The branching structure of uniform recursive trees is investigated in this paper.Using the method of sums for a sequence of independent random variables, the distribution law of ηn, the number of branches of the unif...The branching structure of uniform recursive trees is investigated in this paper.Using the method of sums for a sequence of independent random variables, the distribution law of ηn, the number of branches of the uniform recursive tree of size n are given first. It is shown that the strong law of large numbers, the central limit theorem and the law of iterated logarithm for ηn follow easily from this method. Next it is shown that ηn and ξn, the depth of vertex n, have the same distribution, and the distribution law of ζn,m, the number of branches of size m, is also given, whose asymptotic distribution is the Poisson distribution with parameter λ = 1/m. In addition, the joint distribution and the asymptotic joint distribution of the numbers of various branches are given. Finally, it is proved that the size of the biggest branch tends to infinity almost sure as n -→∞.展开更多
The carburizing process is the enrichment of the depth of low carbon steels with carbon. It leads to samples with a combination of high surface hardness and high core toughness and to an impact strength that is requir...The carburizing process is the enrichment of the depth of low carbon steels with carbon. It leads to samples with a combination of high surface hardness and high core toughness and to an impact strength that is required for many engineering parts. The material studied is a low carbon steel. The carbon content is little in this type of steel （wc = 0.2 ~）. The calculation of case depth is very important for cementation steels that are hardened in the carburi- zing process. The effective case depth is defined as the perpendicular distance from the surface to a place at which the hardness is HV 550. Nowadays, a great number of studies have been carried out on the simulation of effective case depth, but no studies have been conducted to determine the numerical relation between the total case depth on one hand and the carburizing time and the effective case depth on the other hand. The steel specimens were subjected to graphite powder. Then, they were heat treated at 925 ~C for about 3, 5, 8 and 12 h, respectively. Then, these parts were quenched in oil. To determine the effective case depth, the micr0hardness test was performed on the cross-section of specimens. Plotting the case depth vs carburizing time, the required conditions for obtaining the specified case depth were determined. Also, the comparison between the case depths in numerical solution and the actual position in pack carburizing was performed.展开更多
基金This work was supported by the National Natural Science Foundation(Grant No.10371117)Special Foundation of USTC.
文摘The branching structure of uniform recursive trees is investigated in this paper.Using the method of sums for a sequence of independent random variables, the distribution law of ηn, the number of branches of the uniform recursive tree of size n are given first. It is shown that the strong law of large numbers, the central limit theorem and the law of iterated logarithm for ηn follow easily from this method. Next it is shown that ηn and ξn, the depth of vertex n, have the same distribution, and the distribution law of ζn,m, the number of branches of size m, is also given, whose asymptotic distribution is the Poisson distribution with parameter λ = 1/m. In addition, the joint distribution and the asymptotic joint distribution of the numbers of various branches are given. Finally, it is proved that the size of the biggest branch tends to infinity almost sure as n -→∞.
文摘The carburizing process is the enrichment of the depth of low carbon steels with carbon. It leads to samples with a combination of high surface hardness and high core toughness and to an impact strength that is required for many engineering parts. The material studied is a low carbon steel. The carbon content is little in this type of steel （wc = 0.2 ~）. The calculation of case depth is very important for cementation steels that are hardened in the carburi- zing process. The effective case depth is defined as the perpendicular distance from the surface to a place at which the hardness is HV 550. Nowadays, a great number of studies have been carried out on the simulation of effective case depth, but no studies have been conducted to determine the numerical relation between the total case depth on one hand and the carburizing time and the effective case depth on the other hand. The steel specimens were subjected to graphite powder. Then, they were heat treated at 925 ~C for about 3, 5, 8 and 12 h, respectively. Then, these parts were quenched in oil. To determine the effective case depth, the micr0hardness test was performed on the cross-section of specimens. Plotting the case depth vs carburizing time, the required conditions for obtaining the specified case depth were determined. Also, the comparison between the case depths in numerical solution and the actual position in pack carburizing was performed.
