Combinatorial proofs of some properties of tangent and Genocchi numbers

The tangent number $T_{2n+1}$ is equal to the number of increasing labelled complete binary trees with $2n+1$ vertices. This combinatorial interpretation immediately proves that $T_{2n+1}$ is divisible by $2^n$. However, a stronger divisibility property is known in the studies of Bernoulli and Genocchi numbers, namely, the divisibility of $(n+1)T_{2n+1}$ by $2^{2n}$. The traditional proofs of this fact need significant calculations. In the present paper, we provide a combinatorial proof of the latter divisibility by using the hook length formula for trees. Furthermore, our method is extended to $k$-ary trees, leading to a new generalization of the Genocchi numbers

Three Essays on the Understanding of Urban Development

Cities started the unprecedented growth about one hundred years ago. Their importance and significance are reflected by their high productivities and spatial concentrations. The understanding on urban development would help improve urban management and policies and increase wellbeing of urban residents. The three related essays in this dissertation try to improve the understanding from the perspectives of employment centers and agglomeration economies, interactions between labor and housing markets, and the behavior of local governments. The first essay examines the role of employment centers on economic development. The theoretical literature suggests that agglomeration economies are the main force behind the formation and evolution of employment centers, as well as behind economic growth in general. Applying the birth model to employment centers in Maryland, I find agglomeration effects are increased by the centers, particularly those with high employment size or industrial diversity. Ignoring employment centers may overestimate the agglomeration effects when using the fixed distance measurement. Policy implications are local officials may use employment centers as a vehicle to promote economic growth. In the second essay I test the impact of job loss on housing foreclosures. A great challenge in this study, as well as in interactions between labor and housing markets in general, is the geographic mismatch between employment and residential locations. This partially explains the mixed effects of job loss on foreclosures found in the literature. In order to gauge this effect, I develop a job loss vulnerability index using home-work commuting pairs. After fixing the attenuation bias from measurement errors, I find that job loss plays an important role in foreclosure decisions. This essay provides evidence for impact from labor market bust to housing market depression. The third essay estimates the spending pattern of off-budget revenues. The literature assumes different spending preferences of budgetary and off-budget revenues, but empirical evidence are scarce due to the lack of off-budget data. I use land revenues to proxy off-budget revenues in Chinese cities. I find that off-budget revenues do not crowd out budgetary expenditures, and they tend to support visible and tangible projects, rather than some other traditional public spending items that are not quite obvious

An Accelerated DC Programming Approach with Exact Line Search for The Symmetric Eigenvalue Complementarity Problem

In this paper, we are interested in developing an accelerated Difference-of-Convex (DC) programming algorithm based on the exact line search for efficiently solving the Symmetric Eigenvalue Complementarity Problem (SEiCP) and Symmetric Quadratic Eigenvalue Complementarity Problem (SQEiCP). We first proved that any SEiCP is equivalent to SEiCP with symmetric positive definite matrices only. Then, we established DC programming formulations for two equivalent formulations of SEiCP (namely, the logarithmic formulation and the quadratic formulation), and proposed the accelerated DC algorithm (BDCA) by combining the classical DCA with inexpensive exact line search by finding real roots of a binomial for acceleration. We demonstrated the equivalence between SQEiCP and SEiCP, and extended BDCA to SQEiCP. Numerical simulations of the proposed BDCA and DCA against KNITRO, FILTERED and MATLAB FMINCON for SEiCP and SQEiCP on both synthetic datasets and Matrix Market NEP Repository are reported. BDCA demonstrated dramatic acceleration to the convergence of DCA to get better numerical solutions, and outperformed KNITRO, FILTERED, and FMINCON solvers in terms of the average CPU time and average solution precision, especially for large-scale cases.Comment: 24 page