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Coretractable modules relative to a submodule 相对于子模块的可伸缩模块
Q3 Mathematics Pub Date : 2019-05-07 DOI: 10.13069/JACODESMATH.561322
A. R. M. Hamzekolaee, Y. Talebi
Let $R$ be a ring and $M$ a right $R$-module. Let $N$ be a proper submodule of $M$. We say that $M$ is $N$-coretractable (or $M$ is coretractable relative to $N$) provided that, for every proper submodule $K$ of $M$ containing $N$, there is a nonzero homomorphism $f:M/Krightarrow M$. We present some conditions that a module $M$ is coretractable if and only if $M$ is coretractable relative to a submodule $N$. We also provide some examples to illustrate special cases.
设$R$是环,$M$是右$R$模。设$N$是$M$的真子模。我们说$M$是$N$-可缩的(或者$M$相对于$N$是可缩的),前提是,对于$M$的每一个包含$N$的固有子模块$K$,存在一个非零同态$f:M/K右移M$。给出了一个模$M$可缩当且仅当$M$相对于子模$N$可缩的几个条件。我们还提供了一些例子来说明特殊情况。
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引用次数: 2
Complexity of neural networks on Fibonacci-Cayley tree Fibonacci-Cayley树上神经网络的复杂性
Q3 Mathematics Pub Date : 2019-05-07 DOI: 10.13069/JACODESMATH.560410
Jung-Chao Ban, Chih-Hung Chang
This paper investigates the coloring problem on Fibonacci-Cayley tree, which is a Cayley graph whose vertex set is the Fibonacci sequence. More precisely, we elucidate the complexity of shifts of finite type defined on Fibonacci-Cayley tree via an invariant called entropy. We demonstrate that computing the entropy of a Fibonacci tree-shift of finite type is equivalent to studying a nonlinear recursive system and reveal an algorithm for the computation. What is more, the entropy of a Fibonacci tree-shift of finite type is the logarithm of the spectral radius of its corresponding matrix. We apply the result to neural networks defined on Fibonacci-Cayley tree, which reflect those neural systems with neuronal dysfunction. Aside from demonstrating a surprising phenomenon that there are only two possibilities of entropy for neural networks on Fibonacci-Cayley tree, we address the formula of the boundary in the parameter space.
本文研究了Fibonacci-Cayley树的着色问题,该树是一个顶点集为Fibonacci序列的Cayley图。更准确地说,我们通过一个称为熵的不变量来阐明在Fibonacci-Cayley树上定义的有限类型移位的复杂性。我们证明了计算有限型Fibonacci树移位的熵等价于研究非线性递归系统,并揭示了计算算法。此外,有限型斐波那契树移位的熵是其对应矩阵的谱半径的对数。我们将结果应用于Fibonacci-Cayley树上定义的神经网络,它反映了那些具有神经元功能障碍的神经系统。除了证明Fibonacci-Cayley树上神经网络熵只有两种可能性这一令人惊讶的现象外,我们还讨论了参数空间中的边界公式。
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引用次数: 1
Betweenness centrality in convex amalgamation of graphs 图凸合并中的中间性中心性
Q3 Mathematics Pub Date : 2019-01-19 DOI: 10.13069/JACODESMATH.508983
Sunil Kumar Raghavan Unnithan, K. Balakrishnan
Betweenness centrality measures the potential or power of a node to control the communication over the network under the assumption that information flows primarily over the shortest paths between pair of nodes. The removal of a node with highest betweenness from the network will most disrupt communications between other nodes because it lies on the largest number of paths. A large network can be thought of as inter-connection between smaller networks by means of different graph operations. Thus the structure of a composite graph can be studied by analysing its component graphs. In this paper we present the betweenness centrality of some classes of composite graphs constructed by the graph operation called amalgamation or merging.
在假定信息主要在一对节点之间的最短路径上流动的情况下,中间性中心性衡量节点控制网络通信的潜力或能力。从网络中移除具有最高中间度的节点将最严重地破坏其他节点之间的通信,因为它位于路径数量最多的节点上。一个大的网络可以被认为是通过不同的图运算在较小的网络之间的相互连接。因此,可以通过分析复合图的组成图来研究复合图的结构。本文给出了用合并或合并图操作构造的几类复合图的中间性中心性。
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引用次数: 2
New Linear Codes over $GF(3)$, $GF(11)$, and $GF(13)$ $GF(3)$、$GF(11)$和$GF(13)上的新线性码$
Q3 Mathematics Pub Date : 2019-01-19 DOI: 10.13069/JACODESMATH.508968
N. Aydin, Derek Foret
Explicit construction of linear codes with best possible parameters is one of the major and challenging problems in coding theory. Cyclic codes and their various generalizations, such as quasi-twisted (QT) codes, are known to contain many codes with best known parameters. Despite the fact that these classes of codes have been extensively searched, we have been able to refine existing search algorithms to discover many new linear codes over the alphabets $mathbb{F}_{3}$, $mathbb{F}_{11}$, and $mathbb{F}_{13}$ with better parameters. A total of 38 new linear codes over the three alphabets are presented.
具有最佳可能参数的线性码的显式构造是编码理论中的一个主要且具有挑战性的问题。循环码及其各种推广,如准扭曲(QT)码,已知包含许多具有最佳参数的码。尽管这些代码类别已经被广泛搜索,但我们已经能够改进现有的搜索算法,以发现字母表$mathbb上的许多新的线性代码{F}_{3} $,$mathbb{F}_{11} $和$mathbb{F}_{13} 具有更好参数的$。给出了三个字母表上总共38个新的线性码。
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引用次数: 6
Codes over $mathbb{Z}_{p}[u]/{langle u^r rangle}timesmathbb{Z}_{p}[u]/{langle u^s rangle}$ $mathbb上的代码{Z}_{p} [u]/{langle u^rrangle}timesmathbb{Z}_{p} [u]/{langle u^srangle}$
Q3 Mathematics Pub Date : 2019-01-19 DOI: 10.13069/JACODESMATH.514339
Ismail Aydogdu
{In this paper we generalize $mathbb{Z}_{2}mathbb{Z}_{2}[u]$-linear codes to codes over $mathbb{Z}_{p}[u]/{langle u^r rangle}timesmathbb{Z}_{p}[u]/{langle u^s rangle}$ where $p$ is a prime number and $u^r=0=u^s$. We will call these family of codes as $mathbb{Z}_{p}[u^r,u^s]$-linear codes which are actually special submodules. We determine the standard forms of the generator and parity-check matrices of these codes. Furthermore, for the special case $p=2$, we define a Gray map to explore the binary images of $mathbb{Z}_{2}[u^r,u^s]$-linear codes. Finally, we study the structure of self-dual $mathbb{Z}_{2}[u^2,u^3]$-linear codes and present some examples.
{在本文中,我们推广了$mathbb{Z}_{2} mathbb{Z}_{2} [u]$-线性代码到$mathbb上的代码{Z}_{p} [u]/{langle u^rrangle}timesmathbb{Z}_{p} 其中$p$是素数,$u^r=0=u^s$。我们将这些代码族称为$mathbb{Z}_{p} [u^r,u^s]$线性码实际上是特殊的子模块。我们确定了这些代码的生成器和奇偶校验矩阵的标准形式。此外,对于特殊情况$p=2$,我们定义了一个Gray映射来探索$mathbb的二进制图像{Z}_{2} [u^r,u^s]$线性码。最后,我们研究了自对偶$mathbb的结构{Z}_{2} [u^2,u^3]$线性码,并给出了一些例子。
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引用次数: 1
Weight distribution of a class of cyclic codes of length $2^n$ 一类长度为2^n的循环码的权值分布
Q3 Mathematics Pub Date : 2019-01-19 DOI: 10.13069/JACODESMATH.505364
Manjit Singh, Sudhir Batra
Let $mathbb{F}_q$ be a finite field with $q$ elements and $n$ be a positive integer. In this paper, we determine the weight distribution of a class cyclic codes of length $2^n$ over $mathbb{F}_q$ whose parity check polynomials are either binomials or trinomials with $2^l$ zeros over $mathbb{F}_q$, where integer $lge 1$. In addition, constant weight and two-weight linear codes are constructed when $qequiv3pmod 4$.
让$mathbb{F}_q$是包含$q$元素的有限域,$n$是正整数。在本文中,我们确定了一类长度为$2^n$的循环码在$mathbb上的权重分布{F}_q$的奇偶校验多项式是二进制或三进制,$mathbb上有$2^l$个零{F}_q$,其中整数$lge为1$。此外,当$qequiv3pmod4$时,构造了常权和双权线性码。
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引用次数: 0
Recent results on Choi's orthogonal Latin squares 关于Choi正交拉丁方的最新结果
Q3 Mathematics Pub Date : 2018-12-05 DOI: 10.13069/jacodesmath.1056511
Jon-Lark Kim, D. Ohk, Doo Young Park, Jae Woo Park
Choi Seok-Jeong studied Latin squares at least 60 years earlier than Euler although this was less known. He introduced a pair of orthogonal Latin squares of order 9 in his book. Interestingly, his two orthogonal non-double-diagonal Latin squares produce a magic square of order 9, whose theoretical reason was not studied. There have been a few studies on Choi’s Latin squares of order 9. The most recent one is Ko-Wei Lih’s construction of Choi’s Latin squares of order 9 based on the two 3ˆ3 orthogonal Latin squares. In this paper, we give a new generalization of Choi’s orthogonal Latin squares of order 9 to orthogonal Latin squares of size n2 using the Kronecker product including Lih’s construction. We find a geometric description of Choi’s orthogonal Latin squares of order 9 using the dihedral group D8. We also give a new way to construct magic squares from two orthogonal non-double-diagonal Latin squares, which explains why Choi’s Latin squares produce a magic square of order 9.
崔锡正研究拉丁方块的时间比欧拉早了60年,但这一点并不为人所知。他在书中介绍了一对9阶的正交拉丁方阵。有趣的是,他的两个正交的非双对角线拉丁方产生了一个9阶的魔方,其理论原因没有被研究。关于Choi的拉丁9阶方阵已有一些研究。最近的一个是李高伟(Ko-Wei Lih)在两个3 * 3正交拉丁方阵的基础上构造了9阶的Choi拉丁方阵。本文利用包含Lih构造的Kronecker积,将9阶的Choi正交拉丁平方推广到n2阶的正交拉丁平方。我们用二面体群D8找到了9阶Choi正交拉丁方的几何描述。我们还给出了一种由两个正交的非双对角线拉丁方构造幻方的新方法,这解释了为什么Choi的拉丁方产生一个9阶的幻方。
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引用次数: 0
Characterization of $2times 2$ nil-clean matrices over integral domains 积分域上$2times2$nil干净矩阵的刻画
Q3 Mathematics Pub Date : 2018-09-08 DOI: 10.13069/jacodesmath.451229
K. N. Rajeswari, Umesh Gupta
Let $R$ be any ring with identity. An element $a in R$ is called nil-clean, if $a=e+n$ where $e$ is an idempotent element and $n$ is a nil-potent element. In this paper we give necessary and sufficient conditions for a $2times 2$ matrix over an integral domain $R$ to be nil-clean.
设$R$是任何具有恒等的环。如果$a=e+n$其中$e$是幂等元素而$n$是幂等元素,则$a 在R$中被称为零元素。本文给出了积分域$R$上的$2 × 2$矩阵是零清洁的充要条件。
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引用次数: 0
Game chromatic number of Cartesian and corona product graphs 笛卡尔图和电晕积图的对策色数
Q3 Mathematics Pub Date : 2018-09-08 DOI: 10.13069/JACODESMATH.458240
Syed Ahtsham ul Haq Bokhary, Tanveer Iqbal, Usman Ali
The game chromatic number $chi_g$ is investigated for Cartesian product $Gsquare H$ and corona product $Gcirc H$ of two graphs $G$ and $H$. The exact values for the game chromatic number of Cartesian product graph of $S_{3}square S_{n}$ is found, where $S_n$ is a star graph of order $n+1$. This extends previous results of Bartnicki et al. [1] and Sia [5] on the game chromatic number of Cartesian product graphs. Let $P_m$ be the path graph on $m$ vertices and $C_n$ be the cycle graph on $n$ vertices. We have determined the exact values for the game chromatic number of corona product graphs $P_{m}circ K_{1}$ and $P_{m}circ C_{n}$.
研究了两个图$G$和$H$的笛卡尔积$Gsquare H$和电晕积$Gcirc H$的游戏色数$chi_g$。求出了$S_{3}square S_{n}$笛卡尔积图的游戏色数的精确值,其中$S_n$是阶$n+1$的星图。这扩展了Bartnicki et al.[1]和Sia[5]关于笛卡尔积图的博弈色数的先前结果。设$P_m$为$m$顶点上的路径图,$C_n$为$n$顶点上的循环图。我们已经确定了电晕积图$P_{m}circ K_{1}$和$P_{m}circ C_{n}$的游戏色数的确切值。
{"title":"Game chromatic number of Cartesian and corona product graphs","authors":"Syed Ahtsham ul Haq Bokhary, Tanveer Iqbal, Usman Ali","doi":"10.13069/JACODESMATH.458240","DOIUrl":"https://doi.org/10.13069/JACODESMATH.458240","url":null,"abstract":"The game chromatic number $chi_g$ is investigated for Cartesian product $Gsquare H$ and corona product $Gcirc H$ of two graphs $G$ and $H$. The exact values for the game chromatic number of Cartesian product graph of $S_{3}square S_{n}$ is found, where $S_n$ is a star graph of order $n+1$. This extends previous results of Bartnicki et al. [1] and Sia [5] on the game chromatic number of Cartesian product graphs. Let $P_m$ be the path graph on $m$ vertices and $C_n$ be the cycle graph on $n$ vertices. We have determined the exact values for the game chromatic number of corona product graphs $P_{m}circ K_{1}$ and $P_{m}circ C_{n}$.","PeriodicalId":37029,"journal":{"name":"Journal of Algebra Combinatorics Discrete Structures and Applications","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41980859","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Finite Rogers-Ramanujan type continued fractions 有限Rogers-Ramanujan型连分数
Q3 Mathematics Pub Date : 2018-09-08 DOI: 10.13069/JACODESMATH.451218
H. Prodinger
{"title":"Finite Rogers-Ramanujan type continued fractions","authors":"H. Prodinger","doi":"10.13069/JACODESMATH.451218","DOIUrl":"https://doi.org/10.13069/JACODESMATH.451218","url":null,"abstract":"","PeriodicalId":37029,"journal":{"name":"Journal of Algebra Combinatorics Discrete Structures and Applications","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47218816","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
期刊
Journal of Algebra Combinatorics Discrete Structures and Applications
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