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Littlewood-Paley sets and sums of permuted lacunary sequencesTrudeau, Sidney. January 2009 (has links)
Let {Ij} be an interval partition of the integers, f(x) a function on the circle group T and S(f) = (sum |f j|2)1/2 where fˆ j = fˆ cIj . In their 1995 paper, Hare and Klemes showed that, for fixed p ∈ (1, infinity), there exist lambdap > 1 and Ap, Bp > 0 such that if l(Ij+1)/ l(Ij) ≥ lambdap, where l(Ij) is the length of the interval Ij, then Ap∥ f∥p ≤ ∥S( f)∥p ≤ Bp∥ f∥p. That is, {Ij} is a Littlewood-Paley (p) partition. Since the intervals need not be adjacent, these partitions may be viewed as permutations of lacunary intervals. Partitions like these can be induced by subsets of sums of permuted lacunary sequences. In this thesis, we present two main results. First, complementary to the aforementioned work of Hare and Klemes who proved that sums of permuted lacunary sequences were Littlewood-Paley (p) partitions (for large enough ratio), we prove the surprising result that there are sums of permuted lacunary sequences of fixed ratio that cannot be obtained by iterating sums of permuted lacunary sequences of larger ratio finitely many times. The proof of this statement is based on the ideas developed in the 1989 paper of Hare and Klemes, especially with respect to the definition of a tree and to the theorem on the equivalency of a finitely generated partition and the absence of certain trees. These special sums may then be viewed as the critical test case for further progress on the conjecture of Hare and Klemes that sums of permuted lacunary sequences are Littlewood-Paley (p) partitions for any p. Secondly, we use the non-branching case of the method of Hare and Klemes developed in their 1992 and 1995 papers, and further developed by Hare in a general setting in 1997, to prove a result of Marcinkiewicz on iterated lacunary sequences in the case p = 4. This shows that the method introduced by Hare and Klemes can potentially be adapted to partitions other than those they were originally applied to. As well, in considering the proof given by Hare and Klemes (and by Hare in a general setting) that lacunary sequences are Littlewood-Paley (4) partitions, we present a slight variation on one of the computations which may be useful in regard to sharp versions of some of these computations, but otherwise follows the same pattern as that of the above papers. Finally, we prove an elementary property of the finite union of lacunary sequences.
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Littlewood-Paley sets and sums of permuted lacunary sequencesTrudeau, Sidney. January 2009 (has links)
No description available.
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Non-abelian Littlewood–Offord inequalitiesTiep, Pham H., Vu, Van H. 10 1900 (has links)
In 1943, Littlewood and Offord proved the first anti-concentration result for sums of independent random variables. Their result has since then been strengthened and generalised by generations of researchers, with applications in several areas of mathematics. In this paper, we present the first non-abelian analogue of the Littlewood Offord result, a sharp anti-concentration inequality for products of independent random variables. (C) 2016 Elsevier Inc. All rights reserved.
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Symmetric functions and Macdonald polynomialsLanger, R. January 2008 (has links)
The ring of symmetric functions Λ, with natural basis given by the Schur functions, arise in many different areas of mathematics. For example, as the cohomology ring of the grassmanian, and as the representation ring of the symmetric group. One may define a coproduct on Λ by the plethystic addition on alphabets. In this way the ring of symmetric functions becomes a Hopf algebra. The Littlewood–Richardson numbers may be viewed as the structure constants for the co-product in the Schur basis. The first part of this thesis, inspired by the umbral calculus of Gian-Carlo Rota, is a study of the co-algebra maps of Λ. The Macdonald polynomials are a somewhat mysterious qt-deformation of the Schur functions. The second part of this thesis contains a proof a generating function identity for the Macdonald polynomials which was originally conjectured by Kawanaka.
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Lower bounds for multiparameter square functions /Anderson, Abraham Quillan. January 2000 (has links)
Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, August 2000. / Includes bibliographical references. Also available on the Internet.
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Mapping properties of multi-parameter multipliersBakas, Odysseas January 2017 (has links)
This thesis is motivated by the problem of understanding the endpoint mapping properties of higher-dimensional Marcinkiewicz multipliers. The one-dimensional case was definitively characterised by Tao and Wright. In particular, they proved that Marcinkiewicz multipliers acting on functions over the real line map the Hardy space H¹(ℝ) to L¹;∞(ℝ) and they locally map L log¹/² L to L¹;∞ and that these results are sharp. The classical inequalities of Paley and Zygmund involving lacunary sequences can be regarded as rudimentary prototypes of the aforementioned results of Tao and Wright on the behaviour of Marcinkiewicz multipliers "near" L¹(ℝ). Motivated by this fact, in Chapter 3 we obtain higher-dimensional variants of these two inequalities and we establish sharp multiplier inclusion theorems on the torus and on the real line. In Chapter 4 we extend the multiplier inclusion theorem on T of Chapter 3 to higher dimensions. In the last chapter of this thesis, we study endpoint mapping properties of the classical Littlewood-Paley square function which can essentially be regarded as a model Marcinkiewicz multiplier. More specifically, we give a new proof to a theorem due to Bourgain on the growth of the operator norm of the Littlewood- Paley square function as p → 1+ and then extend this result to higher dimensions. We also obtain sharp weak-type inequalities for the multi-parameter Littlewood- Paley square function and prove that the two-parameter Littlewood-Paley square function does not map the product Hardy space H¹ to L¹;∞.
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Hardy-Littlewood Maximal FunctionsVaughan, David 09 1900 (has links)
<p> The principal object of this study is
to find weak and strong type estimates concerning functions
in weighted Lp spaces and their maximal functions. We
also apply these results to the study of convolution
integrals. </p> / Thesis / Master of Science (MSc)
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Viscosité évanescente dans les équations de la mécanique des Fluides bidimensionnels.Hmidi, Taoufik 10 December 2003 (has links) (PDF)
Ma thèse est consacrée à l'étude de quelques problèmes liés à la stabilité des structures de poches de tourbillon dans les équations de Navier-Stokes incompressibles 2D. Dans le premier chapitre on démontre en particulier que si l'on se donne à l'instant initial un tourbillon valant l'indicatrice d'un domaine borné dont le bord est de classe $C^{1+EE}$ (espace de Hölder), alors son transporté par le flot visqueux préserve en tout temps cette régularité. Dans le deuxième chapitre, on montre que dans le cas des données de type poches de tourbillon à bord de mesure nulle, le tourbillon visqueux converge fortement en norme $L^p$ vers le tourbillon eulérien. Le dernier chapitre est une généralisation pour le système de Navier-Stokes d'un résultat obtenu par J.-Y. Chemin dans le cadre eulerien et concernant les poches de tourbillon singulières. On démontre que si le bord de la poche de tourbillon est régulier en dehors d'un ensemble fermé, alors son transporté par le flot visqueux est régulier en dehors de l'image par le flot de l'ensemble singulier. On prouve également qu'en dehors de cet ensemble la solution du système de Navier-Stokes est lipschitzienne avec un contrôle indépendant de la viscosité.
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A Puzzle-Based Synthesis Algorithm For a Triple Intersection of Schubert VarietiesBrown, Andrew Alexander Harold 28 January 2010 (has links)
This thesis develops an algorithm for the Schubert calculus of the Grassmanian.
Specifically, we state a puzzle-based, synthesis algorithm for a triple
intersection of Schubert varieties. Our algorithm is a reformulation of the
synthesis algorithm by Bercovici, Collins, Dykema, Li, and Timotin. We replace
their combinatorial approach, based on specialized Lebesgue measures,
with an approach based on the puzzles of Knutson, Tao and Woodward.
The use of puzzles in our algorithm is beneficial for several reasons, foremost
among them being the larger body of work exploiting puzzles. To understand
the algorithm, we study the necessary Schubert calculus of the Grassmanian
to define synthesis. We also discuss the puzzle-based Littlewood-Richardson
rule, which connects puzzles to triple intersections of Schubert varieties. We
also survey three combinatorial objects related to puzzles in which we include
a puzzle-based construction, by King, Tollu, and Toumazet, of the well
known Horn inequalities.
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A Puzzle-Based Synthesis Algorithm For a Triple Intersection of Schubert VarietiesBrown, Andrew Alexander Harold 28 January 2010 (has links)
This thesis develops an algorithm for the Schubert calculus of the Grassmanian.
Specifically, we state a puzzle-based, synthesis algorithm for a triple
intersection of Schubert varieties. Our algorithm is a reformulation of the
synthesis algorithm by Bercovici, Collins, Dykema, Li, and Timotin. We replace
their combinatorial approach, based on specialized Lebesgue measures,
with an approach based on the puzzles of Knutson, Tao and Woodward.
The use of puzzles in our algorithm is beneficial for several reasons, foremost
among them being the larger body of work exploiting puzzles. To understand
the algorithm, we study the necessary Schubert calculus of the Grassmanian
to define synthesis. We also discuss the puzzle-based Littlewood-Richardson
rule, which connects puzzles to triple intersections of Schubert varieties. We
also survey three combinatorial objects related to puzzles in which we include
a puzzle-based construction, by King, Tollu, and Toumazet, of the well
known Horn inequalities.
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