Lattice Green Function (at 0) for the 4d Hypercubic Lattice

The generating function for recurrent Polya walks on the four dimensional hypercubic lattice is expressed as a Kampe-de-Feriet function. Various properties of the associated walks are enumerated.Comment: latex, 5 pages, Res. Report 1

The Cauchy-Schlomilch transformation

The Cauchy-Schl\"omilch transformation states that for a function $f$ and $a, \, b > 0$, the integral of $f(x^{2})$ and $af((ax-bx^{-1})^{2}$ over the interval $[0, \infty)$ are the same. This elementary result is used to evaluate many non-elementary definite integrals, most of which cannot be obtained by symbolic packages. Applications to probability distributions is also given

“More than scaling up”: a critical and practical inquiry into operationalizing sustainability competencies

This chapter starts from the UN Decade of Education for Sustainable Development (DESD) Final Report’s call that in Higher Education, ‘more than scaling up of good practice’ and ‘greater attention to systemic approaches to curriculum change and capacity building for leaders will be needed’ (UNESCO 2014a, p. 31). It recognises this need and the additional, rather profound reform and transformation of educational policy and practice that is required to meet the heightened expectations of education in an increasingly volatile, conflict laden, and challenging world. The emphasis is on clarification and framing of work to date and identification of relevant research gaps. In particular, it addresses the current status of the literature on competencies in ESD, which is characterised by a sea of labels, terminological confusion, and relative inattention to pedaogogic implications. The research outlined is both a critical inquiry into the status of work to date on sustainability competencies and a practical inquiry into the possibility of innovative and transformative institutional strategies and pedagogies around a suite of specific competencies. To this end, the early stages of an international and cross-institutional pilot project collaboration designed to help realize the UN’s ambitious Sustainable Development Goals (SDGs) and UNESCO’s Global Action Plan (GAP) (UNESCO 2014b), is described briefl

Generating Diffusion MRI scalar maps from T1 weighted images using generative adversarial networks

Diffusion magnetic resonance imaging (diffusion MRI) is a non-invasive microstructure assessment technique. Scalar measures, such as FA (fractional anisotropy) and MD (mean diffusivity), quantifying micro-structural tissue properties can be obtained using diffusion models and data processing pipelines. However, it is costly and time consuming to collect high quality diffusion data. Here, we therefore demonstrate how Generative Adversarial Networks (GANs) can be used to generate synthetic diffusion scalar measures from structural T1-weighted images in a single optimized step. Specifically, we train the popular CycleGAN model to learn to map a T1 image to FA or MD, and vice versa. As an application, we show that synthetic FA images can be used as a target for non-linear registration, to correct for geometric distortions common in diffusion MRI

‘More than scaling-up’: Sustainability contexts, competencies, and consequences - a critical inquiry

To identify and problematise the key issues characterising the relationship between global sustainability contexts and the limited response of HE to date - with the purpose of unlocking the potential for innovative, replicable efforts to develop sustainability competencies through innovation in curriculum policy and practice, through addressing these sub-aims: How far does HE policy accommodate and reflect the need for sustainability competencies? How can capacity for teaching for competency be built and supported through new policies? How can curricula and pedagogy be better aligned to facilitate the building of sustainability competency in learners and teachers? What effect and influence does education for sustainability competency have in terms of facilitating transformative social learning, supporting systems structure change, and cultivating informed responsibility (in terms of policy and everyday decision making)?PedRI

Green's function of a finite chain and the discrete Fourier transform

A new expression for the Green's function of a finite one-dimensional lattice with nearest neighbor interaction is derived via discrete Fourier transform. Solution of the Heisenberg spin chain with periodic and open boundary conditions is considered as an example. Comparison to Bethe ansatz clarifies the relation between the two approaches.Comment: preprint of the paper published in Int. J. Modern Physics B Vol. 20, No. 5 (2006) 593-60

Exact results for some Madelung type constants in the finite-size scaling theory

A general formula is obtained from which the madelung type constant: $$C(d|\nu)=\int_0^\infty dx x^{d/2-\nu-1}[(\sum_{l=-\infty}^\infty e^{-xl^2})^d-1-(\frac\pi x)^{d/2}]$$ extensively used in the finite-size scaling theory is computed analytically for some particular cases of the parameters $d$ and $\nu$. By adjusting these parameters one can obtain different physical situations corresponding to different geometries and magnitudes of the interparticle interaction.Comment: IOP- macros, 5 pages, replaced with amended version (1 ref. added

Diffusion-Limited One-Species Reactions in the Bethe Lattice

We study the kinetics of diffusion-limited coalescence, A+A-->A, and annihilation, A+A-->0, in the Bethe lattice of coordination number z. Correlations build up over time so that the probability to find a particle next to another varies from \rho^2 (\rho is the particle density), initially, when the particles are uncorrelated, to [(z-2)/z]\rho^2, in the long-time asymptotic limit. As a result, the particle density decays inversely proportional to time, \rho ~ 1/kt, but at a rate k that slowly decreases to an asymptotic constant value.Comment: To be published in JPCM, special issue on Kinetics of Chemical Reaction

A geomorphology based reconstruction of ice volume distribution at the Last Glacial Maximum across the Southern Alps of New Zealand

We present a 3D reconstruction of ice thickness distribution across the New Zealand Southern Alps at the Last Glacial Maximum (LGM, c. 30–18 ka). To achieve this, we used a perfect plasticity model which could easily be applied to other regions, hereafter termed REVOLTA (Reconstruction of Volume and Topography Automation). REVOLTA is driven by a Digital Elevation Model (DEM), which was modified to best represent LGM bed topography. Specifically, we removed contemporary ice, integrated offshore bathymetry and removed contemporary lakes. A review of valley in-fill sediments, uplift and denudation was also undertaken. Down-valley ice extents were constrained to an updated geo-database of LGM ice limits, whilst the model was tuned to best-fit known vertical limits from geomorphological and geochronological dating studies. We estimate a total LGM ice volume of 6,800 km3, characterised predominantly by valley style glaciation but with an ice cap across Fiordland. With a contemporary ice volume of approximately 50 km3, this represents a loss of 99.25% since the LGM. Using the newly created ice surface, equilibrium line altitudes (ELAs) for each glacier were reconstructed, revealing an average ELA depression of approximately 950 m from present. Analysis of the spatial variation of glacier-specific ELAs and their depression relative to today shows that whilst an east-west ELA gradient existed during the LGM it was less pronounced than at present. The reduced ELA gradient is attributed to an overall weakening of westerlies, a conclusion consistent with those derived from the latest independent climate models

Shear modulus of the hadron-quark mixed phase

Robust arguments predict that a hadron-quark mixed phase may exist in the cores of some "neutron" stars. Such a phase forms a crystalline lattice with a shear modulus higher than that of the crust due to the high density and charge separation, even allowing for the effects of charge screening. This may lead to strong continuous gravitational-wave emission from rapidly rotating neutron stars and gravitational-wave bursts associated with magnetar flares and pulsar glitches. We present the first detailed calculation of the shear modulus of the mixed phase. We describe the quark phase using the bag model plus first-order quantum chromodynamics corrections and the hadronic phase using relativistic mean-field models with parameters allowed by the most massive pulsar. Most of the calculation involves treating the "pasta phases" of the lattice via dimensional continuation, and we give a general method for computing dimensionally continued lattice sums including the Debye model of charge screening. We compute all the shear components of the elastic modulus tensor and angle average them to obtain the effective (scalar) shear modulus for the case where the mixed phase is a polycrystal. We include the contributions from changing the cell size, which are necessary for the stability of the lower-dimensional portions of the lattice. Stability also requires a minimum surface tension, generally tens of MeV/fm^2 depending on the equation of state. We find that the shear modulus can be a few times 10^33 erg/cm^3, two orders of magnitude higher than the first estimate, over a significant fraction of the maximum mass stable star for certain parameter choices.Comment: 22 pages, 12 figures, version accepted by Phys. Rev. D, with the corrections to the shear modulus computation and Table I given in the erratu