1,355,233 research outputs found
Lax--Oleinik formula on networks
No full textWe provide a Lax-Oleinik-type representation formula for solutions of
time-dependent Hamilton-Jacobi equations, posed on a network with a rather
general geometry, under standard assumptions on the Hamiltonians. It depends on
a given initial datum at and a flux limiter at the vertices, which both
have to be assigned in order the problem to be uniquely solved. Previous
results in the same direction are solely in the frame of junction, namely
network with a single vertex. An important step to get the result is to define
a suitable action functional and prove existence as well as
Lipschitz-continuity of minimizers between two fixed points of the network in a
given time, despite the fact that the integrand lacks convexity at the
vertices.Comment: 28 page
Hyperbolicity and exponential convergence of the Lax–Oleinik semigroup
No full textAbstractFor a convex superlinear Lagrangian L:TM→R on a compact manifold M it is known that there is a unique number c such that the Lax–Oleinik semigroup Lt+ct:C(M,R)→C(M,R) has a fixed point. Moreover for any u∈C(M,R) the uniform limit u˜=limt→∞Ltu+ct exists.In this paper we assume that the Aubry set consists in a finite number of periodic orbits or critical points and study the relation of the hyperbolicity of the Aubry set to the exponential rate of convergence of the Lax–Oleinik semigroup
An Oleinik-type estimate for a convection–diffusion equation and convergence to N-waves
No full textAbstractIn this article we propose an Oleinik-type estimate for sign-changing solutions to a convection–diffusion equationut+(|u|γ/γ)x=μuxx,u(x,0)=u0(x),u,x∈R,1<γ⩽2,μ,t>0.Since the Oleinik entropy inequality holds for nonnegative solutions or inviscid case (μ=0) only, the theoretical progress for the case was limited. In this paper we show that its solution satisfies an Oleinik-type estimate,t2γux⩽C,1<γ⩽2,t>0,where C=C(u0,γ)>0. Using this estimate, the convergence to an N-wave is proved for sign changing solutions and the theoretical gap in asymptotic convergence of the corresponding problem is filled
Time-asymptotic stability of composite waves of degenerate Oleinik shock and rarefaction for non-convex conservation laws
No full textWe are concerned with the large-time behavior of the solution to one-dimensional (1D) cubic non-convex scalar viscous conservation laws. Due to the inflection point of the cubic non-convex flux, the solution to the corresponding inviscid Riemann problem can be the composite wave of a degenerate Oleinik shock and a rarefaction wave and these two nonlinear waves are always attached together. We give a first proof of the time-asymptotic stability of this composite wave, up to a time-dependent shift to the viscous Oleinik shock, for the viscous equation. The Oleinik shock wave strength can be arbitrarily large. The main difficulty is due to the incompatibility of the time-asymptotic stability proof framework of individual viscous shock by the so-called anti-derivative method and the direct -energy method to rarefaction wave. Here we develop a new type of -contraction method with suitable weight function and the time-dependent shift to the viscous shock, which is motivated by [9,12]. Another difficulty comes from that the Oleinik shock and rarefaction wave are always attached together and their wave interactions are very subtle. Therefore, the same time-dependent shift needs to be equipped to both Oleinik shock and rarefaction wave such that the wave interactions can be treated in our stability proof. Time-asymptotically, this shift function grows strictly sub-linear with respect to the time and then the shifted rarefaction wave is equivalent to the original self-similar rarefaction wave.45 pages, all comments are welcom
The horocycle flows from a Hamiltonian viewpoint
No full textIn questa tesi di laurea Magistrale ci occupiamo del flusso di orociclo e del flusso geodetico sul piano iperbolico e sui suoi quozienti, cercando di fornire una referenza quanto più possibile completa e esaustiva. Indicate con le coordinate standard sul piano iperbolico \HH, il flusso di orociclo può essere visto come il flusso di Eulero-Lagrange al livello di energia della Lagrangiana
dove è una opportuna 1-forma su \HH. Il livello di energia 1/2 è il livello critico di Mané per la Lagrangiana L; quindi abbiamo un cambiamento drastico nella dinamica oltrepassando questo valore. Nella sezione 5.2 dimostriamo infatti che per livelli di energia minori di 1/2 le soluzioni dell'equazione di Eulero-Lagrange associata a L sono chiuse; nella sezione 5.3 invece dimostriamo che il flusso di Eulero-Lagrange per un livello di energia sopracritico k è, a meno di riparametrizzazione, il flusso geodetico definito da una opportuna metrica Finsler. Inoltre, per livelli di energia alti, tale metrica tende a quella iperbolica. A livello critico il flusso di Eulero-Lagrange di L è appunto il flusso di orociclo e l'equazione di Hamilton-Jacobi associata è data da
Nella sezione 5.4 ci occupiamo di determinare le soluzioni di tale PDE utilizzando un approccio geometrico, il quale consiste nel determinare i grafici lagrangiani invarianti contenuti nel livello di energia 1/2 e vedere quali di essi sono esatti, cioè definiti da una 1-forma esatta. Se infatti è un grafico lagrangiano esatto allora è soluzione dell'equazione di Hamilton-Jacobi; utilizzando questo metodo dimostriamo che tutte le soluzioni della PDE sopra sono costanti oppure della forma
In particolare otteniamo la seguente proprietà: se è una 1-forma su \HH tale che il suo grafico è invariante e contenuto nel livello H=1/2 allora è esatta. In questo caso particolare vale dunque il viceversa del generale teorema 3.1 (Hamilton-Jacobi), il quale afferma che se è una sottovarietà Lagrangiana (in particolare quindi un grafico) tale che è costante su allora è invariante (si ricordi che essendo \HH semplicemente connesso, ogni 1-forma chiusa è anche esatta). Nella sezione 5.5, utilizzando un metodo completamente analogo a quello utilizzato in 5.4, determiniamo alcune (a priori non tutte) soluzioni dell'equazione di Hamilton-Jacobi
relativa al flusso geodetico sul piano iperbolico.
I primi capitoli sono dedicati a richiamare la teoria generale; per prima cosa ricordiamo le definizioni di Lagrangiana e Hamiltoniana e introduciamo i rispettivi formalismi e alcuni esempi significativi. Nel secondo capitolo passiamo a sviluppare la teoria dei valori critici di Mané per una Lagrangiana , dove M è una varietà riemanniana connessa e senza bordo e TM è il suo fibrato tangente, e delle loro connessioni con le misure minimizzanti. Nel capitolo 3 invece ci occupiamo del punto di vista Hamiltoniano, introducendo l'equazione di Hamilton-Jacobi, i concetti di soluzione e sottosoluzione, di grafico lagrangiano (esatto); tralasciando alcuni dettagli tecnici forniamo una dimostrazione del seguente importante
Teorema
Sia M un qualsiasi rivestimento di una varietà chiusa e sia c(L) il valore critico di Mané relativo alla Lagrangiana . Allora vale
c(L) = inf {k\in \R | esiste u\in C^\infty(M,\R) tale che H(x,du(x))<k}.
Nel capitolo 4 introduciamo il piano iperbolico e ne riassumiamo le principali proprietà; definiamo quindi il flusso geodetico e il flusso di orociclo sul piano iperbolico e sui suoi quozienti (ovvero sulle superfici iperboliche); infine nella sezione 4.4 ripercorriamo brevemente la dimostrazione del classico
Teorema[Hedlund, 1936]
Se M è una superficie iperbolica compatta allora ogni orbita del flusso di orociclo è densa in
Going Beyond Counting First Authors in Author Co-citation Analysis
No full textThe 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
A quantitative version of the Hopf–Oleinik lemma for a quasilinear non-uniformly elliptic operator
No full textThis paper establishes a quantitative version of the Hopf–Oleinik lemma (HOL) for a quasilinear non-uniformly elliptic operator of the form . One key point in the proof is the passage from non-uniformly elliptic operators to locally uniformly ones via a new, uniform, and, rescaled version of the gradient estimate obtained by Evans and Smart for solutions to a family of non-uniformly quasilinear elliptic operators.
 
Variations on the Author
No full text“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
Herglotz' variational principle and Lax-Oleinik evolution
No full textWe develop an elementary method to give a Lipschitz estimate for the minimizers in the problem of Herglotz’ variational principle proposed in [17] in the time- dependent case. We deduce Erdmann’s condition and the Euler-Lagrange equation sep- arately under different sets of assumptions, by using a generalized du Bois-Reymond lemma. As an application, we obtain a representation formula for the viscosity solution of the Cauchy problem for the Hamilton-Jacobi equation
Dtu(t, x) + H(t, x, Dxu(t, x), u(t, x)) = 0 and study the related Lax-Oleinik evolution
Appropriate Similarity Measures for Author Cocitation Analysis
No full textWe 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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