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gamma-function
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  gamma-function
Another problem, considered in the second part is the analogue inCm of the classical gamma-function and several properties for further consideration.
      
It was shown that these results are applicable to the Goddard Gl-function method, the generalized valence bonds method, and also the Smeyers gamma-function method.
      
These methods were the conventional Hamilton, the gamma-function model, the tanks-in-series model, the simplified gamma method and the Dow formula.
      
The gamma-function method has a tendency to overestimate on the average (+7·1%±1·6%), with an 82% probability of falling within ±20% of the Stewart-Hamilton value.
      
A series representation in terms of Tricomi functions is obtained for generalized exponential integral,Ev{χ}, and related incomplete gamma-function,Γ(a, x).
      
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The authors of the present paper applied the incomplete gamma function in calculating and analysing 175 lactation periods of Holstein-Friesian in Jinshan, Shanghai. The results are given as fallows:1. As the lactation curves in the average month milk production com-ared with the actual lactation curves were estimated by the incomplelte gamma function, RMS was 0.716± 0.274Kg, and the bias was -0.68Kg. It was because the error was inconsiderable that the lactation curves might be analysed by the incomplete gamma...

The authors of the present paper applied the incomplete gamma function in calculating and analysing 175 lactation periods of Holstein-Friesian in Jinshan, Shanghai. The results are given as fallows:1. As the lactation curves in the average month milk production com-ared with the actual lactation curves were estimated by the incomplelte gamma function, RMS was 0.716± 0.274Kg, and the bias was -0.68Kg. It was because the error was inconsiderable that the lactation curves might be analysed by the incomplete gamma function.2. RMS per month on mathematical model was distributed thus: slightly high in the second and ninth milking month, comparatively low from the third to the eighth milking month and low and steady especially from the sixth to the eighth milking month.3. The absolute error would be litte if the 305 days of the milk yield was predicted by means of the mathematical model calculating the 240 days of the milk yield. The errors were 40.01Kg and 36.00Kg respectively, both two values being less than 0.8% according to the calculation of 60 lactation periods of two samples.4. The variance analysis showed that the main factors affecting parameter b were the dairy farm ( P<0.001 ) and the calving month ( P<0.05 ) , while the main factor affecting parameter c was the dairy farm ( P<0.05 ) . The calving month had a significant effect on the milking persistency ( P <0.01 ) , and the dairy farm had a significant effect on the arising of peak milk production period ( P<0.01 ) .5. The heritability of the parameter b was 0.0976, and that of the parameter cwas 0.018. The phenotypic carrelation was very significant between b and c ( P<0.001 ) .

本文作者应用不完全函数计算和分析了上海金山县黑白花乳牛175个泌乳期的资料,其结果如下: 1.不完全函数估计的月平均产乳量泌乳曲线与实际泌乳曲线比较,RMS为0.716±0.274公斤,偏差为0.68公斤,误差很小,说明用不完全部函数分析曲线是可行的。 2.模型的各月RMS分布,以第二和第九个泌乳月稍高,第三至第八个泌乳月较低,尤以第六至第八个泌乳月RMS很低且平稳。 3.采用模型以204天实际产乳量预测305天产乳量,绝对误差很小。根据对二个样本60个泌乳期的计算,误差分别为40.01和36.00公斤,均小于0.8%。 4.方差分析表明,影响参数b的主要因素是牧场(P<0.001)和产犊月份(P<0.05);影响参数C的主要因素是牧场(P<0.05);产犊月份对泌乳持久力的影响显著(P<0.01);牧场对泌乳出现高峰的时间影响显著(P<0.01)。 5.参数b的遗传力为0.0976,C为0.018,b和c的表型相关很显著(P<0.001)。

A calculating method has been inferred for the reactivity ratio of semicontinuous reactors. We sum this problem up as a corresponding differential equation and solve it. The complex solution obtained can be expressed by the incomplete Gamma function through mathematical treatments, and the unigue definite solution can be obtained through 《The Table of Incomplete Gamma Function》. The solution of differential equation for specific types can be integrated directly, and at the sametime we have improved the original...

A calculating method has been inferred for the reactivity ratio of semicontinuous reactors. We sum this problem up as a corresponding differential equation and solve it. The complex solution obtained can be expressed by the incomplete Gamma function through mathematical treatments, and the unigue definite solution can be obtained through 《The Table of Incomplete Gamma Function》. The solution of differential equation for specific types can be integrated directly, and at the sametime we have improved the original method of numerical solution.

本文推导了一种半连续式反应器反应速率的计算方法。将问题归纳成相应的微分方程后求解。所得复杂的解经过数学处理可用不完全Gamma函数表示。查不完全Gamma函数表即得到唯一确定解。特殊类型的微分方程解可直接积分求出。并对原数值解法作了改进。

The total work of compaction of powders, from zero strain(ε= 0) to in-finite strain (ε= ∞), is derived in this paper to be: a_(total)= Mw (1/d_0-1/d_m) Γ(m + 1)where M is the modulus of compaction of powder, w is the weight of powder,d_0 is the initial density of powder, d_m is the theoretical density of densemetal, m is the index of non-linearity and Γ(m + 1) is the gamma functionof (m + 1). Γ(m + 1) = ∫_0~∞ e~(-8)ε~mdεwhere ε is the strain of compaction, ε=ln((d_m-d_0)d)/((d_m-d)d_0)and d is the green density...

The total work of compaction of powders, from zero strain(ε= 0) to in-finite strain (ε= ∞), is derived in this paper to be: a_(total)= Mw (1/d_0-1/d_m) Γ(m + 1)where M is the modulus of compaction of powder, w is the weight of powder,d_0 is the initial density of powder, d_m is the theoretical density of densemetal, m is the index of non-linearity and Γ(m + 1) is the gamma functionof (m + 1). Γ(m + 1) = ∫_0~∞ e~(-8)ε~mdεwhere ε is the strain of compaction, ε=ln((d_m-d_0)d)/((d_m-d)d_0)and d is the green density of powder compact.The actual work of compaction of powders from ε_1 to ε_2 is derived to be: a = Mw (1/d_0-1/d_m)∫ε_1ε~2 e~(-ε)ε~mdεwhere ∫_(ε_1)~(ε_2)e~(-ε)ε~mdεis the incomplete gamma function of (m + 1), the numer-ical value of which can be evaluated by computers. Examples of calculations for the work of compaction on tungsten powderare given and tabulated.

本文导出了粉体从应变为0(ε=0)到应变无穷大(ε=∞)时的压制总功: α_总=MW(1/d_o-1/d_m)Γ(m+1)式中,M是粉末压制模量,W是粉末的重量,d_o是粉末的原始密度,d_m是致密金属的理论密度,Γ(m+1)是m+1的Γ函数, Γ(m+1)=∫_0~ ∞e~(-ε)ε~mdεε是压制应变, ε=ln(d_m-d_o)d/(d_m-d)d_od是压坯密度,m是非线性指数。还导出了应变从ε_1到ε_2时实际的粉末压制功, α=∫_(ε_1)~(ε_2)e~(-ε)ε~mdε式中,∫_(ε_1)~(ε_2)e~(-ε)ε~mdε是m+1的不完全Γ函数,其函数值可由电子计算机近似求得。文中列表给出了钨粉压制功的计算实例。

 
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