Phase transitions near black hole horizons

The Reissner-Nordstrom black hole in four dimensions can be made unstable without violating the dominant energy condition by introducing a real massive scalar with non-renormalizable interactions with the gauge field. New stable black hole solutions then exist with greater entropy for fixed mass and charge than the Reissner-Nordstrom solution. In these new solutions, the scalar condenses to a non-zero value near the horizon. Various generalizations of these hairy black holes are discussed, and an attempt is made to characterize when black hole hair can occur.Comment: 30 pages, 6 figures. v2: minor corrections, references adde

Generic Cosmic Censorship Violation in anti de Sitter Space

We consider (four dimensional) gravity coupled to a scalar field with potential V(\phi). The potential satisfies the positive energy theorem for solutions that asymptotically tend to a negative local minimum. We show that for a large class of such potentials, there is an open set of smooth initial data that evolve to naked singularities. Hence cosmic censorship does not hold for certain reasonable matter theories in asymptotically anti de Sitter spacetimes. The asymptotically flat case is more subtle. We suspect that potentials with a local Minkowski minimum may similarly lead to violations of cosmic censorship in asymptotically flat spacetimes, but we do not have definite results.Comment: 4 pages, v2: minor change

Stability in Designer Gravity

We study the stability of designer gravity theories, in which one considers gravity coupled to a tachyonic scalar with anti-de Sitter boundary conditions defined by a smooth function W. We construct Hamiltonian generators of the asymptotic symmetries using the covariant phase space method of Wald et al.and find they differ from the spinor charges except when W=0. The positivity of the spinor charge is used to establish a lower bound on the conserved energy of any solution that satisfies boundary conditions for which $W$ has a global minimum. A large class of designer gravity theories therefore have a stable ground state, which the AdS/CFT correspondence indicates should be the lowest energy soliton. We make progress towards proving this, by showing that minimum energy solutions are static. The generalization of our results to designer gravity theories in higher dimensions involving several tachyonic scalars is discussed.Comment: 29 page

A tale of two superpotentials: Stability and Instability in Designer Gravity

We investigate the stability of asymptotically anti-de Sitter gravity coupled to tachyonic scalar fields with mass at or slightly above the Breitenlohner-Freedman bound. The boundary conditions in these ``designer gravity'' theories are defined in terms of an arbitrary function W. Previous work had suggested that the energy in designer gravity is bounded below if i) W has a global minimum and ii) the scalar potential admits a superpotential P. More recently, however, certain solutions were found (numerically) to violate the proposed energy bound. We resolve the discrepancy by observing that a given scalar potential can admit two possible branches of the corresponding superpotential, P_{\pm}. When there is a P_- branch, we rigorously prove a lower bound on the energy; the P_+ branch alone is not sufficient. Our numerical investigations i) confirm this picture, ii) confirm other critical aspects of the (complicated) proofs, and iii) suggest that the existence of P_- may in fact be necessary (as well as sufficient) for the energy of a designer gravity theory to be bounded below

Supporting First Grade Students Learning Number Facts Up to 10 Using a Parrot Game

Knowing number facts up to 10 become crucial if we want students to solve addition and subtraction problems using more abbreviated strategies. Otherwise, students will keep counting one-by-one until they get an answer. One of important number facts is number pairs that make ten because it is an important ‘benchmark' that students will refer to constantly. Considering the important of number facts up to 10, we designed a parrot game activity to supports students learning process. We designed an instructional activity based on Realistic Mathematics Education (RME) approach. We tried this instructional activity with twenty seven first grade students in SDN 179 Palembang. As a result, we found that many first grade students were able to know number facts up to 10, but they still need models to support their thinking

Long Range Order at Low Temperature in Dipolar Spin Ice

Recently it has been suggested that long range magnetic dipolar interactions are responsible for spin ice behavior in the Ising pyrochlore magnets ${\rm Dy_{2}Ti_{2}O_{7}}$ and ${\rm Ho_{2}Ti_{2}O_{7}}$. We report here numerical results on the low temperature properties of the dipolar spin ice model, obtained via a new loop algorithm which greatly improves the dynamics at low temperature. We recover the previously reported missing entropy in this model, and find a first order transition to a long range ordered phase with zero total magnetization at very low temperature. We discuss the relevance of these results to ${\rm Dy_{2}Ti_{2}O_{7}}$ and ${\rm Ho_{2}Ti_{2}O_{7}}$.Comment: New version of the manuscript. Now contains 3 POSTSCRIPT figures as opposed to 2 figures. Manuscript contains a more detailed discussion of the (i) nature of long-range ordered ground state, (ii) finite-size scaling results of the 1st order transition into the ground state. Order of authors has been changed. Resubmitted to Physical Review Letters Contact: [email protected]

On Convex Quadratic Approximation

In this paper we prove the counterintuitive result that the quadratic least squares approximation of a multivariate convex function in a finite set of points is not necessarily convex, even though it is convex for a univariate convex function. This result has many consequences both for the field of statistics and optimization. We show that convexity can be enforced in the multivariate case by using semidefinite programming techniques.Convex function;least squares;quadratic interpolation;semidefinite program- ming

Helping Students Acquainted with Multiplication in Rectangular Model

Usually, multiplication is introduced to students to represent quantities that come in group. However there is also rectangular array model which is also related to multiplication. Barmby et al (2009) has shown that the rectangular model such as array representations encourage students to develop their thinking about multiplication as a binary operation with row and column representing two inputs. Considering that finding, this study focusses on a design research that was conducted in Indonesia in which I investigate second grade students' (between 7 and 8 years old) in Madrasah Ibtidaiyah Negeri (MIN) 2 Palembang, Indonesia, ability to structuring the situation and their ability to represent rectangular model into multiplication sentence. The results shows us that students activity to structuring the situation, looking the number of objects in row or in column, lead them to repeated addition and transform it into multiplication sentence

Asymptotic generators of fermionic charges and boundary conditions preserving supersymmetry

We use a covariant phase space formalism to give a general prescription for defining Hamiltonian generators of bosonic and fermionic symmetries in diffeomorphism invariant theories, such as supergravities. A simple and general criterion is derived for a choice of boundary condition to lead to conserved generators of the symmetries on the phase space. In particular, this provides a criterion for the preservation of supersymmetries. For bosonic symmetries corresponding to diffeomorphisms, our prescription coincides with the method of Wald et al. We then illustrate these methods in the case of certain supergravity theories in $d=4$. In minimal AdS supergravity, the boundary conditions such that the supercharges exist as Hamiltonian generators of supersymmetry transformations are unique within the usual framework in which the boundary metric is fixed. In extended ${\mathcal N}=4$ AdS supergravity, or more generally in the presence of chiral matter superfields, we find that there exist many boundary conditions preserving ${\mathcal N}=1$ supersymmetry for which corresponding generators exist. These choices are shown to correspond to a choice of certain arbitrary boundary ``superpotentials,'' for suitably defined ``boundary superfields.'' We also derive corresponding formulae for the conserved bosonic charges, such as energy, in those theories, and we argue that energy is always positive, for any supersymmetry-preserving boundary conditions. We finally comment on the relevance and interpretation of our results within the AdS-CFT correspondence.Comment: 45 pages, Latex, no figures, v2: extended discussion of positive energy theorem and explicit form of fermionic generators, references adde