The level 3 case for Ramanujan-type series has been considered as the most mysterious and the most challenging,out of all possible levels for Ramanujan-type series.This motivates the development of new techniques for ...The level 3 case for Ramanujan-type series has been considered as the most mysterious and the most challenging,out of all possible levels for Ramanujan-type series.This motivates the development of new techniques for constructing Ramanujan-type series of level 3.Chan and Liaw introduced an alternating analogue of the Borwein brothers’identity for Ramanujan-type series of level 3;subsequently,Chan,Liaw,and Tian formulated another proof of the Chan–Liaw identity,via the use of Ramanujan’s class invariant.Using the elliptic lambda function and the elliptic alpha function,we prove,via a limiting case of the Kummer–Goursat transformation,a new identity for evaluating the summands for alternating Ramanujan-type series of level 3,and we apply this new identity to prove three conjectured formulas for quadratic-irrational,Ramanujan-type series that had been discovered via numerical experiments with Maple in 2012 by Aldawoud.We also apply our identity to prove a new Ramanujan-type series of level 3 with a quartic convergence rate and quartic coefficients.展开更多
In a recent study published in Cell by Shen et al.,changes in the cerebrospinal fluid(CSF)proteome in carriers of known mutations for autosomal dominant inherited Alzheimer’s disease(ADAD)led to the identification of...In a recent study published in Cell by Shen et al.,changes in the cerebrospinal fluid(CSF)proteome in carriers of known mutations for autosomal dominant inherited Alzheimer’s disease(ADAD)led to the identification of novel,early biomarkers.Based on the changes of six selected proteins a predictive model for disease onset and progression was developed:1 SPARC Related Modular Calcium Binding 1(SMOC1),SPARC Related Modular Calcium Binding 2(SMOC2),Neuronal Pentraxin 2(NPTX),Proliferation And Apoptosis Adaptor Protein 15(PEA15),Glial Fibrillary Acidic Protein(GFAP),and TNF Receptor Superfamily Member 1B(TNFRSF1B).展开更多
基金supported by a Killam Postdoctoral Fellowship from the Killam Trusts.
文摘The level 3 case for Ramanujan-type series has been considered as the most mysterious and the most challenging,out of all possible levels for Ramanujan-type series.This motivates the development of new techniques for constructing Ramanujan-type series of level 3.Chan and Liaw introduced an alternating analogue of the Borwein brothers’identity for Ramanujan-type series of level 3;subsequently,Chan,Liaw,and Tian formulated another proof of the Chan–Liaw identity,via the use of Ramanujan’s class invariant.Using the elliptic lambda function and the elliptic alpha function,we prove,via a limiting case of the Kummer–Goursat transformation,a new identity for evaluating the summands for alternating Ramanujan-type series of level 3,and we apply this new identity to prove three conjectured formulas for quadratic-irrational,Ramanujan-type series that had been discovered via numerical experiments with Maple in 2012 by Aldawoud.We also apply our identity to prove a new Ramanujan-type series of level 3 with a quartic convergence rate and quartic coefficients.
文摘In a recent study published in Cell by Shen et al.,changes in the cerebrospinal fluid(CSF)proteome in carriers of known mutations for autosomal dominant inherited Alzheimer’s disease(ADAD)led to the identification of novel,early biomarkers.Based on the changes of six selected proteins a predictive model for disease onset and progression was developed:1 SPARC Related Modular Calcium Binding 1(SMOC1),SPARC Related Modular Calcium Binding 2(SMOC2),Neuronal Pentraxin 2(NPTX),Proliferation And Apoptosis Adaptor Protein 15(PEA15),Glial Fibrillary Acidic Protein(GFAP),and TNF Receptor Superfamily Member 1B(TNFRSF1B).
