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Normality and Quasinormality of Zero-free Meromorphic Functions 被引量:11
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作者 Jian Ming CHANG 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2012年第4期707-716,共10页
Let k, K ∈1N and F be a family of zero-free meromorphic functions in a domain D such that for each f ∈ F, f(k) - 1 has at most K zeros, ignoring multiplicity. Then F is quasinormal of order at most v=k/k+1,where ... Let k, K ∈1N and F be a family of zero-free meromorphic functions in a domain D such that for each f ∈ F, f(k) - 1 has at most K zeros, ignoring multiplicity. Then F is quasinormal of order at most v=k/k+1,where v is equal to the largest integer not exceeding k/k+1.In particular, if K = k, then F is normal. The results are sharp. 展开更多
关键词 Meromorphic functions normal family
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一种基于三角模糊数理论的断路器状态评估改进方法 被引量:10
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作者 江洁 苟怀强 +2 位作者 唐道龙 卜京 侯洋 《陕西电力》 2016年第1期32-36,共5页
在断路器状态评价的研究中常采用模糊综合评价模型,其评价指标权重的选择主要依赖于经验,可能出现与实际情况不符的现象。针对这一问题,在原有的模糊综合评价模型基础上,提出了应用三角模糊数理论计算断路器状态评估中各指标权重的改进... 在断路器状态评价的研究中常采用模糊综合评价模型,其评价指标权重的选择主要依赖于经验,可能出现与实际情况不符的现象。针对这一问题,在原有的模糊综合评价模型基础上,提出了应用三角模糊数理论计算断路器状态评估中各指标权重的改进方法。使用"正态云"模型表示各指标的隶属度函数,对各因素的隶属度值进行计算,得到各因素的评估结果;再使用三角模糊数计算各因素的权重,确定断路器在整体运行中的状态和评估结果,由最大隶属度原则及其评语集得出断路器的状态评语。最后,对成都市某变电站内SF_6断路器进行评估,验证了该方法的有效性。 展开更多
关键词 多层模糊评估模型 正态云 隶属度函数 三角模糊数 评估权重
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THE NORMALITY OF ALGEBROID MULTIFUNCTIONS AND THEIR COEFFICIENT FUNCTIONS 被引量:5
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作者 柴富杰 高宗升 《Acta Mathematica Scientia》 SCIE CSCD 2015年第1期121-132,共12页
In this paper, we investigate the normality relationship between algebroid multifunctions and their coefficient functions. We prove that the normality of a k-valued entire algebroid multifunctions family is equivalent... In this paper, we investigate the normality relationship between algebroid multifunctions and their coefficient functions. We prove that the normality of a k-valued entire algebroid multifunctions family is equivalent to their coefficient functions in some conditions. Furthermore, we obtain some new normality criteria for algebroid multifunctions families based on these results. We also provide some examples to expound that some restricted conditions of our main results are necessary. 展开更多
关键词 algebroid multifunctions normal families coefficient functions Hausdorff distance
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Normality concerning shared values 被引量:5
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作者 CHANG JianMing Department of Mathematics, Changshu Institute of Technology, Changshu 215500, China 《Science China Mathematics》 SCIE 2009年第8期1717-1722,共6页
Let $ \mathcal{F} $ be a family of meromorphic functions in a plane domain D, and a and b be finite non-zero complex values such that $ a/b \notin \mathbb{N}\backslash \{ 1\} $ . If for $ f \in \mathcal{F}, f(z) = a \... Let $ \mathcal{F} $ be a family of meromorphic functions in a plane domain D, and a and b be finite non-zero complex values such that $ a/b \notin \mathbb{N}\backslash \{ 1\} $ . If for $ f \in \mathcal{F}, f(z) = a \Rightarrow f'(z) = a $ and $ f'(z) = b \Rightarrow f''(z) = b $ , then $ \mathcal{F} $ is normal. We also construct a non-normal family $ \mathcal{F} $ of meromorphic functions in the unit disk Δ={|z|<1} such that for every $ f \in \mathcal{F}, f(z) = m + 1 \Leftrightarrow f'(z) = m + 1 $ and $ f'(z) = 1 \Leftrightarrow f''(z) = 1 $ in Δ, where m is a given positive integer. This answers Problem 5.1 in the works of Gu, Pang and Fang. 展开更多
关键词 meromorphic function holomorphic functions normal family 30D45
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A Normal Criterion about Two Families of Meromorphic Functions Concerning Shared Values 被引量:5
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作者 Xiao Jun LIU San Hua LI Xue Cheng PANG 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2013年第1期151-158,共8页
In this paper, we mainly discuss the normality of two families of functions concerning shared values and proved: Let F and G be two families of functions meromorphic on a domain D C C, a1, a2, a3, a4 be four distinct... In this paper, we mainly discuss the normality of two families of functions concerning shared values and proved: Let F and G be two families of functions meromorphic on a domain D C C, a1, a2, a3, a4 be four distinct finite complex numbers. If G is normal, and for every f 9~, there exists g C G such that f(z) and g(z) share the values a1, a2, a3, a4, then F is normal on D. 展开更多
关键词 normal family shared values meromorphic functions
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双领导者樽海鞘群算法 被引量:6
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作者 俞家珊 吴雷 《计算机科学》 CSCD 北大核心 2021年第4期254-260,共7页
为了提升樽海鞘群(Salp Swarm Algorithm,SSA)算法的求解精度和全局搜索能力,提出了一种基于正态过程搜索和差分进化(Differential Evolution,DE)算法的改进樽海鞘群算法——双领导者樽海鞘群算法(Two Types of Leaders Salp Swarm Algo... 为了提升樽海鞘群(Salp Swarm Algorithm,SSA)算法的求解精度和全局搜索能力,提出了一种基于正态过程搜索和差分进化(Differential Evolution,DE)算法的改进樽海鞘群算法——双领导者樽海鞘群算法(Two Types of Leaders Salp Swarm Algorithm,TTLSSA)。该算法设置了两类领导者和两种跟随群体,其中执行正态过程搜索的领导者需要进行正态过程游走、交叉、选择等操作,主要用于全局勘探;当前最优解附近的领导者在随迭代次数呈锯齿状变化的参数gap的影响下,兼顾了全局搜索和局部开发两种功能。用18个不同类型的标准测试函数检验所提算法的性能,并与DE、SSA、正弦余弦算法(Sines and Cosines Algorithm,SCA)、灰狼优化(Grey Wolf Optimizer,GWO)算法以及鲸鱼优化算法(Whale Optimization Algorithm,WOA)做对比,TTLSSA在16个测试函数上的平均精度排名第1或并列第1,在2个测试函数上的平均精度排名第2,在6种算法中平均耗时排名第2,说明了TTLSSA在没有增加SSA时间成本的前提下,显著提升了优化能力。 展开更多
关键词 差分进化 樽海鞘群算法 正态过程 测试函数
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涉及高阶导数的分担函数与正规族
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作者 王皓然 杨祺 《商丘师范学院学报》 CAS 2024年第12期1-5,共5页
设F是定义在复平面D上的一族亚纯函数,m≥0, n≥2, k≥1为3个正整数.设φ(z)为区域D上零点重数最少为m的解析函数,且φ(z)≢0.设P是一个多项式,deg(P)≥3,且至少有一个q重零点,m可被n+q整除.若对任意的f∈F, f的零点重数最少是k+m/q,并... 设F是定义在复平面D上的一族亚纯函数,m≥0, n≥2, k≥1为3个正整数.设φ(z)为区域D上零点重数最少为m的解析函数,且φ(z)≢0.设P是一个多项式,deg(P)≥3,且至少有一个q重零点,m可被n+q整除.若对任意的f∈F, f的零点重数最少是k+m/q,并且极点重数最少是m+1.则对每一组(f,g)∈F,若P(f)(f^((k)))^(n)与P(g)(g^(k))^(n)分担φ(z),则F在D上正规. 展开更多
关键词 正规族 分担函数 亚纯函数
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A Normal Criterion Concerning Omitted Holomorphic Function 被引量:2
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作者 Jin Hua YANG Qi YANG Xue Cheng PANG 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2019年第12期1972-1978,共7页
In this paper,we continue to discuss the normality concerning omitted holomorphic function and get the following result.Let F be a family of meromorphic functions on a domain D,k≥4 be a positive integer,and let a(z)a... In this paper,we continue to discuss the normality concerning omitted holomorphic function and get the following result.Let F be a family of meromorphic functions on a domain D,k≥4 be a positive integer,and let a(z)and b(z)be two holomorphic functions on D,where a(z)■0 and f(z)≠∞ whenever a(z)=0.If for any f∈F,f'(z)?a(z)f^k(z)≠b(z),then F is normal on D. 展开更多
关键词 normal FAMILY HOLOMORPHIC functions omitted functions
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亚纯函数涉及微分多项式的正规族
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作者 王皓然 杨祺 《五邑大学学报(自然科学版)》 CAS 2024年第2期9-13,26,共6页
本文研究了一类涉及微分多项式与分担值的正规定则.设n,k为2个正整数,令F是定义在复平面D上的一族亚纯函数,其零点重级至少k+1,a为一非零有穷复数,L (f)是关于f的微分多项式,且L(f)■0,若对■f,g∈F,都有f^(n)L(f)与g^(n)L(g)分担a,则F... 本文研究了一类涉及微分多项式与分担值的正规定则.设n,k为2个正整数,令F是定义在复平面D上的一族亚纯函数,其零点重级至少k+1,a为一非零有穷复数,L (f)是关于f的微分多项式,且L(f)■0,若对■f,g∈F,都有f^(n)L(f)与g^(n)L(g)分担a,则F在D上正规. 展开更多
关键词 正规族 亚纯函数 微分多项式 分担值
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Normal family and the sequence of omitted functions 被引量:4
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作者 CHEN QiaoYu YANG Liu PANG XueCheng 《Science China Mathematics》 SCIE 2013年第9期1821-1830,共10页
Let {fn} be a sequence of meromorphic functions on a plane domain D, whose zeros and poles have multiplicity at least 3. Let {hn} be a sequence of meromorphic functions on D, whose poles are multiple, such that {hn} c... Let {fn} be a sequence of meromorphic functions on a plane domain D, whose zeros and poles have multiplicity at least 3. Let {hn} be a sequence of meromorphic functions on D, whose poles are multiple, such that {hn} converges locally uniformly in the spherical metric to a function h which is meromorphic and zero-free on D.If fn≠hn, then {fn} is normal on D. 展开更多
关键词 meromorphic function normal family the sequence of omitted functions
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正规定则与重值 被引量:4
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作者 徐焱 常建明 《数学学报(中文版)》 SCIE CSCD 北大核心 2011年第2期265-270,共6页
设■为区域D内的一亚纯函数族,ψ(≠0)为D内的全纯函数,k为正整数.如果对每个f∈■,有f(z)≠0,f^((k))(z)≠0及f^((k))(z)-ψ(z)的零点重级至少为(k+2)/k。
关键词 亚纯函数 正规族 例外函数 重数
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户外背包的功能与结构设计 被引量:3
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作者 任蕊 弓太生 金鑫 《中国皮革》 CAS 北大核心 2015年第17期56-59,共4页
应用面料的改性和复合防水处理方法及缝制、外层压胶工艺处理等,通过在背包贴身部分设计空气流通通道,设置密码锁、反光条、报警器等,在包顶、包身、包底和睡袋部位设计扣袋和兜网等,以提高背包的防水性、透气性、安全性、装载性等常规... 应用面料的改性和复合防水处理方法及缝制、外层压胶工艺处理等,通过在背包贴身部分设计空气流通通道,设置密码锁、反光条、报警器等,在包顶、包身、包底和睡袋部位设计扣袋和兜网等,以提高背包的防水性、透气性、安全性、装载性等常规功能,以及保健、背负可调、平衡重心等新型功能。从背包的制作材料、工艺特点、结构设计、用户体验等方面,对国内外现有背包产品进行分析、比较,并提出了未来户外背包的设计发展方向,为户外背包的设计者和生产企业提供参考。 展开更多
关键词 户外背包 工艺 常规功能 新型功能 结构设计
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A Criterion of Normality Based on a Single Holomorphic Function 被引量:2
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作者 Xiao Jun LIU Shahar NEVO 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2011年第1期141-154,共14页
Let F be a family of functions holomorphic on a domain D C C. Let k ≥ 2 be an integer and let h be a holomorphic function on D, all of whose zeros have multiplicity at most k- 1, such that h(z) has no common zeros ... Let F be a family of functions holomorphic on a domain D C C. Let k ≥ 2 be an integer and let h be a holomorphic function on D, all of whose zeros have multiplicity at most k- 1, such that h(z) has no common zeros with any f∈F. Assume also that the following two conditions hold for every f ∈F :(a) f(z) = 0 == f (z) = h(z); and (b) y(z) = h(z) == |f(k)(z)| ≤ c, where c is a constant.Then F is normal on D. 展开更多
关键词 normal family holomorphic functions zero points
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A NEW CHARACTERIZATION OF Q_p~# FUNCTIONS
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作者 乌兰哈斯 李晓南 《Acta Mathematica Scientia》 SCIE CSCD 2004年第4期583-588,共6页
A new characterization of Q#p is given, which implies immediately a known result. Also, the authors consider a class Νp of bounded characteristic with order p, 0 < p < ∞, in the unit disk and give some relatio... A new characterization of Q#p is given, which implies immediately a known result. Also, the authors consider a class Νp of bounded characteristic with order p, 0 < p < ∞, in the unit disk and give some relationship between it and other classes of meromorphic functions. This paper answers partly a question mentioned by Aulaskari and Lappan. 展开更多
关键词 normal functions Q#p bounded characteristic the Green function
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THE NORMAL FAMILITY OF MERMORPHIC FUNCTIONS 被引量:2
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作者 韩明华 顾永兴 《Acta Mathematica Scientia》 SCIE CSCD 2008年第4期759-762,共4页
Let F be a family of mermorphic functions in a domain D, and let a, b, c be complex numbers, a ≠ b. If for each f ∈ F, the zeros of f-c are of multiplicity ≥ k + 1, and -↑Ef(k)(a) belong to -↑Ef (a), -↑Ef... Let F be a family of mermorphic functions in a domain D, and let a, b, c be complex numbers, a ≠ b. If for each f ∈ F, the zeros of f-c are of multiplicity ≥ k + 1, and -↑Ef(k)(a) belong to -↑Ef (a), -↑Ef(k)(b)belong to -↑Ef (b), then F is normal in D. 展开更多
关键词 Mermorphic functions normal family shared values
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亚纯函数的正规定则 被引量:2
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作者 鲁凤年 杨锦华 《新疆师范大学学报(自然科学版)》 2023年第4期43-48,共6页
文章通过幅角原理和Zalcman-Pang引理,利用构造辅助函数的方法,对满足条件f′-a(z)f k≠b(z)的亚纯函数族的正规性进行研究。首先,讨论当k=3,例外函数a(z),b(z)均为全纯函数时,在函数族零点重级的条件下,亚纯函数族是正规的,完善了正规... 文章通过幅角原理和Zalcman-Pang引理,利用构造辅助函数的方法,对满足条件f′-a(z)f k≠b(z)的亚纯函数族的正规性进行研究。首先,讨论当k=3,例外函数a(z),b(z)均为全纯函数时,在函数族零点重级的条件下,亚纯函数族是正规的,完善了正规族已有结论;其次,将例外函数由全纯函数推广为亚纯函数,讨论例外函数a(z),b(z)分别为亚纯函数时的正规性,得到当a(z)是亚纯函数时,全纯函数族是正规的,当b(z)是亚纯函数时,无零点的亚纯函数族是正规的,在一定范围内推广了亚纯函数的现有结果。 展开更多
关键词 亚纯函数 正规定则 例外函数
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一类具有校正隔离率随机SIQS模型的绝灭性与分布 被引量:3
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作者 魏凤英 林青腾 《数学物理学报(A辑)》 CSCD 北大核心 2017年第6期1148-1161,共14页
该文探讨了一类具有校正隔离率的随机传染病模型,得到了该模型存在唯一的全局解.研究表明,当白噪声强度取较大值时,随机模型的解在无病平衡点附近是绝灭的,感染者的密度将指数衰减到零.当白噪声的强度较小时,随机模型的正解在地方病平... 该文探讨了一类具有校正隔离率的随机传染病模型,得到了该模型存在唯一的全局解.研究表明,当白噪声强度取较大值时,随机模型的解在无病平衡点附近是绝灭的,感染者的密度将指数衰减到零.当白噪声的强度较小时,随机模型的正解在地方病平衡点附近服从唯一的平稳分布.进而,若地方病平衡点是稳定的,在适当的条件下,该解渐近服从一个三维正态分布,且得到了均值与方差的表达式.最后,数值模拟图显示了该解的性质并对模型做出了合理的解释. 展开更多
关键词 随机SIQS传染病模型 绝灭 平稳分布 正态分布 李雅谱诺夫函数
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Normal Families and Shared Functions
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作者 WANG Xue-qin LEI Chun-lin 《Chinese Quarterly Journal of Mathematics》 2015年第2期267-273,共7页
Let k be a positive integer,let h be a holomorphic function in a domain D,h■0and let F be a family of nonvanishing meromorphic functions in D.If each pair of functions f and q in F,f^((k)) and g^((k)) share h in D,th... Let k be a positive integer,let h be a holomorphic function in a domain D,h■0and let F be a family of nonvanishing meromorphic functions in D.If each pair of functions f and q in F,f^((k)) and g^((k)) share h in D,then F is normal in D. 展开更多
关键词 normal families meromorphic functions shared functions
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Normal Families of Holomorphic Functions Concerning Shared a Polynomial 被引量:1
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作者 LIU Zhi-hong LI Ying-chun HUANG Bin 《Chinese Quarterly Journal of Mathematics》 CSCD 2011年第3期394-399,共6页
In this paper,we study normal families of holomorphic function concerning shared a polynomial.Let F be a family of holomorphic functions in a domain D,k(2)be a positive integer,K be a positive number andα(z)be a poly... In this paper,we study normal families of holomorphic function concerning shared a polynomial.Let F be a family of holomorphic functions in a domain D,k(2)be a positive integer,K be a positive number andα(z)be a polynomial of degree p(p 1).For each f∈F and z∈D,if f and f sharedα(z)CM and|f(k)(z)|K whenever f(z)-α(z)=0 in D, then F is normal in D. 展开更多
关键词 holomorphic functions normal family shared values POLYNOMIAL
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A Criterion of Normality Concerning Holomorphic Functions Whose Derivative Omits a Function 被引量:2
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作者 Xiaojun LIU Yasheng YE 《Chinese Annals of Mathematics,Series B》 SCIE CSCD 2011年第5期699-710,共12页
The authors discuss the normality concerning holomorphic functions and get the following result. Let F be a family of holomorphic functions on a domain D C C, all of whose zeros have multiplicity at least k, where k ... The authors discuss the normality concerning holomorphic functions and get the following result. Let F be a family of holomorphic functions on a domain D C C, all of whose zeros have multiplicity at least k, where k 〉 2 is an integer. And let h(z)≠ 0 be a holomorphic function on D. Assume also that the following two conditions hold for every f ∈F: (a) f(z) = 0 [f^(k)(z)| 〈 |h(z)|; (b) f^(k)(z)≠ h(z). Then F is normal on 展开更多
关键词 normal family Holomorphic functions Omitted function
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