10,995 research outputs found
Linearly Convergent First-Order Algorithms for Semi-definite Programming
In this paper, we consider two formulations for Linear Matrix Inequalities
(LMIs) under Slater type constraint qualification assumption, namely, SDP
smooth and non-smooth formulations. We also propose two first-order linearly
convergent algorithms for solving these formulations. Moreover, we introduce a
bundle-level method which converges linearly uniformly for both smooth and
non-smooth problems and does not require any smoothness information. The
convergence properties of these algorithms are also discussed. Finally, we
consider a special case of LMIs, linear system of inequalities, and show that a
linearly convergent algorithm can be obtained under a weaker assumption
Slepian Spatial-Spectral Concentration on the Ball
We formulate and solve the Slepian spatial-spectral concentration problem on
the three-dimensional ball. Both the standard Fourier-Bessel and also the
Fourier-Laguerre spectral domains are considered since the latter exhibits a
number of practical advantages (spectral decoupling and exact computation). The
Slepian spatial and spectral concentration problems are formulated as
eigenvalue problems, the eigenfunctions of which form an orthogonal family of
concentrated functions. Equivalence between the spatial and spectral problems
is shown. The spherical Shannon number on the ball is derived, which acts as
the analog of the space-bandwidth product in the Euclidean setting, giving an
estimate of the number of concentrated eigenfunctions and thus the dimension of
the space of functions that can be concentrated in both the spatial and
spectral domains simultaneously. Various symmetries of the spatial region are
considered that reduce considerably the computational burden of recovering
eigenfunctions, either by decoupling the problem into smaller subproblems or by
affording analytic calculations. The family of concentrated eigenfunctions
forms a Slepian basis that can be used be represent concentrated signals
efficiently. We illustrate our results with numerical examples and show that
the Slepian basis indeeds permits a sparse representation of concentrated
signals.Comment: 33 pages, 10 figure
Spectrum and Statistical Entropy of AdS Black Holes
Popular approaches to quantum gravity describe black hole microstates
differently and apply different statistics to count them. Since the
relationship between the approaches is not clear, this obscures the role of
statistics in calculating the black hole entropy. We address this issue by
discussing the entropy of eternal AdS black holes in dimension four and above
within the context of a midisuperspace model. We determine the black hole
eigenstates and find that they describe the quantization in half integer units
of a certain function of the Arnowitt-Deser-Misner (ADM) mass and the
cosmological constant. In the limit of a vanishing cosmological constant (the
Schwarzschild limit) the quantized function becomes the horizon area and in the
limit of a large cosmological constant it approaches the ADM mass of the black
holes. We show that in the Schwarzschild limit the area quatization leads to
the Bekenstein-Hawking entropy if Boltzmann statistics are employed. In the
limit of a large cosmological constant the Bekenstein-Hawking entropy can be
recovered only via Bose statistics. The two limits are separated by a first
order phase transition, which seems to suggest a shift from "particle-like"
degrees of freedom at large cosmological constant to geometric degrees of
freedom as the cosmological constant approaches zero.Comment: 14 pages. No figures. Some references added. Version to appear in
Phys. Rev.
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