title: Models of q-algebra representations: Matrix elements of the q-oscillator algebra

authors: Kalnins, Ernie G.

subjects: Mathematics

abstract: This article continues a study of function space models of irreducible representations of q analogs of Lie enveloping algebras, motivated by recurrence relations satisfied by q-hypergeometric functions. Here a q analog of the oscillator algebra (not a quantum algebra) is considered. It is shown that various q analogs of the exponential function can be used to mimic the exponential mapping from a Lie algebra to its Lie group and the corresponding matrix elements of the ``group operators'' on these representation spaces are computed. This ``local'' approach applies to more general families of special functions, e.g., with complex arguments and parameters, than does the quantum group approach. It is shown that the matrix elements themselves transform irreducibly under the action of the algebra. q analogs of a formula are found for the product of two hypergeometric functions 1F1 and the product of a 1F1 and a Bessel function. They are interpreted here as expansions of the matrix elements of a ``group operator'' (via the exponential mapping) in a tensor product basis (for the tensor product of two irreducible oscillator algebra representations) in terms of the matrix elements in a reduced basis. As a by-product of this analysis an interesting new orthonormal basis was found for a q analog of the Bargmann–Segal Hilbert space of entire functions.

text: Models of q-algebra representations: Matrix elements of the q-oscillator algebra This article continues a study of function space models of irreducible representations of q analogs of Lie enveloping algebras, motivated by recurrence relations satisfied by q-hypergeometric functions. Here a q analog of the oscillator algebra (not a quantum algebra) is considered. It is shown that various q analogs of the exponential function can be used to mimic the exponential mapping from a Lie algebra to its Lie group and the corresponding matrix elements of the ``group operators'' on these representation spaces are computed. This ``local'' approach applies to more general families of special functions, e.g., with complex arguments and parameters, than does the quantum group approach. It is shown that the matrix elements themselves transform irreducibly under the action of the algebra. q analogs of a formula are found for the product of two hypergeometric functions 1F1 and the product of a 1F1 and a Bessel function. They are interpreted here as expansions of the matrix elements of a ``group operator'' (via the exponential mapping) in a tensor product basis (for the tensor product of two irreducible oscillator algebra representations) in terms of the matrix elements in a reduced basis. As a by-product of this analysis an interesting new orthonormal basis was found for a q analog of the Bargmann–Segal Hilbert space of entire functions.

year: 1993-11

title: An intelligent hammer: a novel concept for automating nail and pile driving

authors: Ilanko, Sinniah

subjects: intelligent hammer

abstract: Some interesting novel ideas on how the compliance needs of manipulators may be met by changing the stiffness of a structure have been presented in a paper by Ang and Andeen [1]. The purpose of the present note is to outline a novel concept for another variable-stiffness mechanism that may be useful in automating the process of driving nails or piles.

text: An intelligent hammer: a novel concept for automating nail and pile driving Some interesting novel ideas on how the compliance needs of manipulators may be met by changing the stiffness of a structure have been presented in a paper by Ang and Andeen [1]. The purpose of the present note is to outline a novel concept for another variable-stiffness mechanism that may be useful in automating the process of driving nails or piles.

year: 2005

title: Transformation geometry: Mā te nekehanga, mā te whakaata, mā te hurihanga

authors: Manuel, Hirāni

abstract: The whakataukī (proverb) promotes the notion that navigating obstacles or barriers successfully is a natural part of learning. The learning and teaching of the Māori language and culture are key practices in Māori-medium settings. Teachers in such classrooms are actively promoting academic success as Māori through the medium of te reo Māori (the Māori language). This stance includes the learning of mathematics.

text: Transformation geometry: Mā te nekehanga, mā te whakaata, mā te hurihanga The whakataukī (proverb) promotes the notion that navigating obstacles or barriers successfully is a natural part of learning. The learning and teaching of the Māori language and culture are key practices in Māori-medium settings. Teachers in such classrooms are actively promoting academic success as Māori through the medium of te reo Māori (the Māori language). This stance includes the learning of mathematics.

year: 2015