The log-convexity of the poly-Cauchy numbers

In 2013, Komatsu introduced the poly-Cauchy numbers, which generalize Cauchy numbers. Several generalizations of poly-Cauchy numbers have been considered since then. One particular type of generalizations is that of multiparameter-poly-Cauchy numbers. In this paper, we study the log-convexity of the multiparameter-poly-Cauchy numbers of the first kind and of the second kind. In addition, we also discuss the log-behavior of multiparameter-poly-Cauchy numbers

Two asymptotic expansions for gamma function developed by Windschitl's formula

In this paper, we develop Windschitl's approximation formula for the gamma function to two asymptotic expansions by using a little known power series. In particular, for $n\in \mathbb{N}$ with $n\geq 4$, we have \begin{equation*} \Gamma \left( x+1\right) =\sqrt{2\pi x}\left( \tfrac{x}{e}\right) ^{x}\left( x\sinh \tfrac{1}{x}\right) ^{x/2}\exp \left( \sum_{k=3}^{n-1}\tfrac{\left( 2k\left( 2k-2\right) !-2^{2k-1}\right) B_{2k}}{2k\left( 2k\right) !x^{2k-1}} +R_{n}\left( x\right) \right) \end{equation*} with \begin{equation*} \left| R_{n}\left( x\right) \right| \leq \frac{\left| B_{2n}\right| }{2n\left( 2n-1\right) }\frac{1}{x^{2n-1}} \end{equation*} for all $x>0$, where $B_{2n}$ is the Bernoulli number. Moreover, we present some approximation formulas for gamma function related to Windschitl's approximation one, which have higher accuracy.Comment: 14 page

An accurate approximation formula for gamma function

In this paper, we present a very accurate approximation for gamma function: \begin{equation*} \Gamma \left( x+1\right) \thicksim \sqrt{2\pi x}\left( \dfrac{x}{e}\right) ^{x}\left( x\sinh \frac{1}{x}\right) ^{x/2}\exp \left( \frac{7}{324}\frac{1}{ x^{3}\left( 35x^{2}+33\right) }\right) =W_{2}\left( x\right) \end{equation*} as $x\rightarrow \infty$, and prove that the function $x\mapsto \ln \Gamma \left( x+1\right) -\ln W_{2}\left( x\right)$ is strictly decreasing and convex from $\left( 1,\infty \right)$ onto $\left( 0,\beta \right)$, where \begin{equation*} \beta =\frac{22\,025}{22\,032}-\ln \sqrt{2\pi \sinh 1}\approx 0.00002407. \end{equation*}Comment: 9 page

Integral Reduction by Unitarity Method for Two-loop Amplitudes: A Case Study

In this paper, we generalize the unitarity method to two-loop diagrams and use it to discuss the integral bases of reduction. To test out method, we focus on the four-point double-box diagram as well as its related daughter diagrams, i.e., the double-triangle diagram and the triangle-box diagram. For later two kinds of diagrams, we have given complete analytical results in general (4-2\eps)-dimension.Comment: 52 pages, 1 figur

Optimization tool for transit bus fleet management

Transit agencies face the challenge of being environmentally-friendly, while maintaining cost-effective operation. Many studies have focused on investigating new bus technologies to reduce emissions and cost. However, they ignored the potential environmental and economic gain by improving the fitness and harmony between individual buses and routes in fleets. This dissertation provided, for the first time, a tool for fleet operator to intelligently dispatch buses and select new technologies that are tailored to their needs and business.;One key element in this tool is a bus life cycle cost model that can simulate and predict every capital and operational cost category for different bus technologies. The cost model was funded by Transportation Research Board and developed in Transit Cooperative Research Program (TCRP) C-15 project, the purpose of which was to assess hybrid-electric bus performance in real-world operation. The research team (author as a key member) picked four bus transit agencies among a handful of test sites that were operating hybrid-electric buses and collected 28 month bus operation data at almost all data collection sites. The sophisticated life cycle cost model is the backbone of this tool to calculate cost.;The other key element is a green house gas (GHG) emissions model, which was based on the fuel consumption model in the TCRP C-15 project and GREET model generated at Argonn National Laboratory. The GHG model utilizes fuel consumption data to provide tail pipe GHG emissions and well-to-tank GHG emissions for specific fuels and bus propulsion technologies.;The last key element is the use of genetic algorithm (GA) as a search and optimization scheme in the fleet management tool. A ranking matrix was developed to rate and compare different dispatch strategies on multiple criteria, which can vary in units or scales. When a fleet has large number of buses (dozens to thousands) on multiple routes, the number of all possible bus dispatch strategies becomes tremendously huge and difficult to explore. The GA uses the evolution theory of Only the strongest survive to find the best strategy.;The tool shows that optimization objectives dictated dispatch strategies that are successful in specific applications. For example, in a 35-bus fleet examined in this dissertation, the proposed dispatch strategy could reduce fleet well-to-wheels (WTW) GHG emissions up to 364 metric tons of carbon dioxide equivalents, a 17.5% GHG emissions reduction from the initial dispatch strategy. The same dispatch strategy increased {dollar}75K in annual operation cost, a 7% increase. However, a different dispatch strategy, found for maximizing cost reduction, could save {dollar}90K instead, a 9% reduction in annual operation cost. For the case of reducing operation cost, the operation cost difference between the best and worst dispatch strategy was {dollar}220K a year