刘玉琏,付沛仁编的《数学分析讲义》最新版（1992年7月第三版）练习题9.2（一）第6题（该讲义下册63页）: 证明:若函数级数sum from n=1 to f<sub>n</sub>（x）与sum from n=1 to g<sub>n</sub>（x）在区间I都...刘玉琏,付沛仁编的《数学分析讲义》最新版（1992年7月第三版）练习题9.2（一）第6题（该讲义下册63页）: 证明:若函数级数sum from n=1 to f<sub>n</sub>（x）与sum from n=1 to g<sub>n</sub>（x）在区间I都一致收敛,且函数列{f<sub>n</sub>（x）}与{g<sub>n</sub>（x）}在区间I都一致有界,则函数级数sum from n=1 to f<sub>n</sub>（x）g<sub>n</sub>（x）在区间I一致收敛。 这是历次版本未有的一道新题,遗憾的是它却又是该讲义中少有的一道伪习题。 定理1 上述习题为伪命题 [反例] 取f<sub>n</sub>（x）=（-1）<sup>n-1</sup>1/n<sup>1/2</sup>,g<sub>n</sub>（x）=（-1）<sup>n-1</sup>1/n<sup>1/3</sup>使用莱布尼兹判别法不难验证sum from n=1 to （-1）<sup>n-1</sup>1/n<sup>1/2</sup>与sum from n=1 to （-1）<sup>n-1</sup>1/n<sup>1/2</sup>与sum from n=1 to （-1）<sup>n-1</sup>1/n<sup>1/3</sup>均收敛,由于与x无关,对x当然一致收敛,又,|（-1）<sup>n-1</sup>1/n<sup>1/2</sup>|≤1,与（-1）<sup>n-1</sup>1/n<sup>1/3</sup>≤1（x）即对x一致有界,但是sum from n=1 to ∞1/n<sup>1/2</sup>·1/n<sup>1/3</sup>=sum from n=1 to ∞1/n<sup>5/6</sup>,5/6【1,发散。 因此,上述习题为伪命题展开更多
By proposing a two-dimensional triggering model with concentrically circular closed magnetic field line structure,numerical research is made on the asymmetric propagation feature of coronal mass ejection (CME) in two ...By proposing a two-dimensional triggering model with concentrically circular closed magnetic field line structure,numerical research is made on the asymmetric propagation feature of coronal mass ejection (CME) in two cases emerging at the solar northern latitudes 10° and 45° respectively.The numerical results can qualitatively explain some features of CME event observed by the spacecraft SOHO and show that:(i) In these two cases,the triggering model can initiate CME with an asymmetric closed magnetic field structure.(ii) Closed magnetic structure of CME event will keep deflecting to the current sheet when it propagates away from the sun and this deflecting effect mostly happens within tens of solar radii before CME travels finally along the current sheet.(iii) The triggering model emerging at different locations can introduce CME events with different magnetic shapes.This shape happens to be circular and crescent when the triggering model emerges at the northern latitudes 10° and 45°,respectively.展开更多
文摘刘玉琏,付沛仁编的《数学分析讲义》最新版（1992年7月第三版）练习题9.2（一）第6题（该讲义下册63页）: 证明:若函数级数sum from n=1 to f<sub>n</sub>（x）与sum from n=1 to g<sub>n</sub>（x）在区间I都一致收敛,且函数列{f<sub>n</sub>（x）}与{g<sub>n</sub>（x）}在区间I都一致有界,则函数级数sum from n=1 to f<sub>n</sub>（x）g<sub>n</sub>（x）在区间I一致收敛。 这是历次版本未有的一道新题,遗憾的是它却又是该讲义中少有的一道伪习题。 定理1 上述习题为伪命题 [反例] 取f<sub>n</sub>（x）=（-1）<sup>n-1</sup>1/n<sup>1/2</sup>,g<sub>n</sub>（x）=（-1）<sup>n-1</sup>1/n<sup>1/3</sup>使用莱布尼兹判别法不难验证sum from n=1 to （-1）<sup>n-1</sup>1/n<sup>1/2</sup>与sum from n=1 to （-1）<sup>n-1</sup>1/n<sup>1/2</sup>与sum from n=1 to （-1）<sup>n-1</sup>1/n<sup>1/3</sup>均收敛,由于与x无关,对x当然一致收敛,又,|（-1）<sup>n-1</sup>1/n<sup>1/2</sup>|≤1,与（-1）<sup>n-1</sup>1/n<sup>1/3</sup>≤1（x）即对x一致有界,但是sum from n=1 to ∞1/n<sup>1/2</sup>·1/n<sup>1/3</sup>=sum from n=1 to ∞1/n<sup>5/6</sup>,5/6【1,发散。 因此,上述习题为伪命题
基金This work is jointly supported by the National -Natural Science Foundation of China ( Grant Nos. 49990450, 49925412, 40104008) and the National Key Basic Research Science Foundation (Grant No G2000078405) .
文摘By proposing a two-dimensional triggering model with concentrically circular closed magnetic field line structure,numerical research is made on the asymmetric propagation feature of coronal mass ejection (CME) in two cases emerging at the solar northern latitudes 10° and 45° respectively.The numerical results can qualitatively explain some features of CME event observed by the spacecraft SOHO and show that:(i) In these two cases,the triggering model can initiate CME with an asymmetric closed magnetic field structure.(ii) Closed magnetic structure of CME event will keep deflecting to the current sheet when it propagates away from the sun and this deflecting effect mostly happens within tens of solar radii before CME travels finally along the current sheet.(iii) The triggering model emerging at different locations can introduce CME events with different magnetic shapes.This shape happens to be circular and crescent when the triggering model emerges at the northern latitudes 10° and 45°,respectively.
