A generalized strongly regular graphof grade p,as anew generalization of strongly regular graphs,is a regular graph such that the number of common neighbours of both any two adjacent vertices and any two non-adjacent ...A generalized strongly regular graphof grade p,as anew generalization of strongly regular graphs,is a regular graph such that the number of common neighbours of both any two adjacent vertices and any two non-adjacent vertices takes on p distinct values.For any vertex u of a generalized strongly regular graph of grade 2 with parameters(n,k;a_(1),a_(2);c_(1),c_(2)),if the number of the vertices that are adjacent to u and share ai(i=1,2)common neighbours with u,or are non-adjacent to u and share c,(i=1,2)common neighbours with is independent of the choice of the vertex u,then the generalized strongly regular graph of grade 2 is free.In this paper,we investigate the generalized strongly regular graph of grade 2 with parameters(n,k;k-1,a_(2);k-1,c_(2))and provide the sufficient and necessary conditions for the existence of a family of free generalized strongly regular graphs of grade 2.展开更多
A combinatorial batch code has strong practical motivation in the distributed storage and retrieval of data in a database.In this survey,we give a brief introduction to the combinatorial batch codes and some progress.
Difference systems of sets (DSS) are combinatorial configurations that arise in connection with code synchronization. This paper proposes a new method to construct DSSs, which uses known DSSs to partition some of th...Difference systems of sets (DSS) are combinatorial configurations that arise in connection with code synchronization. This paper proposes a new method to construct DSSs, which uses known DSSs to partition some of the cosets of Zv relative to subgroup of order k, where v = km is a composite number. As applications, we obtain some new optimal DSSs.展开更多
In this paper, the robustness of the orbit structure is investigated for a partially hyperbolic endomorphism f on a compact manifold M. It is first proved that the dynamical structure of its orbit space(the inverse li...In this paper, the robustness of the orbit structure is investigated for a partially hyperbolic endomorphism f on a compact manifold M. It is first proved that the dynamical structure of its orbit space(the inverse limit space) M^f of f is topologically quasi-stable under C^0-small perturbations in the following sense: For any covering endomorphism g C^0-close to f, there is a continuous map φ from M^g to Multiply form -∞ to ∞ M such that for any {y_i }_(i∈Z) ∈φ(M^g), y_(i+1) and f(y_i) differ only by a motion along the center direction. It is then proved that f has quasi-shadowing property in the following sense: For any pseudo-orbit {x_i }_(i∈Z),there is a sequence of points {y_i }_(i∈Z) tracing it, in which y_(i+1) is obtained from f(y_i) by a motion along the center direction.展开更多
In this paper,a definition of entropy for Z+k(k≥2)-actions due to Friedland is studied.Unlike the traditional definition,it may take a nonzero value for actions whose generators have finite(even zero) entropy as...In this paper,a definition of entropy for Z+k(k≥2)-actions due to Friedland is studied.Unlike the traditional definition,it may take a nonzero value for actions whose generators have finite(even zero) entropy as single transformations.Some basic properties are investigated and its value for the Z+k-actions on circles generated by expanding endomorphisms is given.Moreover,an upper bound of this entropy for the Z+k-actions on tori generated by expanding endomorphisms is obtained via the preimage entropies,which are entropy-like invariants depending on the "inverse orbits" structure of the system.展开更多
In this paper,we consider the P-prize-collecting set cover(P-PCSC)problem,which is a generalization of the set cover problem.In this problem,we are given a set system(U,S),where U is a ground set and S⊆2U is a collect...In this paper,we consider the P-prize-collecting set cover(P-PCSC)problem,which is a generalization of the set cover problem.In this problem,we are given a set system(U,S),where U is a ground set and S⊆2U is a collection of subsets satisfying∪S∈SS=U.Every subset in S has a nonnegative cost,and every element in U has a nonnegative penalty cost and a nonnegative profit.Our goal is to find a subcollection C⊆S such that the total cost,consisting of the cost of subsets in C and the penalty cost of the elements not covered by C,is minimized and at the same time the combined profit of the elements covered by C is at least P,a specified profit bound.Our main work is to obtain a 2f+ε-approximation algorithm for the P-PCSC problem by using the primal-dual and Lagrangian relaxation methods,where f is the maximum frequency of an element in the given set system(U,S)andεis a fixed positive number.展开更多
基金supported by National Natural Science Foundation of China(No.11571091)Natural Science Foundation of Hebei Province,China(No.F2019205147)Innovation Program of Hebei Normal University,China(No.CXZZSS2020050).
文摘A generalized strongly regular graphof grade p,as anew generalization of strongly regular graphs,is a regular graph such that the number of common neighbours of both any two adjacent vertices and any two non-adjacent vertices takes on p distinct values.For any vertex u of a generalized strongly regular graph of grade 2 with parameters(n,k;a_(1),a_(2);c_(1),c_(2)),if the number of the vertices that are adjacent to u and share ai(i=1,2)common neighbours with u,or are non-adjacent to u and share c,(i=1,2)common neighbours with is independent of the choice of the vertex u,then the generalized strongly regular graph of grade 2 is free.In this paper,we investigate the generalized strongly regular graph of grade 2 with parameters(n,k;k-1,a_(2);k-1,c_(2))and provide the sufficient and necessary conditions for the existence of a family of free generalized strongly regular graphs of grade 2.
文摘A combinatorial batch code has strong practical motivation in the distributed storage and retrieval of data in a database.In this survey,we give a brief introduction to the combinatorial batch codes and some progress.
基金Supported by Natural Science Foundation of Hebei Province(Grant No.A2013205073)
文摘Difference systems of sets (DSS) are combinatorial configurations that arise in connection with code synchronization. This paper proposes a new method to construct DSSs, which uses known DSSs to partition some of the cosets of Zv relative to subgroup of order k, where v = km is a composite number. As applications, we obtain some new optimal DSSs.
基金supported by the National Natural Science Foundation of China(No.11371120)the High-level Personnel for Institutions of Higher Learning in Hebei Province(No.GCC2014052)the Natural Science Foundation of Hebei Province(No.A2013205148)
文摘In this paper, the robustness of the orbit structure is investigated for a partially hyperbolic endomorphism f on a compact manifold M. It is first proved that the dynamical structure of its orbit space(the inverse limit space) M^f of f is topologically quasi-stable under C^0-small perturbations in the following sense: For any covering endomorphism g C^0-close to f, there is a continuous map φ from M^g to Multiply form -∞ to ∞ M such that for any {y_i }_(i∈Z) ∈φ(M^g), y_(i+1) and f(y_i) differ only by a motion along the center direction. It is then proved that f has quasi-shadowing property in the following sense: For any pseudo-orbit {x_i }_(i∈Z),there is a sequence of points {y_i }_(i∈Z) tracing it, in which y_(i+1) is obtained from f(y_i) by a motion along the center direction.
基金Supported by National Natural Science Foundation of China(Grant No.11071054)the Key Project of Chinese Ministry of Education(Grant No.211020)+1 种基金the Program for New Century Excellent Talents in University(Grant No.11-0935)the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars,State Education Ministry(Grant No.11126011)
文摘In this paper,a definition of entropy for Z+k(k≥2)-actions due to Friedland is studied.Unlike the traditional definition,it may take a nonzero value for actions whose generators have finite(even zero) entropy as single transformations.Some basic properties are investigated and its value for the Z+k-actions on circles generated by expanding endomorphisms is given.Moreover,an upper bound of this entropy for the Z+k-actions on tori generated by expanding endomorphisms is obtained via the preimage entropies,which are entropy-like invariants depending on the "inverse orbits" structure of the system.
基金This work was supported by the National Natural Science Foundation of China(No.11971146)the Natural Science Foundation of Hebei Province of China(Nos.A2019205089 and A2019205092)+1 种基金Hebei Province Foundation for Returnees(No.CL201714)Overseas Expertise Introduction Program of Hebei Auspices(No.25305008).
文摘In this paper,we consider the P-prize-collecting set cover(P-PCSC)problem,which is a generalization of the set cover problem.In this problem,we are given a set system(U,S),where U is a ground set and S⊆2U is a collection of subsets satisfying∪S∈SS=U.Every subset in S has a nonnegative cost,and every element in U has a nonnegative penalty cost and a nonnegative profit.Our goal is to find a subcollection C⊆S such that the total cost,consisting of the cost of subsets in C and the penalty cost of the elements not covered by C,is minimized and at the same time the combined profit of the elements covered by C is at least P,a specified profit bound.Our main work is to obtain a 2f+ε-approximation algorithm for the P-PCSC problem by using the primal-dual and Lagrangian relaxation methods,where f is the maximum frequency of an element in the given set system(U,S)andεis a fixed positive number.
基金Supported by NSFC(Grant Nos.11871192,11471095)the Program for Foreign Experts of Hebei Province(Grant Nos.2019YX002A,2020,2021)the Program for 100 Foreign Experts Plan of Hebei Province(Grant No.2021)。
文摘In this paper we investigate the tr-convexity and the rectangular biconvexity in Euclidean spaces.