Coalgebraic Geometric Logic: Basic Theory

Using the theory of coalgebra, we introduce a uniform framework for adding modalities to the language of propositional geometric logic. Models for this logic are based on coalgebras for an endofunctor on some full subcategory of the category of topological spaces and continuous functions. We investigate derivation systems, soundness and completeness for such geometric modal logics, and we we specify a method of lifting an endofunctor on Set, accompanied by a collection of predicate liftings, to an endofunctor on the category of topological spaces, again accompanied by a collection of (open) predicate liftings. Furthermore, we compare the notions of modal equivalence, behavioural equivalence and bisimulation on the resulting class of models, and we provide a final object for the corresponding category

On Jordan's measurements

The Jordan measure, the Jordan curve theorem, as well as the other generic references to Camille Jordan's (1838-1922) achievements highlight that the latter can hardly be reduced to the "great algebraist" whose masterpiece, the Trait\'e des substitutions et des equations alg\'ebriques, unfolded the group-theoretical content of \'Evariste Galois's work. The present paper appeals to the database of the reviews of the Jahrbuch \"uber die Fortschritte der Mathematik (1868-1942) for providing an overview of Jordan's works. On the one hand, we shall especially investigate the collective dimensions in which Jordan himself inscribed his works (1860-1922). On the other hand, we shall address the issue of the collectives in which Jordan's works have circulated (1860-1940). Moreover, the time-period during which Jordan has been publishing his works, i.e., 1860-1922, provides an opportunity to investigate some collective organizations of knowledge that pre-existed the development of object-oriented disciplines such as group theory (Jordan-H\"older theorem), linear algebra (Jordan's canonical form), topology (Jordan's curve), integral theory (Jordan's measure), etc. At the time when Jordan was defending his thesis in 1860, it was common to appeal to transversal organizations of knowledge, such as what the latter designated as the "theory of order." When Jordan died in 1922, it was however more and more common to point to object-oriented disciplines as identifying both a corpus of specialized knowledge and the institutionalized practices of transmissions of a group of professional specialists

An algebraic approach to Polya processes

P\'olya processes are natural generalization of P\'olya-Eggenberger urn models. This article presents a new approach of their asymptotic behaviour {\it via} moments, based on the spectral decomposition of a suitable finite difference operator on polynomial functions. Especially, it provides new results for {\it large} processes (a P\'olya process is called {\it small} when 1 is simple eigenvalue of its replacement matrix and when any other eigenvalue has a real part $\leq 1/2$; otherwise, it is called large)