We further study the kinetic behavior of the exchange-driven growth withbirth and death for the case of birth rate kernel being less than that of death based on themean-Geld theory. The symmetric exchange rate kernel ...We further study the kinetic behavior of the exchange-driven growth withbirth and death for the case of birth rate kernel being less than that of death based on themean-Geld theory. The symmetric exchange rate kernel is K(k,j) = K′(k,j) = Ikj~v, and the birth anddeath rates are proportional to the aggregate's size. The long time asymptotic behavior of theaggregate size distribution a_k(t) is found to obey a much unusual scaling law with an exponentiallygrowing scaling function Φ(x) = exp(x).展开更多
Abstract Let x = （xn）n≥1 be a martingale on a noncommutative probability space （М,τ） and （Wn）n≥1 a sequence of positive numbers such that Wn =∑^n_k=1 wk→∞ as n→∞. We prove that x = （Xn）n≥1 converges...Abstract Let x = （xn）n≥1 be a martingale on a noncommutative probability space （М,τ） and （Wn）n≥1 a sequence of positive numbers such that Wn =∑^n_k=1 wk→∞ as n→∞. We prove that x = （Xn）n≥1 converges bilaterally almost uniformly （b.a.u.） if and only if the weighted average （σan（x））n≥1 of x converges b.a.u, to the same limit under some condition, where σn（x） is given by σn（x）=1/Wn ^n∑_k=1 wkxk,n=1,2,… Furthermore, we prove that x = （xn）n≥1 converges in Lp（М） if and only if （σ＇n（x））n≥1 converges in Lp（М）, where 1 ≤p 〈 ∞ .We also get a criterion of uniform integrability for a family in L1（М）.展开更多
文摘We further study the kinetic behavior of the exchange-driven growth withbirth and death for the case of birth rate kernel being less than that of death based on themean-Geld theory. The symmetric exchange rate kernel is K(k,j) = K′(k,j) = Ikj~v, and the birth anddeath rates are proportional to the aggregate's size. The long time asymptotic behavior of theaggregate size distribution a_k(t) is found to obey a much unusual scaling law with an exponentiallygrowing scaling function Φ(x) = exp(x).
基金supported by National Natural Science Foundation of China (Grant No.11071190)
文摘Abstract Let x = （xn）n≥1 be a martingale on a noncommutative probability space （М,τ） and （Wn）n≥1 a sequence of positive numbers such that Wn =∑^n_k=1 wk→∞ as n→∞. We prove that x = （Xn）n≥1 converges bilaterally almost uniformly （b.a.u.） if and only if the weighted average （σan（x））n≥1 of x converges b.a.u, to the same limit under some condition, where σn（x） is given by σn（x）=1/Wn ^n∑_k=1 wkxk,n=1,2,… Furthermore, we prove that x = （xn）n≥1 converges in Lp（М） if and only if （σ＇n（x））n≥1 converges in Lp（М）, where 1 ≤p 〈 ∞ .We also get a criterion of uniform integrability for a family in L1（М）.
