The products of graphs discussed in this paper are the following four kinds: the Cartesian product of graphs, the tensor product of graphs, the lexicographic product of graphs and the strong direct product of graphs. ...The products of graphs discussed in this paper are the following four kinds: the Cartesian product of graphs, the tensor product of graphs, the lexicographic product of graphs and the strong direct product of graphs. It is proved that:① If the graphs G 1 and G 2 are the connected graphs, then the Cartesian product, the lexicographic product and the strong direct product in the products of graphs, are the path positive graphs. ② If the tensor product is a path positive graph if and only if the graph G 1 and G 2 are the connected graphs, and the graph G 1 or G 2 has an odd cycle and max{ λ 1μ 1,λ nμ m}≥2 in which λ 1 and λ n [ or μ 1 and μ m] are maximum and minimum characteristic values of graph G 1 [ or G 2 ], respectively.展开更多
Tracy-Widom distribution was rst discovered in the study of largest eigenvalues of high dimensional Gaussian unitary ensembles(GUE),and since then it has appeared in a number of apparently distinct research elds.It is...Tracy-Widom distribution was rst discovered in the study of largest eigenvalues of high dimensional Gaussian unitary ensembles(GUE),and since then it has appeared in a number of apparently distinct research elds.It is believed that Tracy-Widom distribution have a universal feature like classic normal distribution.Airy2 process is de ned through nite dimensional distributions with Tracy-Widom distribution as its marginal distributions.In this introductory survey,we will briey review some basic notions,intuitive background and fundamental properties concerning Tracy-Widom distribution and Airy2 process.For sake of reading,the paper starts with some simple and well-known facts about normal distributions,Gaussian processes and their sample path properties.展开更多
The Hansdorff dimensions of the image and the graph of random fields are given under general conditions.The results can be used to a wider class of self-similar random fields and processes,including Brownian motion,Br...The Hansdorff dimensions of the image and the graph of random fields are given under general conditions.The results can be used to a wider class of self-similar random fields and processes,including Brownian motion,Brownian sheet,fractional Brownian motion,processes with stable or(α,β)-fractional stable components.展开更多
The effects of various cooling paths on uniformity of through-thickness microstructure and mechanical properties of X80 pipeline steel of 22.0 mm in thickness were studied. The finite difference method was employed to...The effects of various cooling paths on uniformity of through-thickness microstructure and mechanical properties of X80 pipeline steel of 22.0 mm in thickness were studied. The finite difference method was employed to calculate the temperature field during cooling. It was confirmed by the experimental result and temperature field calculation that the optimizing process was achieved by the ultra-fast cooling with medium cooling capacity（cooling rate of ＊23 K/s）followed by ultimate cooling capacity（cooling rate of ＊50 K/s）. After optimization, the experimental steel displayed much uniform microstructure and the deviation of through-thickness hardness was controlled within 20 HV. In addition,the yield strength, tensile strength and elongation of the experimental steel were 621, 728 MPa and 21.5%, respectively,meeting the requirements of the API standard for X80 pipeline steels.展开更多
文摘The products of graphs discussed in this paper are the following four kinds: the Cartesian product of graphs, the tensor product of graphs, the lexicographic product of graphs and the strong direct product of graphs. It is proved that:① If the graphs G 1 and G 2 are the connected graphs, then the Cartesian product, the lexicographic product and the strong direct product in the products of graphs, are the path positive graphs. ② If the tensor product is a path positive graph if and only if the graph G 1 and G 2 are the connected graphs, and the graph G 1 or G 2 has an odd cycle and max{ λ 1μ 1,λ nμ m}≥2 in which λ 1 and λ n [ or μ 1 and μ m] are maximum and minimum characteristic values of graph G 1 [ or G 2 ], respectively.
基金the National Natural Science Foundation of China(11731012,11871425)Fundamental Research Funds for Central Universities grant(2020XZZX002-03).
文摘Tracy-Widom distribution was rst discovered in the study of largest eigenvalues of high dimensional Gaussian unitary ensembles(GUE),and since then it has appeared in a number of apparently distinct research elds.It is believed that Tracy-Widom distribution have a universal feature like classic normal distribution.Airy2 process is de ned through nite dimensional distributions with Tracy-Widom distribution as its marginal distributions.In this introductory survey,we will briey review some basic notions,intuitive background and fundamental properties concerning Tracy-Widom distribution and Airy2 process.For sake of reading,the paper starts with some simple and well-known facts about normal distributions,Gaussian processes and their sample path properties.
基金Supported by the National Natural Science Foundation of China.
文摘The Hansdorff dimensions of the image and the graph of random fields are given under general conditions.The results can be used to a wider class of self-similar random fields and processes,including Brownian motion,Brownian sheet,fractional Brownian motion,processes with stable or(α,β)-fractional stable components.
基金supported by the National Natural Science Foundation of China(No.51234002)the Fundamental Research Funds for the Central Universities(N130407001)
文摘The effects of various cooling paths on uniformity of through-thickness microstructure and mechanical properties of X80 pipeline steel of 22.0 mm in thickness were studied. The finite difference method was employed to calculate the temperature field during cooling. It was confirmed by the experimental result and temperature field calculation that the optimizing process was achieved by the ultra-fast cooling with medium cooling capacity（cooling rate of ＊23 K/s）followed by ultimate cooling capacity（cooling rate of ＊50 K/s）. After optimization, the experimental steel displayed much uniform microstructure and the deviation of through-thickness hardness was controlled within 20 HV. In addition,the yield strength, tensile strength and elongation of the experimental steel were 621, 728 MPa and 21.5%, respectively,meeting the requirements of the API standard for X80 pipeline steels.
