Vacuum Instability

NAbstract Following fresh attempts to resolve the problem of the energy density of the vacuum, we reconsider the case where the cosmological constant is derived from a higher-dimensional version of general relativity, and interpret the gauge-dependence of Λ as a dynamical effect. This leads to a relation between the change in Λ and the line element (action) which is independent of gauge choices and fundamental constants: dΛds² = −6. This implies that the (classical) vacuum is unstable, with implications for particle production.     

Zeta regularized determinants for conic manifolds

We study (relative) zeta regularized determinants of Laplace type operators on compact conic manifolds. We establish gluing formulae for relative zeta regularized determinants. For arbitrary self-adjoint extensions of the Laplace-Beltrami operator, we express the relative ζ-determinants for these as a ratio of the determinants of certain finite matrices. For the self-adjoint extensions corresponding to Dirichlet and Neumann conditions, the formula is particularly simple and elegant.     

Metric spaces with discrete topological fundamental group

With a certain natural topology, the fundamental group of a locally path connected metric space X becomes discrete if and only if X is semilocally simply connected. We include some related material and offer a counterexample to a similar result.     

How to Exhibit Toroidal Maps in Space

Steinitz's theorem states that a graph is the 1-skeleton of a convex polyhedron if and only if it is 3-connected and planar. The polyhedron is called a geometric realization of the embedded graph. Its faces are bounded by convex polygons whose points are coplanar. A map on the torus does not necessarily have such a geometric realization. In this paper we relax the condition that faces are the convex hull of coplanar points. We require instead that the convex hull of the points on a face can be projected onto a plane so that the boundary of the convex hull of the projected points is the image of the boundary of the face. We also require that the interiors of the convex hulls of different faces do not intersect. Call this an exhibition of the map. A map is polyhedral if the intersection of any two closed faces is simply connected. Our main result is that every polyhedral toroidal map can be exhibited. As a corollary, every toroidal triangulation has a geometric realization.     

When maths really is rocket science

Two problems in projectiles are discussed which generate interestingloci.     

Steady state analysis of an M/G/1 queue with two phase service and Bernoulli vacation schedule under multiple vacation policy

This paper examines the steady state behaviour of a batch arrival queue with two phases of heterogeneous service along and Bernoulli schedule vacation under multiple vacation policy, where after two successive phases service or first vacation the server may go for further vacations until it finds a new batch of customer in the system. We carry out an extensive stationary analysis of the system, including existence of stationary regime, queue size distribution of idle period process, embedded Markov chain steady state distribution of stationary queue size, busy period distribution along with some system characteristics.     

A comparative electrochemical study of diffusion in room temperature ionic liquid solvents versus acetonitrile.

Measurements on the diffusion coefficient of the neutral molecule N,N,N',N'-tetramethyl-para-phenylenediamine and the radical cation and dication generated by its one- and two-electron oxidation, respectively, are reported over the range 298-348 K in both acetonitrile and four room temperature ionic liquids (RTILs). Data were collected using single and double potential step chronoamperometry at a gold disk electrode of micrometer dimension, and analysed via fitting to the appropriate analytical expression or, where necessary, to simulation. The variation of diffusion coefficient with temperature was found to occur in an Arrhenius-type manner for all combinations of solute and solvent. For a given ionic liquid, the diffusional activation energies of each species were not only closely equivalent to each other, but also to the RTIL's activation energy of viscous flow. In acetonitrile supported with 0.1 M tetrabutylammonium perchlorate, the ratio in diffusion coefficients of the radical cation and dication to the neutral molecule were calculated as 0.89 +/- 0.05 and 0.51 +/- 0.03, respectively. In contrast, amongst the ionic liquids the same ratios were determined to be on average 0.53 +/- 0.04 and 0.33 +/- 0.03. The consequences of this dissimilarity are considered in terms of the modelling of voltammetric data gathered within ionic liquid solvents.