title: Computing L-Invariants for the Symmetric Square of an Elliptic Curve

authors: Delbourgo, Daniel

subjects: Science & Technology

abstract: Let E be an elliptic curve over Q, and p≠2 a prime of good ordinary reduction. The p-adic L-function for Sym²E always vanishes at s = 1, even though the complex L-function does not have a zero there. The L-invariant itself appears on the right-hand side of the formula
ddsLp(Sym²E,s)∣∣∣s=1=Lp(Sym²E)×(1−α⁻²ᵖ)(1−pα⁻²ᵖ)×L∞(Sym²E,1)(2πi)⁻¹Ω+EΩ-E
where X²−aᵖ(E)X+p=(X−αᵖ)(X−βᵖ) with αp∈Z×ᵖ. We first devise a method to calculate Lp(Sym²E) effectively, then show it is non-trivial for all elliptic curves E of conductor NE≤300 with 4|NE, and almost all ordinary primes p < 17. Hence, in these cases at least, the order of the zero in Lp(Sym²E,s) at s = 1 is exactly one.

text: Computing L-Invariants for the Symmetric Square of an Elliptic Curve Let E be an elliptic curve over Q, and p≠2 a prime of good ordinary reduction. The p-adic L-function for Sym²E always vanishes at s = 1, even though the complex L-function does not have a zero there. The L-invariant itself appears on the right-hand side of the formula
ddsLp(Sym²E,s)∣∣∣s=1=Lp(Sym²E)×(1−α⁻²ᵖ)(1−pα⁻²ᵖ)×L∞(Sym²E,1)(2πi)⁻¹Ω+EΩ-E
where X²−aᵖ(E)X+p=(X−αᵖ)(X−βᵖ) with αp∈Z×ᵖ. We first devise a method to calculate Lp(Sym²E) effectively, then show it is non-trivial for all elliptic curves E of conductor NE≤300 with 4|NE, and almost all ordinary primes p < 17. Hence, in these cases at least, the order of the zero in Lp(Sym²E,s) at s = 1 is exactly one.

year: 2019

title: Reliability-based design optimisation of structural systems using high-order analytical moments

authors: Rajan, Arvind

subjects: Science & Technology

abstract: Reliability-based design optimisation paradigm has become increasingly popular to achieve economical yet safer structural designs. However, within the iterative optimisation procedure, it is a challenging problem to simultaneously satisfy both, accuracy and computational efficiency of reliability analysis, particularly for nonlinear or large design problems. Addressing the shortcomings of the mainstream reliability estimation methods, this paper presents a new reliability-based design optimisation method that combines an analytical high-order moment-based uncertainty evaluation with an efficient response surface modelling. The proposed framework allows for the precise calculation of reliability through accurate high-order moments. The fast and accurate reliability estimation in combination with efficient sampling, in turn, reduces the total number of finite element analysis in achieving the final design. The proposed moment-based design optimisation methodology was tested on problems ranging from the design of a simple cantilever to a three-dimensional multistorey steel structure. It outperforms the mainstream methods with higher accuracy and lower computational burden especially when applied to a highly nonlinear numerical problem.

text: Reliability-based design optimisation of structural systems using high-order analytical moments Reliability-based design optimisation paradigm has become increasingly popular to achieve economical yet safer structural designs. However, within the iterative optimisation procedure, it is a challenging problem to simultaneously satisfy both, accuracy and computational efficiency of reliability analysis, particularly for nonlinear or large design problems. Addressing the shortcomings of the mainstream reliability estimation methods, this paper presents a new reliability-based design optimisation method that combines an analytical high-order moment-based uncertainty evaluation with an efficient response surface modelling. The proposed framework allows for the precise calculation of reliability through accurate high-order moments. The fast and accurate reliability estimation in combination with efficient sampling, in turn, reduces the total number of finite element analysis in achieving the final design. The proposed moment-based design optimisation methodology was tested on problems ranging from the design of a simple cantilever to a three-dimensional multistorey steel structure. It outperforms the mainstream methods with higher accuracy and lower computational burden especially when applied to a highly nonlinear numerical problem.

year: 2020

title: A late-Holocene and prehistoric record of environmental change from Lake Waikaremoana, New Zealand

authors: Newnham, Rewi M.

subjects: Palynology

abstract: Further evidence in support of a late pre-European (Polynesian) settlement of New Zealand is provided by an 1850-year-long tephropalynological record from a remote region in New Zealand's North Island. The earliest unequivocally anthropogenic forest clearance is estimated from sedimentation rates to have occurred c. 375 14C years BP (c. ad 1523–1631), although the radiocarbon chronology, shown by tephrochron ology to be erroneous due to hard-water effects, suggested this occurred c. 900 years earlier. Delineation of the anthropogenic era, and the distinction between human activity and other agents of environmental change in the pollen/spore diagram, are supported by cluster analysis and detrended correspondence analysis. Two distinct phases of forest clearance are evident during the pre-European era, reflecting local changes either in population pressure or settlement patterns. We note that the lull between the two phases of forest clearance coincides with the maximum of the ‘Little Ice Age’ within the period c. late ad 1600s to early 1800s.

text: A late-Holocene and prehistoric record of environmental change from Lake Waikaremoana, New Zealand Further evidence in support of a late pre-European (Polynesian) settlement of New Zealand is provided by an 1850-year-long tephropalynological record from a remote region in New Zealand's North Island. The earliest unequivocally anthropogenic forest clearance is estimated from sedimentation rates to have occurred c. 375 14C years BP (c. ad 1523–1631), although the radiocarbon chronology, shown by tephrochron ology to be erroneous due to hard-water effects, suggested this occurred c. 900 years earlier. Delineation of the anthropogenic era, and the distinction between human activity and other agents of environmental change in the pollen/spore diagram, are supported by cluster analysis and detrended correspondence analysis. Two distinct phases of forest clearance are evident during the pre-European era, reflecting local changes either in population pressure or settlement patterns. We note that the lull between the two phases of forest clearance coincides with the maximum of the ‘Little Ice Age’ within the period c. late ad 1600s to early 1800s.

year: 1998