Effective action for scalar fields and generalised zeta-function regularisation

Motivated by the study of quantum fields in a Friedman-Robertson-Walker (FRW) spacetime, the one-loop effective action for a scalar field defined in the ultrastatic manifold $R\times H^3/\Gamma$, $H^3/\Gamma$ being the finite volume, non-compact, hyperbolic spatial section, is investigated by a generalisation of zeta-function regularisation. It is shown that additional divergences may appear at one-loop level. The one-loop renormalisability of the model is discussed and making use of a generalisation of zeta-function regularisation, the one-loop renormalisation group equations are derived.Comment: Latex, 16 pages, no figures; Latex mistakes corrected; accepted for publication in Physical Review

Zeta function determinant of the Laplace operator on the DD-dimensional ball

We present a direct approach for the calculation of functional determinants of the Laplace operator on balls. Dirichlet and Robin boundary conditions are considered. Using this approach, formulas for any value of the dimension, $D$, of the ball, can be obtained quite easily. Explicit results are presented here for dimensions $D=2,3,4,5$ and $6$.Comment: 22 pages, one figure appended as uuencoded postscript fil

Mechanical versus thermodynamical melting in pressure-induced amorphization: the role of defects

We study numerically an atomistic model which is shown to exhibit a one--step crystal--to--amorphous transition upon decompression. The amorphous phase cannot be distinguished from the one obtained by quenching from the melt. For a perfectly crystalline starting sample, the transition occurs at a pressure at which a shear phonon mode destabilizes, and triggers a cascade process leading to the amorphous state. When defects are present, the nucleation barrier is greatly reduced and the transformation occurs very close to the extrapolation of the melting line to low temperatures. In this last case, the transition is not anticipated by the softening of any phonon mode. Our observations reconcile different claims in the literature about the underlying mechanism of pressure amorphization.Comment: 7 pages, 7 figure

Functional determinants for general self-adjoint extensions of Laplace-type operators resulting from the generalized cone

In this article we consider the zeta regularized determinant of Laplace-type operators on the generalized cone. For {\it arbitrary} self-adjoint extensions of a matrix of singular ordinary differential operators modelled on the generalized cone, a closed expression for the determinant is given. The result involves a determinant of an endomorphism of a finite-dimensional vector space, the endomorphism encoding the self-adjoint extension chosen. For particular examples, like the Friedrich's extension, the answer is easily extracted from the general result. In combination with \cite{BKD}, a closed expression for the determinant of an arbitrary self-adjoint extension of the full Laplace-type operator on the generalized cone can be obtained.Comment: 27 pages, 2 figures; to appear in Manuscripta Mathematic

One loop photon-graviton mixing in an electromagnetic field: Part 2

In part 1 of this series compact integral representations had been obtained for the one-loop photon-graviton amplitude involving a charged spin 0 or spin 1/2 particle in the loop and an arbitrary constant electromagnetic field. In this sequel, we study the structure and magnitude of the various polarization components of this amplitude on-shell. Explicit expressions are obtained for a number of limiting cases.Comment: 31 pages, 3 figure

Spectral analysis and zeta determinant on the deformed spheres

We consider a class of singular Riemannian manifolds, the deformed spheres $S^N_k$, defined as the classical spheres with a one parameter family $g[k]$ of singular Riemannian structures, that reduces for $k=1$ to the classical metric. After giving explicit formulas for the eigenvalues and eigenfunctions of the metric Laplacian $\Delta_{S^N_k}$, we study the associated zeta functions $\zeta(s,\Delta_{S^N_k})$. We introduce a general method to deal with some classes of simple and double abstract zeta functions, generalizing the ones appearing in $\zeta(s,\Delta_{S^N_k})$. An application of this method allows to obtain the main zeta invariants for these zeta functions in all dimensions, and in particular $\zeta(0,\Delta_{S^N_k})$ and $\zeta'(0,\Delta_{S^N_k})$. We give explicit formulas for the zeta regularized determinant in the low dimensional cases, $N=2,3$, thus generalizing a result of Dowker \cite{Dow1}, and we compute the first coefficients in the expansion of these determinants in powers of the deformation parameter $k$.Comment: 1 figur

A flexible approach to introductory programming : engaging and motivating students

© 2019 Copyright is held by the owner/author(s). In this paper, we consider an approach to supporting students of Computer Science as they embark upon their university studies. The transition to Computer Science can be challenging for students, and equally challenging for those teaching them. Issues that are unusual – if not unique – to teaching computing at this level include • the wide variety in students background, varying from no prior experience to extensive development practice; • the positives and negatives of dealing with self-taught hobbyists who may developed buggy mental models of the task in hand and are not aware of the problem; • the challenge of getting students to engage with material that includes extensive practical element; • the atypical profile of a computing cohort, with typically 80%+ male students. The variation in background includes the style of prior academic experience, with some students coming from traditional level 3 (i.e. A-levels), some through more vocational routes (e.g. B-Tech, though these have changed in recent years), through to those from experiential (work based) learning. Technical background varies from science, mathematical and computing experience, to no direct advanced technical or scientific experience. A further issue is students’ attainment and progression within higher education, where the success and outcomes in computer science has been identified as particularly problematic. Computer Science has one the worst records for retention (i.e. students leaving with no award, or a lower award than that originally applied for), and the second worst for attainment (i.e. achieving a good degree, that being defined as a first or a 2:1). One way to attempt to improve these outcomes is by identifying effective ways to improve student engagement. This can be through appropriate motivators – though then the balance of extrinsic versus intrinsic motivation becomes critical. In this paper, we consider how to utilize assessment – combining the formative and summative aspects - as a substitute for coarser approaches based on attendance monitoring

A flexible approach to introductory programming : engaging and motivating students

© 2019 Copyright is held by the owner/author(s). In this paper, we consider an approach to supporting students of Computer Science as they embark upon their university studies. The transition to Computer Science can be challenging for students, and equally challenging for those teaching them. Issues that are unusual – if not unique – to teaching computing at this level include • the wide variety in students background, varying from no prior experience to extensive development practice; • the positives and negatives of dealing with self-taught hobbyists who may developed buggy mental models of the task in hand and are not aware of the problem; • the challenge of getting students to engage with material that includes extensive practical element; • the atypical profile of a computing cohort, with typically 80%+ male students. The variation in background includes the style of prior academic experience, with some students coming from traditional level 3 (i.e. A-levels), some through more vocational routes (e.g. B-Tech, though these have changed in recent years), through to those from experiential (work based) learning. Technical background varies from science, mathematical and computing experience, to no direct advanced technical or scientific experience. A further issue is students’ attainment and progression within higher education, where the success and outcomes in computer science has been identified as particularly problematic. Computer Science has one the worst records for retention (i.e. students leaving with no award, or a lower award than that originally applied for), and the second worst for attainment (i.e. achieving a good degree, that being defined as a first or a 2:1). One way to attempt to improve these outcomes is by identifying effective ways to improve student engagement. This can be through appropriate motivators – though then the balance of extrinsic versus intrinsic motivation becomes critical. In this paper, we consider how to utilize assessment – combining the formative and summative aspects - as a substitute for coarser approaches based on attendance monitoring

p-forms on d-spherical tessellations

The spectral properties of p-forms on the fundamental domains of regular tesselations of the d-dimensional sphere are discussed. The degeneracies for all ranks, p, are organised into a double Poincare series which is explicitly determined. In the particular case of coexact forms of rank (d-1)/2, for odd d, it is shown that the heat--kernel expansion terminates with the constant term, which equals (-1)^{p+1}/2 and that the boundary terms also vanish, all as expected. As an example of the double domain construction, it is shown that the degeneracies on the sphere are given by adding the absolute and relative degeneracies on the hemisphere, again as anticipated. The eta invariant on a fundamental domain is computed to be irrational. The spectral counting function is calculated and the accumulated degeneracy give exactly. A generalised Weyl-Polya conjecture for p-forms is suggested and verified.Comment: 23 pages. Section on the counting function adde

On Love-type waves in a finitely deformed magnetoelastic layered half-space

In this paper, the propagation of Love-type waves in a homogeneously and finitely deformed layered half-space of an incompressible non-conducting magnetoelastic material in the presence of an initial uniform magnetic field is analyzed. The equations and boundary conditions governing linearized incremental motions superimposed on an underlying deformation and magnetic field for a magnetoelastic material are summarized and then specialized to a form appropriate for the study of Love-type waves in a layered half-space. The wave propagation problem is then analyzed for different directions of the initial magnetic field for two different magnetoelastic energy functions, which are generalizations of the standard neo-Hookean and Mooney–Rivlin elasticity models. The resulting wave speed characteristics in general depend significantly on the initial magnetic field as well as on the initial finite deformation, and the results are illustrated graphically for different combinations of these parameters. In the absence of a layer, shear horizontal surface waves do not exist in a purely elastic material, but the presence of a magnetic field normal to the sagittal plane makes such waves possible, these being analogous to Bleustein–Gulyaev waves in piezoelectric materials. Such waves are discussed briefly at the end of the paper