Spectral behaviour of a simple non-self-adjoint operator

We investigate the spectrum of a typical non-self-adjoint differential operator $AD=-d^2/dx^2\otimes A$ acting on \Lp(0,1)\otimes \mathbb{C}^2, where $A$ is a $2\times 2$ constant matrix. We impose Dirichlet and Neumann boundary conditions in the first and second coordinate respectively at both ends of $[0,1]\subset\mathbb{R}$. For $A\in \mathbb{R}^{2\times 2}$ we explore in detail the connection between the entries of $A$ and the spectrum of $AD$, we find necessary conditions to ensure similarity to a self-adjoint operator and give numerical evidence that suggests a non-trivial spectral evolution.Comment: 42 pages, 6 figure

Changes in invertebrate assemblage composition in benthic and hyporheic zones during a severe supraseasonal drought

Droughts are unpredictable disturbances characterized in streams by declining flow, reduced habitat availability, and deteriorating abiotic conditions. Such events typically reduce benthic invertebrate taxon richness and modify assemblage composition, but little is known about how hyporheic invertebrate assemblages respond to drought or how these responses relate to changes in benthic assemblages. We hypothesized that taxon richness (diversity) and variability (as within-site diversity) in benthic assemblage composition would decline as drought proceeded, whereas concurrent changes in hyporheic assemblages would be lower in this more stable environment. We predicted that benthic assemblage composition between sites would converge as epigean taxa were selectively eliminated, whereas between-site hyporheic diversity would change little. We sampled benthic and hyporheic invertebrates concurrently from 4 sites along a groundwater-fed stream during the final stages of a severe supraseasonal drought punctuated by a record heat wave. Abiotic conditions in benthic habitats deteriorated as flow declined, but changes were less pronounced in the hyporheic zone. Benthic diversity declined during drought, whereas hyporheic diversity changed little. However, benthic within-site diversity increased as the drought progressed because of localized variation in the abundance of common taxa. Temporal trends in hyporheic diversity were less consistent. Benthic assemblages at individual sites became more similar, especially during the heat wave, reflecting low diversity and abundance. Hyporheic assemblages changed markedly because of temporary increases in abundances of epigean and hypogean amphipods. These contrasting responses of benthic and hyporheic assemblages to drought should be recognized when developing management strategies for drought-impacted streams

On the spectrum of a matrix model for the D=11 supermembrane compactified on a torus with non-trivial winding

The spectrum of the Hamiltonian of the double compactified D=11 supermembrane with non-trivial central charge or equivalently the non-commutative symplectic super Maxwell theory is analyzed. In distinction to what occurs for the D=11 supermembrane in Minkowski target space where the bosonic potential presents string-like spikes which render the spectrum of the supersymmetric model continuous, we prove that the potential of the bosonic compactified membrane with non-trivial central charge is strictly positive definite and becomes infinity in all directions when the norm of the configuration space goes to infinity. This ensures that the resolvent of the bosonic Hamiltonian is compact. We find an upper bound for the asymptotic distribution of the eigenvalues.Comment: 11 pages, LaTe

School-based curriculum development as reflective practice: a case study in Hong Kong

This paper examines a school-based curriculum development (SBCD) experience in Hong Kong. Traditionally, curriculum change in Hong Kong has normally been bureaucratic with teachers’ actions monitored. This qualitative case study investigates the lived experience of an SBCD practice. Semi-structured interviews were utilized to examine teachers’ perceptions of the reflective SBCD experience in their school and what adaptations they had made when delivering the school-based materials. The findings identified that all teachers held a positive attitude towards this reflective approach to SBCD and emphasized artistry in their teaching practice. Teachers also exercised discretion in response to their students’ level and interests when implementing the school-based curriculum at the classroom level. This research concludes that a reflective approach to curriculum planning with a bottom-up implementation can empower teachers reflecting their creativity, artistry, knowledge of the subject and related pedagogy, and knowledge of their students. The findings of this case study thus contrast sharply with previous research relating to Hong Kong government-led SBCD programs which focus more on meeting the requirements of the intended curriculum than on personalizing the curriculum to meet to learners’ needs

Supersymmetric exact sequence, heat kernel and super KdV hierarchy

We introduce the free N=1 supersymmetric derivation ring and prove the existence of an exact sequence of supersymmetric rings and linear transformations. We apply necessary and sufficient conditions arising from this exact supersymmetric sequence to obtain the essential relations between conserved quantities, gradients and the N=1 super KdV hierarchy. We combine this algebraic approach with an analytic analysis of the super heat operator.We obtain the explicit expression for the Green's function of the super heat operator in terms of a series expansion and discuss its properties. The expansion is convergent under the assumption of bounded bosonic and fermionic potentials. We show that the asymptotic expansion when $t\to0^+$ of the Green's function for the super heat operator evaluated over its diagonal generates all the members of the N=1 super KdV hierarchy.Comment: 20 pages, to be published in JM