Let G be a simple connected graph with vertex set V(G).For S■V(G),letπG(S)denote the maximum cardinality of internally disjoint S-paths in G.For an integer k with k≥2,the k-path-connectivityπ_(k)(G)is defined as t...Let G be a simple connected graph with vertex set V(G).For S■V(G),letπG(S)denote the maximum cardinality of internally disjoint S-paths in G.For an integer k with k≥2,the k-path-connectivityπ_(k)(G)is defined as the minimumπ_(G)(S)over all k-subsets S of V(G).It is proved that deciding whetherπ_G(S)≥r is NP-complete problem[Graphs Combin.37(2021)2521-2533].The hypercube Qn is the famous Cayley graph,which is widely studied in the research of developing multiprocessor systems.The hierarchical cubic network HCN_(n)is given in[IEEE TPDS 6(1995)427-435]which takes Q_(n)as building clusters and emulates the desirable properties very efficiently.In this paper,we consider the 3-path-connectivity of HCN_(n)and prove thatπ_(3)(HCN_(n))=[(3n+2)/4]for n≥2 by constructing multiple internally disjoint S-paths.This result improves the 3-tree-connectivity[Discrete Appl.Math.322(2022)203-209]from trees to paths.展开更多
Given two Banach spaces E, F, let B（E, F） be the set of all bounded linear operators from E into F, ∑r the set of all operators of finite rank r in B（E, F）, and ∑r^# the number of path connected components of ∑...Given two Banach spaces E, F, let B（E, F） be the set of all bounded linear operators from E into F, ∑r the set of all operators of finite rank r in B（E, F）, and ∑r^# the number of path connected components of ∑r. It is known that ∑r is a smooth Banach submanifold in B（E,F） with given expression of its tangent space at each A ∈ ∑r. In this paper, the equality ∑r^# = 1 is proved. Consequently, the following theorem is obtained： for any nonnegative integer r,∑r is a smooth and path connected Banach submanifold in B（E, F） with the tangent space TA∑r = {B E B（E,F） ： BN（A） belong to R（A）} at each A ∈ ∑r if dim F = ∞. Note that the routine method can hardly be applied here. So in addition to the nice topological and geometric property of ∑r the method presented in this paper is also interesting. As an application of this result, it is proved that if E = R^n and F = R^m, then ∑r is a smooth and path connected submanifold of B（R^n,R^m） and its dimension is dim ∑r = （m ＋ n）r- r^2 for each r, 0≤r 〈 min{n,m}.展开更多
Based on the problems found in the study of historical blocks, the paper puts forward the quantifiable standards for maintaining the spatial characteristics of historical blocks by analyzing the key factors of update ...Based on the problems found in the study of historical blocks, the paper puts forward the quantifiable standards for maintaining the spatial characteristics of historical blocks by analyzing the key factors of update strategy that influence the spatial characteristics of historical blocks. Combined with the update planning of the north section of Chuancheng Street, a historic block in Handan City, the block space is digitally simulated by space syntax software Depthmap, to study the behavior path and visual field in the renewal of historic blocks. The development strategy of both protection and renewal is put forward, in order to accumulate experience for the sustainable development of historic blocks.展开更多
Environmental impact assessment and environmental sanitation permits are important systems for protecting the ecological and natural environment in our country. They are not only powerful tools for protecting the ecol...Environmental impact assessment and environmental sanitation permits are important systems for protecting the ecological and natural environment in our country. They are not only powerful tools for protecting the ecological and natural environment, but also powerful pillars for promoting sound economic and social development. In recent years, under the background of rapid economic and social development, our environment and air quality have deteriorated, and there is a direct and effective link between environmental impact assessment and environmental sanitation permits to avoid causing damage to the environment and human body.展开更多
The authors give the condition that the Bergman kernel function on the first type of Cartan-Hartogs domain exists zeros.If the Bergman kernel function of this type of domain has zeros,the zero set is composed of sever...The authors give the condition that the Bergman kernel function on the first type of Cartan-Hartogs domain exists zeros.If the Bergman kernel function of this type of domain has zeros,the zero set is composed of several path-connected branches,and there exists a continuous curve to connect any two points in the non-zero set.展开更多
基金supported by the National Natural Science Foundation of China(Nos.12471321 and 12331013)supported by Beijing Natural Science Foundation(No.1244047)+1 种基金China Postdoctoral Science Foundation(No.2023M740207)supported by the Basic Science Research Program through the National Research Foundation of Korea(NRF)funded by the Ministry of Education(2018R1D1A1B05048450)。
文摘Let G be a simple connected graph with vertex set V(G).For S■V(G),letπG(S)denote the maximum cardinality of internally disjoint S-paths in G.For an integer k with k≥2,the k-path-connectivityπ_(k)(G)is defined as the minimumπ_(G)(S)over all k-subsets S of V(G).It is proved that deciding whetherπ_G(S)≥r is NP-complete problem[Graphs Combin.37(2021)2521-2533].The hypercube Qn is the famous Cayley graph,which is widely studied in the research of developing multiprocessor systems.The hierarchical cubic network HCN_(n)is given in[IEEE TPDS 6(1995)427-435]which takes Q_(n)as building clusters and emulates the desirable properties very efficiently.In this paper,we consider the 3-path-connectivity of HCN_(n)and prove thatπ_(3)(HCN_(n))=[(3n+2)/4]for n≥2 by constructing multiple internally disjoint S-paths.This result improves the 3-tree-connectivity[Discrete Appl.Math.322(2022)203-209]from trees to paths.
基金Supported by the National Science Foundation of China (Grant No.10671049 and 10771101).
文摘Given two Banach spaces E, F, let B（E, F） be the set of all bounded linear operators from E into F, ∑r the set of all operators of finite rank r in B（E, F）, and ∑r^# the number of path connected components of ∑r. It is known that ∑r is a smooth Banach submanifold in B（E,F） with given expression of its tangent space at each A ∈ ∑r. In this paper, the equality ∑r^# = 1 is proved. Consequently, the following theorem is obtained： for any nonnegative integer r,∑r is a smooth and path connected Banach submanifold in B（E, F） with the tangent space TA∑r = {B E B（E,F） ： BN（A） belong to R（A）} at each A ∈ ∑r if dim F = ∞. Note that the routine method can hardly be applied here. So in addition to the nice topological and geometric property of ∑r the method presented in this paper is also interesting. As an application of this result, it is proved that if E = R^n and F = R^m, then ∑r is a smooth and path connected submanifold of B（R^n,R^m） and its dimension is dim ∑r = （m ＋ n）r- r^2 for each r, 0≤r 〈 min{n,m}.
基金Sponsored by Graduate Demonstration Course Construction Project of Hebei Province(KCJSX2020081)。
文摘Based on the problems found in the study of historical blocks, the paper puts forward the quantifiable standards for maintaining the spatial characteristics of historical blocks by analyzing the key factors of update strategy that influence the spatial characteristics of historical blocks. Combined with the update planning of the north section of Chuancheng Street, a historic block in Handan City, the block space is digitally simulated by space syntax software Depthmap, to study the behavior path and visual field in the renewal of historic blocks. The development strategy of both protection and renewal is put forward, in order to accumulate experience for the sustainable development of historic blocks.
文摘Environmental impact assessment and environmental sanitation permits are important systems for protecting the ecological and natural environment in our country. They are not only powerful tools for protecting the ecological and natural environment, but also powerful pillars for promoting sound economic and social development. In recent years, under the background of rapid economic and social development, our environment and air quality have deteriorated, and there is a direct and effective link between environmental impact assessment and environmental sanitation permits to avoid causing damage to the environment and human body.
基金supported by the National Natural Science Foundation of China(No.11871044)the Natural Science Foundation of Hebei Province(No.A2019106037)
文摘The authors give the condition that the Bergman kernel function on the first type of Cartan-Hartogs domain exists zeros.If the Bergman kernel function of this type of domain has zeros,the zero set is composed of several path-connected branches,and there exists a continuous curve to connect any two points in the non-zero set.
