A Transcendental Invariant of Pseudo-Anosov Maps

For each pseudo-Anosov map $\phi$ on surface $S$, we will associate it with a $\mathbb{Q}$-submodule of $\mathbb{R}$, denoted by $A(S,\phi)$. $A(S,\phi)$ is defined by an interaction between the Thurston norm and dilatation of pseudo-Anosov maps. We will develop a few nice properties of $A(S,\phi)$ and give a few examples to show that $A(S,\phi)$ is a nontrivial invariant. These nontrivial examples give an answer to a question asked by McMullen: the minimal point of the restriction of the dilatation function on fibered face need not be a rational point.Comment: 32 pages, 10 figures, abstract has been modified by following suggestion from Curtis McMulle

Rate of Decay of Stable Periodic Solutions of Duffing Equations

In this paper, we consider the second-order equations of Duffing type. Bounds for the derivative of the restoring force are given that ensure the existence and uniqueness of a periodic solution. Furthermore, the stability of the unique periodic solution is analyzed; the sharp rate of exponential decay is determined for a solution that is near to the unique periodic solution.Comment: Key words: Periodic solution; Stability; Rate of deca

Bank Discrimination in Transition Economies: Ideology, Information or Incentives?

We study bank discrimination against private firms in transition countries. Theoretically, we show that banks may discriminate for non-profit reasons, but this discrimination diminishes with a bank’s incentives and human capital. Employing matching bank-firm data from China, we empirically examine the extent, sources and consequences of discrimination. Our unique survey design allows us to disentangle sample truncation, omitted variable bias, and endogeneity issues. Our empirical findings confirm the theoretical predictions. We also find that as a result of discrimination, private firms resort to more expensive trade credits.http://deepblue.lib.umich.edu/bitstream/2027.42/39902/3/wp517.pd

Stability and exact multiplicity of periodic solutions of Duffing equations with cubic nonlinearities

We study the stability and exact multiplicity of periodic solutions of the Duffing equation with cubic nonlinearities. We obtain sharp bounds for h such that the equation has exactly three ordered T-periodic solutions. Moreover, when h is within these bounds, one of the three solutions is negative, while the other two are positive. The middle solution is asymptotically stable, and the remaining two are unstable.Comment: Keywords: Duffing equation; Periodic solution; Stabilit