title: Complete sets of invariants for dynamical systems that admit a separation of variables

authors: Kalnins, Ernie G.

subjects: Mathematics

abstract: Consider a classical Hamiltonian H in n dimensions consisting of a kinetic energy term plus a potential. If the associated Hamilton–Jacobi equation admits an orthogonal separation of variables, then it is possible to generate algorithmically a canonical basis Q, P where P1 = H, P2, ,Pn are the other second-order constants of the motion associated with the separable coordinates, and {Qi,Qj} = {Pi,Pj} = 0, {Qi,Pj} =  ij. The 2n–1 functions Q2, ,Qn,P1, ,Pn form a basis for the invariants. We show how to determine for exactly which spaces and potentials the invariant Qj is a polynomial in the original momenta. We shed light on the general question of exactly when the Hamiltonian admits a constant of the motion that is polynomial in the momenta. For n = 2 we go further and consider all cases where the Hamilton–Jacobi equation admits a second-order constant of the motion, not necessarily associated with orthogonal separable coordinates, or even separable coordinates at all. In each of these cases we construct an additional constant of the motion.

text: Complete sets of invariants for dynamical systems that admit a separation of variables Consider a classical Hamiltonian H in n dimensions consisting of a kinetic energy term plus a potential. If the associated Hamilton–Jacobi equation admits an orthogonal separation of variables, then it is possible to generate algorithmically a canonical basis Q, P where P1 = H, P2, ,Pn are the other second-order constants of the motion associated with the separable coordinates, and {Qi,Qj} = {Pi,Pj} = 0, {Qi,Pj} =  ij. The 2n–1 functions Q2, ,Qn,P1, ,Pn form a basis for the invariants. We show how to determine for exactly which spaces and potentials the invariant Qj is a polynomial in the original momenta. We shed light on the general question of exactly when the Hamiltonian admits a constant of the motion that is polynomial in the momenta. For n = 2 we go further and consider all cases where the Hamilton–Jacobi equation admits a second-order constant of the motion, not necessarily associated with orthogonal separable coordinates, or even separable coordinates at all. In each of these cases we construct an additional constant of the motion.

year: 2002-07

title: A study of some aspects of the nitrogen economy of New Zealand grassland soils

authors: Steele, K.W.

abstract: Nitrogen deficiency is widespread and is a major limitation to grassland productivity in New Zealand, indicating inadequate nitrogen supply, losses from the nitrogen cycle, or a combination of both. The intensive grassland farming system on which New Zealand agriculture is based is largely dependent on symbiotic fixation of atmospheric nitrogen by clovers, and many estimates indicate that the amount of nitrogen fixed annually should be adequate to maintain a high level of pasture production. It appears, therefore, that losses of nitrogen be the more important factor contributing to nitrogen deficiency in New Zealand pastures. One transformation which will affect the conservation of nitrogen in soil, and which has received little attention in New Zealand, is nitrification. It was the objective of this thesis to investigate some aspects of nitrification in New Zealand grassland soils.

A technique which has recently been applied to studies on nitrogen transformations in soils involves the use of variations in the natural abundance of ¹⁵N. The possible use of this technique for studying nitrogen transformations in New Zealand soils was investigated. 

A perfusion technique suitable for the measurement of inherent oxidation activities of the indigenous nitrifying organisms in New Zealand grassland soils was developed. Soils were perfused with 0.005M (NH₄)₂SO₄ at 25°C, and the rate of oxidation of ammonium to nitrate measured over a sixteen hour period. Rates of nitrification measured over the initial sixteen hours of perfusion, referred to as the Initial Nitrification Activity (INA), ranged from <0.02 μg Noxidised/g soil/hour to 5.70 μg Noxidised/g soil/hour in sixty nine soils selected to provide a representative sample of agricultural soils from the major soil groups of New Zealand.

High rates of INA (>2 μg N oxidised/g soil/hour) were found in yellow-brown loams, some red and brown loams, and soils with a pH above 7.0. All other soils exhibited rates of nitrification <1 μg Noxidised/g soil/hour.

Differences in INA were found to predict the rates of nitrification occurring in the field in two soils using an in-situ incubation technique. The mean rate of nett nitrification measured over a forty seven week period in the 0 - 7.5 cm depth of Wharekohe silt loam (INA= 0.07 μg Noxidised/g soil/hour), was 0.68 x 10⁻² μg Noxidised/g soil/hour, and that in the 0 - 7. 5 cm depth of Kiripaka silt loam (INA = 0.81 μg Noxidised/g soil/hour) was 5.4 x 10⁻² μg Noxidised/g soil/hour. This difference resulted in a different ratio of ammonium to nitrate between the soils. In Wharekohe, the mean values of ammonium and nitrate nitrogen present in the 0 - 7.5 cm depth over the forty seven week experimental period were 15.7 ± 1.7 and 0.9 ± 0.4 kg N/ha respectively. In Kiripaka, the figures were 6.5 ± 1.3 and 6.6 ± 1.4 kg N/ha.

Rates of nett mineralisation of soil organic nitrogen were also determined in Wharekohe silt loam and Kiripaka silt loam by an in-situ incubation technique. In the former soil, the mean rate of nett mineralisation over a forty seven week period was 0.99 ± 0.08 kg N/ha (0 - 7.5 cm)/day, and in the latter soil 0.74 kg N/ha (0 - 7.5 cm)/day. 

When soils were perfused with 0.005M (NH₄)₂SO₄ for up to twenty days, four general patterns of nitrification were observed.
1) Ammonium was rapidly oxidised to nitrate, the rate of oxidation being linear or near linear from the commencement of the perfusion.
2) Ammonium was oxidised only slowly to nitrate.
3) Ammonium was oxidised slowly to nitrate at the commencement of perfusion, but increased logarithmically with time until a steady rate of nitrification was observed.
4) Type 3 nitrification with a temporary accumulation of nitrite during the initial stages of perfusion.

Type 1 nitrification was observed only in soils having a high INA, type 2 nitrification in soils of low pH (mean pH 5.5), and type 4 nitrification only in soils with a pH above 7.0, with the exception of two podzolic soils. Type 3 nitrification was observed in the majority of the soils studied.

Large populations of ammonium oxidisers (1.6- x 10⁴ to 2.2 x 10⁶/g soil) and nitrite oxidisers (5.4 x 10⁴ to 1.9 x 10⁷/g soil) were found in the 0 - 7. 5 cm depth in New Zealand soils under improved grassland. In contrast to these high numbers, a low population of nitrifying organisms (555 ammonium oxidisers and 1434 nitrite oxidisers/g soil) was found in a native forest soil. Populations of nitrifying organisms were largest close to the soil surface and, in general, declined with depth.

Generation times of ammonium oxidisers were estimated to be in the order of three to five days and were dependent on pH. A variation in the rate of ammonium oxidation per ammonium oxidising cell was found between soils, therefore INA could not be considered as indicative of the size of the nitrifying population. 

Multiple linear regression analysis showed both pH and percent total nitrogen to be significantly correlated with INA. A detailed study of the effect of pH on nitrification showed that soils containing allophane, an amorphous aluminium silicate, exhibited higher rates of nitrification at a given pH than other soils. Nitrification was also able to proceed at lower pH values in the former soils than in the latter. This difference may be partly rationalised by theoretical differences in surface pH of soil colloids.

A study was made of the fate of ¹⁵N enriched Ca(NO₃)₂ and (NH₄)₂SO₄ applied to two soils, one of medium (Waimate North clay loam) and the other of low (Wharekohe silt loam) INA. Recovery of Ca(NO₃)₂ and (NH₄)₂SO₄ nitrogen in pasture herbage was 47.4 and 35.8 percent on the former soil, and 20.3 and 42.5 percent on the latter respectively. After soil nitrogen was taken into account there was still up to 66.8 percent of the nitrogen applied not accounted for. It was concluded that leaching was the major mechanism of loss, although some evidence of losses by denitrification from the Wharekohe soil was found.

The differences in rates of nitrification measured in New Zealand grassland soils are of agronomic significance since the rate of nitrification determines the form of inorganic nitrogen available for plant uptake. Ammonium, and not nitrate, is the major form of nitrogen available for assimilation by plants in several yellow-brown earths, and also in some podzolic soils, yellow-brown sands, recent soils from alluvium and yellow-brown pumice soils. Although differences in rates of nitrification do not appear to affect pasture production, they will, of necessity have to be considered if specialised crops which show a preference for assimilation of either ammonium or nitrate ate grown on these soils.

Soils having a high rate of nitrification are associated with high concentrations of nitrate in groundwaters of the Waikato. 

A low rate of nitrification in soils such as Wharekohe silt loam appears to be an important mechanism for the conservation of nitrogen. Any agricultural practice which will increase the rate of nitrification should be carefully considered and the overall effect on the nitrogen economy of the soil evaluated. If increasing the rate of nitrification results in a lower nitrogen status, then increased inputs of nitrogen either by symbiotic nitrogen fixation or fertiliser application will be required to maintain the same level of pasture production. 

In general, it appears that the disadvantages of high rates of nitrification outweigh the advantages, and agricultural practices should therefore be designed to minimise the rate of nitrification in soils. 

The extent of nitrogen isotope discrimination in various reactions occurring in New Zealand soil-pasture systems was determined. Symbiotic fixation of atmospheric nitrogen by clover, assimilation of inorganic nitrogen by Trifolium repens, the oxidation of ammonium to nitrate by nitrifying organisms and the volatilisation of NH₃ from urine spots all showed discrimination in favour of ¹⁴N. A previously unmeasured fractionation in animals was determined. In cattle, urine was depleted by about 2%₀ and faeces enriched by about 2%₀ relative to the animal feed.

δ¹⁵N values were measured in the 0 - 7.5 cm depth of sixty one New Zealand grassland soils. The mean δ¹⁵N value was 3.2%₀ with a range of -1.1 to 6.8%₀. The δ¹⁵N values in two soil profiles showed that δ¹⁵N values increased with depth reaching a maximum value at 20 cm in Wharekohe silt loam and 50 cm in Waimate North clay loam. New Zealand soils appear to be less enriched in ¹⁵N than many American soils, and it is suggested that this reflects the large annual input of symbiotically fixed nitrogen which has a negative δ¹⁵N value, although the mechanism of loss (i.e. via denitrification, leaching or volatilisation of NH₃) of nitrogen from different soils may be important.

δ¹⁵N values were also determined for nitrate nitrogen in ground water collected from a limestone cave under native forest in the Waitomo region, and also for nitrate nitrogen in the groundwater of shallow aquifiers in the Waikato. 

The δ¹⁵N values for two groundwater samples from the Waitomo area were - 1.6 and 3.8%₀ while nine samples collected from the Waikato were in the range of 5.2 to 10.0%₀. It is suggested that the isotopic composition of groundwater nitrogen in the Waikato is a result of a major contribution of nitrogen derived from urine. This suggests that the grazing animal is an important contributor to the high nitrate concentration in groundwaters of the Waikato.

Because of the variability of δ¹⁵N values that were found in grazed soil-pasture systems, and the complexity of fractionations which occur in such systems, small variations in the natural abundance of ¹⁵N appear to be only of limited use for even qualitative studies of nitrogen transformations.

text: A study of some aspects of the nitrogen economy of New Zealand grassland soils Nitrogen deficiency is widespread and is a major limitation to grassland productivity in New Zealand, indicating inadequate nitrogen supply, losses from the nitrogen cycle, or a combination of both. The intensive grassland farming system on which New Zealand agriculture is based is largely dependent on symbiotic fixation of atmospheric nitrogen by clovers, and many estimates indicate that the amount of nitrogen fixed annually should be adequate to maintain a high level of pasture production. It appears, therefore, that losses of nitrogen be the more important factor contributing to nitrogen deficiency in New Zealand pastures. One transformation which will affect the conservation of nitrogen in soil, and which has received little attention in New Zealand, is nitrification. It was the objective of this thesis to investigate some aspects of nitrification in New Zealand grassland soils.

A technique which has recently been applied to studies on nitrogen transformations in soils involves the use of variations in the natural abundance of ¹⁵N. The possible use of this technique for studying nitrogen transformations in New Zealand soils was investigated. 

A perfusion technique suitable for the measurement of inherent oxidation activities of the indigenous nitrifying organisms in New Zealand grassland soils was developed. Soils were perfused with 0.005M (NH₄)₂SO₄ at 25°C, and the rate of oxidation of ammonium to nitrate measured over a sixteen hour period. Rates of nitrification measured over the initial sixteen hours of perfusion, referred to as the Initial Nitrification Activity (INA), ranged from <0.02 μg Noxidised/g soil/hour to 5.70 μg Noxidised/g soil/hour in sixty nine soils selected to provide a representative sample of agricultural soils from the major soil groups of New Zealand.

High rates of INA (>2 μg N oxidised/g soil/hour) were found in yellow-brown loams, some red and brown loams, and soils with a pH above 7.0. All other soils exhibited rates of nitrification <1 μg Noxidised/g soil/hour.

Differences in INA were found to predict the rates of nitrification occurring in the field in two soils using an in-situ incubation technique. The mean rate of nett nitrification measured over a forty seven week period in the 0 - 7.5 cm depth of Wharekohe silt loam (INA= 0.07 μg Noxidised/g soil/hour), was 0.68 x 10⁻² μg Noxidised/g soil/hour, and that in the 0 - 7. 5 cm depth of Kiripaka silt loam (INA = 0.81 μg Noxidised/g soil/hour) was 5.4 x 10⁻² μg Noxidised/g soil/hour. This difference resulted in a different ratio of ammonium to nitrate between the soils. In Wharekohe, the mean values of ammonium and nitrate nitrogen present in the 0 - 7.5 cm depth over the forty seven week experimental period were 15.7 ± 1.7 and 0.9 ± 0.4 kg N/ha respectively. In Kiripaka, the figures were 6.5 ± 1.3 and 6.6 ± 1.4 kg N/ha.

Rates of nett mineralisation of soil organic nitrogen were also determined in Wharekohe silt loam and Kiripaka silt loam by an in-situ incubation technique. In the former soil, the mean rate of nett mineralisation over a forty seven week period was 0.99 ± 0.08 kg N/ha (0 - 7.5 cm)/day, and in the latter soil 0.74 kg N/ha (0 - 7.5 cm)/day. 

When soils were perfused with 0.005M (NH₄)₂SO₄ for up to twenty days, four general patterns of nitrification were observed.
1) Ammonium was rapidly oxidised to nitrate, the rate of oxidation being linear or near linear from the commencement of the perfusion.
2) Ammonium was oxidised only slowly to nitrate.
3) Ammonium was oxidised slowly to nitrate at the commencement of perfusion, but increased logarithmically with time until a steady rate of nitrification was observed.
4) Type 3 nitrification with a temporary accumulation of nitrite during the initial stages of perfusion.

Type 1 nitrification was observed only in soils having a high INA, type 2 nitrification in soils of low pH (mean pH 5.5), and type 4 nitrification only in soils with a pH above 7.0, with the exception of two podzolic soils. Type 3 nitrification was observed in the majority of the soils studied.

Large populations of ammonium oxidisers (1.6- x 10⁴ to 2.2 x 10⁶/g soil) and nitrite oxidisers (5.4 x 10⁴ to 1.9 x 10⁷/g soil) were found in the 0 - 7. 5 cm depth in New Zealand soils under improved grassland. In contrast to these high numbers, a low population of nitrifying organisms (555 ammonium oxidisers and 1434 nitrite oxidisers/g soil) was found in a native forest soil. Populations of nitrifying organisms were largest close to the soil surface and, in general, declined with depth.

Generation times of ammonium oxidisers were estimated to be in the order of three to five days and were dependent on pH. A variation in the rate of ammonium oxidation per ammonium oxidising cell was found between soils, therefore INA could not be considered as indicative of the size of the nitrifying population. 

Multiple linear regression analysis showed both pH and percent total nitrogen to be significantly correlated with INA. A detailed study of the effect of pH on nitrification showed that soils containing allophane, an amorphous aluminium silicate, exhibited higher rates of nitrification at a given pH than other soils. Nitrification was also able to proceed at lower pH values in the former soils than in the latter. This difference may be partly rationalised by theoretical differences in surface pH of soil colloids.

A study was made of the fate of ¹⁵N enriched Ca(NO₃)₂ and (NH₄)₂SO₄ applied to two soils, one of medium (Waimate North clay loam) and the other of low (Wharekohe silt loam) INA. Recovery of Ca(NO₃)₂ and (NH₄)₂SO₄ nitrogen in pasture herbage was 47.4 and 35.8 percent on the former soil, and 20.3 and 42.5 percent on the latter respectively. After soil nitrogen was taken into account there was still up to 66.8 percent of the nitrogen applied not accounted for. It was concluded that leaching was the major mechanism of loss, although some evidence of losses by denitrification from the Wharekohe soil was found.

The differences in rates of nitrification measured in New Zealand grassland soils are of agronomic significance since the rate of nitrification determines the form of inorganic nitrogen available for plant uptake. Ammonium, and not nitrate, is the major form of nitrogen available for assimilation by plants in several yellow-brown earths, and also in some podzolic soils, yellow-brown sands, recent soils from alluvium and yellow-brown pumice soils. Although differences in rates of nitrification do not appear to affect pasture production, they will, of necessity have to be considered if specialised crops which show a preference for assimilation of either ammonium or nitrate ate grown on these soils.

Soils having a high rate of nitrification are associated with high concentrations of nitrate in groundwaters of the Waikato. 

A low rate of nitrification in soils such as Wharekohe silt loam appears to be an important mechanism for the conservation of nitrogen. Any agricultural practice which will increase the rate of nitrification should be carefully considered and the overall effect on the nitrogen economy of the soil evaluated. If increasing the rate of nitrification results in a lower nitrogen status, then increased inputs of nitrogen either by symbiotic nitrogen fixation or fertiliser application will be required to maintain the same level of pasture production. 

In general, it appears that the disadvantages of high rates of nitrification outweigh the advantages, and agricultural practices should therefore be designed to minimise the rate of nitrification in soils. 

The extent of nitrogen isotope discrimination in various reactions occurring in New Zealand soil-pasture systems was determined. Symbiotic fixation of atmospheric nitrogen by clover, assimilation of inorganic nitrogen by Trifolium repens, the oxidation of ammonium to nitrate by nitrifying organisms and the volatilisation of NH₃ from urine spots all showed discrimination in favour of ¹⁴N. A previously unmeasured fractionation in animals was determined. In cattle, urine was depleted by about 2%₀ and faeces enriched by about 2%₀ relative to the animal feed.

δ¹⁵N values were measured in the 0 - 7.5 cm depth of sixty one New Zealand grassland soils. The mean δ¹⁵N value was 3.2%₀ with a range of -1.1 to 6.8%₀. The δ¹⁵N values in two soil profiles showed that δ¹⁵N values increased with depth reaching a maximum value at 20 cm in Wharekohe silt loam and 50 cm in Waimate North clay loam. New Zealand soils appear to be less enriched in ¹⁵N than many American soils, and it is suggested that this reflects the large annual input of symbiotically fixed nitrogen which has a negative δ¹⁵N value, although the mechanism of loss (i.e. via denitrification, leaching or volatilisation of NH₃) of nitrogen from different soils may be important.

δ¹⁵N values were also determined for nitrate nitrogen in ground water collected from a limestone cave under native forest in the Waitomo region, and also for nitrate nitrogen in the groundwater of shallow aquifiers in the Waikato. 

The δ¹⁵N values for two groundwater samples from the Waitomo area were - 1.6 and 3.8%₀ while nine samples collected from the Waikato were in the range of 5.2 to 10.0%₀. It is suggested that the isotopic composition of groundwater nitrogen in the Waikato is a result of a major contribution of nitrogen derived from urine. This suggests that the grazing animal is an important contributor to the high nitrate concentration in groundwaters of the Waikato.

Because of the variability of δ¹⁵N values that were found in grazed soil-pasture systems, and the complexity of fractionations which occur in such systems, small variations in the natural abundance of ¹⁵N appear to be only of limited use for even qualitative studies of nitrogen transformations.

year: 1977

title: Systemic racism in clinical psychology: Symposia

authors: Wairoa-Harrison, Sophia

subjects: Racial oppression

abstract: This study constitutes a part of the Working to End Racial Oppression (WERO) project on Systemic Racism in Health Education, Training, and Practice. The focus is on the three dimensions of racism in psychology in Aotearoa New Zealand: its costs, systems and the potential responses that exist. Content: To set the scene, the session will begin with a 30-minute presentation that outlines evidence from three WERO studies on different levels of racism affecting indigenous Māori and minoritised groups. First study presents findings from a desktop analysis of papers across university psychology departments. The findings will be compared with previous studies (Levy, 2007; Levy & Waitoki, 2015) to identify the changes in inclusion of Māori and culturally focused paper at different timepoints. The second study involves semi-structured interviews with Māori students enrolled in clinical psychology training programmes (n = 19) to understand the extent of cultural responsiveness, support, disparities, and representation within programmes. The third study draws on preliminary survey findings with directors of training programmes to depict the progress towards including taha Māori and cultural competency in the psychology curriculum since the studies by Abbott and Durie (1987) and Nathan (1999). The rest of the session will be conducted as a wānanga or a forum that includes a panel discussion amongst WERO research team while allowing for a reciprocal exchange of knowledge with the attendees. Opportunities will be provided for attendees to ask questions, provide feedback to the research analyses and findings, and share their experiences of witnessing or encountering racism in psychology. The session will be opened with a karakia (prayer) and attendees to be offered manaakitanga (hospitality) and tikanga (custom) to ensure cultural safety. Goals: In this session, the research team will bring the evidence derived from robust analyses on existence and ramifications of racism in psychology to the fore. These findings will be integrated with the Waitangi Tribunal claim (Levy, 2018) to shed light on the manifestations of racism across systems and agents responsible for the regulation, training and employment of psychologists. Findings will be used to inform ways for all in psychology to embark on Te Tiriti-focused, culturally-competent and equitable approaches in the quest of indigenising the psychology discipline and advancing it to be relevant to the multicultural nature of Aotearoa populations.

text: Systemic racism in clinical psychology: Symposia This study constitutes a part of the Working to End Racial Oppression (WERO) project on Systemic Racism in Health Education, Training, and Practice. The focus is on the three dimensions of racism in psychology in Aotearoa New Zealand: its costs, systems and the potential responses that exist. Content: To set the scene, the session will begin with a 30-minute presentation that outlines evidence from three WERO studies on different levels of racism affecting indigenous Māori and minoritised groups. First study presents findings from a desktop analysis of papers across university psychology departments. The findings will be compared with previous studies (Levy, 2007; Levy & Waitoki, 2015) to identify the changes in inclusion of Māori and culturally focused paper at different timepoints. The second study involves semi-structured interviews with Māori students enrolled in clinical psychology training programmes (n = 19) to understand the extent of cultural responsiveness, support, disparities, and representation within programmes. The third study draws on preliminary survey findings with directors of training programmes to depict the progress towards including taha Māori and cultural competency in the psychology curriculum since the studies by Abbott and Durie (1987) and Nathan (1999). The rest of the session will be conducted as a wānanga or a forum that includes a panel discussion amongst WERO research team while allowing for a reciprocal exchange of knowledge with the attendees. Opportunities will be provided for attendees to ask questions, provide feedback to the research analyses and findings, and share their experiences of witnessing or encountering racism in psychology. The session will be opened with a karakia (prayer) and attendees to be offered manaakitanga (hospitality) and tikanga (custom) to ensure cultural safety. Goals: In this session, the research team will bring the evidence derived from robust analyses on existence and ramifications of racism in psychology to the fore. These findings will be integrated with the Waitangi Tribunal claim (Levy, 2018) to shed light on the manifestations of racism across systems and agents responsible for the regulation, training and employment of psychologists. Findings will be used to inform ways for all in psychology to embark on Te Tiriti-focused, culturally-competent and equitable approaches in the quest of indigenising the psychology discipline and advancing it to be relevant to the multicultural nature of Aotearoa populations.

year: 2023-03-25