title: Determination of the rank of an integration lattice

authors: Joe, Stephen

subjects: mathematics

abstract: The continuing and widespread use of lattice rules for high-dimensional numerical quadrature is driving the development of a rich and detailed theory. Part of this theory is devoted to computer searches for rules, appropriate to particular situations. 
In some applications, one is interested in obtaining the (lattice) rank of a lattice rule Q(Λ) directly from the elements of a generator matrix B (possibly in upper triangular lattice form) of the corresponding dual lattice Λ⊥. We treat this problem in detail, demonstrating the connections between this (lattice) rank and the conventional matrix rank deficiency of modulo p versions of B.

text: Determination of the rank of an integration lattice The continuing and widespread use of lattice rules for high-dimensional numerical quadrature is driving the development of a rich and detailed theory. Part of this theory is devoted to computer searches for rules, appropriate to particular situations. 
In some applications, one is interested in obtaining the (lattice) rank of a lattice rule Q(Λ) directly from the elements of a generator matrix B (possibly in upper triangular lattice form) of the corresponding dual lattice Λ⊥. We treat this problem in detail, demonstrating the connections between this (lattice) rank and the conventional matrix rank deficiency of modulo p versions of B.

year: 2006

title: Phytoplankton dynamics in North Island lakes, New Zealand

authors: Ryan, E.F.

subjects: Freshwater phytoplankton New Zealand Waikato

abstract: The dynamics of phytoplankton were investigated in 40 lakes of the North Island, New Zealand. A range of environmental variables was examined on different temporal and spatial scales in relation to the surface and DCM ( deep chlorophyll maximum) phytoplankton communities in the lakes. The environmental variables that were considered relevant to these communities included turbulent mixing, limitation by light and nutrients, rates of sedimentation or ascent, and zooplankton grazing. 
Cylindrospermopsis raciborskii is a potentially toxic species of cyanobacteria, that in summer 2002-3 formed dense water blooms (> 100 OOO cells/ml) in shallow (max. depth :5 Sm) Waikato peat lakes and a hydro-electric dam on the Waikato River. Net rates of growth calculated within individual lakes demonstrated that C. raciborskii populations grew at rates close to maximum values observed under optimal conditions in laboratory cultures. Changes in species diversity in the lakes prior to and following the invasion of C. raciborskii were assessed. During occasions of C. raciborskii water blooms there was decreased diversity and the phytoplankton assemblage became nearly mono-specific. The shallow lakes in this study have high concentrations of nutrients and low water clarity, which are ideal conditions for growth of C. raciborskii, as it has a competitive advantage at low irradiance but limited capacity for buoyancy. The ability of C. raciborskii to fix nitrogen may also have enhanced its competitive ability amongst other phytoplankton species and, while nutrients were generally considered to be replete, nitrogen was more likely to limit growth of phytoplankton than phosphorus in the lakes studied. 
Lake Tarawera, part of the Rotorua lakes region (area 41.6 km2 and mean depth 50 m), was chosen to investigate variability of phytoplankton biomass with depth and between embayments and a mid-lake station. The variability of DCM depth was contributed by internal seiching, which was most prominent at the edge of the lake. A single mid-lake station was considered representative of lake-wide biomass at the DCM as the differences between stations were small. However, during a bloom of Anabaena lemmermanni, there was large inter-station variability in phytoplankton biomass in the surface mixed layer. Therefore, when quantifying surface populations, particularly buoyant cyanobacterial species which tend to accumulate in bays, discrete samples at a single station and depth may not adequately represent lake-wide biomass. During periods when bloom-forming species are present, several sampling stations should be used to adequately quantify biomass and variability of phytoplankton in the surface mixed layer. 
A population-dynamics theory of sinking phytoplankton, that utilizes interactions between growth limitation by light and nutrients, sinking rates and turbulent diffusion rates, was used to quantify factors contributing to DCM formation in Lake Tarawera. Five 'Modes' of vertical phytoplankton of biomass were categorised relating to four forcing functions. The first mode was a surface chlorophyll maximum (SCM); Mode 2 was a DCM; Mode 3 was a simultaneous double peak of buoyant cyanobacteria and a DCM assemblage; Mode 4 was a linear vertical distribution and Mode 5 was an exponential increase in biomass with depth. The incidence of these distributions and the existence of a DCM were examined in relation to different turbulent diffusion rates{< 0.1 to 10 cm2 s-1 ), sinking rates {+10 to -4 m daf1), and light and nutrient limitation. A DCM was simulated when net growth of the population superceded losses from turbulent diffusion, sinking and zooplankton grazing. Light climate strongly influenced net rates of growth of the DCM assemblage and its position; surface blooms of cyanobacteria could obliterate the DCM by shading cells and suppressing light available for their growth. This finding has important implications for eutrophication of lakes with DCMs, as relatively small changes in surface phytoplankton communities, for example through additional nutrient inputs, could obliterate the DCM. The relevance of the diatom DCM phytoplankton is that it has important ecosystem-level effects through sedimentation of nutrients and organic matter, and transfer of biomass to higher trophic levels resulting from preferential grazing. 
Further examination of DCM communities was carried out in Lake Tikitapu, a small lake (mean depth 18 m, area 1.5 km2 ) in the Rotorua basin. The DCM assemblage in this lake comprised dinoflagellate species. The DCM persisted over the period of thermal stratification, in the metalimnion, and also at a depth where irradiance was approximately 2-3 % of the surface value. Statistical analysis found that the light level at the DCM was more important than the depth of thermocline in determining the depth of the DCM. There was no evidence of diurnal vertical migration of the DCM in Lake Tikitapu, but dispersion of the DCM varied seasonally, whereby the width was strongly positively correlated to the width of the metalimnion (r2 = 0.89, p < 0.01). Light and nutrient incubation experiments provided further evidence for the dominant role of light in influencing the DCM position. There was a biomass decrease in DCM populations that were incubated at high irradiance ( 40 % of surface irradiance ), suggesting that surface irradiances may be damaging or inhibitory to growth. 
A wide range of North Island lakes was investigated in an attempt to delineate the roles of lake mixing, trophic state and light climate in the composition of lake phytoplankton assemblages. For 40 lakes mixing regimes were divided into three classes; stable seasonal stratification, intermittent stratification and mixed, and trophic states were separated in oligotrophic, mesotrophic and eutrophic. In addition, average photosynthetically available radiation experienced by phytoplankton was determined based on integrated levels over the surface mixed layer, and values were divided into three regimes where 5 % or less corresponded to low light, 5 - 20 % to medium light, and > 20 % to high light. Non-metric multi-dimensional scaling (MDS) and analysis of similarities (ANOSIM) were used to detect patterns in phytoplankton composition between lakes and to infer which environmental variables were associated with underlying trends in composition. MDS showed a clear separation of stratified from intermittently stratified and mixed lakes based on phytoplankton composition, and phytoplankton assemblages in stably stratified lakes were significantly different from those in intermittently stratified lakes (p < 0.05) and mixed lakes (p < 0.05). However, results of ANOSIM indicated that trophic state was not a statistically significant factor influencing phytoplankton assemblage composition. Mixing regime therefore provided a more accurate predictor of summer phytoplankton composition than trophic state or light climate, as lakes of different trophic state but similar mixing regime had similar phytoplankton assemblages.
In this study, field work and modelling were used to contribute to the current understanding of spatial and temporal dynamics of phytoplankton in lakes, the roles and interactions of various environmental variables in governing populations, and implications of blooms on assemblage diversity. Phytoplankton assemblage composition and biomass are highly variable, but much of this variation may be explained through detailed analysis. Investigations into the vertical distributions of phytoplankton communities to ascertain the presence of a SCM or DCM, along with quantification of vertical stratification/mixing regimes in lakes, and interactions of buoyancy with horizontal transport in the surface mixed layer, will greatly facilitate in explaining the variability.

text: Phytoplankton dynamics in North Island lakes, New Zealand The dynamics of phytoplankton were investigated in 40 lakes of the North Island, New Zealand. A range of environmental variables was examined on different temporal and spatial scales in relation to the surface and DCM ( deep chlorophyll maximum) phytoplankton communities in the lakes. The environmental variables that were considered relevant to these communities included turbulent mixing, limitation by light and nutrients, rates of sedimentation or ascent, and zooplankton grazing. 
Cylindrospermopsis raciborskii is a potentially toxic species of cyanobacteria, that in summer 2002-3 formed dense water blooms (> 100 OOO cells/ml) in shallow (max. depth :5 Sm) Waikato peat lakes and a hydro-electric dam on the Waikato River. Net rates of growth calculated within individual lakes demonstrated that C. raciborskii populations grew at rates close to maximum values observed under optimal conditions in laboratory cultures. Changes in species diversity in the lakes prior to and following the invasion of C. raciborskii were assessed. During occasions of C. raciborskii water blooms there was decreased diversity and the phytoplankton assemblage became nearly mono-specific. The shallow lakes in this study have high concentrations of nutrients and low water clarity, which are ideal conditions for growth of C. raciborskii, as it has a competitive advantage at low irradiance but limited capacity for buoyancy. The ability of C. raciborskii to fix nitrogen may also have enhanced its competitive ability amongst other phytoplankton species and, while nutrients were generally considered to be replete, nitrogen was more likely to limit growth of phytoplankton than phosphorus in the lakes studied. 
Lake Tarawera, part of the Rotorua lakes region (area 41.6 km2 and mean depth 50 m), was chosen to investigate variability of phytoplankton biomass with depth and between embayments and a mid-lake station. The variability of DCM depth was contributed by internal seiching, which was most prominent at the edge of the lake. A single mid-lake station was considered representative of lake-wide biomass at the DCM as the differences between stations were small. However, during a bloom of Anabaena lemmermanni, there was large inter-station variability in phytoplankton biomass in the surface mixed layer. Therefore, when quantifying surface populations, particularly buoyant cyanobacterial species which tend to accumulate in bays, discrete samples at a single station and depth may not adequately represent lake-wide biomass. During periods when bloom-forming species are present, several sampling stations should be used to adequately quantify biomass and variability of phytoplankton in the surface mixed layer. 
A population-dynamics theory of sinking phytoplankton, that utilizes interactions between growth limitation by light and nutrients, sinking rates and turbulent diffusion rates, was used to quantify factors contributing to DCM formation in Lake Tarawera. Five 'Modes' of vertical phytoplankton of biomass were categorised relating to four forcing functions. The first mode was a surface chlorophyll maximum (SCM); Mode 2 was a DCM; Mode 3 was a simultaneous double peak of buoyant cyanobacteria and a DCM assemblage; Mode 4 was a linear vertical distribution and Mode 5 was an exponential increase in biomass with depth. The incidence of these distributions and the existence of a DCM were examined in relation to different turbulent diffusion rates{< 0.1 to 10 cm2 s-1 ), sinking rates {+10 to -4 m daf1), and light and nutrient limitation. A DCM was simulated when net growth of the population superceded losses from turbulent diffusion, sinking and zooplankton grazing. Light climate strongly influenced net rates of growth of the DCM assemblage and its position; surface blooms of cyanobacteria could obliterate the DCM by shading cells and suppressing light available for their growth. This finding has important implications for eutrophication of lakes with DCMs, as relatively small changes in surface phytoplankton communities, for example through additional nutrient inputs, could obliterate the DCM. The relevance of the diatom DCM phytoplankton is that it has important ecosystem-level effects through sedimentation of nutrients and organic matter, and transfer of biomass to higher trophic levels resulting from preferential grazing. 
Further examination of DCM communities was carried out in Lake Tikitapu, a small lake (mean depth 18 m, area 1.5 km2 ) in the Rotorua basin. The DCM assemblage in this lake comprised dinoflagellate species. The DCM persisted over the period of thermal stratification, in the metalimnion, and also at a depth where irradiance was approximately 2-3 % of the surface value. Statistical analysis found that the light level at the DCM was more important than the depth of thermocline in determining the depth of the DCM. There was no evidence of diurnal vertical migration of the DCM in Lake Tikitapu, but dispersion of the DCM varied seasonally, whereby the width was strongly positively correlated to the width of the metalimnion (r2 = 0.89, p < 0.01). Light and nutrient incubation experiments provided further evidence for the dominant role of light in influencing the DCM position. There was a biomass decrease in DCM populations that were incubated at high irradiance ( 40 % of surface irradiance ), suggesting that surface irradiances may be damaging or inhibitory to growth. 
A wide range of North Island lakes was investigated in an attempt to delineate the roles of lake mixing, trophic state and light climate in the composition of lake phytoplankton assemblages. For 40 lakes mixing regimes were divided into three classes; stable seasonal stratification, intermittent stratification and mixed, and trophic states were separated in oligotrophic, mesotrophic and eutrophic. In addition, average photosynthetically available radiation experienced by phytoplankton was determined based on integrated levels over the surface mixed layer, and values were divided into three regimes where 5 % or less corresponded to low light, 5 - 20 % to medium light, and > 20 % to high light. Non-metric multi-dimensional scaling (MDS) and analysis of similarities (ANOSIM) were used to detect patterns in phytoplankton composition between lakes and to infer which environmental variables were associated with underlying trends in composition. MDS showed a clear separation of stratified from intermittently stratified and mixed lakes based on phytoplankton composition, and phytoplankton assemblages in stably stratified lakes were significantly different from those in intermittently stratified lakes (p < 0.05) and mixed lakes (p < 0.05). However, results of ANOSIM indicated that trophic state was not a statistically significant factor influencing phytoplankton assemblage composition. Mixing regime therefore provided a more accurate predictor of summer phytoplankton composition than trophic state or light climate, as lakes of different trophic state but similar mixing regime had similar phytoplankton assemblages.
In this study, field work and modelling were used to contribute to the current understanding of spatial and temporal dynamics of phytoplankton in lakes, the roles and interactions of various environmental variables in governing populations, and implications of blooms on assemblage diversity. Phytoplankton assemblage composition and biomass are highly variable, but much of this variation may be explained through detailed analysis. Investigations into the vertical distributions of phytoplankton communities to ascertain the presence of a SCM or DCM, along with quantification of vertical stratification/mixing regimes in lakes, and interactions of buoyancy with horizontal transport in the surface mixed layer, will greatly facilitate in explaining the variability.

year: 2006

title: Legal responses to mental health: Is therapeutic jurisprudence the answer – the experience in New Zealand

authors: Toki, Valmaine

abstract: The orthodox understanding of ‘ethics’ encompasses a system of moral principles that define good and bad behavior. When applied to mental health, various jurisdictions, including New Zealand, have codified these principles together with the obligations on health service providers to meet these codified principles and standards. Underpinning this relationship is one of mutual trust where the physician has a responsibility, for example, to secure informed consent prior to any treatment. The existence of this therapeutic relationship can be applied when an offender who presents with a mental illness is subject to the criminal justice system.

text: Legal responses to mental health: Is therapeutic jurisprudence the answer – the experience in New Zealand The orthodox understanding of ‘ethics’ encompasses a system of moral principles that define good and bad behavior. When applied to mental health, various jurisdictions, including New Zealand, have codified these principles together with the obligations on health service providers to meet these codified principles and standards. Underpinning this relationship is one of mutual trust where the physician has a responsibility, for example, to secure informed consent prior to any treatment. The existence of this therapeutic relationship can be applied when an offender who presents with a mental illness is subject to the criminal justice system.

year: 2017