title: Separation equations for 2D superintegrable systems on constant curvature spaces

authors: Escobar-Ruiz, M.A.

subjects: Science & Technology

abstract: Second-order conformal quantum superintegrable systems in two dimensions are Laplace equations on a manifold with an added scalar potential and three independent 2nd order conformal symmetry operators. They encode all the information about 2D Helmholtz or time-independent Schrödinger superintegrable systems in an efficient manner: each of these systems admits a quadratic symmetry algebra (not usually a Lie algebra) and is multiseparable. We study the separation equations for the systems as a family rather than separate cases. We show that the separation equations comprise all of the various types of hypergeometric and Heun equations in full generality. In particular, they yield all of the 1D Schrödinger exactly solvable (ES) and quasi-exactly solvable (QES) systems related to the Heun operator. We focus on complex constant curvature spaces and show explicitly that there are eight pairs of Laplace separation types and these types account for all separable coordinates on the 20 flat space and 9 2-sphere Helmholtz superintegrable systems, including those for the constant potential case. The different systems are related by Stäckel transforms, by the symmetry algebras and by Böcher contractions of the conformal algebra so(4, C) to itself, which enables all systems to be derived from a single one: the generic potential on the complex 2-sphere. This approach facilitates a unified view of special function theory, incorporating hypergeometric and Heun functions in full generality.

text: Separation equations for 2D superintegrable systems on constant curvature spaces Second-order conformal quantum superintegrable systems in two dimensions are Laplace equations on a manifold with an added scalar potential and three independent 2nd order conformal symmetry operators. They encode all the information about 2D Helmholtz or time-independent Schrödinger superintegrable systems in an efficient manner: each of these systems admits a quadratic symmetry algebra (not usually a Lie algebra) and is multiseparable. We study the separation equations for the systems as a family rather than separate cases. We show that the separation equations comprise all of the various types of hypergeometric and Heun equations in full generality. In particular, they yield all of the 1D Schrödinger exactly solvable (ES) and quasi-exactly solvable (QES) systems related to the Heun operator. We focus on complex constant curvature spaces and show explicitly that there are eight pairs of Laplace separation types and these types account for all separable coordinates on the 20 flat space and 9 2-sphere Helmholtz superintegrable systems, including those for the constant potential case. The different systems are related by Stäckel transforms, by the symmetry algebras and by Böcher contractions of the conformal algebra so(4, C) to itself, which enables all systems to be derived from a single one: the generic potential on the complex 2-sphere. This approach facilitates a unified view of special function theory, incorporating hypergeometric and Heun functions in full generality.

year: 2017

title: The right to life and law enforcement activities: The approrpriateness of a military response ot the problem of terrorism

authors: Chevalier-Watts, Juliet

abstract: This paper considers the issue of the right to life in several lethal force cases heard in the European Court of Human Rights as a result of State activity in response to terrorist and armed activity. In light of extensive judicial scrutiny, and the recent erroneous shooting dead of Brazilian electrician, Jean Charles de Menezes on the London Underground by police, this paper submits that a military response may balance the dichotomy of terrorism and the right to life. Article 2, the right to life, is the most fundamental of all the provisions of the European Convention on Human Rights; without it, all other provisions are redundant. However, this very right, in the context of lethal force and terrorism, was brought sharply into focus when Strasbourg heard the ground breaking Death on the Rock case, where terrorist suspects were shot dead by Special Forces soldiers in Gibraltar. Since this case, a number of lethal force cases have been heard by the European Court in relation to military and police terrorist operations. These cases have been subject to detailed scrutiny as to the application of the provisions of Article 2 of the Convention, and have produced divergent results. As a result, this paper submits that, in Europe at least, a military response acknowledges a State's authority to protect its populous and its agents, whilst at the same time entrenching the fundamental principle of the right to life, thus balancing two theoretically opposing concepts: the authority of the State and the right to life.

text: The right to life and law enforcement activities: The approrpriateness of a military response ot the problem of terrorism This paper considers the issue of the right to life in several lethal force cases heard in the European Court of Human Rights as a result of State activity in response to terrorist and armed activity. In light of extensive judicial scrutiny, and the recent erroneous shooting dead of Brazilian electrician, Jean Charles de Menezes on the London Underground by police, this paper submits that a military response may balance the dichotomy of terrorism and the right to life. Article 2, the right to life, is the most fundamental of all the provisions of the European Convention on Human Rights; without it, all other provisions are redundant. However, this very right, in the context of lethal force and terrorism, was brought sharply into focus when Strasbourg heard the ground breaking Death on the Rock case, where terrorist suspects were shot dead by Special Forces soldiers in Gibraltar. Since this case, a number of lethal force cases have been heard by the European Court in relation to military and police terrorist operations. These cases have been subject to detailed scrutiny as to the application of the provisions of Article 2 of the Convention, and have produced divergent results. As a result, this paper submits that, in Europe at least, a military response acknowledges a State's authority to protect its populous and its agents, whilst at the same time entrenching the fundamental principle of the right to life, thus balancing two theoretically opposing concepts: the authority of the State and the right to life.

year: 2013

title: Students’ Social Interest and University Partnership with Local Indigenous People

authors: Theodorus, Edward

subjects: social interest

abstract: According to Alfred Adler, people will have a meaningful life if they contribute to the realisation of an ideal cooperating community. The psychological process of “contributing” is captured in the Adlerian concept of “social interest.” This study investigates students’ social interest in university partnership with local indigenous people. The partnership between the University of Waikato and local tribes of Tainui is a particular concern of this study, with Kīngitanga Day as the specific phenomenon. Mixed methods–descriptive statistics and thematic analysis–were employed to investigate students’ social interest. Data collection was done in three phases. From data analysis, I found eight primary themes of students’ social interest, namely knowledgeability, significance to self, identification, awareness of community context, gradation of sympathetic concern, willingness, action, and reflection. How the findings relate to the literature on social interest and university-community partnership is discussed. A new model of applied social interest is developed. This thesis ends with elaboration on limitations and recommendations for future research.

text: Students’ Social Interest and University Partnership with Local Indigenous People According to Alfred Adler, people will have a meaningful life if they contribute to the realisation of an ideal cooperating community. The psychological process of “contributing” is captured in the Adlerian concept of “social interest.” This study investigates students’ social interest in university partnership with local indigenous people. The partnership between the University of Waikato and local tribes of Tainui is a particular concern of this study, with Kīngitanga Day as the specific phenomenon. Mixed methods–descriptive statistics and thematic analysis–were employed to investigate students’ social interest. Data collection was done in three phases. From data analysis, I found eight primary themes of students’ social interest, namely knowledgeability, significance to self, identification, awareness of community context, gradation of sympathetic concern, willingness, action, and reflection. How the findings relate to the literature on social interest and university-community partnership is discussed. A new model of applied social interest is developed. This thesis ends with elaboration on limitations and recommendations for future research.

year: 2015