Some identities involving the k-th power complements

The main purpose of this paper is using the elementary method to study the calculating problem of one kind infinite series involving the k-th power complements, and obtain several interesting identities

An improved largest gap routing heuristic for order picking

The largest-gap policy is a routing heuristic for order picking systems. In this paper we develop an improved largest gap routing method. A simulation approach is used to demonstrate the superior performance of the improved largest gap routing over traditional largest gap. Moreover, this paper tests the performance impact of storage assignment rules on largest gap routings. Several scenarios with various order sizes and different item popularity proportions are tested. Monte-Carlo simulation is used to carry out the experiments. The numerical results from the computational analysis show that our improved largest gap routing always outperforms the traditional largest gap routing, i.e. for all order sizes. The effect is the most distinct when the order size is smaller. Finally the study demonstrates that the optimal storage assignment rule to be combined with largest gap routing is within-aisle storage

Row Spacing Effects on Biomass Components and Accumulated Energy in \u3cem\u3eElymus sibiricus\u3c/em\u3e L.

Elymus sibiricus L. is widely distributed in the eastern Tibetan Plateau of China. Current research on E. sibiricus mainly focuses on resource assessment, breeding, cultivation and dynamics of forage yield. Little is known about the effects of row spacing on the biomass components and accumulated energy of E. sibiricus. We conducted a study to determine the effects of row spacing on biomass components and accumulated energy in E. sibiricus. These data are important to optimize forage and seed production of E. sibiricus, and can assist in developing animal husbandry, structurally adjusting the agricultural industry and protecting the environment

Convergence analysis of multigrid methods with residual scaling techniques

AbstractIn this paper, multigrid methods with residual scaling techniques for symmetric positive definite linear systems are considered. The idea of perturbed two-grid methods proposed in [7] is used to estimate the convergence factor of multigrid methods with residual scaled by positive constant scaling factors. We will show that if the convergence factors of the two-grid methods are uniformly bounded by σ (σ<0.5), then the convergence factors of the W-cycle multigrid methods are uniformly bounded by σ/(1−σ), whether the residuals are scaled at some or all levels. This result extends Notay’s Theorem 3.1 in [7] to more general cases. The result also confirms the viewpoint that the W-cycle multigrid method will converge sufficiently well as long as the convergence factor of the two-grid method is small enough. In the case where the convergence factor of the two-grid method is not small enough, by appropriate choice of the cycle index γ, we can guarantee that the convergence factor of the multigrid methods with residual scaling techniques still has a uniform bound less than σ/(1−σ). Numerical experiments are provided to show that the performance of multigrid methods can be improved by scaling the residual with a constant factor. The convergence rates of the two-grid methods and the multigrid methods show that the W-cycle multigrid methods perform better if the convergence rate of the two-grid method becomes smaller. These numerical experiments support the proposed theoretical results in this paper

Fabrication of a microresonator-fiber assembly maintaining a high-quality factor by CO2 laser welding

We demonstrate fabrication of a microtoroid resonator of a high-quality (high-Q) factor using femtosecond laser three-dimensional (3D) micromachining. A fiber taper is reliably assembled to the microtoroid using CO2 laser welding. Specifically, we achieve a high Q-factor of 2.12*10^6 in the microresonator-fiber assembly by optimizing the contact position between the fiber taper and the microtoroid.Comment: 7 pages, 5 figure