On the structure of tensor operators in SU3

A global algebraic formulation of SU3 tensor operator structure is achieved. A single irreducible unitary representation (irrep),V, ofso(6, 2) is constructed which contains every SU3 irrep precisely once. An algebra of polynomial differential operatorsA acting onV is given. The algebraA is shown to consist of linear combinations of all SU3 tensor operators with polynomial invariant operators as coefficients. By carrying out an analysis ofA, the multiplicity problem for SU3 tensor operators is resolved.Supported in part by the National Science Foundation     

Letter: Active Gravitational Mass and the Invariant Characterization of Reissner–Nordström Spacetime

We analyse the concept of active gravitational mass for Reissner-Nordström spacetime in terms of scalar polynomial invariants and the Karlhede classification. We show that while the Kretschmann scalar does not produce the expected expression for the active gravitational mass, both scalar polynomial invariants formed from the Weyl tensor, and the Cartan scalars, do.     

Local obstructions to projective surfaces admitting skew-symmetric Ricci tensor

The equation determining whether a projective structure admits a connection in its given projective class that has skew-symmetric Ricci tensor is an overdetermined system of semi-linear partial differential equations which we call the projective Einstein–Weyl (pEW) equation. In 2-dimensions, we give local obstructions for projective surfaces to admit such a connection in its projective class. The obstructions are the resultants of polynomial equations that have to be satisfied for there to admit any pEW solution.     

Three-dimensional space-times

A real version of the Newman-Penrose formalism is developed for (2+1)-dimensional space-times. The complete algebraic classification of the (Ricci) curvature is given. The field equations of Deser, Jackiw, and Templeton, expressing balance between the Einstein and Bach tensors, are reformulated in triad terms. Two exact solutions are obtained, one characterized by a null geodesic eigencongruence of the Ricci tensor, and a second for which all the polynomial curvature invariants are constant.     

Discriminating the Weyl type in higher dimensions using scalar curvature invariants

We consider higher dimensional Lorentzian spacetimes which are currently of interest in theoretical physics. It is possible to algebraically classify any tensor in a Lorentzian spacetime of arbitrary dimensions using alignment theory. In the case of the Weyl tensor, and using bivector theory, the associated Weyl curvature operator will have a restricted eigenvector structure for algebraic types II and D, which leads to necessary conditions on the discriminants of the associated characteristic equation which can be manifestly expressed in terms of polynomial scalar curvature invariants. The use of such necessary conditions in terms of scalar curvature invariants will be of great utility in the study and classification of black hole solutions in more than four dimensions.     

The regularity of static spherically cylindrically and plane symmetric spacetimes at the origin

We find the necessary and sufficient conditions for the regularity of all scalar invariants polynomial in the Riemann tensor at the origin of spherically, cylindrically and plane symmetric static spacetimes under the assumption that the metric functions are sufficiently smooth there. These conditions turn out to be simple enough to allow a check for regularity by inspection.     

Generalized Kerr–NUT–de Sitter metrics in all dimensions

We classify all spacetimes with a closed rank-2 conformal Killing–Yano tensor. They give a generalization of Kerr–NUT–de Sitter spacetimes. The Einstein condition is explicitly solved and written as an indefinite integral. It is characterized by a polynomial in the integrand. We briefly discuss the smoothness conditions of the Einstein metrics over compact Riemannian manifolds.     

Cyclic monodromy matrices forsl(n) trigonometricR-matrices

The algebra of monodromy matrices forsl(n) trigonometricR-matrix is studied. It is shown that a generic finite-dimensional polynomial irreducible representation of this algebra is equivalent to a tensor product ofL-operators. Cocommutativity of representations is discussed and intertwiners for factorizable representations are written through the Boltzmann weights of thesl(n) chiral Potts model.     

The elastic dielectric

Macroscopic field equations, boundary conditions and equations of state are derived for the non-linear, macroscopic elastic and dielectric response of an insulator. A centrosymmetric polynomial representation of order four is introduced for the energy density; the equations of state for the electric field and stress tensor are then deduced as polynomials of degree three in the displacement gradients and electric displacement field. The results are applied to the special case of m3m material symmetry.

A finite, point-charge model of a centrosymmetric ionic crystal is introduced and used to determine 0°K microscopic expressions for the electric field and stress tensor equation of state coefficients introduced in the macroscopic analysis. The results are used to calculate the full set of second and third-order non-linear coefficients for NaI, based on a Born-Mayer potential and the 4·2°K elastic stiffness data of Claytor and Marshall.     

Invariants of the Riemann tensor: A classical approach

This paper describes the start of an investigation into the application of classical invariant theory to scalar polynomial invariants of the Riemann tensor. In particular, the classical methods of enumerating invariants are discussed with the aim, not achieved in this paper, of verifying Sneddon's result, obtained by explicit calculation of the invariants that the dimension of a Hilbert basis is 38.     

Affine and polynomial mutual information coregistration for artifact elimination in diffusion tensor imaging of newborns

We have investigated the use of two different image coregistration algorithms for identifying local regions of erroneously high fractional anisotropy (FA) as derived from diffusion tensor imaging (DTI) data sets in newborns. The first algorithm uses conventional affine registration of each of the diffusion-weighted images to the unweighted (b = 0) image for each slice, while the second algorithm uses second-order polynomial warping. Similarity between images was determined using the mutual information (MI) criterion, which is the preferred 'cost' criterion for coregistration of images with significantly different image intensity distributions. We have found that subtle differences exist in the FA values resulting from affine and second-order polynomial coregistration and demonstrate that nonlinear distortions introduce artifacts of spatial extent similar to real white matter structures in the newborn subcortex. We show that polynomial coregistration systematically reduces the presence of erroneous regions of high FA and that such artifacts can be identified by visual inspection of FA maps resulting from affine and polynomial coregistrations. Furthermore, we show that nonlinear distortions may be particularly pronounced when acquiring image slices of axial orientation at the height of the nasal cavity. Finally, we show that third-order polynomial MI coregistration (using the images resulting from second-order coregistration as input) has no observable effect on the resulting FA maps.     

Universal Star Products

One defines the notion of universal deformation quantization: given any manifold M, any Poisson structure Λ on M and any torsionfree linear connection ? on M, a universal deformation quantization associates to this data a star product on (M, Λ) given by a series of bidifferential operators whose corresponding tensors are given by universal polynomial expressions in the Poisson tensor Λ, the curvature tensor R and their covariant iterated derivatives. Such universal deformation quantization exist. We study their unicity at order 3 in the deformation parameter, computing the appropriate universal Poissoncohomology.     

Some observations on degenerate metrics

There are some polynomial formulations of Einstein's equations in which the metric is allowed to become degenerate. We examine some known exact solutions to see whether they may be smoothly joined to solutions with degenerate metrics. If one uses a lapse function which is a spatial scalar, this is very easy. If the lapse function has a small and negative tensor density weight, the joining together may take place across the horizons in the Schwarzschild and Kerr solutions. For large and negative weights, we have been unable to find any examples.     

The( p,q) inflation model

In this paper we propose a new inflation model named( p, q) inflation model in which the inflaton potential contains both positive and negative powers of inflaton field in the polynomial form. We derive the accurate predictions of the canonical single-field slow-roll inflation model. Using these formula, we show that our inflation model can easily generate a large amplitude of tensor perturbation and a negative running of spectral index with large absolute value.     

Unitary group decomposition of hamiltonian operators. II. SU (4) irreducible tensors,norms and their energy variation and symmetry breaking

SU(6) ? SU(4) tensor decomposition of effective interactions in the 2s-1d shell has been carried out to examine the relative importance of the various irreducible tensors in many-particle spaces. For this purpose norms of the irreducible tensors are evaluated in many-particle spaces. Variation of the expectation value of the square of the irreducible tensor parts with excitation energy has also been examined using the polynomial expansion method. A new measure of symmetry breaking that is theoretically more sound is derived which includes in its definition partial width as well as internal width. This is used to study SU(4) symmetry mixing in nuclei.     

B-Spline Gaussian Collocation Software for Two-Dimensional Parabolic PDEs

In this paper we describe new B-spline Gaussian collocation software for solving two-dimensional parabolic partial differential equations (PDEs) defined over a rectangular region. The numerical solution is represented as a bi-variate piecewise polynomial (using a tensor product B-spline basis) with time-dependent unknown coefficients. These coefficients are determined by imposing collocation conditions: the numerical solution is required to satisfy the PDE and boundary conditions at images of the Gauss points mapped onto certain subregions of the spatial domain. This leads to a large system of time-dependent differential algebraic equations (DAEs) which is solved using the DAE solver, DASPK. We provide numerical results in which we use the new software, called BACOL2D, to solve three test problems.     

Ellipsometry of anisotropic materials: A new efficient polynomial approach

Dielectric multilayer stacks can exhibit anisotropy along the normal to the substrate. We present a polynomial approach for the spectroscopic ellipsometry of anisotropic multilayer structure in the cases where the off-diagonal elements of the dielectric tensor are zero. The ellipsometric parameters ψ and Δ of a stratified anisotropic planar structure can be written in a form so simple that they can be given directly without any calculation for any number of interfaces. The variation of ψ and Δ of two simple structures with the incidence angle is shown. The numerical results reveal an exact match with the well known traditional formalism.     

On the riemannian metrics in which admit all hyperplanes as minimal hypersurfaces

Inspired by a result of Bekkar (1991), Robert Lutz raised the following problem: determine the riemannian metrics in domains of ⁿ which admit all hyperplanes as minimal hypersurfaces. We solve the problem giving a formula which expresses its solutions in terms of the non-degenerate quadratic first integrals of the geodesic motion in the euclidean space (second-order Killing tensor fields). Then, we prove that for n = 3 the non-flat polynomial solutions of the problem are the left invariant riemannian metrics on the Heisenberg group.