The bipartite Turán number of a graph H, denoted by ex(m,n;H), is the maximum number of edges in any bipartite graph G=(A,B;E(G))with | A |=mand | B |=nwhich does not contain H as a subgraph. Whenmin{ m,n }>2t...The bipartite Turán number of a graph H, denoted by ex(m,n;H), is the maximum number of edges in any bipartite graph G=(A,B;E(G))with | A |=mand | B |=nwhich does not contain H as a subgraph. Whenmin{ m,n }>2t, the problem of determining the value of ex(m,n;Km−t,n−t)has been solved by Balbuena et al. in 2007, whose proof focuses on the structural analysis of bipartite graphs. In this paper, we provide a new proof on the value of ex(m,n;Km−t,n−t)by virtue of algebra method with the tool of adjacency matrices of bipartite graphs, which is inspired by the method using { 0,1 }-matrices due to Zarankiewicz [Problem P 101. Colloquium Mathematicum, 2(1951), 301].展开更多
Let F be a graph and H be a hypergraph.We say that H contains a Berge-F If there exists a bijectionψ:E(F)→E(H)such that for Ve E E(F),e C(e),and the Turan number of Berge-F is defined to be the maximum number of edg...Let F be a graph and H be a hypergraph.We say that H contains a Berge-F If there exists a bijectionψ:E(F)→E(H)such that for Ve E E(F),e C(e),and the Turan number of Berge-F is defined to be the maximum number of edges in an r-uniform hypergraph of order n that is Berge-F-free,denoted by ex,(n,Berge-F).A linear forest is a graph whose connected components are all paths or isolated vertices.Let Ln,k be the family of all linear forests of n vertices with k edges.In this paper,Turan number of Berge-Ln,in an r-uniform hypergraph is studied.When r≥k+1 and 3≤r≤l[]=1,we determine 2 the exact value of ex,(n,Berge-Ln,)respectively.When K-1≤r≤k,we 2 determine the upper bound of ex,(n,Berge-Ln,).展开更多
Let F={H_(1),...,H_(k)}(k≥1)be a family of graphs.The Tur´an number of the family F is the maximum number of edges in an n-vertex{H_(1),...,H_(k)}-free graph,denoted by ex(n,F)or ex(n,{H_(1),H_(2),...,H_(k)}).Th...Let F={H_(1),...,H_(k)}(k≥1)be a family of graphs.The Tur´an number of the family F is the maximum number of edges in an n-vertex{H_(1),...,H_(k)}-free graph,denoted by ex(n,F)or ex(n,{H_(1),H_(2),...,H_(k)}).The blow-up of a graph H is the graph obtained from H by replacing each edge in H by a clique of the same size where the new vertices of the cliques are all different.In this paper we determine the Tur´an number of the family consisting of a blow-up of a cycle and a blow-up of a star in terms of the Tur´an number of the family consisting of a cycle,a star and linear forests with k edges.展开更多
This scientific paper is a comparative analysis of two mathematical conjectures. The newly proposed -3(-n) - 1 Remer conjecture and how it is related to and a proof of the more well known 3n + 1 Collatz conjecture. An...This scientific paper is a comparative analysis of two mathematical conjectures. The newly proposed -3(-n) - 1 Remer conjecture and how it is related to and a proof of the more well known 3n + 1 Collatz conjecture. An overview of both conjectures and their respective iterative processes will be presented. Showcasing their unique properties and behavior to each other. Through a detailed comparison, we highlight the similarities and differences between these two conjectures and discuss their significance in the field of mathematics. And how they prove each other to be true.展开更多
A survey of zoological literature affirmed the wide occurrence of Fibonacci numbers in the organization of acellular and prokaryotic life forms as well as in some eukaryotic protistans and in the embryonic development...A survey of zoological literature affirmed the wide occurrence of Fibonacci numbers in the organization of acellular and prokaryotic life forms as well as in some eukaryotic protistans and in the embryonic development and adult forms of many living and fossil remains of metazoan animals. A detailed comparative analysis of the axial skeleton of a fossil fish and humans revealed a new rule of the “nested triad” of bones organized along the proximal to distal axis of limb appendages. This growth pattern and its ubiquity among living vertebrates appear to underlie a profound rule of pattern formation that is dictated in part by the genetics and epigenetic mechanisms of stem cell clonal development.展开更多
It is shown that the Ramsey number r（K2,s＋1,K1,n）≤n＋√sn＋（s＋3）/2＋o（1）for large n, and r（K2,s＋1, K1,n） ∈{（q-1）^2/s ＋ 1,（q-1）^2/s＋2},where n =（q-1）^2/s -q＋2 and q is a prime power such that s｜...It is shown that the Ramsey number r（K2,s＋1,K1,n）≤n＋√sn＋（s＋3）/2＋o（1）for large n, and r（K2,s＋1, K1,n） ∈{（q-1）^2/s ＋ 1,（q-1）^2/s＋2},where n =（q-1）^2/s -q＋2 and q is a prime power such that s｜（q - 1）.展开更多
The highly （1301） oriented triple system of [CoPt/C]n/Ag films was deposited on glass substrates by DC and RF magnetron sputtering. After annealing at 600℃ for 30 min, thin films become magnetically hard with coerc...The highly （1301） oriented triple system of [CoPt/C]n/Ag films was deposited on glass substrates by DC and RF magnetron sputtering. After annealing at 600℃ for 30 min, thin films become magnetically hard with coercivities in the range of 160-875 kA/m because of high anisotropy associated with the L10 ordered phase. C doping plays an important role in improving （001） texture and reducing the intergrain interactions. The oriented growth of CoPt films was influenced strongly by the number of repetitions （n） of CoPt/C. By controlling the C content and the number of repetitions （n） of CoPt/C, nearly perfect （001） orientation can be obtained in the [CoPt3nm/C3nm]5/Ag50 nm.展开更多
文摘The bipartite Turán number of a graph H, denoted by ex(m,n;H), is the maximum number of edges in any bipartite graph G=(A,B;E(G))with | A |=mand | B |=nwhich does not contain H as a subgraph. Whenmin{ m,n }>2t, the problem of determining the value of ex(m,n;Km−t,n−t)has been solved by Balbuena et al. in 2007, whose proof focuses on the structural analysis of bipartite graphs. In this paper, we provide a new proof on the value of ex(m,n;Km−t,n−t)by virtue of algebra method with the tool of adjacency matrices of bipartite graphs, which is inspired by the method using { 0,1 }-matrices due to Zarankiewicz [Problem P 101. Colloquium Mathematicum, 2(1951), 301].
文摘Let F be a graph and H be a hypergraph.We say that H contains a Berge-F If there exists a bijectionψ:E(F)→E(H)such that for Ve E E(F),e C(e),and the Turan number of Berge-F is defined to be the maximum number of edges in an r-uniform hypergraph of order n that is Berge-F-free,denoted by ex,(n,Berge-F).A linear forest is a graph whose connected components are all paths or isolated vertices.Let Ln,k be the family of all linear forests of n vertices with k edges.In this paper,Turan number of Berge-Ln,in an r-uniform hypergraph is studied.When r≥k+1 and 3≤r≤l[]=1,we determine 2 the exact value of ex,(n,Berge-Ln,)respectively.When K-1≤r≤k,we 2 determine the upper bound of ex,(n,Berge-Ln,).
基金Supported by the National Nature Science Foundation of China(Grant Nos.11871329,11971298)。
文摘Let F={H_(1),...,H_(k)}(k≥1)be a family of graphs.The Tur´an number of the family F is the maximum number of edges in an n-vertex{H_(1),...,H_(k)}-free graph,denoted by ex(n,F)or ex(n,{H_(1),H_(2),...,H_(k)}).The blow-up of a graph H is the graph obtained from H by replacing each edge in H by a clique of the same size where the new vertices of the cliques are all different.In this paper we determine the Tur´an number of the family consisting of a blow-up of a cycle and a blow-up of a star in terms of the Tur´an number of the family consisting of a cycle,a star and linear forests with k edges.
文摘This scientific paper is a comparative analysis of two mathematical conjectures. The newly proposed -3(-n) - 1 Remer conjecture and how it is related to and a proof of the more well known 3n + 1 Collatz conjecture. An overview of both conjectures and their respective iterative processes will be presented. Showcasing their unique properties and behavior to each other. Through a detailed comparison, we highlight the similarities and differences between these two conjectures and discuss their significance in the field of mathematics. And how they prove each other to be true.
文摘A survey of zoological literature affirmed the wide occurrence of Fibonacci numbers in the organization of acellular and prokaryotic life forms as well as in some eukaryotic protistans and in the embryonic development and adult forms of many living and fossil remains of metazoan animals. A detailed comparative analysis of the axial skeleton of a fossil fish and humans revealed a new rule of the “nested triad” of bones organized along the proximal to distal axis of limb appendages. This growth pattern and its ubiquity among living vertebrates appear to underlie a profound rule of pattern formation that is dictated in part by the genetics and epigenetic mechanisms of stem cell clonal development.
文摘It is shown that the Ramsey number r（K2,s＋1,K1,n）≤n＋√sn＋（s＋3）/2＋o（1）for large n, and r（K2,s＋1, K1,n） ∈{（q-1）^2/s ＋ 1,（q-1）^2/s＋2},where n =（q-1）^2/s -q＋2 and q is a prime power such that s｜（q - 1）.
基金This work was financially supported by the National Natural Science Foundation of China (No. 10574085) and the Natural Science Foundation of Shanxi Province, China (No. 20041032)
文摘The highly （1301） oriented triple system of [CoPt/C]n/Ag films was deposited on glass substrates by DC and RF magnetron sputtering. After annealing at 600℃ for 30 min, thin films become magnetically hard with coercivities in the range of 160-875 kA/m because of high anisotropy associated with the L10 ordered phase. C doping plays an important role in improving （001） texture and reducing the intergrain interactions. The oriented growth of CoPt films was influenced strongly by the number of repetitions （n） of CoPt/C. By controlling the C content and the number of repetitions （n） of CoPt/C, nearly perfect （001） orientation can be obtained in the [CoPt3nm/C3nm]5/Ag50 nm.
