title: On λ-invariants attached to cyclic cubic number fields

authors: Delbourgo, Daniel

abstract: We describe an algorithm for finding the coefficients of F(X) modulo powers of p, where p ≠2 is a prime number and F(X) is the power series associated to the zeta function of Kubota and Leopoldt. We next calculate the 5-adic and 7-adic λ-invariants attached to those cubic extensions K/Q with cyclic Galois group A₃ (up to field discriminant <10⁷), and also tabulate the class number of K(e2πi/p) for p=5 and p=7. If the λ-invariant is greater than zero, we then determine all the zeros for the corresponding branches of the p-adic L-function and deduce Λ-monogeneity for the class group tower over the cyclotomic Zp-extension of K.

text: On λ-invariants attached to cyclic cubic number fields We describe an algorithm for finding the coefficients of F(X) modulo powers of p, where p ≠2 is a prime number and F(X) is the power series associated to the zeta function of Kubota and Leopoldt. We next calculate the 5-adic and 7-adic λ-invariants attached to those cubic extensions K/Q with cyclic Galois group A₃ (up to field discriminant <10⁷), and also tabulate the class number of K(e2πi/p) for p=5 and p=7. If the λ-invariant is greater than zero, we then determine all the zeros for the corresponding branches of the p-adic L-function and deduce Λ-monogeneity for the class group tower over the cyclotomic Zp-extension of K.

year: 2015

title: Misconceptions arising from the Infinite Solenoid Magnetic Field Formula

authors: Wilson, Marcus T.

abstract: Many high school and first-year university courses include discussion of the magnetic effect of currents. Frequently discussed textbook examples include long, straight wires, circular current loops, and solenoids, partly because these examples are tractable mathematically. The solenoid naturally leads to discussion on magnetic materials since it is readily demonstrated that a paramagnetic core significantly boosts the strength of an electromagnet. However, magnetic effects of solid and even liquid materials are subtle and confusing¹ and the mathematics is not straightforward. This leads to confusion amongst students (and their teachers), which, when taken to more advanced study, leads to significant misconceptions about the nature of magnetic properties and fields. These misconceptions can become problematic when practical (rather than stereotyped) magnetic design and analysis is required such as for transformers,² magnetic recording materials, geomagnetic sensors,³ or biological stimulators4 to name a few. In this article, I highlight examples of this confusion, in particular the failure in realistic situations of the well-quoted formula for an infinite solenoid with a paramagnetic core, and the physical interpretation of the relative permeability of a material, µr.

text: Misconceptions arising from the Infinite Solenoid Magnetic Field Formula Many high school and first-year university courses include discussion of the magnetic effect of currents. Frequently discussed textbook examples include long, straight wires, circular current loops, and solenoids, partly because these examples are tractable mathematically. The solenoid naturally leads to discussion on magnetic materials since it is readily demonstrated that a paramagnetic core significantly boosts the strength of an electromagnet. However, magnetic effects of solid and even liquid materials are subtle and confusing¹ and the mathematics is not straightforward. This leads to confusion amongst students (and their teachers), which, when taken to more advanced study, leads to significant misconceptions about the nature of magnetic properties and fields. These misconceptions can become problematic when practical (rather than stereotyped) magnetic design and analysis is required such as for transformers,² magnetic recording materials, geomagnetic sensors,³ or biological stimulators4 to name a few. In this article, I highlight examples of this confusion, in particular the failure in realistic situations of the well-quoted formula for an infinite solenoid with a paramagnetic core, and the physical interpretation of the relative permeability of a material, µr.

year: 2021

title: Creating a curriculum for indigenous and community languages: Te reo Māori as an example

authors: Bruce, Ian

subjects: linguistics

abstract: Part 1 of this paper introduces the draft Common European Framework of Reference (CEFR) for languages and describes how it was used in the construction of draft New Zealand curriculum guidelines for French and German. Part 2 discusses how the CEFR has been employed in curriculum development in Te Reo Māori at the University of Waikato and suggests that it could also prove useful in relation to curriculum development projects involving indigenous and community languages more generally.

text: Creating a curriculum for indigenous and community languages: Te reo Māori as an example Part 1 of this paper introduces the draft Common European Framework of Reference (CEFR) for languages and describes how it was used in the construction of draft New Zealand curriculum guidelines for French and German. Part 2 discusses how the CEFR has been employed in curriculum development in Te Reo Māori at the University of Waikato and suggests that it could also prove useful in relation to curriculum development projects involving indigenous and community languages more generally.

year: 2002