On the Hecke Eigenvalues of Maass Forms

Let $\phi$ denote a primitive Hecke-Maass cusp form for $\Gamma_o(N)$ with the Laplacian eigenvalue $\lambda_\phi=1/4+t_{\phi}^2$. In this work we show that there exists a prime $p$ such that $p\nmid N$, $|\alpha_{p}|=|\beta_{p}| = 1$, and $p\ll(N(1+|t_{\phi}|))^c$, where $\alpha _{p},\;\beta _{p}$ are the Satake parameters of $\phi$ at $p$, and $c$ is an absolute constant with $0<c<1$. In fact, $c$ can be taken as $0.27332$. In addition, we prove that the natural density of such primes $p$ ($p\nmid N$ and $|\alpha_{p}|=|\beta_{p}| = 1$) is at least $34/35$.Comment: Version 2: typos corrected and a new section on natural density adde

Spectral characterization of a specific class of trees

In this paper, it is shown that the graph $T_4(p,q,r)$ is determined by its Laplacian spectrum and there are no two non-isomorphic such graphs which are cospectral with respect to adjacency spectrum.Comment: 11 pages, 4 figure

Village Inequality in Western China

Increased regional inequality has been a major concern in many emerging economies like China, India, Vietnam and Thailand. However, even a large inequality is observed within the lagging regions. The objective of this paper is to look into what are the sources of within region inequality using the community surveys and a census type of households in Western China. This snapshot view of inequality within and between rural villages in western China is based on a census-type household survey in three administrative villages and a sampling survey of 286 natural villages in the poor province of Guizhou in 2004. In contrast to coastal regions, nonfarm income is distributed unevenly in this inland western region. This acco unts for the largest share of overall income inequality. But agriculture is still the rural peoples major source of livelihood in this particular location. On the expenditure side, health care is one of the most important sources of inequality. Because rural income is strongly related to human capital, the uneven access to health care will translate into a larger income gap in the long run. The analysis based on the natural village survey indicates that income varies widely across villages. Access to infrastructure and markets, education, and political participation explain most of this variation. These findings have important implications on the future development strategy in promoting lagging regions development and poverty reduction. While the overall economic development will be the main instrument to bring the majority poor out of poverty, a targeted approach has become increasingly crucial in helping the poor villages and households. It is critical to understand why these villages and households can not particulate in the growth process and how development programs and various transfer programs help them to overcome the constraints they face.Rural Development, Poverty, Inequality, Public investment, H54, O47, O53, R11, Community/Rural/Urban Development,