arXiv subjects

J. S. Dowker

Publications and source records attributed to J. S. Dowker.

At least 73 records · Page 4Linked to original sources

arXiv2013

Central differences, Euler numbers and symbolic methods

I relate some coefficients encountered when computing the functional determinants on spheres to the central differentials of nothing. In doing this I use some historic works, in particular transcribing the elegant symbolic formalism of Jeffery (1861) into central difference form which has computational advantages for Euler numbers, as discovered by Shovelton (1915). I derive sum rules for these, and for the central differentials, the proof of which involves an interesting expression for powers of sech x as multiple derivatives. I present a more general, symbolic treatment of central difference calculus which allows known, and unknown, things to be obtained in an elegant and compact fashion gaining, at no cost, the expansion of the powers of the inverse sinh, a basic central function. Systematic use is made of the operator 2 asinh(D/2). Umbral calculus is employed to compress the operator formalism. For example the orthogonality/completeness of the factorial numbers, of the first and second kinds, translates, umbrally, to T(t(x))=x. The classic expansions of multiple angle cosh and sinh in terms of powers of sinh are thence obtained with minimal effort.

math.NA↗

arXiv2013

On Sylvester Waves and restricted partitions

The higher Sylvester waves are discussed. Techniques used involve finite difference operators. For example, using Herschel's theorem, elegant expressions for Euler's rational functions and the Todd operator are found. Derivative expansions are also rapidly treated by the same method. A general form for the wave is obtained using multiplicative series, and comments made on its further reduction. As is known, Dedekind sums arise in the case of coprime components and it is pointed out that Brioschi had this result, but not the terminology, very early on. His proof is repeated. Adding a set of ones to the components of the denumerant corresponds to a succession of (discrete) smoothings and is a Cesaro sum. Using a spectral vocabulary, I take the opportunity to exhibit the finite difference counterparts of some continuous, distributional properties of Riesz typical means.

math.NT↗

arXiv2013

Sphere Renyi entropies

I give some scalar field theory calculations on a d-dimensional lune of arbitrary angle, evaluating, numerically, the effective action which is expressed as a simple quadrature, for conformal coupling. Using this, the entanglement and Renyi entropies are computed. Massive fields are also considered and a renormalisation to make the (one-loop) effective action vanish for infinite mass is suggested and used, not entirely successfully. However a universal coefficient is derived from the large mass expansion. From the deformation of the corresponding lune result, I conjecture that the effective action on all manifolds with a simple conical singularity has an extremum when the singularity disappears. For the round sphere, I show how to convert the quadrature form of the conformal Laplacian determinant into the more usual sum of Riemann zeta functions (and log2).

hep-th↗

arXiv2013

A note on the torque anomaly

I reproduce, in the case of a conical geometry, the torque anomaly recently noted by Fulling, Mera and Trendafilova for the wedge. The expected conservation equation is obtained by a variational method and a mathematical cancellation of the anomaly is exhibited, motivated by the process of truncating the cone at some inner radius.

hep-th↗

arXiv2013

Lens space determinants

More analysis of operator determinants on homogeneous three dimensional lens spaces is presented with the emphasis on numerics so that Laplacians for massive fields can be dealt with. Polyhedral quotients are also briefly considered. Twisted fields, corresponding to flat connections, are looked at and examples of determinants computed. Twisted cyclic quantities are sufficient to determine those for any twisting on any factor. An application to the thermodynamics on sphere quotients is given. Some computations are made for inhomogeneous lens spaces and higher dimensions are commented on. Minimal coupling is also dealt with.

math-ph↗

arXiv2012

Heat-kernels on the discrete circle and interval

As is known, the free heat-kernel on the integers (a modified Bessel function) is turned into the periodic free heat-kernel on the discrete circle by factoring, giving a pre-image sum. I generalise existing treatments by making the functions periodic up to a phase, thus introducing an extra parameter into the analysis. Identifying the classical paths form with the conventional eigenfunction expression, I find a combinatorial trace identity which allows various Bessel identities to be extracted, such as a generalisation of the Jacobi-Anger expansion.The free Dirichlet, Neumann and hybrid Dirichlet-Neumann heat-kernels on a discrete interval are constructed using both modes and images. The Neumann imaging mirror has to be placed at a half-integer. The corresponding lattice Green functions are expressed in terms of Chebyshev polynomials and the Laplacian matrices extracted. The generating functions for circuits with bumps are evaluated.

math-ph↗

arXiv2012

Discrete eigenproblems

Schrodinger eigenproblems on a discrete interval are further investigated with special attention given to test cases such as the linear and Rosen--Morse potentials. In the former case it is shown that the characteristic function determining the eigenvalues is a Lommel polynomial and considerable space is devoted to these objects. For example it is shown that the continuum limit of the determinant is obtained by a transitional limit of the Lommel polynomials for large order and argument. Numerical comparisons between discrete approximations and (known) continuum values for the ratio of functional determinants with and without the potential are made and show good agreement, even for small numbers of vertices. The zero mode problem is also briefly dealt with.

math.SP↗

arXiv2011

Discrete Determinants and the Gel'fand-Yaglom formula

I present a partly pedagogic discussion of the Gel'fand-Yaglom formula for the functional determinant of a one-dimensional, second order difference operator, in the simplest settings. The formula is a textbook one in discrete Sturm-Liouville theory and orthogonal polynomials. A two by two matrix approach is developed and applied to Robin boundary conditions. Euler-Rayleigh sums of eigenvalues are computed. A delta potential is introduced as a simple, non-trivial example and extended, in an appendix, to the general case. The continuum limit is considered in a non--rigorous way and a rough comparison with zeta regularised values is made. Vacuum energies are also considered in the free case. Chebyshev polynomials act as free propagators and their properties are developed using the two-matrix formulation, which has some advantages and appears to be novel. A trace formula, rather than a determinant one, is derived for the Gel'fand-Yaglom function.

math-ph↗

arXiv2011

Relations between the Ehrhart polynomial, the heat kernel and Sylvester waves

I show for the specific case of the scalar field spectrum on regular tessellations of the sphere that the first two terms of the heat--kernel expansion are related to the first two terms of the Ehrhart (quasi)polynomial. In trying to make this relation precise, I consider degeneracies as partition denumerants and show the connection of the group theory expressions with Popoviciu's theorem and with the notion of Sylvester waves. General denumerants are considered and the first wave, i.e. the polynomial part, is written using the A-genus multiplicative sequence. It is pointed out that Sylvester in effect did the same thing and that he had also obtained Ehrhart reciprocity. I derive an algebraically neat form for the second wave which involves the combination of two multiplicative sequences.

math.NT↗

arXiv2011

Vacuum energies on spheres and in cubes

The vacuum energy of a conformally coupled scalar field on the d-dimensional sphere is calculated. On even spheres it is zero and on odd spheres it oscillates in sign. Results for the d-torus and d-cube are also given.

hep-th↗

arXiv2011

Spherical Casimir pistons

A piston is introduced into a spherical lune Casimir cavity turning it into two adjacent lunes separated by the (hemispherical) piston. On the basis of zeta function regularisation, the vacuum energy of the arrangement is finite for conformal propagation in space-time. For even spheres this energy is independent of the angle of the lune. For odd dimensions it is shown that for all Neumann, or all Dirichlet, boundary conditions the piston is attracted or repelled by the nearest wall if d=3,7,... or if d=1,5,..., respectively. For hybrid N-D conditions these requirements are switched. If a mass is added, divergences arise which render the model suspect. The analysis, however, is relatively straightforward and involves the Barnes zeta function. The extension to finite temperatures is made and it is shown that for the 3,7,... series of odd spheres, the repulsion by the walls continues but that, above a certain temperature, the free energy acquires two minima symmetrically placed about the mid point.

hep-th↗

arXiv2010

Entanglement entropy for odd spheres

It is shown, non--rigorously, that the effective action on a Z_q factored odd spheres (lune) has a vanishing derivative at q=1. This leaves the effective action on the ordinary odd d-sphere as (minus) the value of the entanglement entropy associated with a (d-2)-sphere. Some numbers are given.

hep-th↗

arXiv2010

Determinants and conformal anomalies of GJMS operators on spheres

The conformal anomalies and functional determinants of the Branson--GJMS operators, P_{2k}, on the d-dimensional sphere are evaluated in explicit terms for any d and k such that k < d/2+1 (if d is even). The determinants are given in terms of multiple gamma functions and a rational multiplicative anomaly, which vanishes for odd d. Taking the mode system on the sphere as the union of Neumann and Dirichlet ones on the hemisphere is a basic part of the method and leads to a heuristic explanation of the non--existence of `super--critical' operators, 2k>d for even d. Significant use is made of the Barnes zeta function. The results are given in terms of ratios of determinants of operators on a (d+1)-dimensional bulk dual sphere. For odd dimensions, the log determinant is written in terms of multiple sine functions and agreement is found with holographic computations, yielding an integral over a Plancherel measure. The N-D determinant ratio is also found explicitly for even dimensions. Ehrhart polynomials are encountered.

hep-th↗

arXiv2010

Entanglement entropy for even spheres

The coefficient of the logarithmic term in the entropy on even spheres is re-computed by the local technique of integrating the finite temperature energy density up to the horizon on static d--dimensional de Sitter space and thence finding the entropy by thermodynamics. Numeric evaluation yields the known answer i.e. (minus) the conformal anomaly on the d-sphere. The de Sitter quantities are obtained by conformal transformation of the Rindler ones, themselves obtained, for convenience, from those around a cosmic string. The expressions are given in terms of generalised Bernoulli polynomials for which an identity is derived. The arising spherical conformal anomaly is discussed and a formula is given for it for Branson's higher GJMS Laplacian, P_2k, as an oscillating polynomial in the level, k.

hep-th↗

arXiv2010

Hyperspherical entanglement entropy

The coefficient of the log term in the entanglement entropy associated with hyperspherical surfaces in flat space-time is shown to equal the conformal anomaly by conformally transforming Euclideanised space--time to a sphere and using already existing formulae for the relevant heat--kernel coefficients after cyclic factoring. The analytical reason for the result is that the conformal anomaly on the lune has an extremum at the ordinary sphere limit. A proof is given. Agreement with a recent evaluation of the coefficient is found.

hep-th↗

arXiv2009

Harmonics on the factored three-sphere and the Hopf map

Laplacian eigenmodes on homogeneous Clifford--Klein factors of the three--sphere are obtained as pullbacks of harmonics on the orbifolded two--sphere using the Hopf map. A method of obtaining these polyhedral, or crystal, harmonics using binary invariants is presented which has computational advantages over those based on projection techniques, or those using invariants constructed in terms of Cartesian coordinates. In addition, modes transforming according to the irreps of the deck group are found in easy fashion using the covariants already conveniently calculated by Desmier and Sharp and by Bellon. Agreement is found with existing results.

gr-qc↗

arXiv2009

Group theory aspects of spectral problems on spherical factors

The Ray-Singer isospectral theorem (1971) is applied to a general spectral function for Laplacians of twisted p-forms (say) on homogeneous Clifford-Klein factors of the three-sphere. The inducing formulae necessary to express any spectral quantity for any twisting in terms of those for cyclic subgroups of the tetrahedral, octahedral and icosahedral deck groups are detailed. Further, Artin's theorem allows the McKay correspondence to be obtained. The isospectral theorem is shown to yield a derivation of the Sunada construction which is equivalent to the later one by Pesce.

math.DG↗

arXiv2009

Analytic torsion on spherical factors and tessellations

The analytic torsion is computed on fixed-point free and non fixed-point free factors (tessellations) of the three--sphere. We repeat the standard computation on spherical space forms (Clifford-Klein spaces) by an improved technique. The transformation to a simpler form of the spectral expression of the torsion on spherical factors effected by Ray is shown to be more general than his derivation implies. It effectively allows the eigenvalues to be considered as squares of integers, and applies also to trivial twistings. The analytic torsions compute to algebraic numbers, as expected. In the case of icosahedral space, the quaternion twisting gives a torsion proportional to the fundamental unit of Q(5^(1/2)). As well as a direct calculation, the torsions are obtained from the lens space values by a character inducing procedure.On tessellations, terms occur due to edge conical singularities.

math.DG↗