We construct two algebraic versions of homotopy theory of rational disconnected topological spaces, one based on differential graded commutative associative algebras and the other one on complete differential graded Lie algebras. As an application of the developed technology we obtain results on the structure of Maurer-Cartan spaces of complete differential graded Lie algebras.

The mixed variational formulation of many elliptic boundary value problems involves vector valued function spaces, like, in three dimensions, H(curl; Ω) and H(Div;Ω). Thus finite element subspaces of these function spaces are indispensable for effective finite element discretization schemes. Given a simplicial triangulation of the computational domain Ω, among others, Raviart, Thomas and Nédéle...

The (classical) modulation spaces, as introduced by Feichtinger during the 80’s, consist of all tempered distributions whose short-time Fourier transforms (STFT) have finite mixed (weighted) Lebesgue norm. By choosing the Lebesgue parameters and weight functions in appropriate ways, one may quantify the degrees of asymptotic decay and singularity of the distributions in a ”detailed way”. A majo...

The problem of controllability of linear and nonlinear systems represented by ordinary differential equations in finite dimensional spaces has been extensively studied. Several authors [5,6,12-14] have extended the concept to infinite dimensional systems in Banach spaces with bounded operators. Triggiani [17] established sufficient conditions for controllability of linear and nonlinear systems ...

In this paper, we show that the category of Mackey-complete, separated, topological convex bornological vector spaces and bounded linear maps is a differential category. Such spaces were introduced by Frölicher and Kriegl, where they were called convenient vector spaces. While much of the structure necessary to demonstrate this observation is already contained in Frölicher and Kriegl’s book, we...

In the case ω = 1, this space is denoted W (Ω). Sobolev spaces without weights occur as spaces of solutions for elliptic and parabolic partial differential equations. In various applications, we can meet boundary value problems for elliptic equations whose ellipticity is “disturbed” in the sense that some degeneration or singularity appears. This “bad” behaviour can be caused by the coefficient...

In this paper, we consider fuzzy fractional partial differential equations under Caputo generalized Hukuhara differentiability. Some new results on the existence and uniqueness of two types of fuzzy solutions are studied via  weakly contractive mapping in the partially ordered metric space. Some application examples are presented to illustrate our main results.

For differential calculi on noncommutative algebras, we construct a twisted de Rham cohomology using flat connections on modules. This has properties similar, in some respects, to sheaf cohomology on topological spaces. We also discuss generalised mapping properties of these theories, and relations of these properties to corings. Using this, we give conditions for the Serre spectral sequence to...