The rings of n-dimensional polytopes

Points of an orbit of a finite Coxeter group G, generated by n reflections starting from a single seed point, are considered as vertices of a polytope (G-polytope) centered at the origin of a real n-dimensional Euclidean space. A general efficient method is recalled for the geometric description of G- polytopes, their faces of all dimensions and their adjacencies. Products and symmetrized powers of G-polytopes are introduced and their decomposition into the sums of G-polytopes is described. Several invariants of G-polytopes are found, namely the analogs of Dynkin indices of degrees 2 and 4, anomaly numbers and congruence classes of the polytopes. The definitions apply to crystallographic and non-crystallographic Coxeter groups. Examples and applications are shown.Comment: 24 page

Vacuum Plane Waves in 4+1 D and Exact solutions to Einstein's Equations in 3+1 D

In this paper we derive homogeneous vacuum plane-wave solutions to Einstein's field equations in 4+1 dimensions. The solutions come in five different types of which three generalise the vacuum plane-wave solutions in 3+1 dimensions to the 4+1 dimensional case. By doing a Kaluza-Klein reduction we obtain solutions to the Einstein-Maxwell equations in 3+1 dimensions. The solutions generalise the vacuum plane-wave spacetimes of Bianchi class B to the non-vacuum case and describe spatially homogeneous spacetimes containing an extremely tilted fluid. Also, using a similar reduction we obtain 3+1 dimensional solutions to the Einstein equations with a scalar field.Comment: 16 pages, no figure

Lie group weight multiplicities from conformal field theory

Dominant weight multiplicities of simple Lie groups are expressed in terms of the modular matrices of Wess-Zumino-Witten conformal field theories, and related objects. Symmetries of the modular matrices give rise to new relations among multiplicities. At least for some Lie groups, these new relations are strong enough to completely fix all multiplicities.Comment: 12 pages, Plain TeX, no figure

Homogeneous Plane-wave Spacetimes and their Stability

We consider the stability of spatially homogeneous plane-wave spacetimes. We carry out a full analysis for plane-wave spacetimes in (4+1) dimensions, and find there are two cases to consider; what we call non-exceptional and exceptional. In the non-exceptional case the plane waves are stable to (spatially homogeneous) vacuum perturbations as well as a restricted set of matter perturbations. In the exceptional case we always find an instability. Also we consider the Milne universe in arbitrary dimensions and find it is also stable provided the strong energy condition is satisfied. This implies that there exists an open set of stable plane-wave solutions in arbitrary dimensions.Comment: 15 pages, no figures; minor changes, new references, to appear in CQ

Exact Solutions of a (2+1)-Dimensional Nonlinear Klein-Gordon Equation

The purpose of this paper is to present a class of particular solutions of a C(2,1) conformally invariant nonlinear Klein-Gordon equation by symmetry reduction. Using the subgroups of similitude group reduced ordinary differential equations of second order and their solutions by a singularity analysis are classified. In particular, it has been shown that whenever they have the Painlev\'e property, they can be transformed to standard forms by Moebius transformations of dependent variable and arbitrary smooth transformations of independent variable whose solutions, depending on the values of parameters, are expressible in terms of either elementary functions or Jacobi elliptic functions.Comment: 16 pages, no figures, revised versio

Essential Constants for Spatially Homogeneous Ricci-flat manifolds of dimension 4+1

The present work considers (4+1)-dimensional spatially homogeneous vacuum cosmological models. Exact solutions -- some already existing in the literature, and others believed to be new -- are exhibited. Some of them are the most general for the corresponding Lie group with which each homogeneous slice is endowed, and some others are quite general. The characterization ``general'' is given based on the counting of the essential constants, the line-element of each model must contain; indeed, this is the basic contribution of the work. We give two different ways of calculating the number of essential constants for the simply transitive spatially homogeneous (4+1)-dimensional models. The first uses the initial value theorem; the second uses, through Peano's theorem, the so-called time-dependent automorphism inducing diffeomorphismsComment: 26 Pages, 2 Tables, latex2

Phase field modeling of electrochemistry I: Equilibrium

A diffuse interface (phase field) model for an electrochemical system is developed. We describe the minimal set of components needed to model an electrochemical interface and present a variational derivation of the governing equations. With a simple set of assumptions: mass and volume constraints, Poisson's equation, ideal solution thermodynamics in the bulk, and a simple description of the competing energies in the interface, the model captures the charge separation associated with the equilibrium double layer at the electrochemical interface. The decay of the electrostatic potential in the electrolyte agrees with the classical Gouy-Chapman and Debye-H\"uckel theories. We calculate the surface energy, surface charge, and differential capacitance as functions of potential and find qualitative agreement between the model and existing theories and experiments. In particular, the differential capacitance curves exhibit complex shapes with multiple extrema, as exhibited in many electrochemical systems.Comment: v3: To be published in Phys. Rev. E v2: Added link to cond-mat/0308179 in References 13 pages, 6 figures in 15 files, REVTeX 4, SIUnits.sty. Precedes cond-mat/030817

Kaon decay interferometry as meson dynamics probes

We discuss the time dependent interferences between $K_L$ and $K_S$ in the decays in $3\pi$ and $\pi\pi\gamma$, to be studied at interferometry machines such as the $\phi$-factory and LEAR. We emphasize the possibilities and the advantages of using interferences, in comparison with width measurements, to obtain information both on $CP$ conserving and $CP$ violating amplitudes. Comparison with present data and suggestions for future experiments are made.Comment: 15 pages, in RevTex, Report INFNNA-IV-93-31, UTS-DFT-93-2

Three dimensional quantum algebras: a Cartan-like point of view

A perturbative quantization procedure for Lie bialgebras is introduced and used to classify all three dimensional complex quantum algebras compatible with a given coproduct. The role of elements of the quantum universal enveloping algebra that, analogously to generators in Lie algebras, have a distinguished type of coproduct is discussed, and the relevance of a symmetrical basis in the universal enveloping algebra stressed. New quantizations of three dimensional solvable algebras, relevant for possible physical applications for their simplicity, are obtained and all already known related results recovered. Our results give a quantization of all existing three dimensional Lie algebras and reproduce, in the classical limit, the most relevant sector of the complete classification for real three dimensional Lie bialgebra structures given by X. Gomez in J. Math. Phys. Vol. 41. (2000) 4939.Comment: LaTeX, 15 page

Extended calibration range for prompt photon emission in ion beam irradiation

Monitoring the dose delivered during proton and carbon ion therapy is still a matter of research. Among the possible solutions, several exploit the measurement of the single photon emission from nuclear decays induced by the irradiation. To fully characterize such emission the detectors need development, since the energy spectrum spans the range above the MeV that is not traditionally used in medical applications. On the other hand, a deeper understanding of the reactions involving gamma production is needed in order to improve the physic models of Monte Carlo codes, relevant for an accurate prediction of the prompt-gamma energy spectrum.This paper describes a calibration technique tailored for the range of energy of interest and reanalyzes the data of the interaction of a 80MeV/u fully stripped carbon ion beam with a Poly-methyl methacrylate target. By adopting the FLUKA simulation with the appropriate calibration and resolution a significant improvement in the agreement between data and simulation is reported.Comment: 4 pages, 7 figures, Submitted to JINS