Two remarkable doubly exponential series transformations of Ramanujan

The purpose of this note is to prove two doubly exponential series transformations found in Ramanujan’s second notebook. Dedicated to the memory of Professor K G Ramanathan     

Derivation of the probability distribution functions for succession quota random variables

The probability distribution functions (pdf's) of the sooner and later waiting time random variables (rv's) for the succession quota problem (k successes and r failures) are derived presently in the case of a binary sequence of order k. The probability generating functions (pgf's) of the above rv's are then obtained directly from their pdf's. In the case of independent Bernoulli trials, expressions for the pdf's in terms of binomial coefficients are also established.     

Process scale high performance liquid chromatography part IV—Examples of reversed phase separations

Summary The flexibility of process scale high performance liquid chromatography is demonstrated by three examples of reversed-phase separations (a) a standard purification (b) isolation of trace compounds, and (c) recovery from crystallisation mother liquors. It was shown that a material, thought to be acceptably pure, contained a toxic impurity, and a remaining impurity, a previously unseen component of potential therapeutic interest. Favourable cost data were indicated for example (3). This work was conducted in the Chemical Technology Unit, University of Manchester Institute of Science and Technology, PO Box 88, Manchester M6O 1OD, UK.     

Ligand Size Effect on PdLn Oxidative Addition with Aryl Bromide: A DFT Study

采用密度泛函理论的COSMO模型模拟了在DMF溶剂中三种不同体积的钯配体Pd(PR₃)_n (n=1,2) (PR₃=PH₃, PMe₃, PtBu₃)与溴苯氧化加成的反应过程. 对比双配体Pd(PR₃)₂的裂解能以及双配体钯Pd(PR₃)₂和单配体钯Pd(PR₃)氧化加成过程的能垒;结果表明,小体积的配体PH₃和PMe₃是以双配位的形式参加氧化加成过程的,而较大体积的配体PtBu₃是以单配位钯的形式参加氧化加成过程的.     

Thermal stability of ZnO thin film prepared by RF-magnetron sputtering evaluated by thermal desorption spectroscopy

Thermal stability of sputter deposited ZnO thin films was evaluated by thermal desorption spectroscopy (TDS). Desorption of Zn was mainly observed from the films deposited at low O₂/Ar gas ratio and low RF power. In contrast, O₂ desorption was mainly observed from the films deposited at high O₂/Ar gas ratio and high RF power. The amount of desorbed O₂ from the film increased with increasing the O₂/Ar gas flow ratio and the RF power. Furthermore, the desorption temperature of O₂ increased with increasing the RF power during the deposition. Thermal stability of the ZnO films was controlled not only by the O₂/Ar gas flow ratio, but also applied RF power to the target.     

Some results for the q-Bernoulli and q-Euler polynomials

We show some results for the q-Bernoulli and q-Euler polynomials. The formulas in series of the Carlitz's q-Stirling numbers of the second kind are also considered. The q-analogues of well-known formulas are derived from these results.     

Models for concurrent product and process design

We propose procedures to address product design and manufacturing process configurations concurrently in environments characterized by large degrees of product proliferation. Exploiting the intrinsic flexibility of product and process design, we present two approaches that synchronize production flows through the manufacturing system. These approaches integrate product and manufacturing system design decisions with operational concerns and provide powerful means for managing production in environments characterized by a proliferation of products. Experimental results show that the proposed methods can substantially reduce manufacturing lead times, work in process (WIP), and overall system complexity.     

Development of the parametric tolerance modeling and optimization schemes and cost-effective solutions

Most of previous research on tolerance optimization seeks the optimal tolerance allocation with process parameters such as fixed process mean and variance. This research, however, differs from the previous studies in two ways. First, an integrated optimization scheme is proposed to determine both the optimal settings of those process parameters and the optimal tolerance simultaneously which is called a parametric tolerance optimization problem in this paper. Second, most tolerance optimization models require rigorous optimization processes using numerical methods, since closed-form solutions are rarely found. This paper shows how the Lambert W function, which is often used in physics, can be applied efficiently to this parametric tolerance optimization problem. By using the Lambert W function, one can express the optimal solutions to the parametric tolerance optimization problem in a closed-form without resorting to numerical methods. For verification purposes, numerical examples for three cases are conducted and sensitivity analyses are performed.     

Coupled Galerkin and parameterization methods for optimal control of discretely connected parallel beams

Galerkin and wavelet methods for optimal boundary control of a couple of discretely connected parallel beams are proposed. First, the problem with boundary controls is converted into a problem with distributed controls. The problem is, then, reduced by a Galerkin-based approach into determining the optimal control of a linear time-invariant lumped parameter system, which will be solved by a wavelet-based method using Legendre wavelets. The integration-operational matrix and Kronecker product are utilized to significantly simplify the optimization problem into a system of linear equations. A numerical example is presented to demonstrate the applicability and the efficiency of the proposed method.