Topological and algebraic structures on the ring of Fermat reals

The ring of Fermat reals is an extension of the real field containing nilpotent infinitesimals, and represents an alternative to Synthetic Differential Geometry in classical logic. In the present paper, our first aim is to study this ring from using standard topological and algebraic structures. We present the Fermat topology, generated by a complete pseudo-metric, and the omega topology, generated by a complete metric. The first one is closely related to the differentiation of (non standard) smooth functions defined on open sets of Fermat reals. The second one is connected to the differentiation of smooth functions defined on infinitesimal sets. Subsequently, we prove that every (proper) ideal is a set of infinitesimals whose order is less than or equal to some real number. Finally, we define and study roots of infinitesimals. A computer implementation as well as an application to infinitesimal Taylor formulas with fractional derivatives are presented.Comment: 43 page

Categorical frameworks for generalized functions

We tackle the problem of finding a suitable categorical framework for generalized functions used in mathematical physics for linear and non-linear PDEs. We are looking for a Cartesian closed category which contains both Schwartz distributions and Colombeau generalized functions as natural objects. We study Fr\"olicher spaces, diffeological spaces and functionally generated spaces as frameworks for generalized functions. The latter are similar to Fr\"olicher spaces, but starting from locally defined functionals. Functionally generated spaces strictly lie between Fr\"olicher spaces and diffeological spaces, and they form a complete and cocomplete Cartesian closed category. We deeply study functionally generated spaces (and Fr\"olicher spaces) as a framework for Schwartz distributions, and prove that in the category of diffeological spaces, both the special and the full Colombeau algebras are smooth differential algebras, with a smooth embedding of Schwartz distributions and smooth pointwise evaluations of Colombeau generalized functions.Comment: 38 page

The spatial sorting and matching of skills and firms

Using a matched employer-employee database for Italy we look at the spatial distribution of wages across provinces. This rich database allows us to contribute at opening the black box of agglomeration economies exploiting the micro dimension of the interaction among economic agents, both individuals and firms. We provide evidence that firm size and particularly skills are sorted across space, and explain a large portion of the spatial wage variation that could otherwise be attributed to aggregate proxies of agglomeration externalities. Our data further support the assortative matching hypothesis, that we show not to be driven by co-location of "good" workers and firms. Finally, we point out that this assortative matching is negatively related to local market size.Spatial Externalities; Panel-Data; Skills; Firms' Heterogeneity; Sorting; Matching

Oxygen control and improved denitrification efficiency by means of a post-anoxic reactor

The presence of dissolved oxygen (DO) in biological denitrification reactors determines inhibition effects on the denitrification rate. The article shows the results of an experimental study to control the DO concentration in the pre-denitrification stage by a post-anoxic reactor. The results demonstrate that the post-anoxic reactor is very effective in improving the nitrogen removal efficiency because it causes a considerable reduction of the DO content in the mixed liquor recycle sent to the pre-denitrification reactor. This reduction is influenced by both the retention time and the F:M ratio (referred to the denitrification and the oxidation-nitrification volume). In fact, a retention time and a F:M ratio equal to 1.5 h and 0.130 kgBOD5 kgMLVSS−1·day−1, respectively, allow to limit DO in the post-anoxic reactor at 0.31 mgO2·L−1. Such concentration determines a DO concentration of 0.11 mgO2·L−1 in the pre-denitrification reactor and, consequently, a denitrification efficiency of 91%. Moreover, the contribution of the endogenous denitrification to the whole denitrification efficiency was found negligible. The paper contributes to the progress in nitrogen removal from sewage, a fundamental issue for a sustainable management of water resources