The selection of the truncation level （TL） and the control of boundary effect （BE） are critical in regional geomagnetic field models that are based on data fitting. We combine Taylor and Legendre polynomials to mo...The selection of the truncation level （TL） and the control of boundary effect （BE） are critical in regional geomagnetic field models that are based on data fitting. We combine Taylor and Legendre polynomials to model geomagnetic data over China's Mainland for years 1960, 1970, 1990, and 2000. To tackle the TL and BE problems, we first determine the range of TL by calculating the root-mean-square error （RMSE） of the models. Next, we determine the optimum TL using the Akaike information criterion （AIC） and the normalized root- mean-square error （NRMSE）. We use the regional anomaly addition （RAA） and the uniform addition （UA） method to add supplementary point outside the national boundary, and find that the intensities of extreme points gradually decrease and stabilize. The UA method better controls BEs over China, whereas the RAA method does a better job at smaller scales. In summary, we rely on a three-step method to determine the optimum TL and propose criteria to determine the optimum number of supplementary points.展开更多
We investigate Bell polynomials, also called Touchard polynomials or exponential polynomials, by using and without using umbral calculus. We use three different formulas in order to express various known families of p...We investigate Bell polynomials, also called Touchard polynomials or exponential polynomials, by using and without using umbral calculus. We use three different formulas in order to express various known families of polynomials such as Bernoulli polynomials, poly-Bernoulli polynomials, Cauchy polynomials and falling factorials in terms of Bell polynomials and vice versa. In addition, we derive several properties of Bell polynomials along the way.展开更多
基金supported by the National Natural Science Foundation of China(Grant Nos.41404053 and 41174165)Special Project for Meteo-Scientific Research in the Public Interest(Grant No.GYHY201306073)Natural Science Foundation of Higher Education Institutions of Jiangsu Province,China(Grant No.14KJB170012)
文摘The selection of the truncation level （TL） and the control of boundary effect （BE） are critical in regional geomagnetic field models that are based on data fitting. We combine Taylor and Legendre polynomials to model geomagnetic data over China's Mainland for years 1960, 1970, 1990, and 2000. To tackle the TL and BE problems, we first determine the range of TL by calculating the root-mean-square error （RMSE） of the models. Next, we determine the optimum TL using the Akaike information criterion （AIC） and the normalized root- mean-square error （NRMSE）. We use the regional anomaly addition （RAA） and the uniform addition （UA） method to add supplementary point outside the national boundary, and find that the intensities of extreme points gradually decrease and stabilize. The UA method better controls BEs over China, whereas the RAA method does a better job at smaller scales. In summary, we rely on a three-step method to determine the optimum TL and propose criteria to determine the optimum number of supplementary points.
文摘We investigate Bell polynomials, also called Touchard polynomials or exponential polynomials, by using and without using umbral calculus. We use three different formulas in order to express various known families of polynomials such as Bernoulli polynomials, poly-Bernoulli polynomials, Cauchy polynomials and falling factorials in terms of Bell polynomials and vice versa. In addition, we derive several properties of Bell polynomials along the way.
