Homographic solutions of the curved 3-body problem

In the 2-dimensional curved 3-body problem, we prove the existence of Lagrangian and Eulerian homographic orbits, and provide their complete classification in the case of equal masses. We also show that the only non-homothetic hyperbolic Eulerian solutions are the hyperbolic Eulerian relative equilibria, a result that proves their instability.Comment: 34 pages, 7 figures, new introductio

A local first law for black hole thermodynamics

We first show that stationary black holes satisfy an extremely simple local form of the first law \delta E=\kappa(l) \delta A/(8 \pi) where the thermodynamical energy E=A/(8\pi l) and (local) surface gravity \kappa(l)=1/l, where A is the horizon area and l is a proper length characterizing the distance to the horizon of a preferred family of local observers suitable for thermodynamical considerations. Our construction is extended to the more general framework of isolated horizons. The local surface gravity is universal. This has important implications for semiclassical considerations of black hole physics as well as for the fundamental quantum description arising in the context of loop quantum gravity.Comment: The derivation of the main result for the special case of isolated horizons has been considerably strengthe

Transparency and Accountability in Bolivia: Does Voice Matter?

Using a survey of clients, doctors and nurses in thirty municipal hospitals, this study examines the scope and determinants of corruption in Bolivia`s public health sector. The authors find that `voice` mechanisms such as citizen activism and oversight are significant in exposing bribery and deterring the overpricing of medical supplies. The existence of `exit` options, such as recourse to private health service providers, additionally serves to deter corruption. Hierarchical controls, such as administrative regulations and procedures, seem to have little impact on hospital corruption, though further research is needed to verify this finding.

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Statistical Entropy of a BTZ Black Hole from Loop Quantum Gravity

We compute the statistical entropy of a BTZ black hole in the context of three-dimensional Euclidean loop quantum gravity with a cosmological constant $\Lambda$. As in the four-dimensional case, a quantum state of the black hole is characterized by a spin network state. Now however, the underlying colored graph $\Gamma$ lives in a two-dimensional spacelike surface $\Sigma$, and some of its links cross the black hole horizon, which is viewed as a circular boundary of $\Sigma$. Each link $\ell$ crossing the horizon is colored by a spin $j_\ell$ (at the kinematical level), and the length $L$ of the horizon is given by the sum $L=\sum_\ell L_\ell$ of the fundamental length contributions $L_\ell$ carried by the spins $j_\ell$ of the links $\ell$. We propose an estimation for the number $N^\text{BTZ}_\Gamma(L,\Lambda)$ of the Euclidean BTZ black hole microstates (defined on a fixed graph $\Gamma$) based on an analytic continuation from the case $\Lambda>0$ to the case $\Lambda<0$. In our model, we show that $N^\text{BTZ}_\Gamma(L,\Lambda)$ reproduces the Bekenstein-Hawking entropy in the classical limit. This asymptotic behavior is independent of the choice of the graph $\Gamma$ provided that the condition $L=\sum_\ell L_\ell$ is satisfied, as it should be in three-dimensional quantum gravity.Comment: 14 pages. 1 figure. Paragraph added on page 7 to clarify the horizon conditio