title: Separation of variables for the Hamilton-Jacobi equation on complex projective spaces

authors: Boyer, C.P.

subjects: Science & Technology

abstract: The additive separation of variables in the Hamilton-Jacobi equation and the multiplicative separation of variables in the Laplace-Beltrami equation are studied for the complex projective space C Pⁿ considered as a Riemannian Einstein space with the standard Fubini-Study metric. The isometry group of C Pⁿ is SU(ⁿ + 1) and its Cartan subgroup is used to generate n ignorable variables (variables not figuring in the metric tensor). A one-to-one correspondence is established between separable coordinate systems on S and separable systems with n ignorable variables on C P. The separable coordinates in C Pⁿ are characterized by 2n integrals of motion in involution: n of them are elements of the Cartan subalgebra of SU(n + 1) and the remaining n are linear combinations of the Casimir operators of n(n + 1)/2 different su(2) subalgebras of su(n + 1). Each system of 2n integrals of motion in involution, and hence each separable system of coordinates on CPⁿ, thus provides a completely integrable Hamiltonian system. For n= 2 it is shown that only two separable systems on CP² exist, both nonorthogonal with two ignorable variables, coming from spherical and elliptic coordinates on S², respectively.

text: Separation of variables for the Hamilton-Jacobi equation on complex projective spaces The additive separation of variables in the Hamilton-Jacobi equation and the multiplicative separation of variables in the Laplace-Beltrami equation are studied for the complex projective space C Pⁿ considered as a Riemannian Einstein space with the standard Fubini-Study metric. The isometry group of C Pⁿ is SU(ⁿ + 1) and its Cartan subgroup is used to generate n ignorable variables (variables not figuring in the metric tensor). A one-to-one correspondence is established between separable coordinate systems on S and separable systems with n ignorable variables on C P. The separable coordinates in C Pⁿ are characterized by 2n integrals of motion in involution: n of them are elements of the Cartan subalgebra of SU(n + 1) and the remaining n are linear combinations of the Casimir operators of n(n + 1)/2 different su(2) subalgebras of su(n + 1). Each system of 2n integrals of motion in involution, and hence each separable system of coordinates on CPⁿ, thus provides a completely integrable Hamiltonian system. For n= 2 it is shown that only two separable systems on CP² exist, both nonorthogonal with two ignorable variables, coming from spherical and elliptic coordinates on S², respectively.

year: 1985

title: Queering the Real: Antisociality and the impossible nothing of queer

authors: Jackson, Shaynah

subjects: Queer theory

abstract: This project charts a logic of queer antisociality along literary, cinematic, and theoretical works. Reading across a collection of textual case studies from Gertrude Stein, Herman Melville, and Samuel Beckett, as well as films from Derek Jarman and David Cronenberg, this thesis thinks through antisociality, employing it as a critical approach to reconcile moments of textual disorientation as queer encounter. The ambition of this thesis is not to designate certain works as queer texts, or to suggest a unifying queer thread across them; it is the very nature of queer as resistant to coherence and continuity that such a task could only ever fail. Rather, reading through antisociality offers a way to think queerness beyond a rubric of sexual difference as what frustrates or works against established forms of knowledge and trace modes of queer negation along inconsistent lines as aporetic junctures, opacity, and incoherence.

This approach is developed in conversation with Lee Edelman’s Bad Education where he conceptualises queer aligned to the incomprehensibility of the Lacanian Real – as being outside of identity markers or systems of recognition. Queer as such does not locate an ontology or a sexuality, but rather resists determinate meaning and troubles categorisation; queer is that which disturbs coherence. To think queerness then, Edelman suggests, is to think beyond the structural limits of language. Following this invocation to think queerly beyond structures of knowledge, this project contemplates its application and details what it might look like to do so. Each chapter then illustrates forms of queer negation that resist the controls immanent in language, and articulates antisociality as a way of thinking, and indeed a way of reading, attuned to semiological abstractions and improper grammar as queer methods of resistance.

Since queer cannot be secured within language, this thesis works to reframe incoherence as queer encounter that changes shape across whatever given literary or visual landscapes. Antisociality then operates as a conceptual lens to locate queer not as content but as form: as structural idiosyncrasies that frustrate comprehension and unsettle conventional logic. Writing this thesis and working across different texts and mediums then allowed for a mapping of the different ways that queer is discursively constructed while simultaneously conceptualising modes of resistance that work against these constructions.

text: Queering the Real: Antisociality and the impossible nothing of queer This project charts a logic of queer antisociality along literary, cinematic, and theoretical works. Reading across a collection of textual case studies from Gertrude Stein, Herman Melville, and Samuel Beckett, as well as films from Derek Jarman and David Cronenberg, this thesis thinks through antisociality, employing it as a critical approach to reconcile moments of textual disorientation as queer encounter. The ambition of this thesis is not to designate certain works as queer texts, or to suggest a unifying queer thread across them; it is the very nature of queer as resistant to coherence and continuity that such a task could only ever fail. Rather, reading through antisociality offers a way to think queerness beyond a rubric of sexual difference as what frustrates or works against established forms of knowledge and trace modes of queer negation along inconsistent lines as aporetic junctures, opacity, and incoherence.

This approach is developed in conversation with Lee Edelman’s Bad Education where he conceptualises queer aligned to the incomprehensibility of the Lacanian Real – as being outside of identity markers or systems of recognition. Queer as such does not locate an ontology or a sexuality, but rather resists determinate meaning and troubles categorisation; queer is that which disturbs coherence. To think queerness then, Edelman suggests, is to think beyond the structural limits of language. Following this invocation to think queerly beyond structures of knowledge, this project contemplates its application and details what it might look like to do so. Each chapter then illustrates forms of queer negation that resist the controls immanent in language, and articulates antisociality as a way of thinking, and indeed a way of reading, attuned to semiological abstractions and improper grammar as queer methods of resistance.

Since queer cannot be secured within language, this thesis works to reframe incoherence as queer encounter that changes shape across whatever given literary or visual landscapes. Antisociality then operates as a conceptual lens to locate queer not as content but as form: as structural idiosyncrasies that frustrate comprehension and unsettle conventional logic. Writing this thesis and working across different texts and mediums then allowed for a mapping of the different ways that queer is discursively constructed while simultaneously conceptualising modes of resistance that work against these constructions.

year: 2025

title: Te Raupatu o Tauranga Moana = The confiscation of Tauranga lands. [Volume 1]

authors: Stokes, Evelyn

subjects: Tauranga lands

abstract: A report providing a historical and geographical overview on the confiscation of Tauranga lands. In two volumes, volume one comprises a narrative of the events described as the raupatu, the confiscation of lands in the Tauranga Moana tribal area under the New Zealand Settlements Act 1863.  Volume two is a collection of documents, edited and annotated which were compiled in support of the report. These documents include personal accounts, tribal history, land purchases, lands returned and crown transactions.

text: Te Raupatu o Tauranga Moana = The confiscation of Tauranga lands. [Volume 1] A report providing a historical and geographical overview on the confiscation of Tauranga lands. In two volumes, volume one comprises a narrative of the events described as the raupatu, the confiscation of lands in the Tauranga Moana tribal area under the New Zealand Settlements Act 1863.  Volume two is a collection of documents, edited and annotated which were compiled in support of the report. These documents include personal accounts, tribal history, land purchases, lands returned and crown transactions.

year: 1990