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Some extremal results on hypergraph Turán problems

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Science China Mathematics 2022年第8期65卷 1765-1774页

作者： Zixiang Xu Tao Zhang gennian ge School of Mathematical Sciences Capital Normal UniversityBeijing 100048China School of Mathematics and Information Science Guangzhou UniversityGuangzhou 510006China

For two r-graphs T and H,let ex_(r)(n,T,H)be the maximum number of copies of T in an n-vertex H *** determination of the Turán number ex_(r)(n,T,H)has become the fundamental core problem in extremal graph theory ever since the pioneering work of Turán’s theorem was published in *** we have some rich results for the simple graph case,only sporadic results have been known for the hypergraph Turán *** this paper,we mainly focus on the function ex_(r)(n,T,H)when H is one of two different hypergraph extensions of the complete bipartite graph K_(s,t).The first extension is the complete bipartite r-graph K^((r))_(s,t),which was introduced by Mubayi and Verstraëte(2004).Using the powerful random algebraic method,we show that if s is sufficiently larger than t,then ex_(r)(n,T,K^((r))_(s,t))=Ω(n^(v−e/t)),where T is an r-graph with v vertices and e *** particular,when T is an edge or some specified complete bipartite r-graph,we can determine their *** second important extension is the complete r-partite r-graph K^((r))_(s1,s2,…,sr,)which has been widely *** r=3,we provide an explicit construction giving ex_(3)(n,K^((3))_(2,2,7))≥1/27n19/7+o(n19/7).Our construction is based on the norm graph,and improves the lower boundΩ(n73/27)obtained by the probabilistic method.

关键词： hypergraph Turán problem random algebraic construction explicit construction

Some intriguing upper bounds for separating hash families

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Science China Mathematics 2019年第2期62卷 269-282页

作者： gennian ge Chong Shangguan Xin Wang School of Mathematical Sciences Capital Normal University School of Mathematical Sciences Zhejiang University Department of Mathematics Soochow University

An N ×n matrix on q symbols is called {w_1,...,w_t}-separating if for arbitrary t pairwise disjoint column sets C_1,..., C_t with |C_i|=w_i for 1 ≤i≤t, there exists a row f such that f(C_1),...,f(C_t) are also pairwise disjoint, where f(C_i) denotes the collection of componentn of C_i restricted to row f. Given integers N, q and w_1,...,w_t, denote by C(N,q,{w_1,...,w_t}) the maximal a such that a corresponding matrix does *** determination of C(N,q,{w_1,...,w_t}) has received remarkable attention during the recent years. The main purpose of this paper is to introduce two novel methodologies to attack the upper bound of C(N, q, {w_1,...,w_t}).The first one is a combination of the famous graph removal lemma in extremal graph theory and a Johnson-type recursive inequality in coding theory, and the second onc is the probabilistic method. As a consequence, we obtain several intriguing upper bounds for some parameters of C(N,q,{w_1,...,w_t}), which significantly improve the previously known results.

关键词： separating hash families Johnson-type recursive bound graph removal lemma probabilistic method

Inverse problems of the Erdos-Ko-Rado type theorems for families of vector spaces and permutations

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Science China Mathematics 2022年第5期65卷 1081-1108页

作者： Xiangliang Kong Yuanxiao Xi Bingchen Qian gennian ge School of Mathematical Sciences Capital Normal UniversityBeijing 100048China School of Mathematical Sciences Zhejiang UniversityHangzhou 310027China

Ever since the famous Erd os-Ko-Rado theorem initiated the study of intersecting families of subsets,extremal problems regarding intersecting properties of families of various combinatorial objects have been extensively *** them,studies about families of subsets,vector spaces and permutations are of particular ***,we proposed a new quantitative intersection problem for families of subsets:For F([n]k),define its total intersection number as I(F)=ΣF1;F2∈F|F1∩F2|.Then,what is the structure of F when it has the maximal total intersection number among all the families in([n]k)with the same family size?In a recent paper,Kong and ge(2020)studied this problem and characterized extremal structures of families maximizing the total intersection number of given *** this paper,we consider the analogues of this problem for families of vector spaces and *** certain ranges of family sizes,we provide structural characterizations for both families of subspaces and families of permutations having maximal total intersection *** some extent,these results determine the unique structure of the optimal family for some certain values of jFj and characterize the relationship between having the maximal total intersection number and being ***,we also show several upper bounds on the total intersection numbers for both families of subspaces and families of permutations of given sizes.

关键词： total intersection number vector spaces permutations

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稀疏超图:从理论到应用

中国科学：数学 2023年第2期53卷 187-216页

作者：上官冲葛根年山东大学数学与交叉科学研究中心青岛266237 首都师范大学数学科学学院北京100048

给定正整数r、e和v,如果某个r-一致超图的任意e条不同边的并都包含至少v+1个顶点,则称其是(v, e)-自由(free)或者(v, e)-稀疏的.稀疏超图的概念由Brown、Erd?s和Sós在20世纪70年代提出.目前,研究给定顶点数的稀疏超图所能包含最大边数... 详细信息

给定正整数r、e和v,如果某个r-一致超图的任意e条不同边的并都包含至少v+1个顶点,则称其是(v, e)-自由(free)或者(v, e)-稀疏的.稀疏超图的概念由Brown、Erd?s和Sós在20世纪70年代提出.目前,研究给定顶点数的稀疏超图所能包含最大边数的上下界已成为极值组合学研究领域内的核心问题之一.该问题的研究方法丰富多变,涉及组合、概率、代数和数论等多个领域.本文介绍Brown、Erd?os和S′os关于稀疏超图的两个重要猜想的最新研究进展以及稀疏超图在极值组合与信息科学中的若干应用,包括朱烈曾作出突出贡献的完美哈希(Hash)矩阵、可分哈希矩阵等几类信息安全中的研究问题.此外,本文在某些参数下给出完美哈希矩阵与求并-自由(union-free)超图的新构造.本文的构造改进了相应问题的已知最优下界.

关键词：稀疏超图 Brown-Erd?s-Sós猜想完美哈希矩阵可消去(cancellative)超图求并-自由超图集中式编码缓存组合列表译码局部可修复码

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四元域上最优局部可修复码的分类

中国科学：数学 2023年第2期53卷 339-368页

作者：奚元霄孔祥粱葛根年浙江大学数学科学学院杭州310058 首都师范大学数学科学学院北京100048

近年来,为了提高分布式存储系统的容错性和可靠性,编码学家们引入了几类新的编码方案,其中局部可修复码(locally repairable codes,LRC)起到了重要的作用.对于一个线性码,若它的一个码字符号能通过其他至多r个码字符号修复,则称其具有... 详细信息

近年来,为了提高分布式存储系统的容错性和可靠性,编码学家们引入了几类新的编码方案,其中局部可修复码(locally repairable codes,LRC)起到了重要的作用.对于一个线性码,若它的一个码字符号能通过其他至多r个码字符号修复,则称其具有局部性参数r.码长为n、维数为k、局部性参数为r的LRC((n,k,r)-LRC),其极小距离d满足Singleton型界d≤n-k-[k/r]+2.自LRC被提出以来,有许多工作研究小域上达到Singleton型界的码类.本文从码的校验矩阵角度出发,利用组合设计和有限几何的工具,研究了达到Singleton型界的最优四元LRC.本文证明了在四元域上共有27类最优的LRC,并且给出了这些最优码的构造.不仅如此,利用有限几何工具,本文还引入了判断最优LRC存在的新方法.

关键词：局部可修复码组合设计有限几何大数据存储

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不可扩展直积基的新进展

中国科学：数学 2021年第10期51卷 1699-1714页

作者：张一炜石飞张先得杨亦挺葛根年山东大学教育部密码技术与信息安全重点实验室山东大学网络空间安全学院青岛266237 中国科学技术大学网络空间安全学院合肥230026 中国科学技术大学数学科学学院合肥230026 同济大学数学科学学院上海200092 首都师范大学数学科学学院北京100048

不可扩展直积基(unextendible product bases, UPB)是量子信息中的重要概念,在量子信息的诸多领域有着广泛的应用. UPB的构造与组合数学有着密切的联系,著名组合学家Alon和Lovász利用一系列图论工具率先刻画了一组UPB态的数目达到平凡... 详细信息

不可扩展直积基(unextendible product bases, UPB)是量子信息中的重要概念,在量子信息的诸多领域有着广泛的应用. UPB的构造与组合数学有着密切的联系,著名组合学家Alon和Lovász利用一系列图论工具率先刻画了一组UPB态的数目达到平凡下界时的充分必要条件,进而冯克勤先生将图的1-因子分解等工具引入到此问题的研究之中.本文继续利用图论工具,在部分参数下得到了UPB最小态数目问题的一系列新结果.此外,本文对C^(2)■C^(2)■C^(k)中的所有UPB态的数目的可能取值做了近乎完全的刻画.

关键词：不可扩展直积基量子信息正交表示连通性循环图 1-因子分解

Longitudial observation of dynamic changes in cortical function and white matter fibrous structure of patients with visual pathway lesions by blood oxygenation level dependent-functional magnetic resonance imaging combined with diffusion tensor imaging

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Neural Regeneration Research 2007年第8期2卷 449-454页

作者： Hui Xiao Ziqian Chen gennian Qian Ping Ni Youqiang Ye Jinhua Chen Department of MedicalRadiology Fuzhou GeneralHospital of Nanjing MilitaryArea Command of ChinesePLA Fuzhou 350025Fujian Province China

BACKGROUND： Functional magnetic resonance imaging （fMRI） is initially used for visual cortex location. However, the application of fMRI in investigating the development of visual pathway lesions needs to be further observed. OBJECTIVE： This study is to longitudially observe the dynamic changes in cortical function and white matter fibrous structure of patients with visual pathway lesions by blood oxygenation level dependent-functional magnetic resonance imaging （BOLD-fMRI） combined with diffusion tensor imaging （DTI）, and to analyze the characteristics of brain function and structural recombination at convalescent period of lesions. DESIGN： Randomized controlled observation. SETTING： Department of Radiology, the general Hospital of Nanjing Military Area Command of Chinese PLA. PARTICIPANTS： Eight patients with unilateral or bilateral visual disorder caused by visual pathway lesions, who admitted to Department of Radiology, the general Hospital of Nanjing Military Area Command of Chinese PLA from January to September 2006 were involved, and served as experimental subjects. The patients, 6 males and 2 females, were aged 16 - 67 years. They had visual disorder confirmed by clinical examination, i.e. visual pathway lesion, which was further diagnosed by MR or CT. Another 12 subjects generally matching to those patients of experimental group in gender, age and sight, who received health examination in synchronization were involved and served as controls. The subjects had no history of eye diseases. Their binocular visual acuity （or corrected visual acuity） was over 1.0. Both routine examination of ophthalmology and examination of fundus were normal. Informed consents of detected items were obtained from all the subjects. METHODS： Signa Excite HD 1.5T magnetic resonance imaging system with 16 passages （ge Company, USA） and coil with 8 passages were used; brain functional stimulus apparatus （SAV-8800. Meide Company） was used for showing experimental mission. At the early s

关键词： magnetic resonance imaging vision rehabilitation

MRI and diffusion tensor imaging in assessing correlation of activation of cortical motor function and manifestations of corticospinal tract with muscle strength in patients with ischemic stroke

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Neural Regeneration Research 2006年第1期1卷 56-59页

作者： Ziqian Chen Hui Xiao Biyun Zhang gennian Qian Ping Ni Xizhang Yang Department of Medical Imaging Fuzhou General Hospital of Nanjing Military Area Command of Chinese PLA Fuzhou 350025Fujian Province China

BACKGROUND： Ischemic stroke is often followed by the abnormalities of neurons and corticospinal tract, which can lead to corresponding clinical symptoms and signs. Recently, with the continuous perfection of high-field MRI instrument, it becomes possible to assess and investigate the cortical function and structural reconstruction following stroke by using MRI and diffusion tensor imaging （DTI）. OBJECTIVE： To observe the cortical motor function and changes of corticospinal tracts by using MRI and DTI in the patients with ischemic stroke at acute period, compare with the normal subjects, and assess the damage of corticospinal tract and muscle strength.DESIGN ： A case-control observation.SETTING： Department of Medical Imaging, Fuzhou general Hospital of Nanjing Military Area Command of Chinese PLA. PARTICIPANTS ： Nine inpatients （5 males and 4 females） with injury of motor function induced by acute ischemic stroke were selected from Fuzhou general Hospital of Nanjing Military Area Command of Chinese PLA between August and December in 2005, they aged 16-87 years with an average of 51 years old, and those with obvious conscious disturbances and severe cognitive disorders were excluded. At the same time, nine healthy right-handed physical examinees matched by age and gender with the patients were also selected, and they all had no nervous disease, epilepsy, mental diseases, cerebrovascular abnormalities and injury history, etc. All the subjects were informed with the detected items and agreed to participate in. METHODS： All the 9 patients with ischemic stroke at acute period and 9 healthy subjects were examined with MRI and DTI.① A block-based design was used in the MRI, the passive finger-to-finger exercise was used as the stimulative task, and the static condition was taken as the baseline task. The ge 1.5T MRI system was used, all the data were processed after off-line, and analyzed with the SPM2 software, the association between the activated area and local

关键词： cortical tensor assessing cortex rehabilitation handed exercise epilepsy atter interrupted

创意产业集群、外部性与城市创新——基于空间杜宾模型

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科技与经济 2017年第6期30卷 31-35页

作者：钟琴葛家玮黄明均唐根年浙江工业大学经贸管理学院杭州310023

基于2005—2014年长三角26个市的统计数据,建立空间杜宾模型,运用matlab研究外部性,创意产业集群对城市创新的影响及空间溢出效应。实证结果表明金融与技术外部性不仅有效推动该市的创新,还存在较强的空间溢出效应,带动邻近城市的创新... 详细信息

基于2005—2014年长三角26个市的统计数据,建立空间杜宾模型,运用matlab研究外部性,创意产业集群对城市创新的影响及空间溢出效应。实证结果表明金融与技术外部性不仅有效推动该市的创新,还存在较强的空间溢出效应,带动邻近城市的创新。创意产业集群对本市创新直接效应为正,对相邻城市创新间接效应为负。长三角各市之间存在较强的空间依赖性,城市创新互动明显,这为长三角地区合理利用外部性与创意产业集群大力提升城市创新水平提供政策依据。

关键词：空间效应外部性创意产业集群城市创新