The Infinity Tree: Representing Infinities of Real Numbers with Countably Infinite Tree Structures

This paper discusses how the infinite set of real numbers between 0 and 1 could be represented by a countably infinite tree structure which would avoid Cantor’s diagonalization argument that the set of real numbers is not countably infinite. Likewise, countably infinite tree structures could represent all real numbers, and all points in any number of dimensions in multi-dimensional spaces. The objective of this paper is not to overturn previous research based on Cantor’s argument, but to suggest that this situation may be treated as a definitional or axiomatic choice. This paper proposes a “non-Cantorian” branch of cardinality theory, representing all these infinities with countably infinite tree structures. This approach would be consistent with the Continuum Hypothesis.

Vertex-Neighbor-Scattering Number Of Trees

A vertex subversion strategy of a graph G=(V,E) is a set of vertices S V(G) whose closed neighborhood is deleted from G . The survival subgraph is denoted by G/S . We call S a cut-strategy of G if G/S is disconnected, or is a clique, or is φ . The vertex-neighbor scattering number of G is defined to be VNS(G)=max{ω(G/S)-|S|} , where S is any cut-strategy of G , and ω(G/G) is the number of the components of G/S . It has been proved that the computing problem of this parameter is NP–complete, so we discuss the properties of vertex-neighbor-scattering number of trees in this paper.

Independence Numbers in Trees

The independence number of a graph G is the maximum cardinality among all independent sets of G. For any tree T of order n ≥ 2, it is easy to see that . In addition, if there are duplicated leaves in a tree, then these duplicated leaves are all lying in every maximum independent set. In this paper, we will show that if T is a tree of order n ≥ 4 without duplicated leaves, then . Moreover, we constructively characterize the extremal trees T of order n ≥ 4, which are without duplicated leaves, achieving these upper bounds.

Edge-face Chromatic Number of 2-connected 1-tree with △(G) = 5

Wang Vei-fan proved that the edge-face chromatic number of a 2-connected1-tree with the maximum degree is not less than 6 is its maximum degree, and he conjecturedthat it is true when the maximum degree is 5. This paper proves the conjecture.

Combinatorial Interpretation of Raney Numbers and Tree Enumerations

A new combinatorial interpretation of Raney numbers is proposed. We apply this combinatorial interpretation to solve several tree enumeration counting problems. Further a generalized Catalan triangle is introduced and some of its properties are proved.

Clique-transversal number of graphs whose clique-graphs are trees

Given a graph G, a subgraph C is called a clique of G if C is a complete subgraph of G maximal under inclusion and ｜C｜ ≥2. A clique-transversal set S of G is a set of vertices of G such that S meets all cliques of G. The clique-transversal number, denoted as τC（G）, is the minimum cardinality of a clique-transversal set in G. The clique-graph of G, denoted as K（G）, is the graph obtained by taking the cliques of G as vertices, and two vertices are adjacent if and only if the corresponding cliques in G have nonempty intersection. Let F be a class of graphs G such that F = {G｜ K（G） is a tree}. In this paper the graphs in F having independent clique-transversal sets are shown and thus τC（G）/｜G｜ ≤ 1/2 for all G ∈F.

STRONG LAW OF LARGE NUMBERS AND ASYMPTOTIC EQUIPARTITION PROPERTY FOR NONSYMMETRIC MARKOV CHAIN FIELDS ON CAYLEY TREES

Some strong laws of large numbers for the frequencies of occurrence of states and ordered couples of states for nonsymmetric Markov chain fields （NSMC） on Cayley trees are studied. In the proof, a new technique for the study of strong limit theorems of Markov chains is extended to the case of Markov chain fields, The asymptotic equipartition properties with almost everywhere （a,e.） convergence for NSMC on Cayley trees are obtained,

STRONG LAW OF LARGE NUMBERS AND SHANNON-MCMILLAN THEOREM FOR MARKOV CHAINS FIELD ON CAYLEY TREE

This paper studies the strong law of large numbers and the Shannom-McMillan theorem for Markov chains field on Cayley tree. The authors first prove the strong law of large number on the frequencies of states and orderd couples of states for Markov chains field on Cayley tree. Then they prove the Shannon-McMillan theorem with a.e. convergence for Markov chains field on Cayley tree. In the proof, a new technique in the study the strong limit theorem in probability theory is applied.

一种高效能可重构1024位大数乘法器的设计

在SM9加密等算法中经常使用大数乘法,为了解决大数乘法中关键电路延迟过高、能耗过大的问题,设计了一种基于流水线的可重构1024位乘法器。使用64位乘法单元和128位先行进位加法单元,分20个周期流水产生最终结果,缓解了传统乘法器中加法部分的延时,实现电路复用,有效减小能耗。在SMIC 0.18μm工艺库下,关键电路延迟2.5 ns,电路面积7.03 mm^(2),能耗576 mW。

给定独立数的树的倒数度距离

设G是一个简单的无向连通图,T_(n,α)是顶点数为n独立数为α的所有树的集合。本文主要讨论了在集合T_(n,α)中的最大倒数度距离,并刻画了唯一对应的极图。

Motzkin树叶点的计数

本文主要研究了n条边的Motzkin树的叶点总数.分别利用符号化方法和双射证明了n条边的Motzkin树的叶点总数与长度为n的自由的Motzkin路的个数相等.利用这个双射也可以得到n个内点的完全二元树的叶点总数,并且给出了半长为n的且有k个峰的Dyck路的个数是Narayana数这一结论一个新的证明.

基于α-截集三角模糊数和攻击树的CTCS网络安全风险评估方法

针对工业控制系统网络安全风险评估影响因素的不确定性和指标量化困难问题,提出一种基于模糊理论和攻击树的方法评估工业控制系统风险,并将它应用于中国列车控制系统(CTCS)的风险评估。首先,基于CTCS可能面临的网络安全威胁和系统自身的脆弱性建立攻击树模型,使用α-截集三角模糊数(TFN)计算攻击树叶节点和攻击路径的区间概率;其次,利用层次分析法(AHP)建立安全事件损失数学模型,最终得出风险评估值。实验结果表明,所提方法可以有效评估系统风险,预测攻击路径,降低主观因素对风险评估过程的影响,使评估结果更契合实际,为安全防护策略的选择提供参考和依据。

长螺旋钻机成桩混合料堵管的模糊故障树分析

在钻探工程施工过程中,不可避免地会出现机械故障。采用故障树分析其原因是常用的方法之一。但工程施工中,往往数据不足或不注意保存这些数据,实际应用时难以赋值,只能进行定性分析。针对工程施工机械出现故障的模糊性和不确定性,用模糊数描述底事件的发生概率,提出一种基于三角模糊数的工程施工机械故障树分析方法。该方法通过专家、现场技术人员及操作人员的经验构造三角模糊数描述各底事件发生的概率区间,通过计算三角模糊数的中位数重要度,按重要度大小对故障原因排序,作为制定预防措施的依据。最后将其应用于长螺旋钻机成桩混合料堵管的原因分析,按模糊重要度对故障有针对性地制定预防措施。结果表明:重要影响因素排序与现场技术人员及操作人员的定性分析与判断相符。

基于最小生成树的数字图像多层次分割方法研究

如今,由于图像离散化造成分割的误差大,因此,提出基于最小生成树的数字图像多层次分割方法的研究。首先,基于最小生成树进行数字图像预处理,对图像进行整体特征提取,其次,构建出图像多层次分割标准,以不同的场景需求为依据,来对权值进行定义,确保最终的分割结果在内部具有高度的相似性,最后,实现数字图像多层次分割。为验证文章方法的有效性需要进行对比测试,实验结果:基于最小生成树的数字图像多层次分割方法的区域数目较少更符合人类视觉特征,分割效果更加理想,具有较好的性能。

K-th Number Query问题的改进算法研究

K-th number query是计算机算法中的一个基础问题,被广泛作为很多算法实现的重要步骤。对该问题进行了深入研究,并找到了单询问渐近时间复杂度最优的算法。目前一般对于多询问的K-th number query问题使用平衡二叉树解决,询问的时间复杂度为O(lbn)。但该算法实现比较复杂,并且常系数较大,提出了基于Bit Indexed Tree数据结构的算法解决,在同等时间复杂度的前提下,实现简单,隐含的常系数很小。最后进行了实验测试,分析显示该新算法不论在时间上还是空间上都优于现有的算法。

基于模糊综合评价的水下采油树结构可靠性分析

针对水下采油树结构可靠性分析,提出一种基于模糊综合评价理论的水下采油树结构可靠性分析方法。首先,通过分析不同厂家水下采油树构型,确定水下采油树的典型结构构型及组成,分析水下采油树在运行过程中所受的外载荷;其次,采用专家打分以及模糊综合评价法求得故障树底事件失效率,通过定量分析求解顶事件失效率、可靠度及可靠寿命曲线等可靠性指标;最后,对故障树进行重要度分析及失效率变化对系统结构可靠性的影响,结果表明:“树本体内磨损”的关键重要度最高,对系统结构可靠性的影响最大,是结构中的薄弱环节,应予以关注。研究成果可为水下采油树结构的风险评估提供技术参考,对水下采油树的设计制造提供一定的指导意义。

存储遥感影像的一种多层AVL tree结构

针对遥感影像数量庞大不利于发布和管理的特点,提出了一种用于离线卫星地图的空间索引结构,采用AVL tree把所需要的遥感影像数据存放在本地若干文件中,能以极快的响应速度将所需要的数据提供给桌面应用平台。

Assessment of gas and dust explosion in coal mines by means of fuzzy fault tree analysis

During the past decade, coal dust and gas explosions have been the most two serious types of disasters in China, threatening the lives of miners and causing significant losses in terms of national property. In this paper, an evaluation model of coal dust and gas explosions was constructed based on a fuzzy fault tree by taking the Xingli Coal Mine as a research site to identify the risk factors of coal dust and gas explosions.Furthermore, the hazards associated with such explosions were evaluated for this particular coal mine.After completing an on-site investigation, the fuzzy probabilities of basic events were obtained through expert scoring, and these expert opinions were then aggregated as trapezoidal fuzzy numbers to calculate the degrees of importance of all basic events. Finally, these degrees of importance were sorted. According to the resulting order, the basic events with higher probabilities were determined to identify key hazards in the daily safety management of this particular coal mine. Moreover, effective measures for preventing gas and coal dust explosions were derived. The fuzzy fault tree analysis method is of high significance in the analysis of accidental coal mine explosions and provides theoretical guidance for improving the efficiency of coal mine safety management in a scientific and feasible manner.

Reliability Analysis of Hydraulic System of Anchor Drilling Rigs Based on Fuzzy Fault Tree

Considering that the fault phenomenon of the power head cannot feed under actual working conditions,fuzzy mathematics theory is introduced into fault tree analysis(FTA)according to the structural characteristics of hydraulic system of anchor drilling rigs in this paper.The triangle fuzzy number is used to describe the fault probability of the basic event,the fuzzy probability importance of the basic event is calculated,and the basic events are sorted by comparing the magnitude of the fuzzy probability importance.The results show that the gear wear of auxiliary oil pump,suction phenomena of gear pump,wear and leakage of No.1 and No.3 pumps are the key factors leading to power head failure.In order to improve the overall reliability of the hydraulic system,fault diagnosis and maintenance decisions provide a theoretical basis.

Uniquely Tree Colorable Graphs

In this paper, the concepts of tree chromatic numbers and uniquely tree colorable graphs are introduced. After discussion some fundamental properties, three necessary conditions for a simple graph to be uniquely tree colorable are given. Moreover, a series of uniquely tree colorable graphs are constructed.