7,399 research outputs found
A counterexample to a conjecture of Larman and Rogers on sets avoiding distance 1
For we construct a measurable subset of the unit ball in
that does not contain pairs of points at distance 1 and whose
volume is greater than times the volume of the ball. This disproves a
conjecture of Larman and Rogers from 1972.Comment: 3 pages, 1 figure; final version to appear in Mathematik
"Inflation Targeting in Brazil"
The purpose of this paper is to examine inflation targeting (IT) in emerging countries by concentrating essentially on the case of Brazil. The IT monetary policy regime has been adopted by a significant number of countries. While the focus of this paper is on Brazil, which began inflation targeting in 1999, we also examine the experience of other countries, both for comparative purposes and for evidence of the extent of this "new" economic policy's success. In addition, we compare the experience of Brazil with that of non-IT countries, and ask the question of whether adopting IT makes a difference in the fight against inflation.
The positive semidefinite Grothendieck problem with rank constraint
Given a positive integer n and a positive semidefinite matrix A = (A_{ij}) of
size m x m, the positive semidefinite Grothendieck problem with
rank-n-constraint (SDP_n) is
maximize \sum_{i=1}^m \sum_{j=1}^m A_{ij} x_i \cdot x_j, where x_1, ..., x_m
\in S^{n-1}.
In this paper we design a polynomial time approximation algorithm for SDP_n
achieving an approximation ratio of
\gamma(n) = \frac{2}{n}(\frac{\Gamma((n+1)/2)}{\Gamma(n/2)})^2 = 1 -
\Theta(1/n).
We show that under the assumption of the unique games conjecture the achieved
approximation ratio is optimal: There is no polynomial time algorithm which
approximates SDP_n with a ratio greater than \gamma(n). We improve the
approximation ratio of the best known polynomial time algorithm for SDP_1 from
2/\pi to 2/(\pi\gamma(m)) = 2/\pi + \Theta(1/m), and we show a tighter
approximation ratio for SDP_n when A is the Laplacian matrix of a graph with
nonnegative edge weights.Comment: (v3) to appear in Proceedings of the 37th International Colloquium on
Automata, Languages and Programming, 12 page
Grothendieck inequalities for semidefinite programs with rank constraint
Grothendieck inequalities are fundamental inequalities which are frequently
used in many areas of mathematics and computer science. They can be interpreted
as upper bounds for the integrality gap between two optimization problems: a
difficult semidefinite program with rank-1 constraint and its easy semidefinite
relaxation where the rank constrained is dropped. For instance, the integrality
gap of the Goemans-Williamson approximation algorithm for MAX CUT can be seen
as a Grothendieck inequality. In this paper we consider Grothendieck
inequalities for ranks greater than 1 and we give two applications:
approximating ground states in the n-vector model in statistical mechanics and
XOR games in quantum information theory.Comment: 22 page
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