4,343 research outputs found
Singularity Classes of Special 2-Flags
In the paper we discuss certain classes of vector distributions in the
tangent bundles to manifolds, obtained by series of applications of the
so-called generalized Cartan prolongations (gCp). The classical Cartan
prolongations deal with rank-2 distributions and are responsible for the
appearance of the Goursat distributions. Similarly, the so-called special
multi-flags are generated in the result of successive applications of gCp's.
Singularities of such distributions turn out to be very rich, although without
functional moduli of the local classification. The paper focuses on special
2-flags, obtained by sequences of gCp's applied to rank-3 distributions. A
stratification of germs of special 2-flags of all lengths into singularity
classes is constructed. This stratification provides invariant geometric
significance to the vast family of local polynomial pseudo-normal forms for
special 2-flags introduced earlier in [Mormul P., Banach Center Publ., Vol. 65,
Polish Acad. Sci., Warsaw, 2004, 157-178]. This is the main contribution of the
present paper. The singularity classes endow those multi-parameter normal
forms, which were obtained just as a by-product of sequences of gCp's, with a
geometrical meaning
On Lie Algebras Generated by Few Extremal Elements
We give an overview of some properties of Lie algebras generated by at most 5
extremal elements. In particular, for any finite graph {\Gamma} and any field K
of characteristic not 2, we consider an algebraic variety X over K whose
K-points parametrize Lie algebras generated by extremal elements. Here the
generators correspond to the vertices of the graph, and we prescribe
commutation relations corresponding to the nonedges of {\Gamma}. We show that,
for all connected undirected finite graphs on at most 5 vertices, X is a
finite-dimensional affine space. Furthermore, we show that for
maximal-dimensional Lie algebras generated by 5 extremal elements, X is a
point. The latter result implies that the bilinear map describing extremality
must be identically zero, so that all extremal elements are sandwich elements
and the only Lie algebra of this dimension that occurs is nilpotent. These
results were obtained by extensive computations with the Magma computational
algebra system. The algorithms developed can be applied to arbitrary {\Gamma}
(i.e., without restriction on the number of vertices), and may be of
independent interest.Comment: 19 page
Fiber Composite Sandwich Thermostructural Behavior: Computational Simulation
Several computational levels of progressive sophistication/simplification are described to computationally simulate composite sandwich hygral, thermal, and structural behavior. The computational levels of sophistication include: (1) three-dimensional detailed finite element modeling of the honeycomb, the adhesive and the composite faces; (2) three-dimensional finite element modeling of the honeycomb assumed to be an equivalent continuous, homogeneous medium, the adhesive and the composite faces; (3) laminate theory simulation where the honeycomb (metal or composite) is assumed to consist of plies with equivalent properties; and (4) derivations of approximate, simplified equations for thermal and mechanical properties by simulating the honeycomb as an equivalent homogeneous medium. The approximate equations are combined with composite hygrothermomechanical and laminate theories to provide a simple and effective computational procedure for simulating the thermomechanical/thermostructural behavior of fiber composite sandwich structures
A Distinguished Vacuum State for a Quantum Field in a Curved Spacetime: Formalism, Features, and Cosmology
We define a distinguished "ground state" or "vacuum" for a free scalar
quantum field in a globally hyperbolic region of an arbitrarily curved
spacetime. Our prescription is motivated by the recent construction of a
quantum field theory on a background causal set using only knowledge of the
retarded Green's function. We generalize that construction to continuum
spacetimes and find that it yields a distinguished vacuum or ground state for a
non-interacting, massive or massless scalar field. This state is defined for
all compact regions and for many noncompact ones. In a static spacetime we find
that our vacuum coincides with the usual ground state. We determine it also for
a radiation-filled, spatially homogeneous and isotropic cosmos, and show that
the super-horizon correlations are approximately the same as those of a thermal
state. Finally, we illustrate the inherent non-locality of our prescription
with the example of a spacetime which sandwiches a region with curvature
in-between flat initial and final regions
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