title: A standardised sampling protocol for robust assessment of reach-scale fish community diversity in wadeable New Zealand streams

authors: David, Bruno O.

subjects: standard methods

abstract: The New Zealand fish fauna contains species that are affected not only by river system connectivity, but also by catchment and local-scale changes in landcover, water quality and habitat quality. Consequently, native fish have potential as multi-scale bioindicators of human pressure on stream ecosystems, yet no standardised, repeatable and scientifically defensible methods currently exist for effectively quantifying their abundance or diversity in New Zealand stream reaches. Here we report on the testing of a back-pack electrofishing method, modified from that used by the United States Environmental Protection Agency, on a wide variety of wadeable stream reaches throughout New Zealand. Seventy-three first- to third-order stream reaches were fished with a single pass over 150-345 m length. Time taken to sample a reach using single-pass electrofishing ranged from 1-8 h. Species accumulation curves indicated that, irrespective of location, continuous sampling of 150 stream metres is required to accurately describe reach-scale fish species richness using this approach. Additional species detection beyond 150 m was rare (<10%) with a single additional species detected at only two out of the 17 reaches sampled beyond this distance. A positive relationship was also evident between species detection and area fished, although stream length rather than area appeared to be the better predictor. The method tested provides a standardised and repeatable approach for regional and/or national reporting on the state of New Zealand's freshwater fish communities and trends in richness and abundance over time.

text: A standardised sampling protocol for robust assessment of reach-scale fish community diversity in wadeable New Zealand streams The New Zealand fish fauna contains species that are affected not only by river system connectivity, but also by catchment and local-scale changes in landcover, water quality and habitat quality. Consequently, native fish have potential as multi-scale bioindicators of human pressure on stream ecosystems, yet no standardised, repeatable and scientifically defensible methods currently exist for effectively quantifying their abundance or diversity in New Zealand stream reaches. Here we report on the testing of a back-pack electrofishing method, modified from that used by the United States Environmental Protection Agency, on a wide variety of wadeable stream reaches throughout New Zealand. Seventy-three first- to third-order stream reaches were fished with a single pass over 150-345 m length. Time taken to sample a reach using single-pass electrofishing ranged from 1-8 h. Species accumulation curves indicated that, irrespective of location, continuous sampling of 150 stream metres is required to accurately describe reach-scale fish species richness using this approach. Additional species detection beyond 150 m was rare (<10%) with a single additional species detected at only two out of the 17 reaches sampled beyond this distance. A positive relationship was also evident between species detection and area fished, although stream length rather than area appeared to be the better predictor. The method tested provides a standardised and repeatable approach for regional and/or national reporting on the state of New Zealand's freshwater fish communities and trends in richness and abundance over time.

year: 2010

title: L-invariants and congruences for Galois representations of dimension 3, 4, and 8

authors: Gilmore, Hamish Julian

abstract: The arithmetic of Galois representations plays a central role in modern number theory. In this thesis we consider representations arising from tensor products of the two-dimensional representations attached to modular forms by Deligne. In particular, we shall study the Iwasawa theory of the adjoint representation, as well as certain double and triple products of Deligne's representations.

In the first half we will undertake a computational study of L-invariants attached to symmetric squares of modular forms. Let ƒ be a primitive modular form of weight 𝘬 and level 𝘕, and 𝘱 ∤ 𝘕 a prime greater than two for which the attached representation is ordinary. The 𝘱-adic 𝘓-function Sym²ƒ 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
	
	\frac{\text{\rm d}\;}{\text{\rm d}s}\mathbf{L}_p(\text{\rm Sym}^2f,s)
	\Bigg|_{s=k-1}=
	\mathcal{L}_p(\text{\rm Sym}^2f) \times 
	 (1-\alpha_p^{-2}p^{k-2})(1-\alpha_p^{-2}p^{k-1}) \times
	\frac{L_{\infty}(\text{\rm Sym}^2f,k-1)}{\pi^{k-1}\langle f,f\rangle_N}
	
where X²-aₚ(ƒ)𝘟 + p = (𝘟-αₚ)(𝘟-βₚ) with αₚ ∈ Zˣₚ.

Now let 𝘌 be an elliptic curve over Q with associated modular form ƒᴇ, and 𝘱 ≠ 2 a prime of good ordinary reduction. We devise a method to calculate Lₚ(Sym²ƒᴇ) effectively, then show it is non-trivial for almost all pairs of elliptic curves 𝘌 of conductor 𝘕ᴇ ≤ 300, with 4|𝘕ᴇ, and ordinary primes 𝘱 < 17. Hence, in these cases at least, the order of the zero in Lₚ(Sym²ƒᴇ, 𝘴) at 𝘴 = 1 is exactly one. We also generalise this method to compute symmetric square L-invariants for modular forms of weight 𝘬 > 2.

In the second half we will establish congruences between 𝘱-adic 𝘓-functions. In the late 1990s, Vatsal showed that a congruence modulo 𝘱 ᵛ between two newforms implied a congruence between their respective 𝘱-adic 𝘓-functions. We shall prove an analogous statement for both the double product and triple product 𝘱-adic 𝘓-functions, Lₚ(f⊗g) and Lₚ (f⊗g⊗h): the former is cyclotomic in its nature, while the latter is over the weight space.

As a corollary, we derive transition formulae relating analytic  λ-invariants for pairs of congruent Galois representations for 𝘝𝒻 ⊗ 𝘝𝓰, and for 𝘝𝒻 ⊗ 𝘝𝓰⊗ 𝘝ₕ.

text: L-invariants and congruences for Galois representations of dimension 3, 4, and 8 The arithmetic of Galois representations plays a central role in modern number theory. In this thesis we consider representations arising from tensor products of the two-dimensional representations attached to modular forms by Deligne. In particular, we shall study the Iwasawa theory of the adjoint representation, as well as certain double and triple products of Deligne's representations.

In the first half we will undertake a computational study of L-invariants attached to symmetric squares of modular forms. Let ƒ be a primitive modular form of weight 𝘬 and level 𝘕, and 𝘱 ∤ 𝘕 a prime greater than two for which the attached representation is ordinary. The 𝘱-adic 𝘓-function Sym²ƒ 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
	
	\frac{\text{\rm d}\;}{\text{\rm d}s}\mathbf{L}_p(\text{\rm Sym}^2f,s)
	\Bigg|_{s=k-1}=
	\mathcal{L}_p(\text{\rm Sym}^2f) \times 
	 (1-\alpha_p^{-2}p^{k-2})(1-\alpha_p^{-2}p^{k-1}) \times
	\frac{L_{\infty}(\text{\rm Sym}^2f,k-1)}{\pi^{k-1}\langle f,f\rangle_N}
	
where X²-aₚ(ƒ)𝘟 + p = (𝘟-αₚ)(𝘟-βₚ) with αₚ ∈ Zˣₚ.

Now let 𝘌 be an elliptic curve over Q with associated modular form ƒᴇ, and 𝘱 ≠ 2 a prime of good ordinary reduction. We devise a method to calculate Lₚ(Sym²ƒᴇ) effectively, then show it is non-trivial for almost all pairs of elliptic curves 𝘌 of conductor 𝘕ᴇ ≤ 300, with 4|𝘕ᴇ, and ordinary primes 𝘱 < 17. Hence, in these cases at least, the order of the zero in Lₚ(Sym²ƒᴇ, 𝘴) at 𝘴 = 1 is exactly one. We also generalise this method to compute symmetric square L-invariants for modular forms of weight 𝘬 > 2.

In the second half we will establish congruences between 𝘱-adic 𝘓-functions. In the late 1990s, Vatsal showed that a congruence modulo 𝘱 ᵛ between two newforms implied a congruence between their respective 𝘱-adic 𝘓-functions. We shall prove an analogous statement for both the double product and triple product 𝘱-adic 𝘓-functions, Lₚ(f⊗g) and Lₚ (f⊗g⊗h): the former is cyclotomic in its nature, while the latter is over the weight space.

As a corollary, we derive transition formulae relating analytic  λ-invariants for pairs of congruent Galois representations for 𝘝𝒻 ⊗ 𝘝𝓰, and for 𝘝𝒻 ⊗ 𝘝𝓰⊗ 𝘝ₕ.

year: 2020

title: Consequences of cultural satisfaction at work: A study of New Zealand Māori

authors: Haar, Jarrod M.

subjects: cultural satisfaction at work

abstract: To expand our understanding of indigenous workers and the importance of indigenous culture in the workplace, we tested the outcomes of cultural satisfaction at work using structural equation modelling with a sample of 174 Māori employees. We show that, consistent with social exchange theory, Māori who are more satisfied with the level of understanding of their cultural values in the workplace report better job outcomes. Cultural satisfaction at work directly predicted loyalty and organisational citizenship behaviours (OCBs). Furthermore, direct-effects and mediation models were tested, showing that loyalty fully mediated the influence of cultural satisfaction at work on OCBs. The implications for HRM are that indigenous workers who are more satisfied with the way their cultural beliefs are valued in the workplace are likely to be more loyal and may be superior performance. The findings highlight the importance for employers of having regard to the culture of their indigenous workers.

text: Consequences of cultural satisfaction at work: A study of New Zealand Māori To expand our understanding of indigenous workers and the importance of indigenous culture in the workplace, we tested the outcomes of cultural satisfaction at work using structural equation modelling with a sample of 174 Māori employees. We show that, consistent with social exchange theory, Māori who are more satisfied with the level of understanding of their cultural values in the workplace report better job outcomes. Cultural satisfaction at work directly predicted loyalty and organisational citizenship behaviours (OCBs). Furthermore, direct-effects and mediation models were tested, showing that loyalty fully mediated the influence of cultural satisfaction at work on OCBs. The implications for HRM are that indigenous workers who are more satisfied with the way their cultural beliefs are valued in the workplace are likely to be more loyal and may be superior performance. The findings highlight the importance for employers of having regard to the culture of their indigenous workers.

year: 2011