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: Comment

authors: Mitchell, Linda

abstract: This issue of Early Childhood Folio ranges from a big-picture and readable review of the main debates within the literature about play in Western settings, to a small, qualitative, in-depth case study of a single child and the use of narrative inquiry to explore concepts of wellbeing and its link to personal identity. All of the authors draw out clear and sometimes challenging implications for the role of the teacher, making the collection highly relevant to practitioners and those working in teacher education and professional development.

text: Comment This issue of Early Childhood Folio ranges from a big-picture and readable review of the main debates within the literature about play in Western settings, to a small, qualitative, in-depth case study of a single child and the use of narrative inquiry to explore concepts of wellbeing and its link to personal identity. All of the authors draw out clear and sometimes challenging implications for the role of the teacher, making the collection highly relevant to practitioners and those working in teacher education and professional development.

year: 2012

title: Indigenous languages shaping multi-lingual interfaces

authors: Keegan, Te Taka Adrian Gregory

subjects: computer science

abstract: This paper reports on an investigation into the indigenous language usage of two bilingual/multilingual digital libraries. Results show that the indigenous language was significantly used by clients and indicate why clients chose to use the indigenous language. Feedback from clients has suggested how the interface should be improved to assist both indigenous and non-indigenous language usage. These results serve as an example of how indigenous languages are shaping multilingual interfaces.

text: Indigenous languages shaping multi-lingual interfaces This paper reports on an investigation into the indigenous language usage of two bilingual/multilingual digital libraries. Results show that the indigenous language was significantly used by clients and indicate why clients chose to use the indigenous language. Feedback from clients has suggested how the interface should be improved to assist both indigenous and non-indigenous language usage. These results serve as an example of how indigenous languages are shaping multilingual interfaces.

year: 2008