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The g-Good-Neighbor Connectivity of Some Cartesian Product Graphs 被引量:1
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作者 Yinkui Li Ting Xie Xiaoxiao Qin 《Open Journal of Discrete Mathematics》 2023年第1期27-37,共11页
The g-good-neighbor connectivity of G is a generalization of the concept of connectivity, which is just for, and an important parameter in measuring the fault tolerance and reliability of interconnection network. Many... The g-good-neighbor connectivity of G is a generalization of the concept of connectivity, which is just for, and an important parameter in measuring the fault tolerance and reliability of interconnection network. Many well-known networks can be constructed by the Cartesian products of some simple graphs. In this paper, we determine the g-good-neighbor connectivity of some Cartesian product graphs. We give the exact value of g-good-neighbor connectivity of the Cartesian product of two complete graphs and for , mesh for , cylindrical grid and torus for . 展开更多
关键词 CONNECTIVITY The g-Good-Neighbor Connectivity cartesian product
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Rupture Degree of Some Cartesian Product Graphs
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作者 Yinkui Li Tingyi Zhu 《Open Journal of Discrete Mathematics》 2023年第1期16-26,共11页
The rupture degree of a noncomplete-connected graph G is defined by , where is the number of components of and is the order of the largest component of. In this paper, we determine the rupture degree of some Cartesian... The rupture degree of a noncomplete-connected graph G is defined by , where is the number of components of and is the order of the largest component of. In this paper, we determine the rupture degree of some Cartesian product graphs. 展开更多
关键词 The Rupture Degree cartesian product The Vulnerability
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Cartesian product over interval valued intuitionistic fuzzy sets 被引量:1
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作者 Jianming Xie Sanyang Liu 《Journal of Systems Engineering and Electronics》 SCIE EI CSCD 2017年第6期1152-1161,共10页
The intuitionistic fuzzy set(IFS) based on fuzzy theory,which is of high efficiency to solve the fuzzy problem, has been introduced by Atanassov. Subsequently, he pushed the research one step further from the IFS to t... The intuitionistic fuzzy set(IFS) based on fuzzy theory,which is of high efficiency to solve the fuzzy problem, has been introduced by Atanassov. Subsequently, he pushed the research one step further from the IFS to the interval valued intuitionistic fuzzy set(IVIFS). On the basis of fuzzy set(FS), the IFS is a generalization concept. And the IFS is generalized to the IVIFS.In this paper, the definition of the sixth Cartesian product over IVIFSs is first introduced and its some properties are explored.We prove some equalities based on the operation and the relation over IVIFSs. Finally, we present one geometric interpretation and a numerical example of the sixth Cartesian product over IVIFSs. 展开更多
关键词 intuitionistic fuzzy sets(IFS) cartesian product ope ration geometric interpretation interval valued intuitionistic fuzzy set(IVIFS)
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Linear arboricity of Cartesian products of graphs
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作者 陶昉昀 林文松 《Journal of Southeast University(English Edition)》 EI CAS 2013年第2期222-225,共4页
A linear forest is a forest whose components are paths. The linear arboricity la (G) of a graph G is the minimum number of linear forests which partition the edge set E(G) of G. The Cartesian product G□H of two g... A linear forest is a forest whose components are paths. The linear arboricity la (G) of a graph G is the minimum number of linear forests which partition the edge set E(G) of G. The Cartesian product G□H of two graphs G and H is defined as the graph with vertex set V(G□H) = {(u, v)| u ∈V(G), v∈V(H) } and edge set E(G□H) = { ( u, x) ( v, Y)|u=v and xy∈E(H), or uv∈E(G) and x=y}. Let Pm and Cm,, respectively, denote the path and cycle on m vertices and K, denote the complete graph on n vertices. It is proved that (Km□Pm)=[n+1/2]for m≥2,la(Km□Cm)=[n+2/2],and la(Km□Km)=[n+m-1/2]. The methods to decompose these graphs into linear forests are given in the proofs. Furthermore, the linear arboricity conjecture is true for these classes of graphs. 展开更多
关键词 linear forest linear arboricity cartesian product
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Embedding Cartesian Product of Some Graphs in Books
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作者 YANG JIAO SHAO ZE-LING LI ZHI-GUO 《Communications in Mathematical Research》 CSCD 2018年第3期253-260,共8页
The book embedding of a graph G consists of placing the vertices of G in a line called spine and assigning edges of the graph to pages so that the edges assigned to the same page do not intersect. The number of pages ... The book embedding of a graph G consists of placing the vertices of G in a line called spine and assigning edges of the graph to pages so that the edges assigned to the same page do not intersect. The number of pages is the minimum number in which the graph can be embedded. In this paper, we study the book embedding of the Cartesian product Pm × Sn, Pm × Wn, Cn × Sm, Cn × Wm, and get an upper bound of their pagenumber. 展开更多
关键词 book embedding cartesian product pagenumber
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Cartesian Product Based Transfer Learning Implementation for Brain Tumor Classification
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作者 Irfan Ahmed Usmani Muhammad Tahir Qadri +2 位作者 Razia Zia Asif Aziz Farheen Saeed 《Computers, Materials & Continua》 SCIE EI 2022年第11期4369-4392,共24页
Knowledge-based transfer learning techniques have shown good performance for brain tumor classification,especially with small datasets.However,to obtain an optimized model for targeted brain tumor classification,it is... Knowledge-based transfer learning techniques have shown good performance for brain tumor classification,especially with small datasets.However,to obtain an optimized model for targeted brain tumor classification,it is challenging to select a pre-trained deep learning(DL)model,optimal values of hyperparameters,and optimization algorithm(solver).This paper first presents a brief review of recent literature related to brain tumor classification.Secondly,a robust framework for implementing the transfer learning technique is proposed.In the proposed framework,a Cartesian product matrix is generated to determine the optimal values of the two important hyperparameters:batch size and learning rate.An extensive exercise consisting of 435 simulations for 11 state-of-the-art pre-trained DL models was performed using 16 paired hyperparameters from the Cartesian product matrix to input the model with the three most popular solvers(stochastic gradient descent with momentum(SGDM),adaptive moment estimation(ADAM),and root mean squared propagation(RMSProp)).The 16 pairs were formed using individual hyperparameter values taken from literature,which generally addressed only one hyperparameter for optimization,rather than making a grid for a particular range.The proposed framework was assessed using a multi-class publicly available dataset consisting of glioma,meningioma,and pituitary tumors.Performance assessment shows that ResNet18 outperforms all other models in terms of accuracy,precision,specificity,and recall(sensitivity).The results are also compared with existing state-of-the-art research work that used the same dataset.The comparison was mainly based on performance metric“accuracy”with support of three other parameters“precision,”“recall,”and“specificity.”The comparison shows that the transfer learning technique,implemented through our proposed framework for brain tumor classification,outperformed all existing approaches.To the best of our knowledge,the proposed framework is an efficient framework that helped reduce the computational complexity and the time to attain optimal values of two important hyperparameters and consequently the optimized model with an accuracy of 99.56%. 展开更多
关键词 Deep transfer learning cartesian product hyperparameter optimization magnetic resonance imaging(MRI) brain tumor classification
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Geodetic Number and Geo-Chromatic Number of 2-Cartesian Product of Some Graphs
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作者 Medha Itagi Huilgol B. Divya 《Open Journal of Discrete Mathematics》 2022年第1期1-16,共16页
A set <em>S ⊆ V (G)</em> is called a geodetic set if every vertex of <em>G</em> lies on a shortest <em>u-v</em> path for some <em>u, v ∈ S</em>, the minimum cardinality... A set <em>S ⊆ V (G)</em> is called a geodetic set if every vertex of <em>G</em> lies on a shortest <em>u-v</em> path for some <em>u, v ∈ S</em>, the minimum cardinality among all geodetic sets is called geodetic number and is denoted by <img src="Edit_82259359-0135-4a65-9378-b767f0405b48.png" alt="" />. A set <em>C ⊆ V (G)</em> is called a chromatic set if <em>C</em> contains all vertices of different colors in<em> G</em>, the minimum cardinality among all chromatic sets is called the chromatic number and is denoted by <img src="Edit_d849148d-5778-459b-abbb-ff25b5cd659b.png" alt="" />. A geo-chromatic set<em> S</em><sub><em>c</em></sub><em> ⊆ V (G</em><em>)</em> is both a geodetic set and a chromatic set. The geo-chromatic number <img src="Edit_505e203c-888c-471c-852d-4b9c2dd1a31c.png" alt="" /><em> </em>of<em> G</em> is the minimum cardinality among all geo-chromatic sets of<em> G</em>. In this paper, we determine the geodetic number and the geo-chromatic number of 2-cartesian product of some standard graphs like complete graphs, cycles and paths. 展开更多
关键词 cartesian product Grid Graphs Geodetic Set Geodetic Number Chromatic Set Chromatic Number Geo-Chromatic Set Geo-Chromatic Number
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The Signed Domination Number of Cartesian Product of Two Paths 被引量:1
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作者 Mohammad Hassan Muhsin Al Hassan Mazen Mostafa 《Open Journal of Discrete Mathematics》 2020年第2期45-55,共11页
Let G be a finite connected simple graph with vertex set V(G) and edge set E(G). A function f:V(G) → {1,1} is a signed dominating function if for every vertex v∈V(G), the closed neighborhood of v contains more verti... Let G be a finite connected simple graph with vertex set V(G) and edge set E(G). A function f:V(G) → {1,1} is a signed dominating function if for every vertex v∈V(G), the closed neighborhood of v contains more vertices with function values 1 than with &#8722;1. The signed domination number γs(G) of G is the minimum weight of a signed dominating function on G. In this paper, we calculate The signed domination numbers of the Cartesian product of two paths Pm and Pn for m = 3, 4, 5 and arbitrary n. 展开更多
关键词 PATH cartesian product SIGNED Dominating Function SIGNED DOMINATION NUMBER
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Domination Number of Square of Cartesian Products of Cycles
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作者 Morteza Alishahi Sakineh Hoseini Shalmaee 《Open Journal of Discrete Mathematics》 2015年第4期88-94,共7页
A set ?is a dominating set of G if every vertex of ?is adjacent to at least one vertex of S. The cardinality of the smallest dominating set of G is called the domination number of G. The square G2 of a graph G is obta... A set ?is a dominating set of G if every vertex of ?is adjacent to at least one vertex of S. The cardinality of the smallest dominating set of G is called the domination number of G. The square G2 of a graph G is obtained from G by adding new edges between every two vertices having distance 2 in G. In this paper we study the domination number of square of graphs, find a bound for domination number of square of Cartesian product of cycles, and find the exact value for some of them. 展开更多
关键词 DOMINATION NUMBER SQUARE of a GRAPH cartesian product
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On the Signed Domination Number of the Cartesian Product of Two Directed Cycles
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作者 Ramy Shaheen 《Open Journal of Discrete Mathematics》 2015年第3期54-64,共11页
Let D be a finite simple directed graph with vertex set V(D) and arc set A(D). A function ?is called a signed dominating function (SDF) if ?for each vertex . The weight ?of f is defined by . The signed domination numb... Let D be a finite simple directed graph with vertex set V(D) and arc set A(D). A function ?is called a signed dominating function (SDF) if ?for each vertex . The weight ?of f is defined by . The signed domination number of a digraph D is . Let Cm × Cn denotes the cartesian product of directed cycles of length m and n. In this paper, we determine the exact values of gs(Cm × Cn) for m = 8, 9, 10 and arbitrary n. Also, we give the exact value of gs(Cm × Cn) when m, ?(mod 3) and bounds for otherwise. 展开更多
关键词 Directed GRAPH Directed CYCLE cartesian product SIGNED Dominating Function SIGNED DOMINATION NUMBER
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Full Friendly Index Sets of Cartesian Products of Two Cycles 被引量:3
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作者 Wai chee SHIU 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2010年第7期1233-1244,共12页
Let G =(V, E) be a connected simple graph. A labeling f : V → Z2 induces an edge labeling f* : E → Z2 defined by f*(xy) = f(x) +f(y) for each xy ∈ E. For i ∈ Z2, let vf(i) = |f^-1(i)| and ef(i... Let G =(V, E) be a connected simple graph. A labeling f : V → Z2 induces an edge labeling f* : E → Z2 defined by f*(xy) = f(x) +f(y) for each xy ∈ E. For i ∈ Z2, let vf(i) = |f^-1(i)| and ef(i) = |f*^-1(i)|. A labeling f is called friendly if |vf(1) - vf(0)| ≤ 1. For a friendly labeling f of a graph G, we define the friendly index of G under f by if(G) = e(1) - el(0). The set [if(G) | f is a friendly labeling of G} is called the full friendly index set of G, denoted by FFI(G). In this paper, we will determine the full friendly index set of every Cartesian product of two cycles. 展开更多
关键词 vertex labeling friendly labeling friendly index set cartesian product of two cycles
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The Crossing Number of the Cartesian Products of Wm with Pn 被引量:6
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作者 WANG Jing LV Sheng Xiang HUANG Yuan Qiu 《Journal of Mathematical Research and Exposition》 CSCD 2009年第2期362-366,共5页
Most results on crossing numbers of graphs focus on some special graphs, such as the Cartesian products of small graphs with path, star and cycle. In this paper, we obtain the crossing number formula of Cartesian prod... Most results on crossing numbers of graphs focus on some special graphs, such as the Cartesian products of small graphs with path, star and cycle. In this paper, we obtain the crossing number formula of Cartesian products of wheel Wm with path Pn for arbitrary m ≥ 3 and n ≥ 1. 展开更多
关键词 DRAWING crossing number WHEEL PATH cartesian product.
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Dimensional Results for Cartesian Products of Homogeneous Moran Sets 被引量:1
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作者 Li CAO Xing-Gang HE 《Acta Mathematicae Applicatae Sinica》 SCIE CSCD 2012年第4期673-680,共8页
M(J, {ms * ns}, {Cs}) be the collection of Cartesian products of two homogenous Moran sets with the same ratios {cs} Where J = [0, 1] × [0, 1]. Then the maximal and minimal values of the Hausdorff dimensions f... M(J, {ms * ns}, {Cs}) be the collection of Cartesian products of two homogenous Moran sets with the same ratios {cs} Where J = [0, 1] × [0, 1]. Then the maximal and minimal values of the Hausdorff dimensions for the elements in M are obtained without any restriction on {msns} or {cs}. 展开更多
关键词 Homogeneous Moran sets cartesian product Hausdorff dimension
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L(2, 1)-Circular Labelings of Cartesian Products of Complete Graphs 被引量:2
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作者 LV Da Mei LIN Wen Song SONG Zeng Min 《Journal of Mathematical Research and Exposition》 CSCD 2009年第1期91-98,共8页
For positive integers j and k with j ≥ k, an L(j, k)-labeling of a graph G is an assignment of nonnegative integers to V(G) such that the difference between labels of adjacent vertices is at least j, and the diff... For positive integers j and k with j ≥ k, an L(j, k)-labeling of a graph G is an assignment of nonnegative integers to V(G) such that the difference between labels of adjacent vertices is at least j, and the difference between labels of vertices that are distance two apart is at least k. The span of an L(j, k)-labeling of a graph G is the difference between the maximum and minimum integers it uses. The λj, k-number of G is the minimum span taken over all L(j, k)-labelings of G. An m-(j, k)-circular labeling of a graph G is a function f : V(G) →{0, 1, 2,..., m - 1} such that |f(u) - f(v)|m ≥ j if u and v are adjacent; and |f(u) - f(v)|m 〉 k ifu and v are at distance two, where |x|m = min{|xl|, m-|x|}. The minimum integer m such that there exists an m-(j, k)-circular labeling of G is called the σj,k-number of G and is denoted by σj,k(G). This paper determines the σ2,1-number of the Cartesian product of any three complete graphs. 展开更多
关键词 λ2 1-number σ2 1-number cartesian product.
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Adjacent Vertex Distinguishing Incidence Coloring of the Cartesian Product of Some Graphs 被引量:1
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作者 Qian WANG Shuang Liang TIAN 《Journal of Mathematical Research and Exposition》 CSCD 2011年第2期366-370,共5页
An adjacent vertex distinguishing incidence coloring of graph G is an incidence coloring of G such that no pair of adjacent vertices meets the same set of colors.We obtain the adjacent vertex distinguishing incidence ... An adjacent vertex distinguishing incidence coloring of graph G is an incidence coloring of G such that no pair of adjacent vertices meets the same set of colors.We obtain the adjacent vertex distinguishing incidence chromatic number of the Cartesian product of a path and a path,a path and a wheel,a path and a fan,and a path and a star. 展开更多
关键词 cartesian product incidence coloring adjacent vertex distinguishing incidence coloring adjacent vertex distinguishing incidence chromatic number
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On total{k}-domatic number of Cartesian and direct product of graphs
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作者 LIANG Yong PEI Lidan +1 位作者 HU Futao HOU Xinmin 《中国科学技术大学学报》 CAS CSCD 北大核心 2018年第8期605-611,共7页
For a positive integer k,the total{k}-dominating function(T{k}DF)of a graph G without isolated vertices is a function f from the vertex set V(G)to the set{0,1,2,…,k}such that for each vertex v∈V(G),the sum of the va... For a positive integer k,the total{k}-dominating function(T{k}DF)of a graph G without isolated vertices is a function f from the vertex set V(G)to the set{0,1,2,…,k}such that for each vertex v∈V(G),the sum of the values of all its neighbors assigned by f is at least k.A set{f_(1),f_(2),…,f_(d)}of pairwise different T{k}DF s of G with the property that∑d i=1 f_(i)(v)≤k for each v∈V(G),is called a total{k}-dominating family(T{k}D family)of G.The total{k}-domatic number of a graph G,denoted by d^({k})_(t)(G),is the maximum number of functions in T{k}D family.In 2013,Aram et al.proposed a problem that whether or not d^({k})_(t)(C_(m)□C_(n))=3 when 4 nmk,and d^({k})_(t)(C m□C n)=4 when 4|nmk.It was shown that d^({k})_(t)(C_(m)□C_(n))=3 if 4 nmk and k≥2 or 4|nmk and 2 nk,which partially answered the above problem.In addition,the total{k}-domatic number of the direct product of a cycle and a path,two paths,and two cycles was studied,respectively. 展开更多
关键词 total{k}-domatic number cartesian product direct product
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The Crossing Numbers of Cartesian Products of Stars with a 5-Vertex Graph
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作者 苏振华 黄元秋 《Journal of Mathematical Research and Exposition》 CSCD 2009年第4期580-586,共7页
In this paper, we compute the crossing number of a specific graph Hn, and then by contraction, we obtain the conclusion that cr(G13 × Sn) = 4[n/2] [n-1/2]+[n/2] . The result fills up the blank of the crossing ... In this paper, we compute the crossing number of a specific graph Hn, and then by contraction, we obtain the conclusion that cr(G13 × Sn) = 4[n/2] [n-1/2]+[n/2] . The result fills up the blank of the crossing numbers of Cartesian products of stars with all 5-vertex graphs presented by Marian Klesc. 展开更多
关键词 GRAPH DRAWING crossing number cartesian products star.
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Paired Domination of Cartesian Products of Graphs
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作者 Xin Min HOU Fan JIANG 《Journal of Mathematical Research and Exposition》 CSCD 2010年第1期181-185,共5页
Let γpr(G) denote the paired domination number and G □ H denote the Cartesian product of graphs G and H. In this paper we show that for all graphs G and H without isolated vertex, γpr(G)γpr(H)≤ 7γpr (G ... Let γpr(G) denote the paired domination number and G □ H denote the Cartesian product of graphs G and H. In this paper we show that for all graphs G and H without isolated vertex, γpr(G)γpr(H)≤ 7γpr (G □H). 展开更多
关键词 DOMINATION paired domination cartesian product.
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集合论、笛卡尔乘积和多项式定理在生命表研究中的应用:以取食玉米粒和茶树叶片的棉铃虫生命表为例
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作者 陈珍珍 边亚楠 +1 位作者 齐心 许永玉 《昆虫学报》 CAS CSCD 北大核心 2024年第10期1388-1403,共16页
【目的】自我重复取样技术(bootstrap technique)广泛用于计算生命表各参数的方差、标准误和置信区间(confidence interval),而配对自我重复取样检验(paired bootstrap test,PBT)则用于检测处理间的生命表参数差异。本研究以取食玉米粒... 【目的】自我重复取样技术(bootstrap technique)广泛用于计算生命表各参数的方差、标准误和置信区间(confidence interval),而配对自我重复取样检验(paired bootstrap test,PBT)则用于检测处理间的生命表参数差异。本研究以取食玉米粒和茶树叶片的棉铃虫Helicoverpa armigera生命表为例,进一步解释集合论、笛卡尔乘积和多项式定理在害虫种群生物学研究中的应用。【方法】利用年龄-龄期两性生命表对取食玉米粒和茶树叶片的棉铃虫种群参数进行分析,利用集合论、笛卡尔乘积和多项式定理将种群统计分析以明确数学的方法呈现,以检测所有可能的自我重复取样样本、精确计算种群参数置信区间以及处理之间差异的置信区间,计算可育和不育样本的概率。【结果】取食茶树叶片的棉铃虫其成虫前期存活率、内禀增长率(r)、周限增长率(λ)、净增殖率(R_(0))、平均世代周期(T)均显著低于取食玉米粒的,说明茶树叶片不是棉铃虫的适宜寄主植物。因取食茶树叶片棉铃虫种群不育样本数较多,利用bootstrap重复取样接受可育与不育样本时,其结果与不用bootstrap技术计算的差异小于5%;若取样时仅接受可育样本,结果与原数值差异显著,误差较大(>5%)。使用笛卡尔配对检验(Cartesian paired test,CPT)比较取食两种食物的棉铃虫种群的R0的差异时,可计算两种处理bootstrap抽样结果所有可能差异的精确置信区间,而使用自我重复取样检验通常会导致置信区间的估值过高或过低,尤其是自我重复取样数较小时。多项式定理可以揭示可育和不育样本的自我重复取样结果。自我重复取样抽样记录必须保留,以便在后续分析中进一步应用。【结论】本研究进一步明确了两性生命表理论的数学基础,也为昆虫学研究中的生命表技术的应用提供数学支撑。 展开更多
关键词 棉铃虫 生命表 集合论 笛卡尔乘积 多项式定理 不育自我重复取样样本
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具有优异汉明相关特性的混沌跳频序列构造方法
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作者 周军 刘星 +1 位作者 肖飞 程石磊 《通信技术》 2024年第9期892-896,共5页
跳频通信具有码分多址、频带共享、抗干扰和抗截获的能力,在无线通信系统中得到了广泛的应用。目前基于混沌方法构造的跳频序列大多具有较高的线性复杂度,所得到的跳频序列虽然具有良好的随机性,但是其汉明相关性还不够。在选取具有优... 跳频通信具有码分多址、频带共享、抗干扰和抗截获的能力,在无线通信系统中得到了广泛的应用。目前基于混沌方法构造的跳频序列大多具有较高的线性复杂度,所得到的跳频序列虽然具有良好的随机性,但是其汉明相关性还不够。在选取具有优异汉明相关特性的m序列的基础上,结合混沌序列,利用笛卡尔积构造得到了具有优异汉明自相关性的跳频序列。该跳频序列具备与混沌序列一致的线性复杂度,而且可以保证其周期不小于m序列的周期,汉明自相关值不超过m序列的汉明自相关值。 展开更多
关键词 跳频序列 M序列 汉明自相关 笛卡尔积 高线性复杂度
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