13,622 research outputs found
Hodge polynomials of the moduli spaces of pairs
Let be a smooth projective curve of genus over the complex
numbers. A holomorphic pair on is a couple , where is a
holomorphic bundle over of rank and degree , and is
a holomorphic section. In this paper, we determine the Hodge polynomials of the
moduli spaces of rank 2 pairs, using the theory of mixed Hodge structures. We
also deal with the case in which has fixed determinant.Comment: 23 pages, typos added, minor change
Converging shocks in elastic-plastic solids
We present an approximate description of the behavior of an elastic-plastic material processed by a cylindrically or spherically symmetric converging shock, following Whitham's shock dynamics theory. Originally applied with success to various gas dynamics problems, this theory is presently derived for solid media, in both elastic and plastic regimes. The exact solutions of the shock dynamics equations obtained reproduce well the results obtained by high-resolution numerical simulations. The examined constitutive laws share a compressible neo-Hookean structure for the internal energy e = e_(s)(I_1)+e_(h)(Ï,Ï), where e_(s) accounts for shear through the first invariant of the CauchyâGreen tensor, and e_(h) represents the hydrostatic contribution as a function of the density Ï and entropy Ï. In the strong-shock limit, reached as the shock approaches the axis or origin r=0, we show that compression effects are dominant over shear deformations. For an isothermal constitutive law, i.e., e_(h) = e_(h)(Ï), with a power-law dependence e_(h) â Ï_(α), shock dynamics predicts that for a converging shock located at r=R(t) at time t, the Mach number increases as M â [log(1/R)]^α, independently of the space index s, where s=2 in cylindrical geometry and 3 in spherical geometry. An alternative isothermal constitutive law with p(Ï) of the arctanh type, which enforces a finite density in the strong-shock limit, leads to M â R^(â(sâ1)) for strong shocks. A nonisothermal constitutive law, whose hydrostatic part eh is that of an ideal gas, is also tested, recovering the strong-shock limit MâR^(â(sâ1)/n(Îł)) originally derived by Whitham for perfect gases, where Îł is inherently related to the maximum compression ratio that the material can reach, (Îł+1)/(Îłâ1). From these strong-shock limits, we also estimate analytically the density, radial velocity, pressure, and sound speed immediately behind the shock. While the hydrostatic part of the energy essentially commands the strong-shock behavior, the shear modulus and yield stress modify the compression ratio and velocity of the shock far from the axis or origin. A characterization of the elastic-plastic transition in converging shocks, which involves an elastic precursor and a plastic compression region, is finally exposed
Spectroscopic investigation of a 'Virgin of Sorrows' canvas painting : a multi-method approach
Boost invariant marginally trapped surfaces in Minkowski 4-space
The extremal and partly marginally trapped surfaces in Minkowski 4-space,
which are invariant under the group of boost isometries, are classified.
Moreover, it is shown that there do not exist extremal surfaces of this kind
with constant Gaussian curvature. A procedure is given in order to construct a
partly marginally trapped surface by gluing two marginally trapped surfaces
which are invariant under the group of boost isometries. As an application, a
proper star-surface is constructed.Comment: 13 pages, comment added in section
D 4.1: Protocol for converting data sources into common standards for input into WP3 and WP9 : Version 3.0 Document date: 2009-10-10
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