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Hamiltonian Systems and Gibbs Measures
Get PDFConsider then cubic defocusing nonlinear wave equation on three dimensional Euclidean space, with radial initial data. In this thesis, we construct a Gibbs measure for this system and prove its invariance under the flow of the NLW. In particular, we are in the infinite volume setting.For the finite volume analogue, specifically on the unit ball with zero boundary values, an invariant Gibbs measure was constructed by Burq, Tvetkov, and de Suzzoni as a Borel measure on super-critical Sobolev spaces.We first show that this finite volume Gibbs measure is supported on a space of weighted Holder continuous functions. Next, we show that the NLW is locally well-posed there, a counter-point to the Sobolev super-criticality noted by Burq and Tzvetkov. Furthermore, the flow of the NLW leaves this measure invariant.We use a multi-time Feynman--Kac formula to construct the infinite volume limit measure by computing the asymptotics of the fundamental solution of an appropriate parabolic PDE. We use finite speed of propagation and results from descriptive set theory to establish invariance of the infinite volume measure
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Some variable-coefficient nonlinear Schrödinger equations at critical regularity
Get PDFWe present several large-data results for nontranslation-invariant analogues of the energy-critical and mass-critical nonlinear Schrödinger equations, obtained by introducing external potentials or non-Euclidean geometries
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Local existence and breakdown of scattering behavior for semilinear Schrödinger equations
Get PDFIn this thesis, we study the behavior of solutions to some semilinear Schr\"odinger equations at short and long time scales. We first consider the nonlinear Schr\"odinger equations with power-type nonlinearity in three dimensions with periodic boundary conditions. We show that this equation is locally well-posed in critically scaling Sobolev spaces H^s(\bb{T}^3). We then investigate the long-time asymptotic behavior of solutions to NLS in Euclidean space with defocusing, mass-subcritical power-type and Hartree nonlinearities. We discuss the divide between the wealth of results on the scattering theory for these equations in weighted spaces and the paucity of analogous results in L^2(\bb{R}^d). To explain this, we show that the scattering problems for these equations are well-posed in weighted spaces in the sense that the scattering operators attain their natural and maximal regularity. Furthermore, we show that these scattering problems are ill-posed in in the sense that the scattering operators cannot be extended to all of without losing a positive (and, in the case of Hartree, infinite) amount of regularity
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Doubly Critical Semilinear Schrödinger Equations.
Get PDFWe present several new results regarding the two and three dimensional energy-critical non- linear Schro ̈dinger equation in the presence of a second critical nonlinearity
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Strichartz estimates for the Schrödinger flow on compact symmetric spaces
Get PDFThis thesis studies scaling critical Strichartz estimates for the Schrödinger flow on compact symmetric spaces. A general scaling critical Strichartz estimate (with an ε-loss, respectively) is given conditional on a conjectured dispersive estimate (with an ε-loss, respectively) on general compact symmetric spaces. The dispersive estimate is then proved for the special case of connected compact Lie groups. Slightly more generally, for products of connected compact Lie groups and spheres of odd dimension, the dispersive estimate is proved with an ε-loss
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Symplectic non-squeezing for the Korteweg--de Vries flow on the line
Get PDFThe main goal of this work is to prove a symplectic non-squeezing result for the Korteweg--de Vries (KdV) equation on the line . This is achieved via a finite-dimensional approximation argument. Our choice of finite-dimensional Hamiltonian system that effectively approximates the KdV flow is inspired by the recent breakthrough in the well-posedness theory of KdV in low regularity spaces \cite{KV18}, relying on its completely integrable structure. The employment of our methods also provides us with a new concise proof of symplectic non-squeezing for the same equation on the circle \T, recovering the result of \cite{CKSTT}
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Get PDF“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Appropriate Similarity Measures for Author Cocitation Analysis
Get PDFWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
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