2,255 research outputs found
Generalized Finite Element Systems for smooth differential forms and Stokes problem
We provide both a general framework for discretizing de Rham sequences of
differential forms of high regularity, and some examples of finite element
spaces that fit in the framework. The general framework is an extension of the
previously introduced notion of Finite Element Systems, and the examples
include conforming mixed finite elements for Stokes' equation. In dimension 2
we detail four low order finite element complexes and one infinite family of
highorder finite element complexes. In dimension 3 we define one low order
complex, which may be branched into Whitney forms at a chosen index. Stokes
pairs with continuous or discontinuous pressure are provided in arbitrary
dimension. The finite element spaces all consist of composite polynomials. The
framework guarantees some nice properties of the spaces, in particular the
existence of commuting interpolators. It also shows that some of the examples
are minimal spaces.Comment: v1: 27 pages. v2: 34 pages. Numerous details added. v3: 44 pages. 8
figures and several comments adde
Hidden Quantum Gravity in 3d Feynman diagrams
In this work we show that 3d Feynman amplitudes of standard QFT in flat and
homogeneous space can be naturally expressed as expectation values of a
specific topological spin foam model. The main interest of the paper is to set
up a framework which gives a background independent perspective on usual field
theories and can also be applied in higher dimensions. We also show that this
Feynman graph spin foam model, which encodes the geometry of flat space-time,
can be purely expressed in terms of algebraic data associated with the Poincare
group. This spin foam model turns out to be the spin foam quantization of a BF
theory based on the Poincare group, and as such is related to a quantization of
3d gravity in the limit where the Newton constant G_N goes to 0. We investigate
the 4d case in a companion paper where the strategy proposed here leads to
similar results.Comment: 35 pages, 4 figures, some comments adde
Algebraic Topology
The chapter provides an introduction to the basic concepts of Algebraic
Topology with an emphasis on motivation from applications in the physical
sciences. It finishes with a brief review of computational work in algebraic
topology, including persistent homology.Comment: This manuscript will be published as Chapter 5 in Wiley's textbook
\emph{Mathematical Tools for Physicists}, 2nd edition, edited by Michael
Grinfeld from the University of Strathclyd
Three-dimensional alpha shapes
Frequently, data in scientific computing is in its abstract form a finite
point set in space, and it is sometimes useful or required to compute what one
might call the ``shape'' of the set. For that purpose, this paper introduces
the formal notion of the family of -shapes of a finite point set in
\Real^3. Each shape is a well-defined polytope, derived from the Delaunay
triangulation of the point set, with a parameter \alpha \in \Real controlling
the desired level of detail. An algorithm is presented that constructs the
entire family of shapes for a given set of size in time , worst
case. A robust implementation of the algorithm is discussed and several
applications in the area of scientific computing are mentioned.Comment: 32 page
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