On minimal energies of trees of a prescribed diameter

The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. Let denote the set of trees on n vertices and diameter d, . Yan and Ye [Appl. Math. Lett. 18 (2005) 1046–1052] have recently determined the unique tree in with minimal energy. In this article, the trees in with second-minimal energy are characterizedAMS Subject Classification: 05C50, 05C35     

空中加油问题

对空中加油问题的前两个问题进行了深入系统的研究,发现并证明了与最优解相关的若干事实,对后续的问题求解具有重要的意义.利用得出的结论,加以推导得出求解rn的递推公式,并由此设计了类似于动态规划的循环递推算法.引入“虚拟基地”和“一次性加油”的概念,通过推导得到rn的上界和下界,得出rn与n的渐进关系是对数关系.最后,又提出将问题转化成为二维平面问题,建立一个二叉树模型,通过求解线性规划得到最优解.     

Optimal Relay Location for Resource-limited Energy-efficient Wireless Communication

In the design of wireless networks, techniques for improving energy efficiency and extending network lifetime have great importance, particularly for defense and civil/rescue applications where resupplying transmitters with new batteries is not feasible. In this paper we study a method for improving the lifetime of wireless networks by minimizing the length of the longest edge in the interconnecting tree by deploying additional relay nodes at specific locations. This optimization problem, known as the Bottleneck Steiner Tree Problem (BSTP), asks to find a Steiner tree for n terminals with at most k Steiner points such that the length of the longest edge in the tree is minimized. We present a ratio- polynomial time approximation algorithm for BSTP, where is an arbitrary positive number.     

Stability of the phase transition of critical-field Ising model on Cayley trees under inhomogeneous external fields

We consider the ferromagnetic Ising model with spatially dependent external fields on a Cayley tree, and we investigate the conditions for the existence of the phase transition for a class of external fields, asymptotically approaching a homogeneous critical external field. Our results extend earlier results by Rozikov and Ganikhodjaev.     

Characterization of polysaccharides purified from Lacquer tree seed cakes and its antioxidant activity

Three polysaccharides, LTPS-1, LTPS-21 and LTPS-31 were isolated and purified from the seed cakes of lacquer tree using DEAE-52 cellulose and Sephadex G-200 column chromatography. The total sugar contents of LTPS-1, LTPS-21 and LTPS-31 were 931.8, 958.2 and 895.1 g kg^?1, respectively. LTPS-1 (3.48 kDa) was mainly composed of rhamnose, arabinose, glucose and galactose in a ratio of 35.36:5.06:1:2. LTPS-21 (11.4 kDa) was mainly composed of rhamnose, arabinose, mannose and galactose in a ratio of 41.93:21.8:1.01:9.24. LTPS-31 (19.49 kDa) was mainly composed of rhamnose, arabinose and mannose in a ratio of 38.31:16.44:1.1. IR analysis suggested they contained lower sulphuric acids, the LTPS-21 and LTPS-31 belonged to β-type polysaccharide. Among the three polysaccharides, LTPS-21 exhibited the strongest reducing power, scavenging activity on ABTS and hydroxyl radicals. These findings suggested that polysaccharides from the seed cakes could be potentially developed as natural functional ingredients in the food and cosmetic industry.     

Mining information from time series in the form of sentences of natural language

The goal of this paper is to provide a more detailed explanation of the principles how special formulas that characterize properties of trend of time series can be formed and how they are interpreted. Then we show how these formulas can be used in a tectogrammatical tree that construes special sentences of natural language, using which information on behavior of time series is provided. We also outline the principles of mining this information. The last part is devoted to application of the theory of intermediate quantifiers to mining summarized information on time series also in sentences of natural language.     

Computational Tools for Evaluating Phylogenetic and Hierarchical Clustering Trees

This article presents an alternative derivation of the generalized approximate crossvalidation (GACV) score of Xiang and Wahba (1996) for smoothing parameter selection in penalized likelihood regression. The new derivation suggests a simple numerical solution that is stable for all sample sizes. Also suggested is a variant of the score that can be computationally more convenient. Simple simulations are presented to illustrate the effectiveness of the scores.     

Patterns in treeshelves

We study the distribution and the popularity of left children on sets of treeshelves avoiding a pattern of size three. (Treeshelves are ordered binary increasing trees where every child is connected to its parent by a left or a right link.) The considered patterns are sub-treeshelves, and for each such a pattern we provide exponential generating function for the corresponding distribution and popularity. Finally, we present constructive bijections between treeshelves avoiding a pattern of size three and some classes of simpler combinatorial objects.