Statistical Optimization of Flavonoid and Antioxidant Recovery from Macerated Chinese and Malaysian Lotus Root (Nelumbo nucifera) Using Response Surface Methodology

In this study, the combination of parameters required for optimal extraction of anti-oxidative components from the Chinese lotus (CLR) and Malaysian lotus (MLR) roots were carefully investigated. Box–Behnken design was employed to optimize the pH (X₁: 2–3), extraction time (X₂: 0.5–1.5 h) and solvent-to-sample ratio (X₃: 20–40 mL/g) to obtain a high flavonoid yield with high % DPPH_sc free radical scavenging and Ferric-reducing power assay (FRAP). The analysis of variance clearly showed the significant contribution of quadratic model for all responses. The optimal conditions for both Chinese lotus (CLR) and Malaysian lotus (MLR) roots were obtained as: CLR: X₁ = 2.5; X₂ = 0.5 h; X₃ = 40 mL/g; MLR: X₁ = 2.4; X₂ = 0.5 h; X₃ = 40 mL/g. These optimum conditions gave (a) Total flavonoid content (TFC) of 0.599 mg PCE/g sample and 0.549 mg PCE/g sample, respectively; (b) % DPPH_sc of 48.36% and 29.11%, respectively; (c) FRAP value of 2.07 mM FeSO₄ and 1.89 mM FeSO₄, respectively. A close agreement between predicted and experimental values was found. The result obtained succinctly revealed that the Chinese lotus exhibited higher antioxidant and total flavonoid content when compared with the Malaysia lotus root at optimum extraction condition.     

A Necessary and Sufficient Condition for the Existence of a Heterochromatic Spanning Tree in a Graph

We prove the following theorem. An edge-colored (not necessary to be proper) connected graph G of order n has a heterochromatic spanning tree if and only if for any r colors (1≤r≤n−2), the removal of all the edges colored with these r colors from G results in a graph having at most r+1 components, where a heterochromatic spanning tree is a spanning tree whose edges have distinct colors.     

时变线性系统特征根判稳的推广

本文研究了二维时变线性系统零解的稳定性，给出了一些允许系数矩阵主对角元变号，系数矩阵特征根变号的判定稳定性的充分条件。     

The semi‐iterative method applied to the hyper‐power iteration

The semi‐iterative method (SIM) is applied to the hyper‐power (HP) iteration, and necessary and sufficient conditions are given for the convergence of the semi‐iterative–hyper‐power (SIM–HP) iteration. The root convergence rate is computed for both the HP and SIM–HP methods, and the quotient convergence rate is given for the HP iteration. Copyright © 2005 John Wiley & Sons, Ltd.     

Cyclic permutations and evolutionary trees

Given a tree with leaf set X, there are certain ways of arranging the elements of X in a circular order so that can be embedded in the plane and ‘preserve’ this ordering. We investigate some new combinatorial properties of these ‘circular orderings.’ We then use these properties to establish two results concerning dissimilarity maps on X that are induced by edge-weighted trees with leaf set X.     

中国邮路问题的一个解法

本给出中国邮路问题的一个解法。虽然它仍只是初始方案，但因考虑了问题的足够信息，因而这种方案至少拉近了与最优解的距离。     

一种基于图论与熵的专家判断客观可信度的确定方法

本文给出一种群决策中确定专家判断可信度的方法，其主要思路是首先通过图论中最小生成树的方法提取专家判断矩阵的全部信息，其次，使用相对熵指标确定获得专家判断的最终结果，并同时衡量专家自身判断的统一程度，从而确定专家判断的相对客观可信度。最后，文章给出一个典型的算例以说明该方法的可行性和有效性。     

Searching in trees

In (Discrete Math. 17 (1977)181) Rivest introduced the search complexity of binary trees and proved that among all binary trees with a fixed search complexity the smallest ones are the so-called Fibonacci trees. This result is extended for q-trees. The structure of the smallest q-trees is again Fibonacci-like but more complicated than in the binary case. In addition an upper bound for the asymptotic growth of these trees is given.     

A greedy algorithm for multicut and integral multiflow in rooted trees

We present an O(min(Kn,n²)) algorithm to solve the maximum integral multiflow and minimum multicut problems in rooted trees, where K is the number of commodities and n is the number of vertices. These problems are NP-hard in undirected trees but polynomial in directed trees. In the algorithm we propose, we first use a greedy procedure to build the multiflow then we use duality properties to obtain the multicut and prove the optimality.     

Stability Conditions of the MMAP [ K ]/ G [ K ]/1/ LCFS Preemptive Repeat Queue

In this paper, we study the stability conditions of the MMAP[K]/G[K]/1/LCFS preemptive repeat queue. We introduce an embedded Markov chain of matrix M/G/1 type with a tree structure and identify conditions for the Markov chain to be ergodic. First, we present three conventional methods for the stability problem of the queueing system of interest. These methods are either computationally demanding or do not provide accurate information for system stability. Then we introduce a novel approach that develops two linear programs whose solutions provide sufficient conditions for stability or instability of the queueing system. The new approach is numerically efficient. The advantages and disadvantages of the methods introduced in this paper are analyzed both theoretically and numerically.