title: Lie theory and separation of variables. 7. The harmonic oscillator in elliptic coordinates and Ince polynomials

authors: Boyer, C.P.

subjects: Mathematics

abstract: As a continuation of Paper 6 we study the separable basis eigenfunctions and their relationships for the harmonic oscillator Hamiltonian in two space variables with special emphasis on products of Ince polynomials, the eigenfunctions obtained when one separates variables in elliptic coordinates. The overlaps connecting this basis to the polar and Cartesian coordinate bases are obtained by computing in a simpler Bargmann Hilbert space model of the problem. We also show that Ince polynomials are intimately connected with the representation theory of SU (2), the group responsible for the eigenvalue degeneracy of the oscillator Hamiltonian.

text: Lie theory and separation of variables. 7. The harmonic oscillator in elliptic coordinates and Ince polynomials As a continuation of Paper 6 we study the separable basis eigenfunctions and their relationships for the harmonic oscillator Hamiltonian in two space variables with special emphasis on products of Ince polynomials, the eigenfunctions obtained when one separates variables in elliptic coordinates. The overlaps connecting this basis to the polar and Cartesian coordinate bases are obtained by computing in a simpler Bargmann Hilbert space model of the problem. We also show that Ince polynomials are intimately connected with the representation theory of SU (2), the group responsible for the eigenvalue degeneracy of the oscillator Hamiltonian.

year: 1975-03

title: The use of positive and negative penalty functions in solving constrained optimization problems and partial differential equations

authors: Monterrubio Salazar, Luis Emilio

subjects: Rayleigh-Ritz Method

abstract: The Rayleigh-Ritz Method together with the Penalty Function Method is used to investigate the use of different types of penalty parameters. The use of artificial springs as penalty parameters is a very well established procedure to model constraints in the Rayleigh-Ritz Method, the Finite Element Method and other numerical methods. Historically, large positive values were used to define the stiffness coefficient of artificial springs, until recent publications demonstrated that it is possible to use negative values to define the stiffness coefficients of the springs. Furthermore, recent publications show that constraints can be enforced using positive and negative mass or inertia in vibration problems and in a more generic sense using eigenpenalty parameters which are penalty parameters in the matrix associated with the eigenvalue. Before the commencement of this thesis, solutions using artificial inertia were published only for beams and simple spring-mass systems.
In this thesis the use of all possible types of penalty parameters are investigated in vibration problems of Euler-Bernoulli beams, thin plates and shallow shells and in elastic stability analysis of Euler-Bernoulli beams, including penalty parameters associated with the geometrical stiffness matrix. The study includes the use of penalty parameters for both enforcing support boundary conditions and continuity conditions along structural joints.
This investigation started with the selection of the set of admissible functions that would: (a) allow modelling of beams, plates and shells in completely free boundary conditions; (b) not present any limitation in the number of functions that can be used in the solution. This gives the possibility to converge to the constraint solution and to model any type of boundary conditions.
The procedure proposed in this work combines several advantages: accuracy of the results, relative fast convergence, simplicity of the set of admissible functions and flexibility to define boundary conditions. While there are other procedures that may give better accuracy for specific cases, the proposed method is more widely applicable.
The procedure used in this work also includes a way to check for round-off errors and ill-conditioning in the results; as well as a way to bracket the  exact  solution with upper and lower-bound results.

text: The use of positive and negative penalty functions in solving constrained optimization problems and partial differential equations The Rayleigh-Ritz Method together with the Penalty Function Method is used to investigate the use of different types of penalty parameters. The use of artificial springs as penalty parameters is a very well established procedure to model constraints in the Rayleigh-Ritz Method, the Finite Element Method and other numerical methods. Historically, large positive values were used to define the stiffness coefficient of artificial springs, until recent publications demonstrated that it is possible to use negative values to define the stiffness coefficients of the springs. Furthermore, recent publications show that constraints can be enforced using positive and negative mass or inertia in vibration problems and in a more generic sense using eigenpenalty parameters which are penalty parameters in the matrix associated with the eigenvalue. Before the commencement of this thesis, solutions using artificial inertia were published only for beams and simple spring-mass systems.
In this thesis the use of all possible types of penalty parameters are investigated in vibration problems of Euler-Bernoulli beams, thin plates and shallow shells and in elastic stability analysis of Euler-Bernoulli beams, including penalty parameters associated with the geometrical stiffness matrix. The study includes the use of penalty parameters for both enforcing support boundary conditions and continuity conditions along structural joints.
This investigation started with the selection of the set of admissible functions that would: (a) allow modelling of beams, plates and shells in completely free boundary conditions; (b) not present any limitation in the number of functions that can be used in the solution. This gives the possibility to converge to the constraint solution and to model any type of boundary conditions.
The procedure proposed in this work combines several advantages: accuracy of the results, relative fast convergence, simplicity of the set of admissible functions and flexibility to define boundary conditions. While there are other procedures that may give better accuracy for specific cases, the proposed method is more widely applicable.
The procedure used in this work also includes a way to check for round-off errors and ill-conditioning in the results; as well as a way to bracket the  exact  solution with upper and lower-bound results.

year: 2009

title: Defence not defiance: The shaping of New Zealand’s volunteer force

authors: Clayton, Garry James

subjects: New Zealand Army history

abstract: In the latter half of the nineteenth century and the first decade of the twentieth New Zealand's land defences were primarily the responsibility of its Volunteer Force. This Force over a period of fifty-two years evolved from ill-considered vigilante bands to a well organised and disciplined Army. This thesis examines the evolution of this Volunteer Army in chronological sequence concentrating on the factors which shaped and developed it.
The preface provides a brief outline of this study on New Zealand's Volunteer Force. The thesis then presents the history of the Force in eight chapters. These chapters are in four basic divisions. The first section, of three chapters. examines the origins of the Force, its role in the New Zealand Wars and its subsequent development in the period of tenuous peace during the 1870's . The second section, comprising of two chapters only, examines the transformation of the Volunteers from a military body which primarily looked inward. into a Force which concentrated on protecting the nation's main ports. The third section , also comprising two chapters , then views the development of this coastal protection Force into a Colonial Army, able and willing to provide for the nation's own security as well as expeditionary units to· serve with the British Imperial Forces ·. The last Section, of one Chapter only, concentrates on the demise of Volunteering and the moves to compulsory military training. The conclusion then draws together the analytical threads of the thesis. 
As the thesis is extensive in its coverage of time, it is only possible to outline the very nature of Volunteering. It will be for Regimental Historians to show the uniqueness of each of the 658 different units that comprised the New Zealand Volunteer Force over its fifty-two years of existence. What this broad survey does do, however, is provide a unique window for the examination of the development of an embryonic Colonial Army in the context of a developing nation.

text: Defence not defiance: The shaping of New Zealand’s volunteer force In the latter half of the nineteenth century and the first decade of the twentieth New Zealand's land defences were primarily the responsibility of its Volunteer Force. This Force over a period of fifty-two years evolved from ill-considered vigilante bands to a well organised and disciplined Army. This thesis examines the evolution of this Volunteer Army in chronological sequence concentrating on the factors which shaped and developed it.
The preface provides a brief outline of this study on New Zealand's Volunteer Force. The thesis then presents the history of the Force in eight chapters. These chapters are in four basic divisions. The first section, of three chapters. examines the origins of the Force, its role in the New Zealand Wars and its subsequent development in the period of tenuous peace during the 1870's . The second section, comprising of two chapters only, examines the transformation of the Volunteers from a military body which primarily looked inward. into a Force which concentrated on protecting the nation's main ports. The third section , also comprising two chapters , then views the development of this coastal protection Force into a Colonial Army, able and willing to provide for the nation's own security as well as expeditionary units to· serve with the British Imperial Forces ·. The last Section, of one Chapter only, concentrates on the demise of Volunteering and the moves to compulsory military training. The conclusion then draws together the analytical threads of the thesis. 
As the thesis is extensive in its coverage of time, it is only possible to outline the very nature of Volunteering. It will be for Regimental Historians to show the uniqueness of each of the 658 different units that comprised the New Zealand Volunteer Force over its fifty-two years of existence. What this broad survey does do, however, is provide a unique window for the examination of the development of an embryonic Colonial Army in the context of a developing nation.

year: 1990