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
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
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.
