Maximisers for Strichartz inequalities on the torus

Nonlinearity ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 311-342
Author(s):  
Oreoluwa Adekoya ◽  
John P Albert

Abstract We study the existence of maximisers for a one-parameter family of Strichartz inequalities on the torus. In general, maximising sequences can fail to be precompact in L 2 ( T ) , and maximisers can fail to exist. We provide a sufficient condition for precompactness of maximising sequences (after translation in Fourier space), and verify the existence of maximisers for a range of values of the parameter. Maximisers for the Strichartz inequalities correspond to stable, periodic (in space and time) solutions of a model equation for optical pulses in a dispersion-managed fiber.

2009 ◽  
Vol 18 (5) ◽  
pp. 691-705 ◽  
Author(s):  
GYÖRGY ELEKES ◽  
MIKLÓS SIMONOVITS ◽  
ENDRE SZABÓ

We give a very general sufficient condition for a one-parameter family of curves not to have n members with ‘too many’ (i.e., a near-quadratic number of) triple points of intersections. As a special case, a combinatorial distinction between straight lines and unit circles will be shown. (Actually, this is more than just a simple application; originally this motivated our results.)


Author(s):  
Héctor Manuel Ramos Romero ◽  
Miguel Ángel Sordo Díaz

<p>El orden de Lorenz es una herramienta adecuada para comparar la desigualdad de dos distribuciones de rentas. En este artículo obtenemos una condición suficiente para que dos distribuciones sean comparables en el orden de Lorenz y aplicamos el resultado para ordenar la familia de distribuciones Gamma triparamétricas.</p><p>Lorenz ordering is an useful tool for comparing inequality of income distributions. In this note, we give a sufficient condition for this order and then apply it to order the three parameter family of Gamma distributions.</p>


1983 ◽  
Vol 245 (4) ◽  
pp. R478-R483 ◽  
Author(s):  
A. Goldbeter ◽  
O. Decroly

The patterns of temporal self-organization in regulated biochemical systems are examined. Simple periodic oscillations are the most frequent type of such organization, as exemplified by glycolytic oscillations in yeast and muscle and by the periodic synthesis of adenosine 3',5'-cyclic monophosphate in Dictyostelium discoideum amoebas. These phenomena originate, respectively, from the periodic operation of the product-activated phosphofructokinase and adenylate cyclase reactions. The analysis of a model for a multiply regulated biochemical system shows more complex oscillatory phenomena, e.g., the coexistence between two stable periodic regimes for the same set of parameter values (birhythmicity) and chaos. The latter phenomenon of aperiodic oscillations occurs in a narrow range of parameter values and is much less frequent than simple or complex periodic behavior. It is suggested that a sufficient condition for the occurrence of birhythmicity and chaos in a regulated biological system subjected to a constant environment (i.e., in the absence of periodic forcing) may be the simultaneous presence and interaction of two mechanisms capable of producing oscillations.


Author(s):  
Dr. Jorge Vaca Uribe

Cualquiera que sea el enfoque adoptado para la enseñanza de la escritura, éste deberá lograr que todos los niños y jóvenes lleguen a dominar el sistema gráfico y ortográfico de su lengua, y para ello la didáctica prevista deberá otorgarle un espacio y un tiempo para su enseñanza, según el desarrollo de los proyectos y las secuencias didácticas planeadas. Comprender ese sistema gráfico y ortográfico es una condición necesaria, aunque no suficiente, para dominarlo, y dicha comprensión no debe confundirse con la memorización de las “reglas de ortografía” ni puede ser reemplazada por la tecnología de apoyo a la escritura, como los correctores ortográficos de los procesadores de texto, mediocres y poco sensibles al contexto sintáctico. Además, aunque se estén extendiendo usos relajados “del código” de escritura mediante ciertas tecnologías de la comunicación (en mensajes por celular, chats, blogs), los usuarios han de conocer todos los “registros” ortográficos y poder usarlos en los contextos apropiados. Finalmente, dicha comprensión es indispensable para entender y tomar una postura frente a las reformas ortográficas, más o menos frecuentes.Por lo anterior, compartimos con los maestros un conjunto de reflexiones y de tipos de actividades útiles para abordar la enseñanza de la ortografía del español en la escuela primaria, actividades cuya eficiencia se probó en la práctica con niños durante casi una década. Estos tipos de ejercicios son sólo una muestra, abierta para que el maestro pueda adaptarla, en contenido y forma, a la población de niños con la que trabaja. AbstractAny approach that could be adopted to teach writing must make that all children master the graphic and orthographic system of his language and, for that, the didactic design must include space and time enough for its teaching in accordance with the projects development and the planed didactic sequences. Understanding that graphic and orthographic system is a necessary but not sufficient condition to master it, and such comprehension must not be confused for the mechanic memorization of any set of "orthographic rules", either replaced by technological supports for writing such as word processors' spell checkers, particularly inefficient (in the case of Spanish writing) because of its lack of sensibility to syntactic context. Even if certain uses of "relaxed" orthographic codes are spread out by some communication platforms like cellular phones, chats and blogs, the users must handle all the orthographic registers and use them in appropriate contexts. Finally, such comprehension is indispensable for the treatment and adoption of a personal posture facing the more than less frequent orthographic reforms.So, we share with teachers a set of reflections and a set of types of activities useful for the teaching of the Spanish spelling at the primary school. The efficiency of those activities has been proved in practice with children during almost a decade.  These kinds of exercises are only an open sample that the teacher could adapt, in form and content, to the population he is working with.Recibido: 05 de octubre de 2012Aceptado: 01 de abril de 2013


Author(s):  
John Albert ◽  
Jaime Angulo Pava

We consider the coupled Schrödinger–Korteweg–de Vries system which arises in various physical contexts as a model for the interaction of long and short nonlinear waves. Ground states of the system are, by definition, minimizers of the energy functional subject to constraints on conserved functionals associated with symmetries of the system. In particular, ground states have a simple time dependence because they propagate via those symmetries. For a range of values of the parameters α, β, γ, δi, ci, we prove the existence and stability of a two-parameter family of ground states associated with a two-parameter family of symmetries.


The Ginzburg-Landau (GL) equation with real coefficients is a model equation appearing in superconductor physics and near-critical hydrodynamic stability problems. The stationary GL equation has a two-parameter ( I 1 , I 2 ) family of spatially quasi-periodic (QP) states with frequencies ( ω 1 , ω 2 ) and frequency map with determinant ∆ K = ∂( ω 1 , ω 2 ) / ∂( I 1 , I 2 ). In this paper the linear stability of these QP states is studied and an expression for the stability exponent is obtained which has a novel geometric interpretation in terms of ∆ K : when ∆ K < 0 the spatially QP state is unstable and ∆ K > 0 is a necessary but not sufficient condition for linear stability. There is an interesting relation between ∆ K and the KAM persistence theorem for invariant toroids.


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