Reflexivity of the translation-dilation algebras on L^2(R)

The hyperbolic algebra A_h, studied recently by Katavolos and Power, is the weak star closed operator algebra on L^2(R) generated by H^\infty(R), as multiplication operators, and by the dilation operators V_t, t \geq 0, given by V_t f(x) = e^{t/2} f(e^t x). We show that A_h is a reflexive operator algebra and that the four dimensional manifold Lat A_h (with the natural topology) is the reflexive hull of a natural two dimensional subspace.Comment: 10 pages, no figures To appear in the International Journal of Mathematic

A molecular theory for two-photon and three-photon fluorescence polarization

In the analysis of molecular structure and local order in heterogeneous samples, multiphoton excitation of fluorescence affords chemically specific information and high-resolution imaging. This report presents the results of an investigation that secures a detailed theoretical representation of the fluorescence polarization produced by one-, two-, and three-photon excitations, with orientational averaging procedures being deployed to deliver the fully disordered limits. The equations determining multiphoton fluorescence response prove to be expressible in a relatively simple, generic form, and graphs exhibit the functional form of the multiphoton fluorescence polarization. Amongst other features, the results lead to the identification of a condition under which the fluorescence produced through the concerted absorption of any number of photons becomes completely unpolarized. It is also shown that the angular variation of fluorescence intensities is reliable indicator of orientational disorder

Interpreting and Reporting Principal Component Analysis in Food Science Analysis and Beyond

Principal component analysis (PCA) is one of the most widely used data mining techniques in sciences and applied to a wide type of datasets (e.g. sensory, instrumental methods, chemical data). However, several questions and doubts on how to interpret and report the results are still asked every day from students and researchers. This brief communication is inspired in relation to those questions asked by colleagues and students. Please note that this article is a focus on the practical aspects, use and interpretation of the PCA to analyse multiple or varied data sets. In summary, the application of the PCA provides with two main elements, namely the scores and loadings. The scores provide with a location of the sample where the loadings indicate which variables are the most important to explain the trends in the grouping of samples

Scaling Symmetries of Scatterers of Classical Zero-Point Radiation

Classical radiation equilibrium (the blackbody problem) is investigated by the use of an analogy. Scaling symmetries are noted for systems of classical charged particles moving in circular orbits in central potentials V(r)=-k/r^n when the particles are held in uniform circular motion against radiative collapse by a circularly polarized incident plane wave. Only in the case of a Coulomb potential n=1 with fixed charge e is there a unique scale-invariant spectrum of radiation versus frequency (analogous to zero-point radiation) obtained from the stable scattering arrangement. These results suggest that non-electromagnetic potentials are not appropriate for discussions of classical radiation equilibrium.Comment: 13 page

A Generalised RBF Finite Difference Approach to Solve Nonlinear Heat Conduction Problems on Unstructured Datasets

Radial Basis Functions have traditionally been used to provide a continuous interpolation of scattered data sets. However, this interpolation also allows for the reconstruction of partial derivatives throughout the solution field, which can then be used to drive the solution of a partial differential equation. Since the interpolation takes place on a scattered dataset with no local connectivity, the solution is essentially meshless. RBF-based methods have been successfully used to solve a wide variety of PDEs in this fashion. Such full-domain RBF methods are highly flexible and can exhibit spectral convergence rates Madych & Nelson (1990). However, in their traditional implementation the fully-populated matrix systems which are produced lead to computational complexities of at least order-N2 with datasets of size N. In addition, they suffer fromincreasingly poor numerical conditioning as the size of the dataset grows, and also with increasingly flat interpolating functions. This is a consequence of ill-conditioning in the determination of RBF weighting coefficients (as demonstrated in Driscoll & Fornberg (2002)), and is described by Robert Schaback Schaback (1995) as the uncertainty relation; better conditioning is associated with worse accuracy, and worse conditioning is associated with improved accuracy. Many techniques have been developed to reduce the effect of the uncertainty relation in the traditional RBF formulation, such as RBF-specific preconditioners Baxter (2002); Beatson et al. (1999); Brown (2005); Ling & Kansa (2005), or adaptive selection of data centres Ling et al. (2006); Ling & Schaback (2004). However, at present the only reliable methods of controlling numerical ill-conditioning and computational cost as problem size increases are domain decomposition Hernandez Rosales & Power (2007); Wong et al. (1999); Zhang (2007); Zhou et al. (2003), or the use of locally supported basis functions Fasshauer (1999); Schaback (1997); Wendland (1995); Wu (1995). In this work the domain decomposition principle is applied, forming a large number of heavily overlapping systems that cover the solution domain. A small RBF collocation system is formed around each global data centre, with each collocation system used to approximate the governing PDE at its centrepoint, in terms of the solution value at surrounding collocation points. This leads to a sparse global linear system which may be solved using a variety of standard solvers. In this way, the proposed formulation emulates a finite difference method, with the RBF collocation systems replacing the polynomial interpolation functions used in traditional finite difference methods. However, unlike such polynomial functions RBF collocation is well suited to scattered data, and the method may be applied to both structured and unstructured datasets without modification. The method is applied here to solve the nonlinear heat conduction equation. In order to reduce the nonlinearity in the governing equation the Kirchhoff integral transformation is applied, and the transformed equation is solved using a Picard iterative process. The application of the Kirchhoff transform necessitates that the thermal property functions be transformed to Kirchhoff space also. If the thermal properties are a known and integrable function of temperature then the transformation may be performed analytically. Otherwise, an integration-interpolation procedure can be performed using 1D radial basis functions, as described in Stevens & Power (2010). In recent years a number of local RBF collocation techniques have been proposed, and applied a wide variety of problems (for example; Divo & Kassab (2007); Lee et al. (2003); Sarler & Vertnik (2006); Wright & Fornberg (2006)). A more comprehensive review of such methods is given in Stevens et al. (2009). Unlike most local RBF collocation methods that are used in the literature, the technique described here utilises the Hermitian RBF collocation formulation (see section 2 for more details), and allows both the PDE-boundary and PDE-governing operators to be included within in the local collocation systems. This inclusion of the governing PDE within the basis functions is shown in Stevens et al. (2009) to significantly improve the accuracy and stability of solutions obtained for linear transport problems. Additionally, the incorporation of information about the convective velocity field into the basis functionswas shown to have a stabilising effect, similar to traditional upwinding methods but without the requirement to alter the stencil configuration based on the local convective field. The standard approach to the solution of linear and nonlinear heat conduction problems is the use of finite difference and finite volume methods with simple polynomial interpolants Bejan (1993); Holman (2002); Kreith & Bohn (2000). Due to the dominance of diffusion in most cases, central differencing techniques are commonly used to compute the heat fluxes. However, limiter methods (such as the unconditionally stable TVD schemes) may be used for nonlinear heat conduction problems where the effective convection term, which results from the non-zero variation of thermal conductivity with temperature, can be expected to approach the magnitude of the diffusive term (see, for example, Shen & Han (2002)). Full-domain RBF methods have also been examined for use with nonlinear heat conduction problems (see Chantasiriwan (2007)), however such methods are restricted to small dataset sizes, due to the computational cost and numerical conditioning experienced by full-domain RBF techniques on large datasets. The present work demonstrates how local RBF collocation may be used as an alternative to traditional finite difference and finite volume methods, for nonlinear heat conduction problems. The described method retains freedom from a volumetric mesh, while allowing solution over unstructured datasets. A central stencil configuration is used in each case, and the solution is stabilised via the inclusion of the governing and boundary PDEs within the local collocation systems (\u201cimplicit upwinding\u201d), rather than by adjusting the stencil configuration based on the local solution field (\u201ctraditional upwinding\u201d). The method is validated using a transient numerical example with a known analytical solution (see section 4), and the ability of the formulation to handle strongly nonlinear problems is demonstrated in the solution of a food freezing problem (see section 5)