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Shadow of a Schwarzschild black hole surrounded by a Bach-Weyl ring
We have studied the shadows of a Schwarzschild black hole surrounded by a
Bach-Weyl ring through the backward ray-tracing method. The presence of
Bach-Weyl ring leads to that the photon dynamical system is non-integrable and
then chaos would appear in the photon motion, which affects sharply the black
hole shadow. The size and shape the black hole shadow depend on the black hole
parameter, the Bach-Weyl ring mass and the Weyl radius between black hole and
ring. Some self-similar fractal structures also appear in the black hole
shadow, which originates from the chaotic lensing. We also study the change of
the image of Bach-Weyl ring with the ring mass and the Weyl radius. Finally, we
analyze the invariant manifolds of Lyapunov orbits near the fixed points and
discuss further the formation of the shadow of a Schwarzschild black hole with
Bach-Weyl ring.Comment: 16 pages,8 figures, the version published in EPJ
The Shadows of a Cycle Cannot All Be Paths
A "shadow" of a subset of Euclidean space is an orthogonal projection of
into one of the coordinate hyperplanes. In this paper we show that it is
not possible for all three shadows of a cycle (i.e., a simple closed curve) in
to be paths (i.e., simple open curves).
We also show two contrasting results: the three shadows of a path in can all be cycles (although not all convex) and, for every ,
there exists a -sphere embedded in whose shadows
have no holes (i.e., they deformation-retract onto a point).Comment: 6 pages, 10 figure
Numerical shadow and geometry of quantum states
The totality of normalised density matrices of order N forms a convex set Q_N
in R^(N^2-1). Working with the flat geometry induced by the Hilbert-Schmidt
distance we consider images of orthogonal projections of Q_N onto a two-plane
and show that they are similar to the numerical ranges of matrices of order N.
For a matrix A of a order N one defines its numerical shadow as a probability
distribution supported on its numerical range W(A), induced by the unitarily
invariant Fubini-Study measure on the complex projective manifold CP^(N-1). We
define generalized, mixed-states shadows of A and demonstrate their usefulness
to analyse the structure of the set of quantum states and unitary dynamics
therein.Comment: 19 pages, 5 figure
The tangent splash in \PG(6,q)
Let B be a subplane of PG(2,q^3) of order q that is tangent to .
Then the tangent splash of B is defined to be the set of q^2+1 points of
that lie on a line of B. In the Bruck-Bose representation of
PG(2,q^3) in PG(6,q), we investigate the interaction between the ruled surface
corresponding to B and the planes corresponding to the tangent splash of B. We
then give a geometric construction of the unique order--subplane determined
by a given tangent splash and a fixed order--subline.Comment: arXiv admin note: substantial text overlap with arXiv:1303.550
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