A topos of presheaves can be seen as an extension of classical set theory, where sets vary over informational states, therefore it is a powerful and expressive mathematical framework. I introduce a suitable topos of presheaves where imprecise probabilities and imprecise probabilistic reasoning can be represented. In this way we obtain a mathematical definition of imprecise-probabilistic sets. A...

Computer-based tools for mathematics are changing how mathematics is researched, taught and communicated to society. Future technology trends point to evermore powerful tools in the future. Computation in mathematics is thus giving rise to a new mode of mathematical research, where algorithms, datasets and public databases are as significant as the resulting theorems, and even the definition of...

Popular finite difference numerical schemes for the resolution of the one-dimensional acoustic wave equation are well-known to be convergent. We present a comprehensive formalization of the simplest scheme and formally prove its convergence in Coq. The main difficulties lie in the proper definition of asymptotic behaviors and the implicit way they are handled in the mathematical pen-and-paper p...

A topos of presheaves can be seen as an extension of classical set theory, where sets vary over informational states, therefore it is a powerful and expressive mathematical framework. I introduce a suitable topos of presheaves where probabilities and probabilistic reasoning can be represented. In this way we obtain a mathematical definition of probabilistic sets. A valid and complete proof syst...

What the algebra of convolution does for linear systems, the algebra of mathematical morphology does for shape. Since shape is a prime carrier of information in machine vision, there should be little surprise about the importance of the mathematical morphology. Now adays the operations of mathematical morphology are very much useful in object or defect identification required in industrial visi...

Generalization is one of the main mathematical activities. Some generalizations turn out to be useful for working mathematics, while many other generalizations have so far been not very useful. E. Bishop believed that most fruitless-so-far generalizations are hopeless, that every mathematical statement has only a few useful generalizations. In this paper, we show that, under a natural definitio...

How should one quantify the strength of association between two random variables without bias for relationships of a specific form? Despite its conceptual simplicity, this notion of statistical "equitability" has yet to receive a definitive mathematical formalization. Here we argue that equitability is properly formalized by a self-consistency condition closely related to Data Processing Inequa...

M athematical ecology has attracted a vibrant community of applied mathematicians. Collaborations with biologists are encouraged, and a variety of nonlinear phenomena have been modeled successfully in several laboratory systems [7]. Examples of “hard science” conducted with mathematical models of field populations and ecosystems are still rather uncommon, however, in part because appropriate da...