title: Lie theory and separation of variables. 5. The equations iUt + Uxx = 0 and iUt + Uxx −c/x2 U = 0

authors: Kalnins, Ernie G.

subjects: Mathematics

abstract: A detailed study of the group of symmetries of the time-dependent free particle Schrödinger equation in one space dimension is presented. An orbit analysis of all first order symmetries is seen to correspond in a well-defined manner to the separation of variables of this equation. The study gives a unified treatment of the harmonic oscillator (both attractive and repulsive), Stark effect, and free particle Hamiltonians in the time dependent formalism. The case of a potential c/x2 is also discussed in the time dependent formalism. Use of representation theory for the symmetry groups permits simple derivation of expansions relating various solutions of the Schrödinger equation, several of which are new.

text: Lie theory and separation of variables. 5. The equations iUt + Uxx = 0 and iUt + Uxx −c/x2 U = 0 A detailed study of the group of symmetries of the time-dependent free particle Schrödinger equation in one space dimension is presented. An orbit analysis of all first order symmetries is seen to correspond in a well-defined manner to the separation of variables of this equation. The study gives a unified treatment of the harmonic oscillator (both attractive and repulsive), Stark effect, and free particle Hamiltonians in the time dependent formalism. The case of a potential c/x2 is also discussed in the time dependent formalism. Use of representation theory for the symmetry groups permits simple derivation of expansions relating various solutions of the Schrödinger equation, several of which are new.

year: 1974-10

title: Theoretical consequences of trait-based environmental filtering for the breadth and shape of the niche: New testable hypotheses generated by the Traitspace model

authors: Laughlin, Daniel C.

subjects: Community assembly

abstract: Every species on Earth fills a unique environmental niche that is driven, in part, by the process of environmental filtering, where the adaptive value of the functional traits of individuals determine their fitness within the given environmental conditions. Despite its long-standing importance in ecology, theoretical investigations of environmental filtering have lagged behind studies of species interactions and neutral dynamics. A new statistical model of trait-based environmental filtering can be a useful tool for exploring the logical consequences of this process while holding all other processes constant. The model uses the logic of objective Bayesian inference to compute the probabilities of species within different environments using two sources of information: the location and dispersion of species within functional trait space, and the statistical relationship between traits and environmental gradients. By varying key parameters in the model, we highlight several testable hypotheses for trait-based ecology. First, niche breadth decreases as intraspecific trait variation decreases, as the strength of the environmental filter increases, and if the trait values do not enhance fitness in any environmental condition in the landscape. Second, niche shape is determined by the form of the trait-environment relationships, where species with extreme trait values are predicted to dominate at the environmental extremes when traits are linearly related to the environment, species with intermediate trait values generally have a selective advantage across a broader environmental range, and bimodal species response curves can occur independently from negative species interactions. The generality of these modelling results can be tested using empirical data from any ecosystem.

text: Theoretical consequences of trait-based environmental filtering for the breadth and shape of the niche: New testable hypotheses generated by the Traitspace model Every species on Earth fills a unique environmental niche that is driven, in part, by the process of environmental filtering, where the adaptive value of the functional traits of individuals determine their fitness within the given environmental conditions. Despite its long-standing importance in ecology, theoretical investigations of environmental filtering have lagged behind studies of species interactions and neutral dynamics. A new statistical model of trait-based environmental filtering can be a useful tool for exploring the logical consequences of this process while holding all other processes constant. The model uses the logic of objective Bayesian inference to compute the probabilities of species within different environments using two sources of information: the location and dispersion of species within functional trait space, and the statistical relationship between traits and environmental gradients. By varying key parameters in the model, we highlight several testable hypotheses for trait-based ecology. First, niche breadth decreases as intraspecific trait variation decreases, as the strength of the environmental filter increases, and if the trait values do not enhance fitness in any environmental condition in the landscape. Second, niche shape is determined by the form of the trait-environment relationships, where species with extreme trait values are predicted to dominate at the environmental extremes when traits are linearly related to the environment, species with intermediate trait values generally have a selective advantage across a broader environmental range, and bimodal species response curves can occur independently from negative species interactions. The generality of these modelling results can be tested using empirical data from any ecosystem.

year: 2015-07-01

title: Rangatahi Wahine: Overcoming the odds – preliminary results

authors: Merritt, Kiri

subjects: Maori

abstract: The purpose of my research was to
identify factors that have contributed to the
development of resilience in young Māori
girls’ lives. Incorporated into this was an
exploration of their psychosocial
competence makeup, to see whether their
sense of self, sense of relationship to the
world, and ability to negotiate life events,
was unique.

text: Rangatahi Wahine: Overcoming the odds – preliminary results The purpose of my research was to
identify factors that have contributed to the
development of resilience in young Māori
girls’ lives. Incorporated into this was an
exploration of their psychosocial
competence makeup, to see whether their
sense of self, sense of relationship to the
world, and ability to negotiate life events,
was unique.

year: 2003