A strongly connected digraph \(D\) is primitive provided the greatest common divisor of the lengths of its directed cycles equals 1. The scrambling index of a primitive digraph \(D\) is the smallest positive integer \(k\) such that for every pair of vertices \(u\) and \(v\), there is a vertex \(w\) such that we can get to \(w\) from \(u\) and \(v\) in \(D\) by directed walks of length \(k\). In this paper, we characterize those primitive doubly symmetric digraphs with the largest scrambling index.

立体几何是历年高考的必考题型,2021年新高考数学Ⅰ卷第20 题,延续历年高考的考点和难度,主要考查学生直观想象等核心素养,该题从三棱锥出发,考察学生对于空间点、线、面的掌握情况,同时又对空间角和立体图形的体积都有一定的涵盖.

新修订的《普通高中数学课程标准》明确提出了要通过数学教学,发展学生数学学科的核心素养,要不断引领学生感悟数学的科学价值、应用价值、文化和审美价值[1] .教师在传授知识和技能的过程中,应该注重知识性、趣味性和思想性的统一,将科学素质教育与人文素质教育有机融合,提高教学的实效性.对数函数是建立在对数基础之上的一类重要的基本初等函数,也是中学数学中函数教学的难点之一.本文试图从科学价值、应用价值、文化和审美价值等方面就对数函数的数学价值做一个简单的分析,使学生感悟对数函数的数学价值,并希望在使学生理解对数与对数函数的概念、把握对数函数的本质的同时,学生的数学核心素养也能得到发展.

本文提出了带两个形状参数的有理二次三角Bézier曲线,由4个控制顶点生成的曲线具有传统有理三次Bézier曲线的几何特性,包括端点性质、对称性、凸包性、几何和仿射不变性、变差缩减性.分析了在权因子固定情形下,通过改变形状参数值可以局部调控曲线形状;也得出当形状参数值都为-1时,曲线可退化为直线段.曲线在适当的控制顶点下,可精确表示椭圆弧和圆弧,从而可方便整圆的表示.在控制顶点和权因子相同的条件下,当形状参数取值在一定范围内,曲线具有比有理三次Bézier曲线对控制多边形更好的逼近.

We prove that for any valuation ring R of Krull dimension <= 1 or any commutative local ring R with Krull dimension 0 and n >= 3, the special linear group SLn(R[x]) coincides with the group of elementary matrices E-n(R[x]), and for an arbitrary arithmetical ring R of Krull dimension <= 1 and n >= 3, the group SLn(R[x]) = SLn(R).E-n(R[x]) . These give partial solutions to two conjectures of Yengui.

In this paper,we give all integal solutions of a family of quartic Thue equations with three parameters x4-4sx3y-(2ab + 4(a + b)s)x2y2-4absxy3 + a2b2y4 =1,s ≥ 1,when a =2,b =1.

一个本原不可幂带号有向图S的基指数l(S)是指使得在S中,从任意一点u到任意一点v都有一对长为l的SSSD途径的最小整数l.本文将完全刘划n阶本原不可幂简单图的基指数集.

一个本原不可幂带号有向图S的基指数l(S)是这样的最小正整数l,使得在S中,从任意一点u到任意一点v都有一对长为l的SSSD途径.本文给出了本原不可幂带号简单图的基指数的最大值,并完全刻划了具有最大基指数的本原不可幂带号简单图.

Accurately extracting and timely updating the information of road network is vital to road planning and vehicle naviga -tion.Currently, the road network’s extraction method for mining urban trunk roads , based on GPS trajectories, commonly uses floating car or taxi .However , the existing methods ignore the automatic extraction of pathway , which is very important for earth-quake relief , community navigation and village tour and other occasions .Therefore , this paper proposes a new method of road network extraction , based on walking GPS trajectories , which consists of three parts:data preprocessing , road centerline genera-tion and road network accuracy evaluation .In this paper , three methods are adopted to generate road centerlines successively , such as trajectories clustering , cluster center segmentation and centerline fitting .Making experiment with self-collected walking GPS data, the results show that the proposed method not only is able to accurately extract the road network , coverage rate can reach 96.21%while error detection rate was 3.26%, but also can extract pathway and update road network .

线性代数是理工科各专业一门重要的基础课。本文结合线性代数课程的基本内容,从数学材料概念化的能力、用数学符号进行运算的能力、思维的逻辑性、思维的创造性、数学记忆能力与空间想象能力等方面阐述了数学能力的培养,并从教学环节方面探讨了提高学生的数学能力的若干途径。

The general theory about Padé approximants was introduced,and a new way of computations was provided,that with extended Euclidean algorithm of an(m,n)-Padé approximant to a formal power series.In addiotions,an application example was given by applying Extended Euclidean Algorithm to obtain Padé approximation.

The base l(S) of a primitive nonpowerfu signed digraph S,is the least integer l such that there exists a pair of SSSD walks of length l from each vertex u to each vertex v in 5.In this paper,we study the bases of primitive nonpowerful symmetric signed digraphs of order n with shortest odd cycle length r,and give the largest value of the bases for such digraphs.

在线性代数课程中引入MATLAB辅助教学,是当今国际上针对这门课程改革的一个重要方面。本文通过一些线性方程组求解的实例,探讨了运用几个常见的MATLAB命令在辅助教学中的作用。

The scrambing index of a primitive matrix A is the smallest positive integer k such that any two rows of Ak have at least one positive element in a coincident position.In this paper,the scrambing index of symmetric primitive matrices by using graph theory were investigated.The upper bound for the scrambing index of zero-trace symmetric primitive matrices was obtained.Moreover,the scrambing index set for this class of matrices was determined,and characterize completely the extremal matrices.

A nonnegative square matrix A is primitive if some power Ak>0 (that is, Ak is entrywise positive). The least such k is called the exponent of A. In [2], Akelbek and Kirkland defined the scrambling index of a primitive matrix A, which is the smallest positive integer k such that any two rows of Ak have at least one positive element in a coincident position. In this paper, we give a relation between the scrambling index and the exponent for symmetric primitive matrices, and determine the scrambling index set for the class of symmetric primitive matrices. We also characterize completely the symmetric primitive matrices in this class such that the scrambling index is equal to the maximum value.

Given c0,an(n,c)-expander is a simple bigraph with bipartition(X_1,X_2) and |X_1|= |X_2| = n,such that |N(S)|≥(1+c)|S| for every S(?) X_i(i=1,2) with |S|≤n/2,where N(S) is the neighbour set of S.In this paper,amenable bounds on the ratios of the number of k-matchings to the number of perfect matchings of an(n,c)-expander are obtained.

Let D be a digraph(loops are permitted but no multiple arcs)of order n, and let X(?)V(D).The set exponent exP_D(X)is defined to be the smallest positive integer p such that for each vertex v of D,there exists a walk of length p from at least one vertex in X to v.If no such p exists,then we define exp_D(X)=∞.Let 1≤k≤n.Then F(D,k): =max{exp_D(X)|X(?)V(D),|X|=k}is called the kth upper generalized exponent of D. D is said to be k-upper primitive if F(D,k)<∞.The k-upper primitive symmetric digraph of order n whose kth upper generalized exponents achieve the maximum value is completely characterized.

Let D = (V,E) be a primitive digraph. The vertex exponent of D at a vertex v ∈ V, denoted by expD(v), is the least integer p such that there is a v → u walk of length p for each u ∈ V. Following Brualdi and Liu, we order the vertices of D so that exp D (v 1) ≤ exp D (v 2) ≤ ... ≤ exp D (v n ). Then exp D (v k ) is called the k-point exponent of D and is denoted by exp D (k), 1 ≤ k ≤ n. In this paper we define e(n, k):=max{exp D (k)|D ∈ PD(n, 2)} and E(n, k):= {exp D (k)|D ∈ PD(n, 2)}, where PD(n, 2) is the set of all primitive digraphs of order n with girth 2. We completely determine e(n, k) and E(n, k) for all n, k with n ≥ 3 and 1 ≤ k ≤ n.