Spherical Tiling by 12 Congruent Pentagons

The tilings of the 2-dimensional sphere by congruent triangles have been extensively studied, and the edge-to-edge tilings have been completely classified. However, not much is known about the tilings by other congruent polygons. In this paper, we classify the simplest case, which is the edge-to-edge tilings of the 2-dimensional sphere by 12 congruent pentagons. We find one major class allowing two independent continuous parameters and four classes of isolated examples. The classification is done by first separately classifying the combinatorial, edge length, and angle aspects, and then combining the respective classifications together.Comment: 53 pages, 40 figures, spherical geometr

China’s new exchange rate regime, optimal basket currency and currency diversification

We build an optimising framework to analyse a class of economies that adopt an ECU-type basket currency while in transition to increased flexibility of the exchange rate regime. Instead of conventional basket pegging, such an economy uses an ECU-type currency index as a benchmark for monitoring and assessing exchange rate movements. This provides an anchoring device for the nations exchange rate regime and allows the home currencys exchange rate to fluctuate. Under the assumption that the central bank is chiefly interested in maintaining stability, the optimal structure of the basket currency is based on its contribution to minimizing the volatility of the countrys external account. A currency invariance index is applied to capture the effect of the countrys exit from exclusive linkage with the US dollar. The approach is illustrated by Chinese exchange rate policy. We find it advisable and viable for China to form a basket currency with a diversified portfolio of currencies. While the portfolios weighting scheme could favour the dollar, euro and Japanese yen, we show that the composition of the basket is open to a wide range of possibilities. Moreover, contrary to general fears, there is considerable potential for China to engage in currency diversification, which will not necessarily affect the dollars position.basket currency; currency diversification; China

Monotonicity Results for Arithmetic Means of Concave and Convex Functions

By majorization approaches, some known results on monotonicity of the arithmetic means of convex and concave functions are proved and generalized once again