Let {X,X n;n≥1} be a strictly stationary sequence of ρ-mixing random variables with mean zero and finite variance. Set S n=n k=1X k,M n=max k≤n｜S k｜,n≥1. Suppose lim n→∞ES2 n/n=∶σ2>0 and ∞...Let {X,X n;n≥1} be a strictly stationary sequence of ρ-mixing random variables with mean zero and finite variance. Set S n=n k=1X k,M n=max k≤n｜S k｜,n≥1. Suppose lim n→∞ES2 n/n=∶σ2>0 and ∞n=1ρ 2/d(2n)<∞, where d=2,if -1<b<0 and d>2(b+1),if b≥0. It is proved that,for any b>-1, limε0ε 2(b+1)∞n=1(loglogn)bnlognP{M n≥εσ2nloglogn}= 2(b+1)πГ(b+3/2)∞k=0(-1)k(2k+1) 2b+2,where Г(·) is a Gamma function.展开更多
By using a Rosenthal type inequality established in this paper, the complete convergence and almost sure summability on the convergence rates with respect to the strong law of large numbers are discussed for *-mixing...By using a Rosenthal type inequality established in this paper, the complete convergence and almost sure summability on the convergence rates with respect to the strong law of large numbers are discussed for *-mixing random fields.展开更多
We give some theorems of strong law of large numbers and complete convergence for sequences of φ-mixing random variables. In particular, Wittmann＇s strong law of large numbers and Teicher＇s strong law of large nnum...We give some theorems of strong law of large numbers and complete convergence for sequences of φ-mixing random variables. In particular, Wittmann＇s strong law of large numbers and Teicher＇s strong law of large nnumbers for independent random variables are generalized to the case of φ -minxing random variables.展开更多
The multiphase reaction process of pressure leaching is mainly carried out in the liquid phase. Therefore, gas holdup is essential for the gas–liquid–solid phase reaction and the extraction rate of valuable metals. ...The multiphase reaction process of pressure leaching is mainly carried out in the liquid phase. Therefore, gas holdup is essential for the gas–liquid–solid phase reaction and the extraction rate of valuable metals. In this paper, a transparent quartz autoclave, a six blades disc turbine-type agitator, and a high-speed camera were used to investigate the gas holdup of the pressure leaching process. Furthermore, experiments determining the effects of agitation rate, temperature, and oxygen partial pressure on gas holdup were carried out. The results showed that when the agitation rate increased from 350 to 600 r/min, the gas holdup increased from 0.10% to 0.64%. When the temperature increased from 363 to 423 K, the gas holdup increased from 0.14% to 0.20%. When the oxygen partial pressure increased from 0.1 to 0.8 MPa, the gas holdup increased from 0.13% to 0.19%. A similar criteria relationship was established by Homogeneous Principle and Buckingham's theorem. Comprehensively, empirical equation of gas holdup was deduced on the basis of experimental data and the similarity theory, where the criterion equation was determined as ε=4.54×10^(-11)n^(3.65)T^(2.08)P_g^(0.18). It can be seen from the formula that agitation rate made the most important impact on gas holdup in the pressure leaching process using the mixed-flow agitator.展开更多
In this paper, the complete convergence and weak law of large numbers are established for ρ-mixing sequences of random variables. Our results extend and improve the Baum and Katz complete convergence theorem and the ...In this paper, the complete convergence and weak law of large numbers are established for ρ-mixing sequences of random variables. Our results extend and improve the Baum and Katz complete convergence theorem and the classical weak law of large numbers, etc. from independent sequences of random variables to ρ-mixing sequences of random variables without necessarily adding any extra conditions.展开更多
Based on the calculating model of metallic melts involving eutectic, the calculating equations of mixing thermodynamic parameters for two phase metallic melts have been formulated in the light of those equations of ho...Based on the calculating model of metallic melts involving eutectic, the calculating equations of mixing thermodynamic parameters for two phase metallic melts have been formulated in the light of those equations of homogeneous solutions. Irrespective as to whether the activity deviation relative to Raoultian behavior is positive or negative, or the deviation is symmetrical or unsym-metrical, the evaluated results not only agree well with experimental values, but also strictly obey the mass action law. This testifies that these equations can authentically reflect the structural reality and mixing thermodynamic characteristics of two-phase metallic melts. The calculating equations of mixing thermodynamic parameters for the model of two phase metallic melts offer two practical criteria (activity and mixing thermodynamic parameters) and one theoretical criterion (the mass action law).展开更多
Based on the coexistence theory of metallic melts involving compound formation, the theoretical calculation equations of mixing thermodynamic parameters are established by giving up some empirical parameters in the as...Based on the coexistence theory of metallic melts involving compound formation, the theoretical calculation equations of mixing thermodynamic parameters are established by giving up some empirical parameters in the associated solution model. For Fe-Al, Mn-Al and Ni-Al, the calculated results agree well with the experimental values, testifying that these equations can exactly embody mixing thermodynamic characteristics of these melts.展开更多
Application of equations of mixing thermodynamic parameters formulated on the basis of the coexistence theory of metallic melts in Ba-Al, Mg-Al, Sr-Al and Cu-Al melts leads to fruitful results that not only the evalua...Application of equations of mixing thermodynamic parameters formulated on the basis of the coexistence theory of metallic melts in Ba-Al, Mg-Al, Sr-Al and Cu-Al melts leads to fruitful results that not only the evaluated mass action concentrations agree well with the measured activities, but also the calculated mixing thermodynamic parameters are quite coincident with the experimental values. Moreover, the calculated mass action concentrations strictly obey the mass action law. The evaluated mixing thermodynamic parameters have very fine regularity: the mixing free energy is composed of standard free formation energies of all compounds and chemical potentials of all structural units at equilibrium; the mixing enthalpy consists of standard formation enthalpies of all compounds; the mixing entropy is composed of standard entropies of all compounds and configuration entropies of all structural units at equilibrium. As the equations of mixing thermodynamic parameters formulated are widely applicable to metallic melts involving compound formation, they can be used as the second practical criterion to determine whether thermodynamic models of metallic melts are correctly formulated.展开更多
Let variables in the {X, Xn, n ≥ 1} be a sequence of strictly stationary φ-mixing positive random domain of attraction of the normal law. Under some suitable conditions the principle for self-normalized products of ...Let variables in the {X, Xn, n ≥ 1} be a sequence of strictly stationary φ-mixing positive random domain of attraction of the normal law. Under some suitable conditions the principle for self-normalized products of partial sums is obtained.展开更多
In this paper, we will present some strong convergence results for sequences of ψ-mixing random variables. The results for sequences of ψ-mixing random variables generalize the corresponding results for independent ...In this paper, we will present some strong convergence results for sequences of ψ-mixing random variables. The results for sequences of ψ-mixing random variables generalize the corresponding results for independent random variable sequences without any extra conditions.展开更多
基金Research supported by the National Natural Science Foundation of China (1 0 0 71 0 72 )
文摘Let {X,X n;n≥1} be a strictly stationary sequence of ρ-mixing random variables with mean zero and finite variance. Set S n=n k=1X k,M n=max k≤n｜S k｜,n≥1. Suppose lim n→∞ES2 n/n=∶σ2>0 and ∞n=1ρ 2/d(2n)<∞, where d=2,if -1<b<0 and d>2(b+1),if b≥0. It is proved that,for any b>-1, limε0ε 2(b+1)∞n=1(loglogn)bnlognP{M n≥εσ2nloglogn}= 2(b+1)πГ(b+3/2)∞k=0(-1)k(2k+1) 2b+2,where Г(·) is a Gamma function.
基金National Natural Science Foundation of China! (No. 19701O11) Foundation of "151 talent project" of Zhejiang provience.
文摘By using a Rosenthal type inequality established in this paper, the complete convergence and almost sure summability on the convergence rates with respect to the strong law of large numbers are discussed for *-mixing random fields.
基金Supported by the National Natural Science Foundation of China (10671149)
文摘We give some theorems of strong law of large numbers and complete convergence for sequences of φ-mixing random variables. In particular, Wittmann＇s strong law of large numbers and Teicher＇s strong law of large nnumbers for independent random variables are generalized to the case of φ -minxing random variables.
基金financially supported by the Joint Funds of the National Natural Science Foundation of China (Nos.U1402271, 51504058, and 51504059)
文摘The multiphase reaction process of pressure leaching is mainly carried out in the liquid phase. Therefore, gas holdup is essential for the gas–liquid–solid phase reaction and the extraction rate of valuable metals. In this paper, a transparent quartz autoclave, a six blades disc turbine-type agitator, and a high-speed camera were used to investigate the gas holdup of the pressure leaching process. Furthermore, experiments determining the effects of agitation rate, temperature, and oxygen partial pressure on gas holdup were carried out. The results showed that when the agitation rate increased from 350 to 600 r/min, the gas holdup increased from 0.10% to 0.64%. When the temperature increased from 363 to 423 K, the gas holdup increased from 0.14% to 0.20%. When the oxygen partial pressure increased from 0.1 to 0.8 MPa, the gas holdup increased from 0.13% to 0.19%. A similar criteria relationship was established by Homogeneous Principle and Buckingham's theorem. Comprehensively, empirical equation of gas holdup was deduced on the basis of experimental data and the similarity theory, where the criterion equation was determined as ε=4.54×10^(-11)n^(3.65)T^(2.08)P_g^(0.18). It can be seen from the formula that agitation rate made the most important impact on gas holdup in the pressure leaching process using the mixed-flow agitator.
文摘In this paper, the complete convergence and weak law of large numbers are established for ρ-mixing sequences of random variables. Our results extend and improve the Baum and Katz complete convergence theorem and the classical weak law of large numbers, etc. from independent sequences of random variables to ρ-mixing sequences of random variables without necessarily adding any extra conditions.
文摘Based on the calculating model of metallic melts involving eutectic, the calculating equations of mixing thermodynamic parameters for two phase metallic melts have been formulated in the light of those equations of homogeneous solutions. Irrespective as to whether the activity deviation relative to Raoultian behavior is positive or negative, or the deviation is symmetrical or unsym-metrical, the evaluated results not only agree well with experimental values, but also strictly obey the mass action law. This testifies that these equations can authentically reflect the structural reality and mixing thermodynamic characteristics of two-phase metallic melts. The calculating equations of mixing thermodynamic parameters for the model of two phase metallic melts offer two practical criteria (activity and mixing thermodynamic parameters) and one theoretical criterion (the mass action law).
文摘Based on the coexistence theory of metallic melts involving compound formation, the theoretical calculation equations of mixing thermodynamic parameters are established by giving up some empirical parameters in the associated solution model. For Fe-Al, Mn-Al and Ni-Al, the calculated results agree well with the experimental values, testifying that these equations can exactly embody mixing thermodynamic characteristics of these melts.
文摘Application of equations of mixing thermodynamic parameters formulated on the basis of the coexistence theory of metallic melts in Ba-Al, Mg-Al, Sr-Al and Cu-Al melts leads to fruitful results that not only the evaluated mass action concentrations agree well with the measured activities, but also the calculated mixing thermodynamic parameters are quite coincident with the experimental values. Moreover, the calculated mass action concentrations strictly obey the mass action law. The evaluated mixing thermodynamic parameters have very fine regularity: the mixing free energy is composed of standard free formation energies of all compounds and chemical potentials of all structural units at equilibrium; the mixing enthalpy consists of standard formation enthalpies of all compounds; the mixing entropy is composed of standard entropies of all compounds and configuration entropies of all structural units at equilibrium. As the equations of mixing thermodynamic parameters formulated are widely applicable to metallic melts involving compound formation, they can be used as the second practical criterion to determine whether thermodynamic models of metallic melts are correctly formulated.
基金National Natural Science Foundation of China(1067117610771192).
文摘Let variables in the {X, Xn, n ≥ 1} be a sequence of strictly stationary φ-mixing positive random domain of attraction of the normal law. Under some suitable conditions the principle for self-normalized products of partial sums is obtained.
基金Supported by the University Students Science Research Training Program of Anhui University(KYXL20110004)
文摘In this paper, we will present some strong convergence results for sequences of ψ-mixing random variables. The results for sequences of ψ-mixing random variables generalize the corresponding results for independent random variable sequences without any extra conditions.
