In the present paper we construct Virtual Element Spaces that are H(div)-conforming and H(curl)-conforming on general polygonal and polyhedral elements; these spaces can be interpreted as a generalization of well known Finite Elements. We moreover present the basic tools needed to make use of these spaces in the approximation of partial differential equations. Finally, we discuss the constructi...

Finite element exterior calculus is an approach to the design and understanding of finite element discretizations for a wide variety of systems of partial differential equations. This approach brings to bear tools from differential geometry, algebraic topology, and homological algebra to develop discretizations which are compatible with the geometric, topological, and algebraic structures which...

The main theme of this workshop (Dagstuhl seminar 04351) is ‘Spatial Representation: Continuous vs. Discrete’. Spatial representation has two contrasting but interacting aspects (i) representation of spaces’ and (ii) representation by spaces. In this paper we will examine two aspects that are common to both interpretations of the theme, namely nerve constructions and refinement. Representations...

We report on our recent breakthrough [9] in the costruction for q > 0 of Hermitean and “tractable” differential operators out of the Uqso(N)covariant differential calculus on the noncommutative manifolds Rq (the socalled “quantum Euclidean spaces”).

We report on our recent breakthrough [9] in the costruction for q > 0 of Hermitean and “tractable” differential operators out of the Uqso(N)covariant differential calculus on the noncommutative manifolds Rq (the socalled “quantum Euclidean spaces”).

Processes are often viewed as coalgebras, with the structure maps specifying the state transitions. In the simplest case, the state spaces are discrete, and the structure map simply takes each state to the next states. But the coalgebraic view is also quite effective for studying processes over structured state spaces, e.g. measurable, or continuous. In the present paper we consider coalgebras ...

In the context of Synthetic Differential Geometry, we discuss vector fields/ordinary differential equations as actions; in particular, we exploit function space formation (exponential spaces) in the category of actions. c © Central European Science Journals. All rights reserved.

The aim of this special issue is to focus on recent developments and achievements in the theory of function spaces, sequences spaces and their geometry, and compact operators and their applications in various fields of applied mathematics, engineering, and other sciences. The theory of sequence spaces is powerful tool for obtaining positive results concerning Schauder basis and plays a fundamen...

We consider discrete pseudo-differential equations with elliptic symbols and the corresponding boundary-value problems in special canonical domains of multidimensional spaces. The solvability such analogs Sobolev–Slobodetsky spaces is examined.

We conservatively extend classical elementary differential calculus to the Cartesian closed category of convergence spaces. By specializing results about the convergence space representation of directed graphs, we use Cayley graphs to obtain a differential calculus on groups, from which we then extract a Boolean differential calculus, in which both linearity and the product rule, also called th...