1,721,119 research outputs found
Well-posedness and uniform decay rates for weak solutions to a von Kármán system with nonlinear dissipative boundary conditions
No full textHorn, Mary Ann; Lasiecka, Irena; Tataru, Daniel. (1993). Well-posedness and uniform decay rates for weak solutions to a von Kármán system with nonlinear dissipative boundary conditions. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/2517
Global existence, uniqueness and regularity of solutions to a Von Kármán system with nonlinear boundary dissipation
No full textFavini, Angelo; Horn, Mary Ann; Lasiecka, Irena; Tataru, Daniel. (1993). Global existence, uniqueness and regularity of solutions to a Von Kármán system with nonlinear boundary dissipation. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/2475
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe 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
Full text link“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
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On Low Regularity Dynamics for Quasilinear Dispersive Equations and Free Boundary Problems
Full text linkThis thesis provides a detailed account of several novel methods and ideas developed by the author and collaborators to study the low regularity dynamics for a diverse selection of nonlinear PDE. In this manuscript, these techniques are applied to resolve several questions concerning the well-posedness of various families of quasilinear dispersive equations and free boundary problems arising in fluid mechanics. \medskipAfter giving a brief overview of the main results in Chapter 1, we begin in Chapter 2 with a systematic analysis of the \emph{incompressible free boundary Euler equations} on a time-dependent, compact fluid domain ,\begin{equation*}
\begin{cases}
&\partial_tv+v\cdot \nabla v=-\nabla p-ge_d \ \ \text{on} \ \ \Omega_t,
\\
&\nabla\cdot v=0 \ \ \text{on} \ \ \Omega_t,
\\
&\partial_t+v\cdot\nabla \ \ \text{is tangent to} \ \bigcup_t \{t\}\times\partial\Omega_t\subseteq\mathbb{R}^{d+1},
\\
&p_{|\partial\Omega_t}=0.
\end{cases}
\end{equation*}
This system models, among other things, the dynamics of a fluid droplet under the influence of gravity. In this chapter, a complete local well-posedness theory in -based Sobolev spaces is developed. Our well-posedness theory includes (i) Local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and the first proof of continuous dependence on the data, all in low regularity Sobolev spaces; (ii) Enhanced uniqueness: Our uniqueness result holds at the level of the Lipschitz norm of the velocity and the regularity of the free surface; (iii) Stability bounds: We construct a nonlinear functional which measures, in a suitable sense, the distance between two solutions (even when defined on different domains) and we show that this distance is propagated by the flow; (iv) Energy estimates: We prove refined, essentially scale invariant energy estimates for solutions, relying on a newly constructed family of elliptic estimates; (v) Continuation criterion: We give the first proof of a sharp continuation criterion in the physically relevant pointwise norms, at the level of scaling. In essence, we show that solutions can be continued as long as the velocity is in and the free surface is in , which is at the same level as the Beale-Kato-Majda criterion for the boundaryless case; (vi) A novel proof of the construction of regular solutions. Our entire approach is in the Eulerian framework and can be adapted to work in more general fluid domains.
\medskipIn Chapter 3, we move to a systematic study of the so-called \emph{general quasilinear ultrahyperbolic Schr\"odinger equation},\begin{equation*}
\begin{cases}
&i\partial_tu+\partial_jg^{jk}(u,\overline{u})\partial_k u=F(u,\overline{u},\nabla u,\nabla\overline{u}),\hspace{5mm} u:\mathbb{R}\times\mathbb{R}^d\to\mathbb{C}^m,
\\
&u(0,x)=u_0(x).
\end{cases}
\end{equation*}
Here, is some real, symmetric, and uniformly non-degenerate metric and is some smooth nonlinear function of its arguments. In this chapter, we develop novel techniques for establishing large data local well-posedness in low regularity Sobolev spaces for this equation. Our main result represents a definitive improvement over the landmark results of Kenig, Ponce, Rolvung, and Vega \cite{kenig2005variable, MR1660933, MR2096797,MR2263709}, as it weakens the regularity and decay assumptions to the same scale of spaces considered by Marzuola, Metcalfe and Tataru in \cite{marzuola2021quasilinear}, but removes the uniform ellipticity assumption on the metric from their result. Our method has the additional benefit of being relatively simple and robust. In particular, it only relies on pseudodifferential calculus for classical symbols.
\medskipFinally, in Chapter 4, we turn our attention to a more specialized quasilinear dispersive model; namely, the \emph{generalized derivative nonlinear Schr\"odinger equation} (GDNLS),\begin{equation*}
\begin{cases}
&i\partial_t u+\partial_x^2u=i|u|^{2\sigma}\partial_xu,
\\
&u(0)=u_0.
\end{cases}
\end{equation*}
We study this equation in the regime \frac{1}{2}<\sigma<1 where the local theory is most difficult, and analyze this equation at both low and high regularity to establish the first global well-posedness result for this problem in spaces. This appears to also be the first result of its kind for any quasilinear dispersive model where the nonlinearity is both rough and lacks the decay necessary for global smoothing-type estimates. These two features pose considerable difficulty when trying to apply standard tools for closing low-regularity estimates, such as Strichartz estimates, gauge transformations or maximal function estimates. To circumvent this issue, several new ideas are developed. In addition to establishing a suitable global theory, we also dramatically improve the local results in the high regularity regime compared to the previous literature
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A Collection of Results on Nonlinear Dispersive Equations, Banach Lattices and Phase Retrieval
Full text linkThis thesis collects various linear and nonlinear techniques developed by the author and his collaborators to attack problems in Function Space Theory, Phase Retrieval and PDE. The thesis begins with an analysis of the \emph{generalized derivative nonlinear Schr\"odinger equation}\begin{equation}\label{gDNLSi}\tag{gDNLS}
\begin{cases}
&i\partial_t u+\partial_x^2u=i|u|^{2\sigma}\partial_xu,
\\
&u(0)=u_0.
\end{cases}
\end{equation}
This is a canonical model of a quasilinear dispersive PDE, i.e., a dispersive PDE where one expects continuity but not uniform continuity of the data-to-solution map. As opposed to the semilinear case where Strichartz and contraction mapping arguments are directly applicable, the well-posedness theory for such quasilinear PDE is largely open. In Chapter 2, we study \eqref{gDNLSi} when \sigma<1, which is the regime where local well-posedness is hardest to establish. Our main result establishes global well-posedness in the energy space , as long as is not too small. In Chapter 3 we transition to the water waves problem. That is, we consider the motion of water when the interface between the water and the air is free to move. In this case, we do not consider the well-posedness problem, but rather the existence of special solutions. Our primary interest is in solitary waves, which are waves that travel across the ocean's surface at constant speed while never changing shape. When modelling water waves, the fundamental physical parameters are the gravity, surface tension, and fluid depth. It is then an interesting question to identify which combinations of parameters lead to a given physical phenomenon. For solitary waves in two dimensions, we discuss the complete solution to the existence/non-existence problem. More specifically, we prove non-existence of solitary waves when surface tension and depth are arbitrary but gravity is zero, which was the only case that had not yet yielded a solution.Chapter 4 is dedicated to the phase retrieval problem; that is, the determination of a function up to unavoidable ambiguity from . In a recent article, Calderbank, Daubechies, Freeman and Freeman dispelled of the prevailing belief that phase retrieval in infinite dimensions is inherently unstable. Motivated by this, Chapter 4 contains an extensive study of the stability of phase retrieval, for both real and complex scalars. In particular, we give the first construction of an infinite-dimensional subspace
with the property that for any ,
if is approximately equal to with respect to
the norm, then there exists a unimodular scalar
such that is approximately equal to . Recall that a basis of a Banach space is a sequence in such that every admits a unique sequence of scalars satisfying The goal of Chapter 5 is to study bases of consisting entirely of non-negative functions. Such non-negative coordinate systems are of relevance in both Functional Analysis and Applied Mathematics. However, constructing them is notoriously difficult, as can be extrapolated from the following fact: For any non-negative basis of there exists a permutation such that is \emph{not} a basis of .
Overcoming this issue, in Chapter 5 we give the first construction of a non-negative basis of .Chapter 6 is devoted to free Banach lattices. Given a Banach space , one can generate a Banach lattice \fbl[E] so that every operator into a Banach lattice uniquely extends to \fbl[E] as a lattice homomorphism of the same norm. The correspondence E\mapsto \fbl[E] provides an indispensable link between Banach space theory and Banach lattice theory. In Chapter 6, we give a convenient functional representation of \fbl[E] and its -convex variants, and then deeply study these spaces. In particular, we study how properties of an operator between Banach spaces transfer to the associated lattice homomorphism \overline{T}:\fbl[E]\rightarrow \fbl[F]. Special consideration is devoted to the case when the operator is an isomorphic embedding, which leads us to examine extension properties of operators into , and several classical Banach space properties such as being a G.T. space. A detailed investigation of basic sequences and sublattices of free Banach lattices is also provided. Among other things, this allows us to settle an a priori unrelated question, providing the first instance of a subspace of a Banach lattice without bibasic sequences. Along the way, a dictionary between Banach space properties of and Banach lattice properties of \fbl[E] is assembled. For example, we characterize the existence of lattice copies of in \fbl[E] and show that \fbl[E] has an upper -estimate if and only if is -summing ()
Appropriate Similarity Measures for Author Cocitation Analysis
Full text linkWe 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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Modified scattering for a scalar quasilinear wave equation satisfying the weak null condition
Full text linkThe objective of this dissertation is to study the long time dynamics of a scalar quasilinear wave equation \begin{align*}g^{\alpha\beta}(u)\partial_\alpha\partial_\beta u=0,\hspace{2em}\text{in }\mathbb{R}^{1+3}_{t,x}.\end{align*} This equation satisfies the weak null condition introduced by Lindblad and Rodnianski \cite{lindrodn,lindrodn2}. Lindblad \cite{lind} proved that, for small and localized initial data, this equation has a global solution. In the present work, we establish a modified scattering theory for the above equation. Such a modified scattering theory provides an accurate description of asymptotic behavior of the global solutions.To study modified scattering, we first identify a notion of asymptotic profile and an associated notion of scattering data. One candidate for the asymptotic profile is given by the asymptotic PDE\begin{align*}2U_{sq}+G(\omega)UU_{qq}=0\end{align*} which was derived by H\"{o}rmander \cite{horm2,horm,horm3}. In Chapter 2, we derive a new reduced system, called the \emph{geometric reduced system}, by modifying H\"{o}rmander's method. In our derivation, we make use of the optical function, i.e.\ a solution to the eikonal equation. In this setting, the scattering data is the initial data for our geometric reduced system, and it is chosen in a way such that the global solution to the quasilinear wave equation and the exact solution to the reduced system match at infinite time. One may infer, from this dissertation, that this new system is more accurate, in that it both describes the long time evolution and contains full information about it. In Chapter 3, we prove the existence of the modified wave operators for the scalar quasilinear wave equation. Fixing a scattering data which is the initial data for the geometric reduced system, we can first construct an approximate solution to the model equation. Then, by studying a backward Cauchy problem, we show that there exists a global solution to the scalar quasilinear wave equation which matches the approximate solution at infinite time.In Chapter 4, we prove the asymptotic completeness for the same equation. Given a global solution to the scalar quasilinear wave equation, we rigorously derive the geometric reduced system with error terms. These allow us to recover the scattering data, as well as to construct a matching exact solution to the reduced system
Dispelling the Myths Behind First-author Citation Counts
Full text linkWe conducted a full-scale evaluative citation analysis study of scholars in the XML research field to explore just how different from each other author rankings resulting from different citation counting methods actually are, and to demonstrate the capability of emerging data and tools on the Web in supporting more realistic citation counting methods. Our results contest some common arguments for the continued
use of first-author citation counts in the evaluation of scholars, such as high correlations between author rankings by first-author citation counts and other citation
counting methods, and high costs of using more realistic citation counting methods that are not well-supported by the ISI databases. It is argued that increasingly available digital full text research papers make it possible for citation analysis studies to go beyond what the ISI databases have directly supported and to employ more
sophisticated methods
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