A generalized form of the error function, Gp(x)=pΓ(1/p)∫0xe−tpdt, which is directly associated with the gamma function, is evaluated for arbitrary real values of p>1and 0x≤+∞by employing a fast-converging power...A generalized form of the error function, Gp(x)=pΓ(1/p)∫0xe−tpdt, which is directly associated with the gamma function, is evaluated for arbitrary real values of p>1and 0x≤+∞by employing a fast-converging power series expansion developed in resolving the so-called Grandi’s paradox. Comparisons with accurate tabulated values for well-known cases such as the error function are presented using the expansions truncated at various orders.展开更多
Grandi’s paradox, which was posed for a real function of the form <span style="white-space:nowrap;">1/(1+ <em>x</em>)</span>, has been resolved and extended to complex valued functio...Grandi’s paradox, which was posed for a real function of the form <span style="white-space:nowrap;">1/(1+ <em>x</em>)</span>, has been resolved and extended to complex valued functions. Resolution of this approximately three-hundred-year-old paradox is accomplished by the use of a consistent truncation approach that can be applied to all the series expansions of Grandi-type functions. Furthermore, a new technique for improving the convergence characteristics of power series with alternating signs is introduced. The technique works by successively averaging a series at different orders of truncation. A sound theoretical justification of the successive averaging method is demonstrated by two different series expansions of the function <span style="white-space:nowrap;">1/(1+ e<sup><em>x</em> </sup>)</span> . Grandi-type complex valued functions such as <span style="white-space:nowrap;">1/(<em>i</em> + <em>x</em>)</span> are expressed as consistently-truncated and convergence-improved forms and Fagnano’s formula is established from the series expansions of these functions. A Grandi-type general complex valued function <img src="Edit_f4efd7cd-6853-4ca4-b4dc-00f0b798c277.png" width="80" height="24" alt="" /> is introduced and expanded to a consistently truncated and successively averaged series. Finally, an unorthodox application of the successive averaging method to polynomials is presented.展开更多
On October,30th,as an important constituent part of Mercedes-Benz China Fashion Week 2018 Spring and Summer Series,Beijing Fashion Forum·Buyer Summit was held in DHUB Activity Area,79 Tank Zone,751D·PARK in ...On October,30th,as an important constituent part of Mercedes-Benz China Fashion Week 2018 Spring and Summer Series,Beijing Fashion Forum·Buyer Summit was held in DHUB Activity Area,79 Tank Zone,751D·PARK in Beijing.The organizers of the fashion industry,experts of fashion market,fashion buyers,principals of fashion industrial parks,展开更多
Two new tirucallane-type triterpenoids, 3a,24β＇,25-trihydroxy-21,21-dimethoxy-23-oxo-tirucall-7-ene （I） and 3a-acetoxy-21fl-methoxy-24,25,26,27-tetranortirucall-7-ene-23（21）-lactone （2）, were isolated from the...Two new tirucallane-type triterpenoids, 3a,24β＇,25-trihydroxy-21,21-dimethoxy-23-oxo-tirucall-7-ene （I） and 3a-acetoxy-21fl-methoxy-24,25,26,27-tetranortirucall-7-ene-23（21）-lactone （2）, were isolated from the stem barks ofAphanamixis grandifblia, and their structures were elucidated on the basis of spectroscopic data analysis includ- ing ID, 2D NMR, IR, and ESI-MS spectral methods. The two isolates were evaluated for their cytotoxicities using NCI-H460 and HeLa cell lines.展开更多
文摘A generalized form of the error function, Gp(x)=pΓ(1/p)∫0xe−tpdt, which is directly associated with the gamma function, is evaluated for arbitrary real values of p>1and 0x≤+∞by employing a fast-converging power series expansion developed in resolving the so-called Grandi’s paradox. Comparisons with accurate tabulated values for well-known cases such as the error function are presented using the expansions truncated at various orders.
文摘Grandi’s paradox, which was posed for a real function of the form <span style="white-space:nowrap;">1/(1+ <em>x</em>)</span>, has been resolved and extended to complex valued functions. Resolution of this approximately three-hundred-year-old paradox is accomplished by the use of a consistent truncation approach that can be applied to all the series expansions of Grandi-type functions. Furthermore, a new technique for improving the convergence characteristics of power series with alternating signs is introduced. The technique works by successively averaging a series at different orders of truncation. A sound theoretical justification of the successive averaging method is demonstrated by two different series expansions of the function <span style="white-space:nowrap;">1/(1+ e<sup><em>x</em> </sup>)</span> . Grandi-type complex valued functions such as <span style="white-space:nowrap;">1/(<em>i</em> + <em>x</em>)</span> are expressed as consistently-truncated and convergence-improved forms and Fagnano’s formula is established from the series expansions of these functions. A Grandi-type general complex valued function <img src="Edit_f4efd7cd-6853-4ca4-b4dc-00f0b798c277.png" width="80" height="24" alt="" /> is introduced and expanded to a consistently truncated and successively averaged series. Finally, an unorthodox application of the successive averaging method to polynomials is presented.
文摘On October,30th,as an important constituent part of Mercedes-Benz China Fashion Week 2018 Spring and Summer Series,Beijing Fashion Forum·Buyer Summit was held in DHUB Activity Area,79 Tank Zone,751D·PARK in Beijing.The organizers of the fashion industry,experts of fashion market,fashion buyers,principals of fashion industrial parks,
文摘Two new tirucallane-type triterpenoids, 3a,24β＇,25-trihydroxy-21,21-dimethoxy-23-oxo-tirucall-7-ene （I） and 3a-acetoxy-21fl-methoxy-24,25,26,27-tetranortirucall-7-ene-23（21）-lactone （2）, were isolated from the stem barks ofAphanamixis grandifblia, and their structures were elucidated on the basis of spectroscopic data analysis includ- ing ID, 2D NMR, IR, and ESI-MS spectral methods. The two isolates were evaluated for their cytotoxicities using NCI-H460 and HeLa cell lines.
