title: The Weibull distribution as an extreme value model for transformed annual maxima

authors: Bardsley, W. Earl

subjects: Weibull distribution

abstract: There is no mathematical difference between theoretical extreme value models of maxima and minima because they link to each other by a simple sign reversal transformation. However, other transformations that change sample maxima to sample minima raise the possibility of alternatives to the generalised extreme value (GEV) distribution for annual maxima, while still maintaining extreme value justification. A general class of transformation is proposed that converts positive annual maxima to lower-bounded minima, then amenable to Weibull extreme value analysis for sufficiently large sample sizes. That is, the Weibull distribution of smallest extremes provides a theoretical extreme value model for the transformed maxima, which holds irrespective of any GEV form of the original maxima. This would apply, for example, to the analysis of reciprocals of discharge or rainfall annual maxima.
A useful feature of the transformation approach is that alternative prediction expressions with extreme value justification can arise for defining event magnitude as a function of return period. These expressions may result in different hydrological conclusions than from a GEV analysis of maxima. A double exponential transformation is introduced and its prediction function for maxima is noted to have capability to mimic equivalent Type 3 extreme value (EV3) distribution expressions, but without having to introduce an upper bound parameter. The new function gives a good fit to apparent EV3 annual flood maxima recorded from two very different catchments: the Yangtze River in China and the upper Whanganui River in New Zealand.

text: The Weibull distribution as an extreme value model for transformed annual maxima There is no mathematical difference between theoretical extreme value models of maxima and minima because they link to each other by a simple sign reversal transformation. However, other transformations that change sample maxima to sample minima raise the possibility of alternatives to the generalised extreme value (GEV) distribution for annual maxima, while still maintaining extreme value justification. A general class of transformation is proposed that converts positive annual maxima to lower-bounded minima, then amenable to Weibull extreme value analysis for sufficiently large sample sizes. That is, the Weibull distribution of smallest extremes provides a theoretical extreme value model for the transformed maxima, which holds irrespective of any GEV form of the original maxima. This would apply, for example, to the analysis of reciprocals of discharge or rainfall annual maxima.
A useful feature of the transformation approach is that alternative prediction expressions with extreme value justification can arise for defining event magnitude as a function of return period. These expressions may result in different hydrological conclusions than from a GEV analysis of maxima. A double exponential transformation is introduced and its prediction function for maxima is noted to have capability to mimic equivalent Type 3 extreme value (EV3) distribution expressions, but without having to introduce an upper bound parameter. The new function gives a good fit to apparent EV3 annual flood maxima recorded from two very different catchments: the Yangtze River in China and the upper Whanganui River in New Zealand.

year: 2019

title: Iwasawa theory over solvable three-dimensional p-adic Lie extensions

authors: Qin, Chao

subjects: Iwasawa theory

abstract: Iwasawa theory is a powerful tool which describes the mysterious relationship between arithmetic objects (motives) and the special values of L-functions. A precise form of this relationship is neatly encoded in the so-called "Iwasawa Main Conjecture". Classically the Main Conjecture (as formulated by Iwasawa himself) involved the behaviour of ideal class groups over cyclotomic Zp-extensions, and related this to the Kubota-Leopoldt p-adic zeta-function. During the last two decades, the main conjecture has been greatly generalized to admissible p-adic Lie extensions, and provides a conjectural relationship between L-values of motives and their associated Selmer groups. A key component of the “Non-commutative Iwasawa Main Conjecture” in [CFK+05] predicts the existence of an analytic p-adic L-function L an M inside K1 ( Zp[[G∞]]S∗ ) . To establish the existence of such an object, we need to be able to do two things: (1) describe K1 ( Zp[[G∞]]S∗ ) in terms of the Artin representations factoring through G∞ using p-adic congruences, and then (2) show for each motive that the abelian fragments satisfy these congruences. This thesis provides a complete answer to the first task (1), in the specific situation where the pro-p-group G∞ has dimension ≤ 3 and is torsion-free. We completely describe K1(Zp[[G∞]]) and its localisations by using an infinite family of p-adic congruences, where G∞ is any solvable p-adic Lie group of dimension 3. This builds on earlier work of Kato when dim(G∞) = 2, and of Daniel Delbourgo and Lloyd Peters when G∞ ∼= Z × p ⋉Zd p with a scalar action of Z × p . The method exploits the classification of 3-dimensional p-adic Lie groups due to González-Sánchez and Klopsch, as well as the fundamental ideas of Kakde, Burns, etc. in non-commutative Iwasawa theory. We also undertake a short study of elliptic curves over GL2(Fp)-extensions, and compile some numerical evidence in support of the first layer congruences predicted by Kakde [Kak17] for non-CM curves.

text: Iwasawa theory over solvable three-dimensional p-adic Lie extensions Iwasawa theory is a powerful tool which describes the mysterious relationship between arithmetic objects (motives) and the special values of L-functions. A precise form of this relationship is neatly encoded in the so-called "Iwasawa Main Conjecture". Classically the Main Conjecture (as formulated by Iwasawa himself) involved the behaviour of ideal class groups over cyclotomic Zp-extensions, and related this to the Kubota-Leopoldt p-adic zeta-function. During the last two decades, the main conjecture has been greatly generalized to admissible p-adic Lie extensions, and provides a conjectural relationship between L-values of motives and their associated Selmer groups. A key component of the “Non-commutative Iwasawa Main Conjecture” in [CFK+05] predicts the existence of an analytic p-adic L-function L an M inside K1 ( Zp[[G∞]]S∗ ) . To establish the existence of such an object, we need to be able to do two things: (1) describe K1 ( Zp[[G∞]]S∗ ) in terms of the Artin representations factoring through G∞ using p-adic congruences, and then (2) show for each motive that the abelian fragments satisfy these congruences. This thesis provides a complete answer to the first task (1), in the specific situation where the pro-p-group G∞ has dimension ≤ 3 and is torsion-free. We completely describe K1(Zp[[G∞]]) and its localisations by using an infinite family of p-adic congruences, where G∞ is any solvable p-adic Lie group of dimension 3. This builds on earlier work of Kato when dim(G∞) = 2, and of Daniel Delbourgo and Lloyd Peters when G∞ ∼= Z × p ⋉Zd p with a scalar action of Z × p . The method exploits the classification of 3-dimensional p-adic Lie groups due to González-Sánchez and Klopsch, as well as the fundamental ideas of Kakde, Burns, etc. in non-commutative Iwasawa theory. We also undertake a short study of elliptic curves over GL2(Fp)-extensions, and compile some numerical evidence in support of the first layer congruences predicted by Kakde [Kak17] for non-CM curves.

year: 2018

title: Introduction to special section on Māori culture and education.

authors: Bishop, Russell

subjects: language policy

abstract: An introduction is presented in which the editor discusses various reports within the issue on topics including the need for coordinated language policy, the Treaty of Waitangi, and monoculturalism.

text: Introduction to special section on Māori culture and education. An introduction is presented in which the editor discusses various reports within the issue on topics including the need for coordinated language policy, the Treaty of Waitangi, and monoculturalism.

year: 2002