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: The sleep cycle modelled as a cortical phase transition

authors: Steyn-Ross, D. Alistair

subjects: cortical model

abstract: We present a mean-field model of the cortex that attempts to describe the gross changes in brain electrical activity for the cycles of natural sleep. We incorporate within the model two major sleep modulatory effects: slow changes in both synaptic efficiency and in neuron resting voltage caused by the ∼90-min cycling in acetylcholine, together with even slower changes in resting voltage caused by gradual elimination during sleep of somnogens (fatigue agents) such as adenosine. We argue that the change from slow-wave sleep (SWS) to rapid-eye-movement (REM) sleep can be understood as a first-order phase transition from a low-firing, coherent state to a high-firing, desychronized cortical state. We show that the model predictions for changes in EEG power, spectral distribution, and correlation time at the SWS-to-REM transition are consistent not only with those observed in clinical recordings of a sleeping human subject, but also with the on-cortex EEG patterns recently reported by Destexhe et al. [J. Neurosci. 19(11), (1999) 4595–4608] for the sleeping cat.

text: The sleep cycle modelled as a cortical phase transition We present a mean-field model of the cortex that attempts to describe the gross changes in brain electrical activity for the cycles of natural sleep. We incorporate within the model two major sleep modulatory effects: slow changes in both synaptic efficiency and in neuron resting voltage caused by the ∼90-min cycling in acetylcholine, together with even slower changes in resting voltage caused by gradual elimination during sleep of somnogens (fatigue agents) such as adenosine. We argue that the change from slow-wave sleep (SWS) to rapid-eye-movement (REM) sleep can be understood as a first-order phase transition from a low-firing, coherent state to a high-firing, desychronized cortical state. We show that the model predictions for changes in EEG power, spectral distribution, and correlation time at the SWS-to-REM transition are consistent not only with those observed in clinical recordings of a sleeping human subject, but also with the on-cortex EEG patterns recently reported by Destexhe et al. [J. Neurosci. 19(11), (1999) 4595–4608] for the sleeping cat.

year: 2005-12

title: Exploring the nature of intimate relationships: A Māori perspective

authors: King, Pita

subjects: Maori

abstract: The 2002 World Report on Violence states that violence occurs in about 70 percent of intimate relationships (Krug, Dahlberg, Mercy, Zwi, & Lozano, 2002). New Zealand research indicates that more than a quarter of relationships have experienced intimate partner violence (IPV), with Māori (Indigenous people of New Zealand) women three times more likely to experience IPV in their lifetime. Utilising Kaupapa Māori (Māori-centred approach) and narrative methodologies, this paper explores the nature of intimate relationships from a Māori perspective, investigating how Māori initiate intimate relationships, attempt to maintain positive intimate relationships and when applicable, exit intimate relationships. The findings from two case studies reveal that intimate relationships involve identity negotiation and an incorporation of cultural values. Communication processes are highlighted as a facilitating factor of intimate relationships.

text: Exploring the nature of intimate relationships: A Māori perspective The 2002 World Report on Violence states that violence occurs in about 70 percent of intimate relationships (Krug, Dahlberg, Mercy, Zwi, & Lozano, 2002). New Zealand research indicates that more than a quarter of relationships have experienced intimate partner violence (IPV), with Māori (Indigenous people of New Zealand) women three times more likely to experience IPV in their lifetime. Utilising Kaupapa Māori (Māori-centred approach) and narrative methodologies, this paper explores the nature of intimate relationships from a Māori perspective, investigating how Māori initiate intimate relationships, attempt to maintain positive intimate relationships and when applicable, exit intimate relationships. The findings from two case studies reveal that intimate relationships involve identity negotiation and an incorporation of cultural values. Communication processes are highlighted as a facilitating factor of intimate relationships.

year: 2012