title: Models of q-algebra representations: q-integral transforms and "addition theorems''

authors: Kalnins, Ernie G.

subjects: Mathematics

abstract: In his classic book on group representations and special functions Vilenkin studied the matrix elements of irreducible representations of the Euclidean and oscillator Lie algebras with respect to countable bases of eigenfunctions of the Cartan subalgebras, and he computed the summation identities for Bessel functions and Laguerre polynomials associated with the addition theorems for these matrix elements. He also studied matrix elements of the pseudo-Euclidean and pseudo-oscillator algebras with respect to the continuum bases of generalized eigenfunctions of the Cartan subalgebras of these Lie algebras and this resulted in realizations of the addition theorems for the matrix elements as integral transform identities for Bessel functions and for confluent hypergeometric functions. Here we work out q analogs of these results in which the usual exponential function mapping from the Lie algebra to the Lie group is replaced by the q-exponential mappings Eq and eq. This study of representations of the Euclidean quantum algebra and the q-oscillator algebra (not a quantum algebra) leads to summation, integral transform, and q-integral transform identities for q analogs of the Bessel and confluent hypergeometric functions, extending the results of Vilenkin for the q=1 case.

text: Models of q-algebra representations: q-integral transforms and "addition theorems'' In his classic book on group representations and special functions Vilenkin studied the matrix elements of irreducible representations of the Euclidean and oscillator Lie algebras with respect to countable bases of eigenfunctions of the Cartan subalgebras, and he computed the summation identities for Bessel functions and Laguerre polynomials associated with the addition theorems for these matrix elements. He also studied matrix elements of the pseudo-Euclidean and pseudo-oscillator algebras with respect to the continuum bases of generalized eigenfunctions of the Cartan subalgebras of these Lie algebras and this resulted in realizations of the addition theorems for the matrix elements as integral transform identities for Bessel functions and for confluent hypergeometric functions. Here we work out q analogs of these results in which the usual exponential function mapping from the Lie algebra to the Lie group is replaced by the q-exponential mappings Eq and eq. This study of representations of the Euclidean quantum algebra and the q-oscillator algebra (not a quantum algebra) leads to summation, integral transform, and q-integral transform identities for q analogs of the Bessel and confluent hypergeometric functions, extending the results of Vilenkin for the q=1 case.

year: 1994-04

title: Sulfide-bridged aggregates from the metalloligand [Pt₂(μ-S)₂(PPh₃)₄] and β-diketonate complexes of cobalt(II) and zinc(II)

authors: Bridson, James Hadley

subjects: Zinc complexes

abstract: Reaction of [Pt₂(μ-S)₂(PPh₃)₄] with a range of zinc(II) and cobalt(II) complexes ML₂, where L is a β-diketonate ligand CH₃COCHCOCH₃, PhCOCHCOPh, CF₃COCHCOTh (Th = 2-thienyl)] permits the synthesis of adducts [Pt₂(μ-S)₂(PPh₃)₄M(diketonate)]⁺, isolated as their BPH₄⁻salts in moderate yields. The cobalt and zinc acetylacetonate complexes were characterised by single-crystal X-ray diffraction studies, which reveal isomorphous structures, with tetrahedral heterometal centres.

text: Sulfide-bridged aggregates from the metalloligand [Pt₂(μ-S)₂(PPh₃)₄] and β-diketonate complexes of cobalt(II) and zinc(II) Reaction of [Pt₂(μ-S)₂(PPh₃)₄] with a range of zinc(II) and cobalt(II) complexes ML₂, where L is a β-diketonate ligand CH₃COCHCOCH₃, PhCOCHCOPh, CF₃COCHCOTh (Th = 2-thienyl)] permits the synthesis of adducts [Pt₂(μ-S)₂(PPh₃)₄M(diketonate)]⁺, isolated as their BPH₄⁻salts in moderate yields. The cobalt and zinc acetylacetonate complexes were characterised by single-crystal X-ray diffraction studies, which reveal isomorphous structures, with tetrahedral heterometal centres.

year: 2006-01-01

title: A study of iwi communication between Te Tau Ihu iwi and intergenerational diasporic whānau

authors: Wright, Karamea Moana

subjects: diaspora

abstract: This study looks at the online communication methods used between Te Tau Ihu iwi and intergenerational uri who live in the diaspora. Preferences and perceptions of methods of communication utilized by both iwi, and tāngata who live in the overseas diaspora were explored. Most important are the implications for uri living away from Aotearoa, and their desire to remain connected to iwi in New Zealand. This is a mixed methods study, comprising of a survey of fifty-eight emigrant respondents living abroad, and interviews with one member of the first, second, and third generations of emigrants to the United States, with affiliation to at least one of the eight tribes who claim tāngata whenua status of Te Tau Ihu.

The implications of the findings from the study provide both iwi and tāngata the opportunity to create a synergistic relationship of communicating. Assisting iwi to develop and implement a strategy for relevant communication, and iwi members engaging in a dialogue with iwi about relevance of communication is critical in strengthening ties between iwi and future generations regardless of location. “He aha te mea nui, he tangata, he tangata.”

text: A study of iwi communication between Te Tau Ihu iwi and intergenerational diasporic whānau This study looks at the online communication methods used between Te Tau Ihu iwi and intergenerational uri who live in the diaspora. Preferences and perceptions of methods of communication utilized by both iwi, and tāngata who live in the overseas diaspora were explored. Most important are the implications for uri living away from Aotearoa, and their desire to remain connected to iwi in New Zealand. This is a mixed methods study, comprising of a survey of fifty-eight emigrant respondents living abroad, and interviews with one member of the first, second, and third generations of emigrants to the United States, with affiliation to at least one of the eight tribes who claim tāngata whenua status of Te Tau Ihu.

The implications of the findings from the study provide both iwi and tāngata the opportunity to create a synergistic relationship of communicating. Assisting iwi to develop and implement a strategy for relevant communication, and iwi members engaging in a dialogue with iwi about relevance of communication is critical in strengthening ties between iwi and future generations regardless of location. “He aha te mea nui, he tangata, he tangata.”

year: 2019