A counterexample to a conjecture of Larman and Rogers on sets avoiding distance 1

For $n \geq 2$ we construct a measurable subset of the unit ball in $\mathbb{R}^n$ that does not contain pairs of points at distance 1 and whose volume is greater than $(1/2)^n$ 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

The Market of Foreign Exchange Hedge in Brazil: Reactions of Financial Institutions to Interventions of the Central Bank

Between 1999 and 2002, Brazil's Central Bank sold expressive amounts of dollar indexed debt and foreign exchange swaps. This paper shows that in periods of high volatility of the exchange rate, first semester of 1999 and second semester of 2002, the Central Bank of Brazil increased the foreign exchange hedge, but the financial institutions used this to reduce their foreign exchange exposure. In contrast, increases in foreign hedge during periods of low volatility of the exchange rate were transferred to the productive sector.foreign exchange swaps, central bank interventions, foreign exchange risk

Is Inflation Persistence Over?

We analyze inflation persistence in several industrial and emerging countries in the recent past by estimating reduced-form models of inflation dynamics. We select a very representative group of 23 industrial and 17 emerging economies. Our sample period is comprised of quarterly data and starts in the first quarter of 1995. Our results show that inflation persistence is low and stable for all countries in our sample. It seems to be lower in industrial relative to emerging countries. Finally, even countries that have had “hyperinflation” experience in the recent past showed low levels of inflation persistence, albeit apparently higher than the other countries in our sample.

An Empirical Analysis of the External Finance Premium of Public Non-Financial Corporations in Brazil

Our objective in this paper is to analyze empirically the relationship between the external finance premium of non-financial corporations in Brazil with their default probability and with their demand for inventories. As for the former relation, we find that corporations that have greater external finance premium have greater probability of default. As for the latter, we find that the external finance premium is positive and statistically significantly correlated. The results confirm previous results of the literature that indicate that the balance sheet channel of monetary policy is relevant in Brazil.

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