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

Applications of Noble Gases in Hydrogeology in Fractured, Fast Infiltration Systems - From the Greenland and Columbia Ice Sheets to Hawaii

Due to their temperature dependency, stable noble gases (He, Ne, Ar, Kr, and Xe) have been routinely used as indicators of past climate in sedimentary systems for over four decades. However, noble gas studies in fractured systems, where infiltration is rapid, remain scarce. These include studies in ice sheets in old cratonic regions (e.g., Antarctica and Greenland), as well as old and recent volcanic areas such as the archipelagos of the Galapagos and Hawaii. Here, noble gas studies in fractured systems are presented. These include two studies in ice-covered regions, one in the Greenland Ice Sheet (GrIS) and the other in the Athabasca Glacier (AG) of the Columbia Icefield in the Canadian Rockies, as well as two studies in a tropical basaltic island, the Island of Maui, Hawaii. Noble gases in the GrIS (Chapter 2) and the AG (Chapter 3) studies are used to constrain glacial meltwater sources, water source altitude and water residence times. In Maui, noble gases are first used to characterize the different water sources contributing to groundwater recharge (e.g., fog, orographic and synoptic-scale rain), and to assess whether timing and location of recharge can be estimated based on atmospheric noble gas signatures (Chapter 4). In Chapter 5, the potential for noble gases to record temporal variations is assessed. Noble gases are used together with oxygen and hydrogen isotopic composition data to further constrain water source altitudes in Maui. Noble gases in the meltwater samples from both the GrIS and the AG are dominated by a partially equilibrated air-saturated water (ASW) component rather than trapped air in the glacial ice. Water source altitudes based on Xe range between 0.8 and 2.4 km for most samples from the GrIS and between 2.5 and 3.5 km for the AG. A crustal He component, observed in almost all samples in both studies, is used to estimate water residence times. Most meltwater samples from the GrIS yield water residence times between ~100 and ~400 years while two samples yield older ages of ~2000 and ~4000 years. In contrast, samples from the AG yield a younger average of ~160 years. Water samples were collected in Maui from rain events, springs from perched aquifers, and wells tapping the basal aquifer in June 2014 and February 2016. All samples are in disequilibrium with the atmosphere at the collection point and do not represent the mean annual air temperature. Distinct noble gas signatures in spring and basal aquifer samples suggest that the two types of aquifers are separate entities. In June 2014, noble gases in rainwater and basal aquifer display an ice-like signature possibly related to synoptic-scale rain. The basal aquifer yields similar noble gas signatures in both sampling seasons, while temporal variations are observed in rainwater and spring samples. A few springs and wells yield samples with a significant mantle He component in both years. A combined dataset of noble gas and water stable isotopic composition yield source altitudes for rainwater samples between 0.1 and 3 km above sea level (asl). Water source altitudes for most groundwater samples range between 1.5 and 5.5 km asl, indicating that the water source contributing to groundwater recharge that originate at higher altitudes in the atmosphere was not sampled. This dissertation has important scientific implications in the fields of glaciology, hydrogeology, and meteorology.PHDEarth and Environmental SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/144162/1/niuyi_1.pd

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