title: Lie theory and separation of variables. 3. The equation ftt−fss =γ2f

authors: Kalnins, Ernie G.

subjects: Mathematics

abstract: Kalnins has related the 11 coordinate systems in which variables separate in the equation ftt−fss = γ 2f to 11 symmetric quadratic operators L in the enveloping algebra of the Lie algebra of the pseudo-Euclidean group in the plane E(1,1). There are, up to equivalence, only 12 such operators and one of them, LE, is not associated with a separation of variables. Corresponding to each faithful unitary irreducible representation of E(1,1) we compute the spectral resolution and matrix elements in an L basis for seven cases of interest and also give overlap functions between different bases: Of the remaining five operators three are related to Mathieu functions and two are related to exponential solutions corresponding to Cartesian type coordinates. We then use these results to derive addition and expansion theorems for special solutions of ftt−fss = 2f obtained via separation of variables, e.g., products of Bessel, Macdonald and Bessel, Airy and parabolic cylinder functions. The exceptional operator LE is also treated in detail.

text: Lie theory and separation of variables. 3. The equation ftt−fss =γ2f Kalnins has related the 11 coordinate systems in which variables separate in the equation ftt−fss = γ 2f to 11 symmetric quadratic operators L in the enveloping algebra of the Lie algebra of the pseudo-Euclidean group in the plane E(1,1). There are, up to equivalence, only 12 such operators and one of them, LE, is not associated with a separation of variables. Corresponding to each faithful unitary irreducible representation of E(1,1) we compute the spectral resolution and matrix elements in an L basis for seven cases of interest and also give overlap functions between different bases: Of the remaining five operators three are related to Mathieu functions and two are related to exponential solutions corresponding to Cartesian type coordinates. We then use these results to derive addition and expansion theorems for special solutions of ftt−fss = 2f obtained via separation of variables, e.g., products of Bessel, Macdonald and Bessel, Airy and parabolic cylinder functions. The exceptional operator LE is also treated in detail.

year: 1974-07

title: Stratigraphy, chronology, and correlation of the Plio-Pleistocene (c. 2.2-0.8 Ma) Kauroa Ash sequence, western central North Island, New Zealand

authors: Horrocks, Joanna L.

abstract: The Kauroa Ash beds (K-beds) comprise a 12-20 m-thick sequence of extremely weathered, clay-rich (40-95% <4 μm clay) beds of tephra and loess, and associated paleosols. Found in isolated remnants throughout the western central North Island, the sequence comprises 15 defined members, with as many as 44 constituent macroscopic beds. The type site, ‘Woodstock’, near Raglan, is the most comprehensive sequence known, but other sites (e.g. Papakura Creek and Tiritirimatangi Peninsula) contain units not found or poorly defined at Woodstock. 

Field properties as well as magnetic susceptibility measurements and particle-size analysis characterise the facies in the sequence. Field properties (in particular colour, consistence, macrofabric) describe the lithostratigraphy. The sequence contains five interpretive (i.e. genetic) ‘facies’: paleosols, primary tephra, very weathered tephra (possibly composite beds), loess and ‘tephric loess’ beds. At least seven loess beds are (newly) identified in the sequence: K4a, K5, K6ai, K8ai, K8bi, K10a and K14ai.

Mass-specific susceptibility and frequency-dependent susceptibility results partly conform to established models (developed mostly on Chinese loess-paleosol deposits) of susceptibility enhancement in paleosols and depletion in loess. Many parts of the sequence do not appear to conform to this model and the results more closely resemble the inverse relationship found on Alaskan loess-paleosol beds. Frequency-dependent susceptibility is reliable in delineating paleosols by their ‘ultrafine’ ferrimagnetic mineral content, and citrate-bicarbonate-dithionite treatments successfully remove all iron oxides so that remeasurement of susceptibility isolates a strictly ‘pedogenic’, rather than lithogenic, fraction. 

Laser diffraction particle-size analysis shows that the Kauroa Ash beds are texturally reasonably homogenous. They have bimodal particle-size distributions with the most dominant mode at around 11.25 ɸ inferred to be the product of intense and prolonged weathering. Other modes are variously centred on 7-8.5 ɸ and, despite weathering and pedogenesis, have some relationship to the original depositional particle-size distributions because trends between facies (i.e. genetic units) are delineated. Principal components analysis objectively characterises these modes as (Wentworth size classes) ‘very fine clay’ and ‘coarse silt’, although there is no proportional relationship between them, supporting a post-depositional origin for the former mode. 

The chronology of the sequence, previously poorly defined, is greatly improved by a combination of tephrochronologic correlations, fission-track dating, and paleomagnetism. Five zircon fission-track dates provide independent age ‘spikes’ and range from 2.24 ± 0.29 Ma in the basal member, K1, to 1.28 ± 0.11 Ma for the distal ignimbrite unit K12a. Paleomagnetism is invaluable in providing additional age information. The top of the sequence, member K15, is dated as >0.78 Ma (Brunhes-Matuyama boundary) because of its reversed polarity; two episodes of normal polarity are found in beds K14b and K2b and are inferred to represent the Jaramillo (1.07-0.99 Ma) and Olduvai (1.95-1.79 Ma) subchrons, respectively. Beds underlying the Kauroa Ash sequence are also of normal polarity, indicating that they were deposited in the Gauss Chron (>2.6 Ma). 

Identification and correlation of tephras by conventional means (fingerprinting by their lithological or geochemical properties) is impossible in the Kauroa Ash sequence because the beds have no remaining volcanic glass, which has instead been altered to an assemblage of authigenic phases (clays) by weathering and pedogenesis. However, a new technique analysing fresh glass found as melt inclusions in quartz grains is successful in circumventing this problem. Inclusions represent samples of non-degassed magma that became entrapped during phenocryst growth prior to eruption. The glass has remained unaltered because it is hermetically sealed in a chemically resistant phenocryst, which has protected it from weathering processes. Electron microprobe analysis of the glass inclusions yield results which are wholly reasonable for glass (totals ranging from 93-97%; low standard deviations of <1 %), and a number of provisional correlations are established by comparing the major element composition of Kauroa Ash tephra beds with those of proximal deposits. The Kauroa Ash sequence may contain deposits correlated with at least seven major TVZ eruptions, in many cases expanding the known extent of the (distal) deposit and, for the first time answering the question as to the origin of the Kauroa Ash beds. 

These correlations, together with an improved chronology, enable the Kauroa Ash sequence to be placed in a regional stratigraphic framework alongside other New Zealand Plio-Pleistocene sequences such as those in the Wanganui Basin, Wairarapa, Cape Kidnappers and Port Waikato. Using paleosols as chrono- and climatostratigraphic entities (correlated to warm periods in global climate), the sequence can also be placed alongside a global reference, the marine oxygen isotope stratigraphy. A further correlation to the Chinese loess-paleosol record suggests that large parts of the Kauroa Ash sequence were deposited in an incremental manner akin to deposition of loess, so that the sequence is not only a record of TVZ volcanism, but also of Plio-Pleistocene paleoclimate.

text: Stratigraphy, chronology, and correlation of the Plio-Pleistocene (c. 2.2-0.8 Ma) Kauroa Ash sequence, western central North Island, New Zealand The Kauroa Ash beds (K-beds) comprise a 12-20 m-thick sequence of extremely weathered, clay-rich (40-95% <4 μm clay) beds of tephra and loess, and associated paleosols. Found in isolated remnants throughout the western central North Island, the sequence comprises 15 defined members, with as many as 44 constituent macroscopic beds. The type site, ‘Woodstock’, near Raglan, is the most comprehensive sequence known, but other sites (e.g. Papakura Creek and Tiritirimatangi Peninsula) contain units not found or poorly defined at Woodstock. 

Field properties as well as magnetic susceptibility measurements and particle-size analysis characterise the facies in the sequence. Field properties (in particular colour, consistence, macrofabric) describe the lithostratigraphy. The sequence contains five interpretive (i.e. genetic) ‘facies’: paleosols, primary tephra, very weathered tephra (possibly composite beds), loess and ‘tephric loess’ beds. At least seven loess beds are (newly) identified in the sequence: K4a, K5, K6ai, K8ai, K8bi, K10a and K14ai.

Mass-specific susceptibility and frequency-dependent susceptibility results partly conform to established models (developed mostly on Chinese loess-paleosol deposits) of susceptibility enhancement in paleosols and depletion in loess. Many parts of the sequence do not appear to conform to this model and the results more closely resemble the inverse relationship found on Alaskan loess-paleosol beds. Frequency-dependent susceptibility is reliable in delineating paleosols by their ‘ultrafine’ ferrimagnetic mineral content, and citrate-bicarbonate-dithionite treatments successfully remove all iron oxides so that remeasurement of susceptibility isolates a strictly ‘pedogenic’, rather than lithogenic, fraction. 

Laser diffraction particle-size analysis shows that the Kauroa Ash beds are texturally reasonably homogenous. They have bimodal particle-size distributions with the most dominant mode at around 11.25 ɸ inferred to be the product of intense and prolonged weathering. Other modes are variously centred on 7-8.5 ɸ and, despite weathering and pedogenesis, have some relationship to the original depositional particle-size distributions because trends between facies (i.e. genetic units) are delineated. Principal components analysis objectively characterises these modes as (Wentworth size classes) ‘very fine clay’ and ‘coarse silt’, although there is no proportional relationship between them, supporting a post-depositional origin for the former mode. 

The chronology of the sequence, previously poorly defined, is greatly improved by a combination of tephrochronologic correlations, fission-track dating, and paleomagnetism. Five zircon fission-track dates provide independent age ‘spikes’ and range from 2.24 ± 0.29 Ma in the basal member, K1, to 1.28 ± 0.11 Ma for the distal ignimbrite unit K12a. Paleomagnetism is invaluable in providing additional age information. The top of the sequence, member K15, is dated as >0.78 Ma (Brunhes-Matuyama boundary) because of its reversed polarity; two episodes of normal polarity are found in beds K14b and K2b and are inferred to represent the Jaramillo (1.07-0.99 Ma) and Olduvai (1.95-1.79 Ma) subchrons, respectively. Beds underlying the Kauroa Ash sequence are also of normal polarity, indicating that they were deposited in the Gauss Chron (>2.6 Ma). 

Identification and correlation of tephras by conventional means (fingerprinting by their lithological or geochemical properties) is impossible in the Kauroa Ash sequence because the beds have no remaining volcanic glass, which has instead been altered to an assemblage of authigenic phases (clays) by weathering and pedogenesis. However, a new technique analysing fresh glass found as melt inclusions in quartz grains is successful in circumventing this problem. Inclusions represent samples of non-degassed magma that became entrapped during phenocryst growth prior to eruption. The glass has remained unaltered because it is hermetically sealed in a chemically resistant phenocryst, which has protected it from weathering processes. Electron microprobe analysis of the glass inclusions yield results which are wholly reasonable for glass (totals ranging from 93-97%; low standard deviations of <1 %), and a number of provisional correlations are established by comparing the major element composition of Kauroa Ash tephra beds with those of proximal deposits. The Kauroa Ash sequence may contain deposits correlated with at least seven major TVZ eruptions, in many cases expanding the known extent of the (distal) deposit and, for the first time answering the question as to the origin of the Kauroa Ash beds. 

These correlations, together with an improved chronology, enable the Kauroa Ash sequence to be placed in a regional stratigraphic framework alongside other New Zealand Plio-Pleistocene sequences such as those in the Wanganui Basin, Wairarapa, Cape Kidnappers and Port Waikato. Using paleosols as chrono- and climatostratigraphic entities (correlated to warm periods in global climate), the sequence can also be placed alongside a global reference, the marine oxygen isotope stratigraphy. A further correlation to the Chinese loess-paleosol record suggests that large parts of the Kauroa Ash sequence were deposited in an incremental manner akin to deposition of loess, so that the sequence is not only a record of TVZ volcanism, but also of Plio-Pleistocene paleoclimate.

year: 2000

title: Investment in Authentic Sustainable Cultural Tourism : Planning for the Future Benefit of Waikato-Tainui along the Waikato River, Case Studies at Tūrangawaewae and Te Awamārahi Marae

authors: Cooper, Mei Te Werawera Te Reokaha

abstract: “Tērā ōkū hoa kei ngā tōpito e whā o te ao - My friends will come from all parts of the world” – Kīngi Tāwhiao 1822-1894

This doctorate is guided by the visions of the second Māori king, Kīngi Tāwhiao, which reflect the importance of global connections, relationships (Turongo House, 2000:109) and investment to achieve the ultimate goal of authentic or ‘true’ sustainability throughout the world.  The strategic process will require a balance between humans and the natural environment to ensure long-term growth and survival.  This research will investigate whether investment in Authentic Sustainable Cultural Tourism (ASCT) will provide long-term and sustainable benefits for tribes throughout New Zealand, using a study of Waikato-Tainui.  ASCT, if practiced correctly, must be driven by whānau of marae, hapū and iwi alongside ideal long-term partners to ultimately achieve indigenous autonomy. ASCT prioritises the wellbeing of indigenous and local communities, cultural preservation and environmental sustainability while providing secure economic returns to key stakeholders. 

Today, tourism is New Zealand’s largest and fastest growing industry; total tourism expenditure is forecast to increase from $34.7b in 2016, an increase of 12.2% from the previous year, to $41b by 2025.  The natural landscape is the main reason why international travellers visit New Zealand, followed by an interest in Māori culture (Tourism Industry Association, 2015).  Unfortunately, there is sparse funding for Māori tourism development (New Zealand Māori Tourism, 2015; Goulter, 2015) compared to mainstream tourism and Māori are the most impoverished people in New Zealand (Statistics New Zealand, 2011).  Cultural tourism has existed in New Zealand for more than 150 years and has a vital role in New Zealand’s tourism industry.  Te Kīngitanga and the Waikato River are the main aspects that make Waikato unique in New Zealand, however, there is very little cultural tourism in the region.  As a descendant of Waikato-Tainui, I work with my people at the marae level to inform tribal governance about the significant opportunities available to whānau compared to current mainstream investments by the tribe. 

Using Kaupapa Māori and critical realism, the methods used in this research include an investigation of government tourism organisations, interviews with Waikato-Tainui governance groups, case studies at Tūrangawaewae and Te Awamārahi Marae, case studies at fourteen successful Māori tourism providers throughout New Zealand, and an international comparison of best practices in indigenous tourism.

text: Investment in Authentic Sustainable Cultural Tourism : Planning for the Future Benefit of Waikato-Tainui along the Waikato River, Case Studies at Tūrangawaewae and Te Awamārahi Marae “Tērā ōkū hoa kei ngā tōpito e whā o te ao - My friends will come from all parts of the world” – Kīngi Tāwhiao 1822-1894

This doctorate is guided by the visions of the second Māori king, Kīngi Tāwhiao, which reflect the importance of global connections, relationships (Turongo House, 2000:109) and investment to achieve the ultimate goal of authentic or ‘true’ sustainability throughout the world.  The strategic process will require a balance between humans and the natural environment to ensure long-term growth and survival.  This research will investigate whether investment in Authentic Sustainable Cultural Tourism (ASCT) will provide long-term and sustainable benefits for tribes throughout New Zealand, using a study of Waikato-Tainui.  ASCT, if practiced correctly, must be driven by whānau of marae, hapū and iwi alongside ideal long-term partners to ultimately achieve indigenous autonomy. ASCT prioritises the wellbeing of indigenous and local communities, cultural preservation and environmental sustainability while providing secure economic returns to key stakeholders. 

Today, tourism is New Zealand’s largest and fastest growing industry; total tourism expenditure is forecast to increase from $34.7b in 2016, an increase of 12.2% from the previous year, to $41b by 2025.  The natural landscape is the main reason why international travellers visit New Zealand, followed by an interest in Māori culture (Tourism Industry Association, 2015).  Unfortunately, there is sparse funding for Māori tourism development (New Zealand Māori Tourism, 2015; Goulter, 2015) compared to mainstream tourism and Māori are the most impoverished people in New Zealand (Statistics New Zealand, 2011).  Cultural tourism has existed in New Zealand for more than 150 years and has a vital role in New Zealand’s tourism industry.  Te Kīngitanga and the Waikato River are the main aspects that make Waikato unique in New Zealand, however, there is very little cultural tourism in the region.  As a descendant of Waikato-Tainui, I work with my people at the marae level to inform tribal governance about the significant opportunities available to whānau compared to current mainstream investments by the tribe. 

Using Kaupapa Māori and critical realism, the methods used in this research include an investigation of government tourism organisations, interviews with Waikato-Tainui governance groups, case studies at Tūrangawaewae and Te Awamārahi Marae, case studies at fourteen successful Māori tourism providers throughout New Zealand, and an international comparison of best practices in indigenous tourism.

year: 2018