Rearrangement transformations on general measure spaces

For a general set transformation ${\cal R}$ between two measure spaces, we define the rearrangement of a measurable function by means of the Layer's cake formula. We study some functional properties of the Lorentz spaces defined in terms of ${\cal R}$, giving a unified approach to the classical rearrangement, Steiner's symmetrization, the multidimensional case, and the discrete setting of trees.Comment: 17 page

Lorentz and Gale-Ryser theorems on general measure spaces

Based on the Gale-Ryser theorem for the existence of suitable $(0,1)$-matrices for different partitions of a natural number, we revisit the classical result of G. G. Lorentz regarding the characterization of a plane measurable set, in terms of its cross sections, and extend it to general measure spaces

Positive isometric averaging operators on l2(Z,µ)

© 2016 Springer International Publishing We show that positive isometric averaging operators on the sequence space (Formula presented.) are determined by very subtle arithmetic conditions on (Formula presented.) (even for very simple examples), contrary to what happens in the continuous case (Formula presented.), where any possible average value is realized by a suitable positive isometry.Postprint (author's final draft

Norm estimates for the hardy operator in terms of B-p weights

First published in Proceedings of the American Mathematical Society in 145 (2017), 2455-246, published by the American Mathematical SocietyWe study the explicit dependence of the Bp-constant of the weight, [w]Bp , in the estimates of the norm of the Hardy operator acting on non-increasing functions in L p (w) or L p,8(w).Postprint (author's final draft

Weak-type boundedness of the Fourier transform on rearrangement invariant function spacest

We study several questions about the weak-type boundedness of the Fourier transform F on rearrangement invariant spaces. In particular, we characterize the action of F as a bounded operator from the minimal Lorentz space ¿(X) into the Marcinkiewicz maximal space M(X), both associated with a rearrangement invariant space X. Finally, we also prove some results establishing that the weak-type boundedness of F, in certain weighted Lorentz spaces, is equivalent to the corresponding strong-type estimates.Postprint (author's final draft

Averaging operators on decreasing or positive functions: equivalence and optimal bounds

We study the optimal bounds for the Hardy operator S minus the identity, as well as S and its dualoperator S*, on the full range 1 = p = 8, for the cases of decreasing, positive or general functions (infact, these two kinds of inequalities are equivalent for the appropriate cone of functions). For 1< p = 2, we prove that all these estimates are the same, but for 2 < p <8, they exhibit a completely different behavior.Postprint (author's final draft

Las necesidades del aprendizaje de español, con fines de uso turístico, de los turistas extranjeros durante sus vacaciones en Benicàssim

Treballs Finals del Màster de Formació de Professor d'Espanyol com a Llengua Estrangera, Facultat d'Educació, Universitat de Barcelona, Curs: 2017-2018, Tutor: Jaume Batlle Rodríguez[spa] La enseñanza de español con fines específicos es una disciplina relativamente joven y tiene unas características que la diferencian de la enseñanza de español en general. Este tipo de enseñanza está cobrando cada vez más importancia por motivos sociopolíticos y porque cada vez hay más personas interesadas en aprender un idioma enfocado a un fin específico. En este trabajo presentamos como objetivo principal indagar sobre las necesidades del aprendizaje de español, con fines de uso turístico, de los turistas extranjeros en Benicàssim durante sus vacaciones. Pretendemos determinar qué tipo de necesidades tienen los turistas en función de sus diferentes perfiles. Para ello creamos un cuestionario y entrevistamos a 120 turistas extranjeros en diferentes localizaciones. Parte del cuestionario presenta una recogida de datos de tipo etnográfico, lo cual nos ha ayudado a interpretar cómo puede influir la nacionalidad, la profesión, la edad e incluso el género, a la hora de tomar una determinada actitud hacia el aprendizaje de español. Los resultados de esta investigación nos han llevado a plantearnos una nueva línea de investigación: una propuesta didáctica para pilotarla en los meses de julio y agosto. El interés de este trabajo radica en nuestra observación directa de la comunicación que se establece entre los turistas extranjeros y los residentes de Benicàssim, especialmente el personal del sector servicios. Esta comunicación, en muchas ocasiones, es bastante complicada y se crean malentendidos debido a aspectos culturales y a la falta de conocimiento de español de muchos de los turistas.[eng] The Spanish teaching for specific purposes is a relatively young discipline and it has many characteristics that differentiate them from the general Spanish teaching. This kind of teaching is taking more and more importance due to sociopolitical issues and because nowadays, more a more people are interested in learning a language focused on specific purposes. In this paper our main goal is to inquire about the Spanish learning needs, for tourism purposes, of the foreigner tourists during their holidays in Benicasim. We pretend to determine what kind of needs they have according to their different profiles. For doing so, we created a questionnaire and we interviewed 120 foreign tourists in different locations. Part of this questionnaire presents an ethnographic data collection, which has helped us interpret how nationality, profession, age and even gender can influence for taking a particular attitude towards the learning of the Spanish language. The outcomes of this research have taken us to arise a new line of investigation: a didactic proposal for putting it into practice next July and August. The interest of this research is originated from our direct observation about the communication that is established between the foreigner tourist and the residents of Benicasim, especially the service sector personnel. This communication, in many occasions, is quite complicated and many misunderstandings are crated due to the lack of Spanish knowledge of many of the tourists

The effect of wall slip on the dewetting of ultrathin films on solid substrates: linear instability and second-order lubrication theory

The influence of wall slip on the instability of a non-wetting liquid film placed on a solid substrate is analyzed in the limit of negligible inertia. In particular, we focus on the stability properties of the film, comparing the performance of the three lubrication models available in the literature, namely, the weak, intermediate, and strong slip models, with the Stokes equations. Since none of the aforementioned leading-order lubrication models is shown to be able to predict the growth rate of perturbations for the whole range of slipping lengths, we develop a parabolic model able to accurately predict the linear dynamics of the film for arbitrary slip lengths.This research was funded by the Spanish MINECO, Subdirección General de Gestión de Ayudas a la Investigación, through Project No. RED2018-102829-T and by the Spanish MCIU-Agencia Estatal de Investigación through Project No. DPI2017-88201-C3-3-R, partly financed through FEDER European funds. A.M.-C. also acknowledges support from the Spanish MECD through Grant No. FPU16/02562.Publicad

Universal Free-Fall Law for Liquid Jets under Fully Developed Injection Conditions

We show that verticals lenderjets of liquid injected in air with a fully developed outlet velocity profile have a universal shape in the common case in which the viscous force is much smaller than the gravitational force. The theory of ideal flows with vorticity provides an analytical solution that, under negligible surface tension forces, predicts RjðZÞ¼½ð1þZ=4Þ1=2−ðZ=4Þ1=21=2, where Rj is the jet radius scaled with the injector radius and Z is the vertical distance scaled with the gravitational length, lg¼u2 o=2g, where uo is the mean velocity at the injector outlet and g is the gravitational acceleration. In contrast with Mariotte’s law, Rj¼ð1þZÞ−1=4, previously reported experiments employing long injectors collapse almost perfectly on to the new solution.The authors thank the Spanish MCIU-Agencia Estatal de Investigación through Project No. PID2020–115655GB-C22, partially financed through FEDER European funds

The role of inertia in the rupture of ultrathin liquid films

Theory and numerical simulations of the Navier–Stokes equations are used to unravel the influence of inertia on the dewetting dynamics of an ultrathin film of Newtonian liquid deposited on a solid substrate. A classification of the self-similar film thinning regimes at finite Ohnesorge numbers is provided, unifying previous findings. We reveal that, for Ohnesorge numbers smaller than one, the structure of the rupture singularity close to the molecular scales is controlled by a balance between liquid inertia and van der Waals forces, leading to a self-similar asymptotic regime with hmin ∝ τ2/5 as τ → 0, where hmin is the minimum film thickness and τ is the time remaining before rupture. The flow exhibits a three-region structure comprising an irrotational core delimited by a pair of boundary layers at the wall and at the free surface. A potential-flow description of the irrotational core is provided, which is matched with the vortical layers, allowing us to present a complete parameter-free asymptotic description of inertia-dominated film rupture.This research was funded by the Spanish MINECO, Subdirección General de Gestión de Ayudas a la Investigación, through Project No. RED2018-102829-T and by the Spanish MCIU-Agencia Estatal de Investigación through Project No. DPI2017-88201-C3-3-R, partly financed through FEDER European funds. A.M.-C. also acknowledges support from the Spanish MECD through the Grant No. FPU16/02562.Publicad