Beginner's guide to Aggregation-Diffusion Equations

The aim of this survey is to serve as an introduction to the different techniques available in the broad field of Aggregation-Diffusion Equations. We aim to provide historical context, key literature, and main ideas in the field. We start by discussing the modelling and famous particular cases: Heat equation, Fokker-Plank, Porous medium, Keller-Segel, Chapman-Rubinstein-Schatzman, Newtonian vortex, Caffarelli-V\'azquez, McKean-Vlasov, Kuramoto, and one-layer neural networks. In Section 4 we present the well-posedness frameworks given as PDEs in Sobolev spaces, and gradient-flow in Wasserstein. Then we discuss the asymptotic behaviour in time, for which we need to understand minimisers of a free energy. We then present some numerical methods which have been developed. We conclude the paper mentioning some related problems

Linear diffusion with singular absorption potential and/or unbounded convective flow: the weighted space approach

In this paper we prove the existence and uniqueness of very weak solutions to linear diffusion equations involving a singular absorption potential and/or an unbounded convective flow on a bounded open set of $\mathbb R^N$. In most of the paper we consider homogeneous Dirichlet boundary conditions but we prove that when the potential function grows faster than the distance to the boundary to the power -2 then no boundary condition is required to get the uniqueness of very weak solutions. This result is new in the literature and must be distinguished from other previous results in which such uniqueness of solutions without any boundary condition was proved for degenerate diffusion operators (which is not our case). Our approach, based on the treatment on some distance to the boundary weighted spaces, uses a suitable regularity of the solution of the associated dual problem which is here established. We also consider the delicate question of the differentiability of the very weak solution and prove that some suitable additional hypothesis on the data is required since otherwise the gradient of the solution may not be integrable on the domain

Proyecto de investigación: modelo de generación de valor teniendo en cuenta una prospectiva financiera basada en el análisis externo e interno organizacional encaminado a la toma de decisiones financieras. estudio de caso: Empresa Guillermo Pulgarin S.S.A. del subsector confección

La presente propuesta de investigación está encaminada al planteamiento de un modelo de generación de valor, donde se sustenta metodológicamente los pasos para el análisis financiero, el análisis externo e interno organizacional a partir del diseño y aplicación de una matriz DOFA sustentada en dos variables cuantitativas: prioridad e impacto, la primera en términos de focalizar las cuentas objeto de análisis y la segunda variable producto de la comparación financiera con empresas del sector a nivel regional y nacional, comparación obtenida a través de la simulación de distribuciones de probabilidad generada a partir de indicadores financieros de las empresas con reporte de estados financieros ante la Superintendencia de Sociedades Colombiana en los años 2013-2015, con el fin de determinar una medida de oportunidad o riesgo para la empresa analizada, el caso de estudio se realizó en la organización Guillermo Pulgarín S. S.A. del subsector confección de Risaralda. Finalmente, a partir del cuadrante de direccionamiento estratégico donde se ubica la empresa se sugiere una serie de tácticas financieras, las cuales son aplicadas y evaluadas en términos de generación de valor. El modelo propuesto aporta a la necesidad de contar con una herramienta encaminada a la toma de decisiones financieras, reduciendo la subjetividad en el direccionamiento financiero para focalizar y priorizar las estrategias empresariales, como respuesta a las dinámicas actuales de mercado y la globalización

Interpreting systems of continuity equations in spaces of probability measures through PDE duality

We introduce a notion of duality solution for a single or a system of transport equations in spaces of probability measures reminiscent of the viscosity solution notion for nonlinear parabolic equations. Our notion of solution by duality is, under suitable assumptions, equivalent to gradient flow solutions in case the single/system of equations has this structure. In contrast, we can deal with a quite general system of nonlinear non-local, diffusive or not, system of PDEs without any variational structure

Interpreting systems of continuity equations in spaces of probability measures through PDE duality

We introduce a notion of duality solution for a single or a system of transport equations in spaces of probability measures reminiscent of the viscosity solution notion for nonlinear parabolic equations. Our notion of solution by duality is, under suitable assumptions, equivalent to gradient flow solutions in case the single/system of equations has this structure. In contrast, we can deal with a quite general system of nonlinear non-local, diffusive or not, system of PDEs without any variational structure

Defining a Benchmark Suite for Evaluating the Import of OWL Lite Ontologies

SemanticWeb tools should be able to correctly interchange ontologies and, therefore, to interoperate. This interchange is not always a straightforward task if tools have different underlying knowledge representation paradigms. This paper describes the process followed to define a benchmark suite for evaluating the OWL import capabilities of ontology development tools in a benchmarking activity in progress in the Knowledge Web European Network of Excellence