Mathematical analysis of a discrete fracture model coupling Darcy flow in the matrix with Darcy-Forchheimer flow in the fracture

We consider a model for flow in a porous medium with a fracture in which the flow in the fracture is governed by the Darcy-Forchheimer law while that in the surrounding matrix is governed by Darcy's law. We give an appropriate mixed, variational formulation and show existence and uniqueness of the solution. To show existence we give an analogous formulation for the model in which the Darcy-Forchheimer law is the governing equation throughout the domain. We show existence and uniqueness of the solution and show that the solution for the model with Darcy's law in the matrix is the weak limit of solutions of the model with the Darcy-Forchheimer law in the entire domain when the Forchheimer coefficient in the matrix tends toward zero

Maggay\u27s Global kingdom, global people: Living faithfully in a multicultural world

Ce numéro de Trivium est consacré à un aspect de la réflexion de l’un des plus grands représentants de la pensée sociale en France, Emile Durkheim (1858-1917). Aussi bien en dehors qu’au dedans des cercles de spécialistes, on attribue communément à Durkheim la fondation de la sociologie comme discipline scientifique autonome ; on l’évoque comme l’un des « pères fondateurs », et aussi comme l’un des plus éminents représentants, de cette branche des sciences sociales. Qu’il existe un rapport ét..

Local statistics of lattice points on the sphere

A celebrated result of Legendre and Gauss determines which integers can be represented as a sum of three squares, and for those it is typically the case that there are many ways of doing so. These different representations give collections of points on the unit sphere, and a fundamental result, conjectured by Linnik, is that under a simple condition these become uniformly distributed on the sphere. In this note we survey some of our recent work, which explores what happens beyond uniform distribution, giving evidence to randomness on smaller scales. We treat the electrostatic energy, local statistics such as the point pair statistic (Ripley's function), nearest neighbour statistics, minimum spacing and covering radius. We briefly discuss the situation in other dimensions, which is very different. In an appendix we compute the corresponding quantities for random pointsComment: 2 figures. Included reviewer comments. Accepted for the Contemporary Mathematics proceedings of Constructive Functions 201

Social Security Investment in Equities I: Linear Case

Social Security trust fund portfolio diversification to include some equities reduces the equity premium by raising the safe real interest rate. This requires changes in taxes. Under the hypothesis of constant marginal returns to risky investments, trust fund diversification lowers the price of land, increases aggregate investment, and raises the sum of household utilities, suitably weighted. It makes workers who do not own equities on their own better off, though it may hurt some others since changed taxes and asset values redistribute wealth across contemporaneous households and across generations. In our companion paper we reconsider the effects of diversification when there are decreasing marginal returns to safe and risky investment. Our analysis uses a two-period overlapping generations general equilibrium model with two types of agents, savers and workers who do not save. The latter represent approximately half of all workers who hold no equities whatsoever.