1,135 research outputs found
Gott Time Machines, BTZ Black Hole Formation, and Choptuik Scaling
We study the formation of BTZ black holes by the collision of point
particles. It is shown that the Gott time machine, originally constructed for
the case of vanishing cosmological constant, provides a precise mechanism for
black hole formation. As a result, one obtains an exact analytic understanding
of the Choptuik scaling.Comment: 6 pages, Late
Thurston's pullback map on the augmented Teichm\"uller space and applications
Let be a postcritically finite branched self-cover of a 2-dimensional
topological sphere. Such a map induces an analytic self-map of a
finite-dimensional Teichm\"uller space. We prove that this map extends
continuously to the augmented Teichm\"uller space and give an explicit
construction for this extension. This allows us to characterize the dynamics of
Thurston's pullback map near invariant strata of the boundary of the augmented
Teichm\"uller space. The resulting classification of invariant boundary strata
is used to prove a conjecture by Pilgrim and to infer further properties of
Thurston's pullback map. Our approach also yields new proofs of Thurston's
theorem and Pilgrim's Canonical Obstruction theorem.Comment: revised version, 28 page
Hamiltonian submanifolds of regular polytopes
We investigate polyhedral -manifolds as subcomplexes of the boundary
complex of a regular polytope. We call such a subcomplex {\it -Hamiltonian}
if it contains the full -skeleton of the polytope. Since the case of the
cube is well known and since the case of a simplex was also previously studied
(these are so-called {\it super-neighborly triangulations}) we focus on the
case of the cross polytope and the sporadic regular 4-polytopes. By our results
the existence of 1-Hamiltonian surfaces is now decided for all regular
polytopes.
Furthermore we investigate 2-Hamiltonian 4-manifolds in the -dimensional
cross polytope. These are the "regular cases" satisfying equality in Sparla's
inequality. In particular, we present a new example with 16 vertices which is
highly symmetric with an automorphism group of order 128. Topologically it is
homeomorphic to a connected sum of 7 copies of . By this
example all regular cases of vertices with or, equivalently, all
cases of regular -polytopes with are now decided.Comment: 26 pages, 4 figure
Photoemission Beyond the Sudden Approximation
The many-body theory of photoemission in solids is reviewed with emphasis on
methods based on response theory. The classification of diagrams into loss and
no-loss diagrams is discussed and related to Keldysh path-ordering
book-keeping. Some new results on energy losses in valence-electron
photoemission from free-electron-like metal surfaces are presented. A way to
group diagrams is presented in which spectral intensities acquire a
Golden-Rule-like form which guarantees positiveness. This way of regrouping
should be useful also in other problems involving spectral intensities, such as
the problem of improving the one-electron spectral function away from the
quasiparticle peak.Comment: 18 pages, 11 figure
New Method for Phase transitions in diblock copolymers: The Lamellar case
A new mean-field type theory is proposed to study order-disorder transitions
(ODT) in block copolymers. The theory applies to both the weak segregation (WS)
and the strong segregation (SS) regimes. A new energy functional is proposed
without appealing to the random phase approximation (RPA). We find new terms
unaccounted for within RPA. We work out in detail transitions to the lamellar
state and compare the method to other existing theories of ODT and numerical
simulations. We find good agreements with recent experimental results and
predict that the intermediate segregation regime may have more than one scaling
behavior.Comment: 23 pages, 8 figure
Shapes of polyhedra, mixed volumes and hyperbolic geometry
We generalize to higher dimensions the Bavard–Ghys construction of the hyperbolic metric on the space of polygons with fixed directions of edges. The space of convex d -dimensional polyhedra with fixed directions of facet normals has a decomposition into type cones that correspond to different combinatorial types of polyhedra. This decomposition is a subfan of the secondary fan of a vector configuration and can be analyzed with the help of Gale diagrams. We construct a family of quadratic forms on each of the type cones using the theory of mixed volumes. The Alexandrov–Fenchel inequalities ensure that these forms have exactly one positive eigenvalue. This introduces a piecewise hyperbolic structure on the space of similarity classes of polyhedra with fixed directions of facet normals. We show that some of the dihedral angles on the boundary of the resulting cone-manifold are equal to π/2
Minkowski-type and Alexandrov-type theorems for polyhedral herissons
Classical H.Minkowski theorems on existence and uniqueness of convex
polyhedra with prescribed directions and areas of faces as well as the
well-known generalization of H.Minkowski uniqueness theorem due to
A.D.Alexandrov are extended to a class of nonconvex polyhedra which are called
polyhedral herissons and may be described as polyhedra with injective spherical
image.Comment: 19 pages, 8 figures, LaTeX 2.0
Square-tiled cyclic covers
A cyclic cover of the complex projective line branched at four appropriate
points has a natural structure of a square-tiled surface. We describe the
combinatorics of such a square-tiled surface, the geometry of the corresponding
Teichm\"uller curve, and compute the Lyapunov exponents of the determinant
bundle over the Teichm\"uller curve with respect to the geodesic flow. This
paper includes a new example (announced by G. Forni and C. Matheus in
\cite{Forni:Matheus}) of a Teichm\"uller curve of a square-tiled cyclic cover
in a stratum of Abelian differentials in genus four with a maximally degenerate
Kontsevich--Zorich spectrum (the only known example found previously by Forni
in genus three also corresponds to a square-tiled cyclic cover
\cite{ForniSurvey}).
We present several new examples of Teichm\"uller curves in strata of
holomorphic and meromorphic quadratic differentials with maximally degenerate
Kontsevich--Zorich spectrum. Presumably, these examples cover all possible
Teichm\"uller curves with maximally degenerate spectrum. We prove that this is
indeed the case within the class of square-tiled cyclic covers.Comment: 34 pages, 6 figures. Final version incorporating referees comments.
In particular, a gap in the previous version was corrected. This file uses
the journal's class file (jmd.cls), so that it is very similar to published
versio
Polydispersity and ordered phases in solutions of rodlike macromolecules
We apply density functional theory to study the influence of polydispersity
on the stability of columnar, smectic and solid ordering in the solutions of
rodlike macromolecules. For sufficiently large length polydispersity (standard
deviation ) a direct first-order nematic-columnar transition is
found, while for smaller there is a continuous nematic-smectic and
first-order smectic-columnar transition. For increasing polydispersity the
columnar structure is stabilized with respect to solid perturbations. The
length distribution of macromolecules changes neither at the nematic-smectic
nor at the nematic-columnar transition, but it does change at the
smectic-columnar phase transition. We also study the phase behaviour of binary
mixtures, in which the nematic-smectic transition is again found to be
continuous. Demixing according to rod length in the smectic phase is always
preempted by transitions to solid or columnar ordering.Comment: 13 pages (TeX), 2 Postscript figures uuencode
Minkowski Tensors of Anisotropic Spatial Structure
This article describes the theoretical foundation of and explicit algorithms
for a novel approach to morphology and anisotropy analysis of complex spatial
structure using tensor-valued Minkowski functionals, the so-called Minkowski
tensors. Minkowski tensors are generalisations of the well-known scalar
Minkowski functionals and are explicitly sensitive to anisotropic aspects of
morphology, relevant for example for elastic moduli or permeability of
microstructured materials. Here we derive explicit linear-time algorithms to
compute these tensorial measures for three-dimensional shapes. These apply to
representations of any object that can be represented by a triangulation of its
bounding surface; their application is illustrated for the polyhedral Voronoi
cellular complexes of jammed sphere configurations, and for triangulations of a
biopolymer fibre network obtained by confocal microscopy. The article further
bridges the substantial notational and conceptual gap between the different but
equivalent approaches to scalar or tensorial Minkowski functionals in
mathematics and in physics, hence making the mathematical measure theoretic
method more readily accessible for future application in the physical sciences
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