse sono approssimativamente differenziabili quasi ovunque, esaminiamo il loro insieme di discontinuità approssimata e la decomposizione della loro derivata distribuzionale. Assumendo un'ipotesi addizionale sullo spazio, che chiamiamo proprietà R, mostriamo che quasi tutti i punti di discontinuità approssimata sono di salto e studiamo una formula per la parte di salto della derivata. Nel Capitolo 3 dimostriamo un teorema di rango uno à la G. Alberti per la derivata distribuzionale di funzioni vettoriali a variazione limitata in una classe di gruppi di Carnot che contiene tutti i gruppi di Heisenberg H^n con n ≥ 2. Uno strumento chiave nella dimostrazione è costituito da alcune proprietà che legano le derivate orizzontali di una funzione a variazione limitata con il suo sottografico. Nel Capitolo 4 dimostriamo un risultato di compattezza per succesioni (u_j) equi-limitate in spazi metrici (X, d_j) quando lo spazio X è fissato ma la metrica può variare con j. Mostriamo inoltre un'applicazione agli spazi di Carnot-Carathéodory. I risultati del Capitolo 4 sono fondamentali per la dimostrazione di alcuni fatti contenuti nel Capitolo 2.

2012/2013
We investigate by different techniques, the solvability of a class of capillarity-type problems, in a bounded N-dimensional domain. Since our approach is variational, the natural context where this problem has to be settled is the space of bounded variation functions. Solutions of our equation are defined as subcritical points of the associated action functional.
We first introduce a lower and upper solution method in the space of bounded variation functions. We prove the existence of solutions in the case where the lower solution is smaller than the upper solution. A solution, bracketed by the given lower and upper solutions, is obtained as a local minimizer of the associated functional without any assumption on the boundedness of the right-hand side of the equation. In this context we also prove order stability results for the minimum and the maximum solution lying between the given lower and upper solutions. Next we develop an asymmetric version of the Poincaré inequality in the space of bounded variation functions. Several properties of the curve C are then derived and basically relying on these results, we discuss the solvability of the capillarity-type problem, assuming a suitable control on the interaction of the supremum and the infimum of the function at the right-hand side with the curve C. Non-existence and multiplicity results are investigated as well. The one-dimensional case, which sometimes presents a different behaviour, is also discussed. In particular, we provide an existence result which recovers the case of non-ordered lower and upper solutions.
XXV Ciclo
1985

The main thread of this work is the bounded variation (BV) functions in abstract Wiener spaces (a topic in infinite-dimensional analysis). In the first Part of this work, we present some known results, and we introduce the concepts of Wiener space, of Sobolev space in Wiener spaces, of BV functions (and finite perimeter sets) in Wiener spaces, and of BV functions in convex sets of Wiener spaces (by following the definition in V. I. Bogachev, A. Y. Pilipenko, A. V. Shaposhnikov, “Sobolev Functions on Infinite-dimensional domains”, J. Math. Anal. Appl., 2014); moreover, we introduce the trace theory on subsets of a Wiener space (by following P. Celada, A. Lunardi, “Traces of Sobolev functions on regular surfaces in infinite dimensions”, J. Funct. Anal., 2014), and the concept of Mosco convergence. In the second Part we present some new results. In Chapter 6, we consider a subset O of a Wiener space which satisfies a regularity condition, and we prove that a function in W^{1,2}(O) has null trace if and only if it is the limit of a sequence of functions with support contained in O. The main chapter is Chapter 7, which is devoted to the extension in the Wiener spaces setting of a result given in the section 8 of (V. Barbu, M. Röckner, “Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative noise”, Arch. Ration. Mech. Anal., 2013): if O is a convex bounded set with regular boundary in R^{d} and L is the Laplace operator in O with null Dirichlet boundary condition, then the normalized resolvent of L is contractive in sense L^1 respect to the gradient. We extend this result to the case of L Ornstein-Uhlenbeck operator in O with null Dirichlet boundary condition, with Gaussian measure (by using the results of Chapter 6): in this case O must satisfy a condition (which we call Gaussian convexity) which takes the place of the convexity in the Gaussian setting. Moreover, we extend the result also to the case of: L Laplace operator in an open convex O with null Neumann boundary condition, with Lebesgue measure; L Ornstein-Uhlenbeck operator in an open convex O with null Neumann boundary condition, with Gaussian measure. In the last part of Chapter 7, we use the preceding results to give an alternative definition of BV function (in the case L^2(O)). In Chapter 8, let X the set of continuous functions on [0,1] with starting point 0, provided with the measure induced by the Brownian motion with starting point 0; it is a Wiener space. For every A subset of X, we define Ξ_A, set of functions in X with image in A. In (M. Hino, H. Uchida, “Reflecting Ornstein–Uhlenbeck processes on pinned path spaces”, Res. Inst. Math. Sci. (RIMS), 2008) it is proved that, if d ≥ 2 and A is an open subset of R^d which satisfies an uniform outer ball condition then Ξ_A has finite perimeter in the sense of Gaussian measure. We present a weaker condition on A (in dimension sufficiently great) such that Ξ_A has finite perimeter: in particular, A can be the complement of a convex unbounded symmetric cone.
L’argomento principale di questo lavoro sono le funzioni a variazione limitata (BV) in spazi di Wiener astratti (un argomento di analisi infinito-dimensionale). Nella prima parte di questo lavoro, presentiamo alcuni risultati noti, e introduciamo i concetti di spazi di Wiener, di classi di Sobolev su spazi di Wiener, di funzioni BV (e insiemi di perimetro finito) in spazi di Wiener, e di funzioni BV in sottoinsiemi convessi di Spazi di Wiener (seguendo la definizione in V. I. Bogachev, A. Y. Pilipenko, A. V. Shaposhnikov, “Sobolev Functions on Infinite-dimensional domains”, J. Math. Anal. Appl., 2014); inoltre, introduciamo la teoria delle tracce su sottoinsiemi di uno spazio di Wiener( seguendo P. Celada, A. Lunardi, “Traces of Sobolev functions on regular surfaces in infinite dimensions”, J. Funct. Anal., 2014), e il concetto di convergenza di Mosco. Nella seconda parte presentiamo alcuni risultati originali. Nel capitolo 6, consideriamo un sottoinsieme O di uno spazio di Wiener che soddisfa a una condizione di regolarità, e proviamo che una funzione in W^{1,2} (O) ha traccia nulla se e solo se è il limite di una sequenza di funzioni con supporto contenuto in O. Il capitolo principale è il 7, che è dedicato all'estensione all'ambito degli spazi di Wiener di un risultato dato nella sezione 8 di (V. Barbu, M. Röckner, “Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative noise”, Arch. Ration. Mech. Anal., 2013): se O è un insieme convesso limitato con frontiera regolare in R^{d} e L è l'operatore di Laplace in O con condizione al bordo di Dirichlet nulla, allora il risolvente normalizzato di L è contrattivo nel senso L^1 rispetto al gradiente. Estendiamo questo risultato al caso di L operatore di Ornstein-Uhlenbeck in O con condizione al bordo di Dirichlet nulla, con misura gaussiana (usando i risultati del Capitolo 6): in questo caso O deve soddisfare una condizione (che chiamiamo convessità Gaussiana) che nel caso gaussiano prende il posto della convessità. Inoltre, estendiamo il risultato anche al caso di: L operatore di Laplace in un insieme aperto e convesso O con condizione al bordo di Neumann nulla, con misura di Lebesgue; L operatore in un insieme aperto e convesso O con condizione al bordo di Neumann nulla, con misura gaussiana. Nell'ultima parte del Capitolo 7, usiamo i precedenti risultati per dare una definizione alternativa di funzione BV in O (nel caso L^2(O) ). Nel Capitolo 8, sia X l'insieme delle funzioni continue in R^d su [ 0,1 ] con punti di partenza nell’origine fornito della misura indotta dal moto browniano con punto di partenza nell’origine; è uno spazio di Wiener. Per ogni A sottoinsieme di X, definiamo Ξ_A, insieme delle funzioni in X con immagine in A. In (M. Hino, H. Uchida, “Reflecting Ornstein–Uhlenbeck processes on pinned path spaces”, Res. Inst. Math. Sci. (RIMS), 2008) viene dimostrato che, se d ≥ 2 e A è un insieme aperto in R^d che soddisfa una condizione di uniforme palla esterna, allora Ξ_A ha perimetro finito nel senso della misura gaussiana. Presentiamo una condizione più debole su A (in dimensione sufficientemente grande) tale che Ξ_A ha perimetro finito: in particolare, A può essere il complementare di un cono convesso illimitato simmetrico.

Cette thèse est consacrée dans un premier temps à l’élaboration d’un modèle variationnel dedébruitage d’ordre deux, faisant intervenir l’espace BV 2 des fonctions à hessien borné. Nous nous inspirons ici directement du célèbre modèle de Rudin, Osher et Fatemi (ROF), remplaçant la minimisation de la variation totale de la fonction par la minimisation de la variation totale seconde, c’est à dire la variation totale de ses dérivées. Le but est ici d’obtenir un modèle aussi performant que le modèle ROF, permettant de plus de résoudre le problème de l’effet staircasing que celui-ci engendre. Le modèle que nous étudions ici semble efficace, entraînant toutefois l’apparition d’un léger effet de flou. C’est afin de réduire cet effet que nous introduisons finalement un modèle mixte, permettant d’obtenir des solutions à la fois non constantes par morceaux et sans effet de flou au niveau des détails. Dans une seconde partie, nous nous intéressons au problème d’extraction de texture. Un modèle reconnu comme étant l’un des plus performants est le modèle T V -L1, qui consiste simplement à remplacer dans le modèle ROF la norme L2 du terme d’attache aux données par la norme L1. Nous proposons ici une méthode originale permettant de résoudre ce problème utilisant des méthodes de Lagrangien augmenté. Pour les mêmes raisons que dans le cas du débruitage, nous introduisons également le modèle T V 2-L1, consistant encore une fois à remplacer la variation totale par la variation totale seconde. Un modèle d’extraction de texture mixte est enfin très brièvement introduit. Ce manuscrit est ponctué d’un vaste chapitre dédié aux tests numériques
This thesis is devoted in a first part to the elaboration of a second order variational modelfor image denoising, using the BV 2 space of bounded hessian functions. We here take a leaf out of the well known Rudin, Osher and Fatemi (ROF) model, where we replace the minimization of the total variation of the function with the minimization of the second order total variation of the function, that is to say the total variation of its partial derivatives. The goal is to get a competitive model with no staircasing effect that generates the ROF model anymore. The model we study seems to be efficient, but generates a blurry effect. In order to deal with it, we introduce a mixed model that permits to get solutions with no staircasing and without blurry effect on details. In a second part, we take an interset to the texture extraction problem. A model known as one of the most efficient is the T V -L1 model. It just consits in replacing the L2 norm of the fitting data term with the L1 norm.We propose here an original way to solve this problem by the use of augmented Lagrangian methods. For the same reason than for the denoising case, we also take an interest to the T V 2-L1 model, replacing again the total variation of the function by the second order total variation. A mixed model for texture extraction is finally briefly introduced. This manuscript ends with a huge chapter of numerical tests

Dissertation for the Degree of Doctor of Philosophy in Mathematics
Fundação para a Ciência e a Tecnologia through the Carnegie Mellon | Portugal Program under Grant SFRH/BD/35695/2007, the Financiamento Base 20010 ISFL–1–297, PTDC/MAT/109973/2009 and UTA

The thesis concerns the mathematical study of the calculus of variations and its application to liquid crystals. In the first chapter we examine vectorial problems in the calculus of variations with an additional pointwise constraint so that any admissible function n ε W^1,1(ΩM), and M is a manifold of suitable regularity. We formulate necessary and sufficient conditions for any given state n to be a strong or weak local minimiser of I. This is achieved using a nearest point projection mapping in order to use the more classical results which apply in the absence of a constraint. In the subsequent chapters we study various static continuum theories of liquid crystals. More specifically we look to explain a particular cholesteric fingerprint pattern observed by HP Labs. We begin in Chapter 2 by focusing on a specific cholesteric liquid crystal problem using the theory originally derived by Oseen and Frank. We find the global minimisers for general elastic constants amongst admissible functions which only depend on a single variable. Using the one-constant approximation for the Oseen-Frank free energy, we then show that these states are global minimisers of the three-dimensional problem if the pitch of the cholesteric liquid crystal is sufficiently long. Chapter 3 concerns the application of the results from the first chapter to the situations investigated in the second. The local stability of the one-dimensional states are quantified, analytically and numerically, and in doing so we unearth potential shortcomings of the classical Oseen-Frank theory. In Chapter 4, we ascertain some equivalence results between the continuum theories of Oseen and Frank, Ericksen, and Landau and de Gennes. We do so by proving lifting results, building on the work of Ball and Zarnescu, which relate the regularity of line and vector fields. The results prove to be interesting as they show that for a director theory to respect the head to tail symmetry of the liquid crystal molecules, the appropriate function space for the director field is S BV² (Ω,S^{2,/sup>). We take this idea and in the final chapter we propose a mathematical model of liquid crystals based upon the Oseen-Frank free energy but using special functions of bounded variation. We establish the existence of a minimiser, forms of the Euler-Lagrange equation, and find solutions of the Euler-Lagrange equation in some simple cases. Finally we use our proposed model to re-examine the same problems from Chapter 2. By doing so we extend the analysis we were able to achieve using Sobolev spaces and predict the existence of multi-dimensional minimisers consistent with the known experimental properties of high-chirality cholesteric liquid crystals.}

In this dissertation we will address two types of homogenization problems. The first one is a spectral problem in the realm of lower dimensional theories, whose physical motivation is the study of waves propagation in a domain of very small thickness and where it is introduced a very thin net of heterogeneities. Precisely, we consider an elliptic operator with "ε-periodic coefficients and the corresponding Dirichlet spectral problem in a three-dimensional bounded domain of small thickness δ. We study the asymptotic behavior of the spectrum as ε and δ tend to zero. This asymptotic behavior depends crucially on whether ε and δ are of the same order (δ ≈ ε), or ε is of order smaller than that of δ (δ = ετ , τ < 1), or ε is of order greater than that of δ (δ = ετ , τ > 1). We consider all three cases. The second problem concerns the study of multiscale homogenization problems with linear growth, aimed at the identification of effective energies for composite materials in the presence of fracture or cracks. Precisely, we characterize (n+1)-scale limit pairs (u,U) of sequences {(uεLN⌊Ω,Duε⌊Ω)}ε>0 ⊂ M(Ω;ℝd) × M(Ω;ℝd×N) whenever {uε}ε>0 is a bounded sequence in BV (Ω;ℝd). Using this characterization, we study the asymptotic behavior of periodically oscillating functionals with linear growth, defined in the space BV of functions of bounded variation and described by n ∈ ℕ microscales

In this thesis we study novel higher order total variation-based variational methods for digital image reconstruction. These methods are formulated in the context of Tikhonov regularisation. We focus on regularisation techniques in which the regulariser incorporates second order derivatives or a sophisticated combination of first and second order derivatives. The introduction of higher order derivatives in the regularisation process has been shown to be an advantage over the classical first order case, i.e., total variation regularisation, as classical artifacts such as the staircasing effect are significantly reduced or totally eliminated. Also in image inpainting the introduction of higher order derivatives in the regulariser turns out to be crucial to achieve interpolation across large gaps. First, we introduce, analyse and implement a combined first and second order regularisation method with applications in image denoising, deblurring and inpainting. The method, numerically realised by the split Bregman algorithm, is computationally efficient and capable of giving comparable results with total generalised variation (TGV), a state of the art higher order method. An additional experimental analysis is performed for image inpainting and an online demo is provided on the IPOL website (Image Processing Online). We also compute and study properties of exact solutions of the one dimensional total generalised variation problem with L^{2} data fitting term, for simple piecewise affine data functions, with or without jumps . This gives an insight on how this type of regularisation behaves and unravels the role of the TGV parameters. Finally, we introduce, study and analyse a novel non-local Hessian functional. We prove localisations of the non-local Hessian to the local analogue in several topologies and our analysis results in derivative-free characterisations of higher order Sobolev and BV spaces. An alternative formulation of a non-local Hessian functional is also introduced which is able to produce piecewise affine reconstructions in image denoising, outperforming TGV.

In this thesis, we study two problems. The first problem is to introduce the general version of Abramovich-Wickstead type spaces and investigate its order properties. In particular, we study the ideals, order bounded sets, disjointness properties, Dedekind completion and the norm properties of this Riesz space. We also define a new concrete example of Riesz space-valued uniformly continuous functions, denoted by CDr0 which generalizes the original Abramovich-Wickstead space. It is also shown that similar spaces CD0 and CDw introduced earlier by Alpay and Ercan are decomposable lattice-normed spaces. The second problem is related to analytic representations of different classes of dominated operators on these spaces. Our main representation theorems say that regular linear operators on CDr0 or linear dominated operators on CD0 may be represented as the sum of integration with respect to operator-valued measure and summation operation. In the case when the operator is order continuous or bo-continuous, then these representations reduce to discrete parts.

Cette thèse est une étude de modèles d'image aléatoires avec des applications en synthèse de texture.Dans la première partie de la thèse, des algorithmes de synthèse de texture basés sur le modèle shot noise sont développés. Dans le cadre discret, deux processus aléatoires, à savoir le shot noise discret asymptotique et le bruit à phase aléatoire, sont étudiés. On élabore ensuite un algorithme rapide de synthèse de texture basé sur ces processus. De nombreuses expériences démontrent que cet algorithme permet de reproduire une certaine classe de textures naturelles que l'on nomme micro-textures. Dans le cadre continu, la convergence gaussienne des modèles shot noise est étudiée d'avantage et de nouvelles bornes pour la vitesse de cette convergence sont établies. Enfin, on présente un nouvel algorithme de synthèse de texture procédurale par l'exemple basé sur le récent modèle Gabor noise. Cet algorithme permet de calculer automatiquement un modèle procédural représentant des micro-textures naturelles.La deuxième partie de la thèse est consacrée à l'étude du processus feuilles mortes transparentes (FMT), un nouveau modèle germes-grains obtenu en superposant des objets semi-transparents. Le résultat principal de cette partie montre que, lorsque la transparence des objets varie, le processus FMT fournit une famille de modèles variant du modèle feuilles mortes à un champ gaussien. Dans la troisième partie de la thèse, les champs aléatoires à variation bornés sont étudiés et on établit des résultats généraux sur le calcul de la variation totale moyenne de ces champs. En particulier, ces résultats généraux permettent de calculer le périmètre moyen des ensembles aléatoires et de calculer explicitement la variation totale moyenne des modèles germes-grains classiques.

Since the time of Cauchy, integration theory has in the main been an attempt to regain the Eden of Newton. In that idyllic time [. . . ] derivatives and integrals were [. . . ] different aspects of the same thing. -Peter Bullen, as quoted in [24] The theory of integration has gone through many changes in the past centuries and, in particular, there has been a tension between the Riemann and the Lebesgue approach to integration. Riemann's definition is often the first integral to be introduced in undergraduate studies, while Lebesgue's integral is more powerful but also more complicated and its methods are often postponed until graduate or advanced undergraduate studies. The integral presented in this paper is due to the work of Ralph Henstock and Jaroslav Kurzweil. By a simple exchange of the criterion for integrability in Riemann's definition a powerful integral with many properties of the Lebesgue integral was found. Further, the generalized Riemann integral expands the class of integrable functions with respect to Lebesgue integrals, while there is a characterization of the Lebesgue integral in terms of absolute integrability. As this definition expands the class of functions beyond absolutely integrable functions, some theorems become more cumbersome to prove in contrast to elegant results in Lebesgue's theory and some important properties in composition are lost. Further, it is not as easily abstracted as the Lebesgue integral. Therefore, the generalized Riemann integral should be thought of as a complement to Lebesgue's definition and not as a replacement.
Ända sedan Cauchys tid har integrationsteori i huvudsak varit ett försök att åter finna Newtons Eden. Under den idylliska perioden [. . . ] var derivator och integraler [. . . ] olika sidor av samma mynt.-Peter Bullen, citerad i [24] Under de senaste århundradena har integrationsteori genomgått många förändringar och framförallt har det funnits en spänning mellan Riemanns och Lebesgues respektive angreppssätt till integration. Riemanns definition är ofta den första integral som möter en student pa grundutbildningen, medan Lebesgues integral är kraftfullare. Eftersom Lebesgues definition är mer komplicerad introduceras den först i forskarutbildnings- eller avancerade grundutbildningskurser. Integralen som framställs i det här examensarbetet utvecklades av Ralph Henstock och Jaroslav Kurzweil. Genom att på ett enkelt sätt ändra kriteriet for integrerbarhet i Riemanns definition finner vi en kraftfull integral med många av Lebesgueintegralens egenskaper. Vidare utvidgar den generaliserade Riemannintegralen klassen av integrerbara funktioner i jämförelse med Lebesgueintegralen, medan vi samtidigt erhåller en karaktärisering av Lebesgueintegralen i termer av absolutintegrerbarhet. Eftersom klassen av generaliserat Riemannintegrerbara funktioner är större än de absolutintegrerbara funktionerna blir vissa satser mer omständiga att bevisa i jämforelse med eleganta resultat i Lebesgues teori. Därtill förloras vissa viktiga egenskaper vid sammansättning av funktioner och även möjligheten till abstraktion försvåras. Integralen ska alltså ses som ett komplement till Lebesgues definition och inte en ersättning.

Some problems concerning the algebra of bounded holomorphic functions from bounded domains in Cn are solved. A bounded domain of holomorphy Q in C2 with nonschlicht i7°°- envelope of holomorphy is constructed and it is shown that there is a point in Q for which Gleason’s Problem for H°°(Q) cannot be solved. If A(f2) is the Banach algebra of functions holomorphic in the bounded domain Q in Cn and continuous on the boundary and if p is a point in Q, then the following problem is known as Gleason’s Problem for A(Q) : Is the maximal ideal in A(Q) consisting of functions vanishing at p generated by (Zl ~Pl) , ■■■ , (Zn - Pn) ? A sufficient condition for solving Gleason’s Problem for A(Q) for all points in Q is given. In particular, this condition is fulfilled by a convex domain Q with Lipi+£-boundary (0 < e < 1) and thus generalizes a theorem of S.L.Leibenzon. One of the ideas in the methods of proof is integration along specific polygonal lines. If Gleason’s Problem can be solved in a point it can be solved also in a neighbourhood of the point. It is shown, that the coefficients in this case depends holomorphically on the points. Defining a projection from the spectrum of a uniform algebra of holomorphic functions to Cn, one defines the fiber in the spectrum over a point as the elements in the spectrum that projects on that point. Defining a kind of maximum modulus property for domains in Cn, some problems concerning the fibers and the number of elements in the fibers in certain algebras of bounded holomorphic functions are solved. It is, for example, shown that the set of points, over which the fibers contain more than one element is closed. A consequence is also that a starshaped domain with the maximum modulus property has schlicht /y°°-envelope of holomorphy. These kind of problems are also connected with Gleason’s problem. A survey paper on general properties of algebras of bounded holomorphic functions of several variables is included. The paper, in particular, treats aspects connecting iy°°-envelopes of holomorphy and some areas in the theory of uniform algebras.

Diss. (sammanfattning) Umeå : Umeå universitet, 1994, härtill 6 uppsatser

digitalisering@umu

La presente tesi tratta due argomenti distinti. Il Capitolo 1 e il Capitolo 2 riguardano problemi di evoluzione di interfacce nel piano. Nel Capitolo 1 viene considerata l’evoluzione di un materiale policristallino con tre (o più) fasi, in presenza di un’anisotropia cristallina (pari) ϕo la cui linea di livello 1, Fϕ :={ϕo ≤1} (Frank diagram), è un poligono regolare di n lati. La funzione duale ϕ : R2 →R definita da ϕ(ξ) := sup{ξ·η : ϕo(η)≤1}´e anch’essa un’anisotropia cristallina e Wϕ := {ϕ ≤ 1} è detta Wulff shape. In particolare, viene studiato il moto per curvatura cristallina di triodi elementari, ossia speciali reti piane di curve che sono frontiere regolari di insiemi rappresentanti tre fasi distinte di un materiale. Un triodo elementare è formato dall’unione di tre curve Lipschitziane, le interfacce, che si intersecano in un unico punto detto giunzione tripla. Ogni interfaccia è l’unione di un segmento di lunghezza finita e di una semiretta che riproduce due lati consecutivi della Wulff shape Wϕ. Viene analizzata l’esitenza locale e globale e la stabilità del flusso. Si dimostra l’esistenza locale di un unico flusso regolare stabile a partire da un dato iniziale regolare stabile: se n, il numero dei lati della Wulff shapeWϕ, è un multiplo di 6 allora il flusso è globale e converge a un flusso omotetico per t →+∞. L’analisi del comportamento del flusso per tempi grandi richiede lo studio della stabilità. La stabilità è l’ingrediente che assicura che nessun segmento si sviluppa dalla giunzione tripla durante il flusso. In generale, il flusso può diventare instabile in un tempo finito: se ciò accade e tutte le lunghezze dei segmenti finiti sono strettamente positive per tale tempo,è possibile costruire un flusso regolare per tempi successivi aggiungendo in corrispondenza della giunzione tripla in una delle tre interfacce un segmento infinitesimo opportuno (o addirittura un arco di curva a curvatura cristallina nulla). ´E anche possibile che durante il flusso uno dei tre segmenti scompaia in un tempo finito. In tal caso, in tale tempo il campo vettoriale di Cahn-Hoffman ha un salto di discontinuità e ai tempi successivi la giunzione tripla si muove traslando lungo la semiretta adiacente. Ognuno di questi flussi ha la proprietà che tutte le curvature cristalline rimangono limitate (persino se un segmento appare o scompare). ´E importante sottolineare che Taylor aveva già predetto la nascita di nuovi segmenti dalla giunzione tripla (senza però dimostrarlo). Viene inoltre considerato il flusso per curvatura cristalina di una partizione regolare stabile formata da due triodi elementari adiacenti. Vengono discussi alcuni esempi di situazioni di colasso che portano a cambi di topologia, come ad esempio la collisione di due giunzioni triple. Questi esempi (come anche il risultato di esistenza per tempi piccoli) mostrano uno dei vantaggi del flusso per curvatura cristallino rispetto, ad esempio, all’usuale moto per curvatura: calcoli espliciti possono essere fatti, e nel caso di non unicità, è possibile confrontare le energie delle diverse evoluzioni (difficile nel caso euclideo). Nel Capitolo 2 viene introdotta, usando la teoria delle funzioni a variazione limitata a valori in S1, la sfera diR2, una nuova classe di funzionali energia definiti su partizioni. Attraverso la variazione prima del funzionale energia, viene fornito un nuovo modello per l’evoluzione di interfacce che parzialmente estende quello introdotto nel Capitolo 1 e che consiste in un problema di frontiera libera definito sulle funzioni a variazione limitata a valori in S1. Questo modello è legato all’evoluzione di materiali policristallini dove è consentito alla Wulff shape di ruotare. Assumendo l’esitenza locale del flusso, si dimostra che durante il flusso curve chiuse convesse rimangono convesse e curve chiuse embedded rimangono embedded. Il secondo argomento della tesi è trattato nel Capitolo 3: l’obiettivo è quello di estendere il metodo delle linee di livello a sistemi di equazioni differenziali alle derivate parziali. Il metodo che viene proposto è consistente con la precedente ricerca portata avanti da Evans per l’equazione del calore e da Giga e Sato per equazioni di Hamilton-Jacobi. Il nostro approccio segue una costruzione geometrica che è legate alla nozione di barriera introdotta da De Giorgi. L’idea principale è quella di forzare un principio di confronto tra varietà di diversa codimensione e richiedere che ogni sottolivello di una soluzione dell’equazione per le linee di livello, detta level set equation, sia una barriera per i grafici di soluzioni del corrispondente sistema. Tale metodo ben si applica a una classe di sistemi di equazioni quasi-lineari del primo ordine. Viene fornita la level set equation associata ad opportuni sitemi di leggi di conservazione del primo ordine, al flusso per curvatura media di una varietà di codimensione arbitraria e a sitemi di equazioni di reazione-diffusione. Infine, viene calcolata la level set equation associata al sistema soddisfatto dalle parametrizzazioni di curve piane che si muovono per curvatura.
The present thesis deals with two different subjects. Chapter 1 and Chapter 2 concern interfaces evolution problems in the plane. In Chapter 1 I consider the evolution of a polycrystalline material with three (or more) phases, in presence of for an even crystalline anisotropy ϕo whose one-sublevel set Fϕ := {ϕo ≤ 1} (the Frank diagram) is a regular polygon of n sides. The dual function ϕ : R2 → R defined by ϕ(ξ) := sup{ξ ·η : ϕo(η) ≤ 1} is crystalline too and Wϕ := {ϕ ≤ 1} is called the Wulff shape. I am particularly interested in the motion by crystalline curvature of special planar networks called elementary triods, namely a regular three-phase boundary given by the union of three Lipschitz curves, the interfaces, intersecting at a point called triple junction. Each interface is the union of a segment of finite length and a half-line, reproducing two consecutive sides of Wϕ. I analyze local and global existence and stability of the flow. I prove that there exists, locally in time, a unique stable regular flow starting from a stable regular initial datum. I show that if n, the number of sides of Wϕ, is a multiple of 6 then the flow is global and converge to a homothetic flow as t → +∞. The analysis of the long time behavior requires the study of the stability. Stability is the ingredient that ensures that no additional segments develop at the triple junction during the flow. In general, the flow may become unstable at a finite time: if this occurs and none of the segments desappears, it is possible to construct a regular flow at subsequent times by adding an infinitesimal segment (or even an arc with zero crystalline curvature) at the triple junction. I also show that a segment may desappear. In such a case, the Cahn-Hoffman vector field Nmin has a jump discontinuity and the triple junction translates along the remaining adjacent half-line at subsequent times. Each of these flows has the property that all crystalline curvatures remain bounded (even if a segment appears or disappears). I want to stress that Taylor already predicted the appearance of new edges from a triple junction. I also consider the crystalline curvature flow starting from a stable ϕ-regular partition formed by two adjacent elementary triods. I discuss some examples of collapsing situations that lead to changes of topology, such as for instance the collision of two triple junctions. These examples (as well as the local in time existence result) show one of the advantages of crystalline flows with respect, for instance, to the usual mean curvature flow: explicit computations can be performed to some extent, and in case of nonuniqueness, a comparison between the energies of different evolutions (difficult in the euclidean case) can be made. In Chapter 2 we introduce, using the theory of S1-valued functions of bounded variations, a class of energy functionals defined on partitions and we produce, through the first variation, a new model for the evolution of interfaces which partially extends the one in Chapter 1 and which consists of a free boundary problem defined on S1-valued functions of bounded variation. This model is related to the evolution of polycrystals where the Wulff shape is allowed to rotate. Assuming the local existence of the flow, we show convexity preserving and embeddedness preserving properties. The second subject of the thesis is considered in Chapter 3 where we aim to extend the level set method to systems of PDEs. The method we propose is consistent with the previous research pursued by Evans for the heat equation and by Giga and Sato for Hamilton-Jacobi equations. Our approach follows a geometric construction related to the notion of barriers introduced by De Giorgi. The main idea is to force a comparison principle between manifolds of different codimension and require each sub-level of a solution of the level set equation to be a barrier for the graph of a solution of the corresponding system. We apply the method for a class of systems of first order quasi-linear equations. We compute the level set equation associated with suitable first order systems of conservation laws, with the mean curvature flow of a manifold of arbitrary codimension and with systems of reaction-diffusion equations. Finally, we provide a level set equation associated with the parametric curvature flow of planar curves.

Some problems concerning holomorphic continuation of the class of bounded holo­morphic functions from bounded domains in Cn that are domains of holomorphy are solved. A bounded domain of holomorphy Ω in C2 with nonschlicht H°°-envelope of holomorphy is constructed and it is shown that there is a point in D for which Glea­son’s Problem for H°°(Ω) cannot be solved. Furthermore a proof of the existence of a bounded domain of holomorphy in C2 for which the volume of the H°°-envelope of holomorphy is infinite is given. The idea of the proof is to put a family of so-called ”Sibony domains” into the unit bidisk by a packing procedure and patch them together by thin neighbourhoods of suitably chosen curves. If H°°(Ω) is the Banach algebra of bounded holomorphic functions on a bounded domain Ω in Cn and if p is a point in Ω, then the following problem is known as Gleason’s Problem for Hoo(Ω) : Is the maximal ideal in H°°(Ω) consisting of functions vanishing at p generated by (z1 -p1) , ... ,   (zn - pn) ? A sufficient condition for solving Gleason’s Problem for 77°° (Ω) for all points in Ω is given. In particular, this condition is fulfilled by a convex domain Ω with Lip1+e boundary (0 < e < 1) and thus generalizes a theorem of S.L.Leibenson. It is also proved that Gleason’s Problem can be solved for all points in certain unions of two polydisks in C2. One of the ideas in the methods of proof is integration along specific polygonal lines. Certain properties of some open sets defined by global plurisubharmonic func­tions in Cn are studied. More precisely, the sets Du = {z e Cn : u(z) < 0} and Eh = {{z,w) e Cn X C : h(z,w) < 1} are considered where ti is a plurisubharmonic function of minimal growth and h≠0 is a non-negative homogeneous plurisubharmonic function. (That is, the functions u and h belong to the classes L(Cn) and H+(Cn x C) respectively.) It is examined how the fact that Eh and the connected components of Du are H°°-domains of holomorphy is related to the structure of the set of disconti­nuity points of the global defining functions and to polynomial convexity. Moreover it is studied whether these notions are preserved under a certain bijective mapping from L(Cn) to H+(Cn x C). Two counterexamples are given which show that polynomial convexity is not preserved under this bijection. It is also proved, for example, that if Du is bounded and if the set of discontinuity points of u is pluripolar then Du is of type H°°. A survey paper on general properties of envelopes of holomorphy is included. In particular, the paper treats aspects of the theory for the bounded holomorphic functions. The results for the bounded holomorphic functions are compared with the corresponding ones for the holomorphic functions.
digitalisering@umu.se

The thesis explores several topics, related to polynomial distribution functions and their densities on [0,1]M, including polynomial copula functions and their densities. The contribution of this work can be subdivided into two areas. - Studying the characterization of the extreme sets of polynomial densities and copulas, which is possible due to the Choquet theorem. - Development of statistical methods that utilize the fact that the density is polynomial (which may or may not be an extreme density). With regard to the characterization of the extreme sets, we first establish that in all dimensions the density of an extreme distribution function is an extreme density. As a consequence, characterizing extreme distribution functions is equivalent to characterizing extreme densities, which is easier analytically. We provide the full constructive characterization of the Choquet-extreme polynomial densities in the univariate case, prove several necessary and sufficient conditions for the extremality of densities in arbitrary dimension, provide necessary conditions for extreme polynomial copulas, and prove characterizing duality relationships for polynomial copulas. We also introduce a special case of reflexive polynomial copulas. Most of the statistical methods we consider are restricted to the univariate case. We explore ways to construct univariate densities by mixing the extreme ones, propose non-parametric and ML estimators of polynomial densities. We introduce a new procedure to calibrate the mixing distribution and propose an extension of the standard method of moments to pinned density moment matching. As an application of the multivariate polynomial copulas, we introduce polynomial coupling and explore its application to convolution of coupled random variables. The introduction is followed by a summary of the contributions of this thesis and the sections, dedicated first to the univariate case, then to the general multivariate case, and then to polynomial copula densities. Each section first presents the main results, followed by the literature review.

Les problèmes inverses consistent à retrouver une donnée qui a été transformée ou perturbée. Ils nécessitent une régularisation puisque mal posés. En traitement d'images, la variation totale en tant qu'outil de régularisation a l'avantage de préserver les discontinuités tout en créant des zones lisses, résultats établis dans cette thèse dans un cadre continu et pour des énergies générales. En outre, nous proposons et étudions une variante de la variation totale. Nous établissons une formulation duale qui nous permet de démontrer que cette variante coïncide avec la variation totale sur des ensembles de périmètre fini. Ces dernières années les méthodes non-locales exploitant les auto-similarités dans les images ont connu un succès particulier. Nous adaptons cette approche au problème de complétion de spectre pour des problèmes inverses généraux. La dernière partie est consacrée aux aspects algorithmiques inhérents à l'optimisation des énergies convexes considérées. Nous étudions la convergence et la complexité d'une famille récente d'algorithmes dits Primal-Dual.

The purpose of this paper is to generalize a theorem that characterizes absolute continuity of bounded finitely additive set functions in the form of an integral approximation. We show that his integral exists if the condition of absolute continuity is removed.

In leading up to the proof, methods for constructing fields and finitely additive set functions are introduced with an application involving the Tagaki function given as an example. Also, non-absolutely continuous set functions are constructed using Banach limits and maximal filters.

We investigate inequalities that can be viewed as generalizations of Hardy's inequality about the Fourier coefficients of a function analytic on the circle. The proof of the Littlewood conjecture opened a wide door in front of questions regarding possible generalizations of Hardy's inequality. The proof of the Littlewood conjecture was based on some constructions of bounded functions having particular properties.
In 1993, I. Klemes investigated one of the constructions (we shall call it the algebraic construction) and proved what is called a mixed norm generalization of Hardy's inequality. It turns out that we can work with the same construction and examine more properties of it in order to get more results.
The objectives of the thesis are to give more detailed properties of the algebraic construction and to use these properties in order to prove various versions of two-sided Hardy inequalities.

Die Dissertation untersucht die Invertierbarkeit im Unendlichen fuer Normgrenzwerte von Bandoperatoren - sogenannte band-dominierte Operatoren. Das dazu verwendete Instrument ist die Methode der Limitoperatoren. Es werden grundlegende Eigenschaften von Limitoperatoren bewiesen, Zusammenhaenge zur Invertierbarkeit im Unendlichen hergeleitet, sowie darueber hinaus gehende Anwendungen, z.B. zur Konvergenz von Projektionsverfahren, studiert.

The purpose of this paper is to examine properties of the mean integral. The mean integral is compared with the regular integral. If [a;b] is an interval, f is quasicontinuous on [a;b] and g has bounded variation on [a;b], then the man integral of f with respect to g exists on [a;b]. The following theorem is proved. If [a*;b*] and [a;b] each is an interval and h is a function from [a*;b*] into R, then the following two statements are equivalent: 1) If f is a function from [a;b] into [a*;b*], gi is a function from [a;b] into R with bounded variation and (m)∫^b_afdg exists then (m)∫^b_ah(f)dg exists. 2) h is continuous.

We present Bessel function and Gaussian beam models for a study of microwave power deposition in bounded and inhomogeneous lossy media. The aim is to develop methods that can accurately simulate practical results commonly found in electromagnetic hyperthermic treatment, which is a noninvasive method. The Bessel function method has a closed form solution and can be used to compute accurate results of electromagnetic fields emanating from applicators with cosinusoidal aperture fields. On the other hand, the Gaussian beam method is approximate but has the capability to simplify boundary value problems and to compute fields in three-dimensions with extremely low CPU time (less than 30 sec). Although the Gaussian beam method is derived from geometrical optics theory, it performs very well in domains outside the realm of geometrical optics which stipulates that aperture dimension/λ ≥ 5 in the design of microwave systems. This condition has no relevance to the Gaussian beam method since the method shows that a limit of aperture dimension/ λ ≥ 0.9 is possible, which is a very important achievement in the design and application of microwave systems. Experimental verifications of the two theoretical models are integral parts of the presentation and show the viability of the methods.

The purpose of this study was to determine discourse and functional motivation for passive word order variation as shown in three of the major Early Old English prose texts, Orosius, Pastoral Care, and Ecclesiastical History of the English People. The main variation of Early Old English passive word orders are of three types, which this study showed to be distinct in the extent to which passive subjects represent information structure. This study further shows that, while thematicity functions as a main motivating factor for the use of passives, positional variation of passive verbal elements is also an important determinant of the degrees of information structure of passive main clause subjects.
Department of English

Thesis (Ed.D.)--Teachers College, Columbia University, 1994.
Typescript; issued also on microfilm. Sponsor: Clifford A. Hill. Dissertation Committee: Franklin E. Horowitz. Includes bibliographical references (leaves 125-128).

On détermine la distribution asymptotique des pôles pour trois types de meilleurs approximants (Padé à l’infini, rationnel en L2 sur le cercle unité, méromorphe dans le disque unité en Lp sur le cercle unité, p>2) de la transformée de Cauchy d’une mesure complexe sous l’hypothèse que le support S de la mesure soit de capacité positive et inclus dans (-1, 1), que la mesure satisfasse une condition de densité et que l’argument de la mesure soit la restriction d’une fonction à variation bornée. Les polynômes dénominateurs des approximants satisfont des relations d’orthogonalité. Au moyen d’un théorème de Kestelman, on obtient des contraintes géométriques pour les zéros qui impliquent que chaque mesure limite faible des mesures de comptage associées à son support inclus dans S. Puis, à l’aide de résultats de la théorie du potentiel dans le plan, on montre que les mesures de comptage convergent faiblement vers la distribution d’équilibre logarithmique respectivement hyperbolique de S
We determine the asymptotic pole distribution for three types of best approximants (Padé at infinity, rational in L2 on the unit circle, meromorphic in the unit disk in Lp on the unit circle, p>2) of the Cauchy transform of a complex measure under the hypothesis that the support S of the measure is of positive capacity and included in (-1 1), that the measure satisfies a density condition and that the argument of the measure is the restriction of a function of bounded variation ? The denominator polynomials of the approximants satisfay orthogonality relations ? By means of a theorem of Kestelman we obtain geometric constraints for the zeros which imply that every weak limit measure of the associated counting measures has support included in S. Then, with the help of results from potential theory in the plane, we show that the counting measures converge weakly to the logarithmic respectively hyperbolic equilibrium distribution of S

Mathematical models are useful for simulation, design, analysis, control, and optimization of complex systems. One important step necessary to create an effective model is designing an experiment from which the unknown model parameter can be accurately identified and then verified. The strategy which one approaches this problem is dependent on the amount of data that can be collected and the assumptions made about the behavior of the error in the statistical model. In this presentation we describe how to approach this problem using a combination of statistical and mathematical theory with reliable computation. More specifically, we present a new approach to bounded error parameter validation that approximates the membership set by solving an inverse problem rather than using the standard forward interval analysis methods. For our method we provide theoretical justification, apply this technique to several examples, and describe how it relates to designing experiments. We also address how to define infinite dimensional designs that can be used to create designs of any finite dimension. In general, finding a good design for an experiment requires a careful investigation of all available information and we provide an effective approach to dthe problem.
Ph. D.

This paper is concerned with the numerical analysis of the autoconvolution equation $x*x=y$ restricted to the interval [0,1]. We present a discrete constrained least squares approach and prove its convergence in $L^p(0,1),1