33,032 research outputs found
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 -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
-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
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