1,721,117 research outputs found
Gradient flows and Evolution Variational Inequalities in metric spaces. I: Structural properties
No full textThis is the first of a series of papers devoted to a thorough analysis of the class of gradient flows in a metric space (X,d) that can be characterized by Evolution Variational Inequalities (EVI). We present new results concerning the structural properties of solutions to the EVI formulation, such as contraction, regularity, asymptotic expansion, precise energy identity, stability, asymptotic behavior and their link with the geodesic convexity of the driving functional. Under the crucial assumption of the existence of an EVI gradient flow, we will also prove two main results: – the equivalence with the De Giorgi variational characterization of curves of maximal slope; – the convergence of the Minimizing Movement-JKO scheme to the EVI gradient flow, with an explicit and uniform error estimate of order 1/2 with respect to the step size, independent of any geometric hypothesis (as upper or lower curvature bounds) on d. In order to avoid any compactness assumption, we will also introduce a suitable relaxation of the Minimizing Movement algorithm obtained by the Ekeland variational principle, and we will prove its uniform convergence as well
A Monte Carlo approach to the worldline formalism in curved space
Full text linkWe present a numerical method to evaluate worldline (WL) path integrals defined on a curved Euclidean space, sampled with Monte Carlo (MC) techniques. In particular, we adopt an algorithm known as YLOOPS with a slight modification due to the introduction of a quadratic term which has the function of stabilizing and speeding up the convergence. Our method, as the perturbative counterparts, treats the non-trivial measure and deviation of the kinetic term from flat, as interaction terms. Moreover, the numerical discretization adopted in the present WLMC is realized with respect to the proper time of the associated bosonic point-particle, hence such procedure may be seen as an analogue of the time-slicing (TS) discretization already introduced to construct quantum path integrals in curved space. As a result, a TS counter-term is taken into account during the computation. The method is tested against existing analytic calculations of the heat kernel for a free bosonic point-particle in a D-dimensional maximally symmetric space
On the equivalence of Sobolev norms in Malliavin spaces
Full text linkWe investigate the problem of the equivalence of L-q-Sobolev norms in Malliavin spaces for q is an element of [1, infinity), focusing on the graph norm of the k-th Malliavin derivative operator and the full Sobolev norm involving all derivatives up to order k, where k is any positive integer. The case q = 1 in the infinite-dimensional setting is challenging, since at such extreme the standard approach involving Meyer's inequalities fails. In this direction, we are able to establish the mentioned equivalence for q = 1 and k = 2 relying on a vector-valued Poincare inequality that we prove independently and that turns out to be new at this level of generality, while for q = 1 and k > 2 the equivalence issue remains open, even if we obtain some functional estimates of independent interest. With our argument (that also resorts to the Wiener chaos) we are able to recover the case q is an element of [1, infinity) in the infinite-dimensional setting; the latter is known since the eighties, however our proof is more direct than those existing in the literature, and allows to give explicit bounds on all the multiplying constants involved in the functional inequalities. Finally, we also deal with the finite-dimensional case for all q is an element of [1, infinity) (where the equivalence, without explicit constants, follows from standard compactness arguments): our proof in such setting is much simpler, relying on Gaussian integration-by-parts formulas and an adaptation of Sobolev inequalities in Euclidean spaces, and it provides again quantitative bounds on the multiplying constants, which however blow up when the dimension diverges to infinity (whence the need for a different approach in the infinite-dimensional setting). (c) 2022 Elsevier Inc. All rights reserved
Wasserstein stability of porous medium-type equations on manifolds with Ricci curvature bounded below
Full text linkGiven a complete, connected Riemannian manifold Mn with Ricci curvature bounded from below, we discuss the stability of the solutions of a porous medium-type equation with respect to the 2-Wasserstein distance. We produce (sharp) stability estimates under negative curvature bounds, which to some extent generalize well-known results by Sturm [35] and Otto-Westdickenberg [32]. The strategy of the proof mainly relies on a quantitative L1–L∞ smoothing property of the equation considered, combined with the Hamiltonian approach developed by Ambrosio, Mondino and Savaré in a metric-measure setting [4]
THE POROUS MEDIUM EQUATION WITH LARGE DATA ON CARTAN-HADAMARD MANIFOLDS UNDER GENERAL CURVATURE BOUNDS
No full textWe consider very weak solutions of the Cauchy problem for the porous medium equation on Cartan-Hadamard manifolds, that are assumed to satisfy general curvature bounds and to be stochastically complete. We identify a class of initial data that can grow at infinity at a prescribed rate, which depends on the assumed curvature bounds through an integral function, such that the corresponding solution exists at least on [0,T] for a suitable T > 0. The maximal existence time T is estimated in terms of a suitable weighted norm of the initial datum. Our results are sharp, in the sense that slower growth rates yield global existence, whereas one can construct data with critical growth for which the corresponding solutions blow up in finite time. Under further assumptions, uniqueness of very weak solutions is also proved, in the same growth class
Nonlinear characterizations of stochastic completeness
Full text linkWe prove that conservation of probability for the free heat semigroup on a Riemannian manifold M (namely stochastic completeness), hence a linear property, is equivalent to uniqueness of positive, bounded solutions to nonlinear evolution equations of fast diffusion type on M of the form ut=Δφ(u), φ being an arbitrary concave, increasing, positive function, regular outside the origin and with φ(0)=0. Either property is also equivalent to nonexistence of nonnegative, nontrivial, bounded solutions to the elliptic equation ΔW=φ−1(W), with φ as above. As a consequence, explicit criteria for uniqueness or nonuniqueness of bounded solutions to fast diffusion-type equations on manifolds are given, these being the first results on such issues
Nonexistence of Radial Optimal Functions for the Sobolev Inequality on Cartan-Hadamard Manifolds
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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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