Mother Operators and their Descendants

A mechanism deriving new well-posed evolutionary equations from given ones is inspected. It turns out that there is one particular spatial operator from which many of the standard evolutionary problems of mathematical physics can be generated by this abstract mechanism using suitable projections. The complexity of the dynamics of the phenomena considered can be described in terms of suitable material laws. The idea is illustrated with a number of concrete examples.Comment: This is a revised version of the earlier posted pre-print. The corrections concern predominantly Corollary 1.7 and Theorem 1.9 and their consequence

Hodge-Helmholtz Decompositions of Weighted Sobolev Spaces in Irregular Exterior Domains with Inhomogeneous and Anisotropic Media

We study in detail Hodge-Helmholtz decompositions in non-smooth exterior domains filled with inhomogeneous and anisotropic media. We show decompositions of alternating differential forms belonging to weighted Sobolev spaces into irrotational and solenoidal forms. These decompositions are essential tools, for example, in electro-magnetic theory for exterior domains. In the appendix we translate our results to the classical framework of vector analysis.Comment: Key Words: Hodge-Helmholtz decompositions, Maxwell's equations, electro-magnetic theory, weighted Sobolev space

Shapes of polyhedra and triangulations of the sphere

The space of shapes of a polyhedron with given total angles less than 2\pi at each of its n vertices has a Kaehler metric, locally isometric to complex hyperbolic space CH^{n-3}. The metric is not complete: collisions between vertices take place a finite distance from a nonsingular point. The metric completion is a complex hyperbolic cone-manifold. In some interesting special cases, the metric completion is an orbifold. The concrete description of these spaces of shapes gives information about the combinatorial classification of triangulations of the sphere with no more than 6 triangles at a vertex.Comment: 39 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTMon1/paper25.abs.htm

Maxwell meets Korn: A New Coercive Inequality for Tensor Fields with Square-Integrable Exterior Derivative

Maxwell meets Korn: A New Coercive Inequality for Tensor Fields with Square-Integrable Exterior DerivativeComment: Key Words: Korn's inequality, theory of generalized Maxwell equations, Helmholtz decomposition, Poincare/Friedrichs-type estimates, incompatible tensor

A tree approach to pp-variation and to integration

We consider a real-valued path; it is possible to associate a tree to this path, and we explore the relations between the tree, the properties of $p$-variation of the path, and integration with respect to the path. In particular, the fractal dimension of the tree is estimated from the variations of the path, and Young integrals with respect to the path, as well as integrals from the rough paths theory, are written as integrals on the tree. Examples include some stochastic paths such as martingales, L\'evy processes and fractional Brownian motions (for which an estimator of the Hurst parameter is given)

Simulating the Influence of Collaborative Networks on the Structure of Networks of Organizations, Employment Structure, and Organization Value

From the perspective of reindustrialization, it is important to understand the evolution of the structure of the network of organizations employment structure, and organization value. Understanding the potential influence of collaborative networks (CNs) on these aspects may lead to the development of appropriate economic policies. In this paper, we propose a theoretical approach to analysis this potential influence, based on a model of dynamic networked ecosystem of organizations encompassing collaboration relations among organization, employment mobility, and organization value. A large number of simulations has been performed to identify factors influencing the structure of the network of organizations employment structure, and organization value. The main findings are that 1) the higher the number of members of CNs, the better the clustering and the shorter the average path length among organizations; 2) the constitution of CNs does not affect neither the structure of the network of organizations, nor the employment structure and the organization value.Comment: 10 pages, 1 figure, conference paper at the 14th IFIP Working Conference on Virtual Enterprises, PRO-VE'13, http://www.pro-ve.org

Representation formulae for the fractional Brownian motion

We discuss the relationships between some classical representations of the fractional Brownian motion, as a stochastic integral with respect to a standard Brownian motion, or as a series of functions with independent Gaussian coefficients. The basic notions of fractional calculus which are needed for the study are introduced. As an application, we also prove some properties of the Cameron-Martin space of the fractional Brownian motion, and compare its law with the law of some of its variants. Several of the results which are given here are not new; our aim is to provide a unified treatment of some previous literature, and to give alternative proofs and additional results; we also try to be as self-contained as possible.Comment: to appear in "S\'eminaire de Probabilit\'es