Integrable (3+1)-dimensional generalization for dispersionless Davey--Stewartson system

This paper introduces a (3+1)-dimensional dispersionless integrable system, utilizing a Lax pair involving contact vector fields, in alignment with methodologies presented by A. Sergyeyev in 2018. Significantly, it is shown that the proposed system serves as an integrable (3+1)-dimensional generalization of the well-studied (2+1)-dimensional dispersionless Davey-Stewartson system. This way, an interesting new example on integrability in higher dimensions is presented, with potential applications in modern mathematical physics. The work lays the foundation for future research into symmetries, conservation laws, and Hamiltonian structures, offering avenues for further exploration

Integration of differential equations by C∞\mathcal{C}^{\infty}-structures

Several integrability problems of differential equations are addressed by using the concept of $\mathcal{C}^{\infty}$-structure, a recent generalization of the notion of solvable structure. Specifically, the integration procedure associated with $\mathcal{C}^{\infty}$-structures is used to integrate to a Lotka-Volterra model and several differential equations that lack sufficient Lie point symmetries and cannot be solved using conventional methods

Reducing depopulation in rural Spain: The impact of immigration

The attraction of foreign-born immigrants to rural areas in developed countries has aroused growing interest in recent years. The central issue in this study is the demographic impact of immigration in rural Spain, focusing on depopulated areas. The economic and demographic consequences of depopulation have become major concerns, and the arrival of international migrants has come to be seen as a possible solution. The aim of this study is to add to a literature in which qualitative research and local or regional perspectives predominate. The present research draws on quantitative findings for a significant part of Spain. The evidence in this study is principally based on population figures for the last years of the 20th century, a period of low immigration to Spain, and the early years of the 21st century, when the inflow of foreign migrants gathered intensity. We also explore the early consequences of the present economic crisis, which began in 2008. The analysis is based on estimates of native and foreign-born population growth for a range of territorial aggregations. Counterfactual techniques are also used. The results show that the arrival of immigrants has so far contributed substantially to reducing and even halting or reversing depopulation. A further series of analyses concentrates on the potential of rural areas to retain immigrants in the long run. The study also recommends a comprehensive policy approach in this regard

C∞\mathcal{C}^{\infty}-structures in the integration of involutive distributions

For a system of ordinary differential equations (ODEs) or, more generally, an involutive distribution of vector fields, the problem of its integration is considered. Among the many approaches to this problem, solvable structures provide a systematic procedure of integration via Pfaffian equations that are integrable by quadratures. In this paper structures more general than solvable structures (named cinf-structures) are considered. The symmetry condition in the concept of solvable structure is weakened for cinf-structures by requiring their vector fields be just cinf-symmetries. For cinf-structures there is also an integration procedure, but the corresponding Pfaffian equations, although completely integrable, are not necessarily integrable by quadratures. The well-known result on the relationship between integrating factors and Lie point symmetries for first-order ODEs is generalized for cinf-structures and involutive distributions of arbitrary corank by introducing symmetrizing factors. The role of these symmetrizing factors on the integrability by quadratures of the Pfaffian equations associated with the \cinf-structure is also established. Some examples that show how these objects and results can be applied in practice are also presented

C∞\mathcal{C}^{\infty}-symmetries of distributions and integrability

An extension of the notion of solvable structure for involutive distributions of vector fields is introduced. The new structures are based on a generalization of the concept of symmetry of a distribution of vector fields, inspired in the extension of Lie point symmetries to $\mathcal{C}^{\infty}$-symmetries for ODEs developed in the recent years. These new objects, named $\mathcal{C}^{\infty}$-structures, play a fundamental role in the integrability of the distribution: the knowledge of a $\mathcal{C}^{\infty}$-structure for a corank $k$ involutive distribution permits to find its integral manifolds by solving $k$ successive completely integrable Pfaffian equations. These results have important consequences for the integrability of differential equations. In particular, we derive a new procedure to integrate an $m$th-order ordinary differential equation by splitting the problem into $m$ completely integrable Pfaffian equations. This step-by-step integration procedure is applied to integrate completely several equations that cannot be solved by standard procedures

Variational formulation of partial differential equations

This paper deals with the study of the variational method for partial differential equations concerning the existence, uniqueness and regularity of the solution. The aim of this work is to give a comprehensive description of the variational method, presenting examples from the simple second order linear elliptic partial differential equations to a most complex first order non-linear partial differential equation. Comments on the adaptability of this method to this kind of equations are given.En este trabajo se presenta una descripción del método variacional, el cual que se utiliza para el estudio cualitativo de ecuaciones diferenciales parciales: existencia, unicidad y regularidad de la solución. Se exhibe como ilustración el análisis de ecuaciones diferenciales parciales elípticas lineales de segundo orden, así como el estudio de una ecuación diferencial parcial hiperbólica no lineal de primer orden, en la cual se muestra la adaptabilidad del método. &nbsp

Variational formulation of partial differential equations

This paper deals with the study of the variational method for partial differential equations concerning the existence, uniqueness and regularity of the solution. The aim of this work is to give a comprehensive description of the variational method, presenting examples from the simple second order linear elliptic partial differential equations to a most complex first order non-linear partial differential equation. Comments on the adaptability of this method to this kind of equations are given.En este trabajo se presenta una descripción del método variacional, el cual que se utiliza para el estudio cualitativo de ecuaciones diferenciales parciales: existencia, unicidad y regularidad de la solución. Se exhibe como ilustración el análisis de ecuaciones diferenciales parciales elípticas lineales de segundo orden, así como el estudio de una ecuación diferencial parcial hiperbólica no lineal de primer orden, en la cual se muestra la adaptabilidad del método. &nbsp

Acacia Horrida (L.) Willd.: refugio de artrópodos benéficos en la costa peruana

El huarango (Acacia horrida (L.) Willd., 1806) es una leguminosa arbustiva utilizada como cerco vivo en áreas agrícolas para prevenir la erosión, mejorar la nutrición del suelo y servir, además, como refugio para artrópodos benéficos, contribuyendo así a la sostenibilidad de los agroecosistemas productivos. Por ello, se quiso conocer las especies de artrópodos benéficos asociados a A. horrida en agroecosistemas de la costa centro y sur del Perú. Para ello, se colectó especímenes en cercos vivos de A. horrida cercanos a cultivos de hortalizas de La Molina (Lima), campos de mandarina y palto en Cañete (Lima) y huertos caseros con camote y frutales en Los Aquijes (Ica). Los resultados obtenidos permitieron encontrar en La Molina arañas Salticidae y Argiope sp., insectos depredadores como Harmonia axyridis Pallas, 1773, Cycloneda sanguinea Linnaeus, 1743, Scymnus rubicundus Erichson, 1847 y parasitoides del género Bracon. En Cañete se encontró la araña Gasteracantha cancriformis Linnaeus, 1758, insectos depredadores como C. sanguinea, S. rubicundus, Ceraeochrysa cincta (Schneider, 1851), Allograpta sp., Tachycompilus sp., y parasitoides como Venturia sp., Campoletis sp. Anomalon sinuatum Morley, 1912, subfamilias Cryptinae, Campopleginae (Ichneumonidae), Braconinae, Microgastrinae, Opiinae (Braconidae) y la familia Eulophidae. En Los Aquijes se encontró C. sanguinea, Hippodamia convergens Guérin-Méneville, 1842, Polistes sp. y parasitodes del género Bracon, siendo estos últimos depredados por arañas de la familia Thomisidae. Se concluye que al menos 22 taxa de artrópodos benéficos están asociados a A. horrida como refugio