44,056 research outputs found

    Shadow of a Schwarzschild black hole surrounded by a Bach-Weyl ring

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    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

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    A "shadow" of a subset SS of Euclidean space is an orthogonal projection of SS 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 R3\mathbb R^3 to be paths (i.e., simple open curves). We also show two contrasting results: the three shadows of a path in R3\mathbb R^3 can all be cycles (although not all convex) and, for every d≥1d\geq 1, there exists a dd-sphere embedded in Rd+2\mathbb R^{d+2} whose d+2d+2 shadows have no holes (i.e., they deformation-retract onto a point).Comment: 6 pages, 10 figure

    Numerical shadow and geometry of quantum states

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    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)

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    Let B be a subplane of PG(2,q^3) of order q that is tangent to ℓ∞\ell_\infty. Then the tangent splash of B is defined to be the set of q^2+1 points of ℓ∞\ell_\infty 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-qq-subplane determined by a given tangent splash and a fixed order-qq-subline.Comment: arXiv admin note: substantial text overlap with arXiv:1303.550
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