On the problem of algebraic completeness for the invariants of the Riemann Tensor: I

We present a new determining set, CZ, of Riemann invariants which possesses the minimum degree property. From an analysis on the possible independence of CZ, we are led to the division of all space-times into two distinct, invariantly characterized, classes: a general class M_G+, and a special, singular class M_S For each class, we provide an independent set of invariants (I_G+) ⊂ CZ and I_S ⊂ CZ, respectively) which, with the results of a sequel paper, will be shown to be algebraically complete.

On the problem of algebraic completeness for the invariants of the Riemann tensor. III.

We study the set CZ of invariants [Zakhary and Carminati, J. Math. Phys. 42, 1474 (2001)] for the class of space-times whose Ricci tensors possess a null eigenvector. We show that all cases are maximally backsolvable, in terms of sets of invariants from CZ, but that some cases are not completely backsolvable and these all possess an alignment between an eigenvector of the Ricci tensor with a repeated principal null vector of the Weyl tensor. We provide algebraically complete sets for each canonically different space-time and hence conclude with these results and those of a previous article [Carminati, Zakhary, and McLenaghan, J. Math. Phys. 43, 492 (2002)] that the CZ set is determining or maximal.

The determination of all syzygies for the dependent polynomial invariants of the Riemann tensor. I. Pure Ricci and pure Weyl invariants

In this paper, we shall consider all pure Ricci and pure Weyl scalar invariants of any degree, in a four-dimensional Lorentzian space. We present a general graph-theoretic based reduction algorithm which decomposes, using syzygies, any pure invariant in terms of the independent base invariants {r₁,r₂,r₃} or {w₁,w₂}

Octabutyl-1k2C,2k2C,3k2C,4k2C-di-µ3-oxo-1:2:3k3O;2:3:4k2O-di-µ2-phenoxy-1:2k2O;3:4k2O-diphenyoxy-1kO,4kO-tetratin(IV)

The dimeric title compound, tetrabutyldiphenoxydistannoxane, [Sn₄(C₄H₉)₈(C₆H₅O)₄O₂], adopts a ladder-type structure, featuring an almost planar inorganic framework with three four-membered Sn₂O₂ rings and four coplanar phenoxy groups.

A dimeric tellurastannoxane carbonate cluster, tetra-tert-butyl-di-µ3-carbonato-tetrakis-[4-(N,N-dimethylamino)phenyl]di-µ-oxo-ditelluriumditin chloroform tetrasolvate

The title compound, [Sn₂Te₂(C₄H₉)₄(CO₃)₂O₂(C₈H₁₀N)₄]·4CHCl₃ or [(p-Me₂NC₆H₄)₂TeOSn^tBu₂CO₃]₂·4CHCl₃, contains an almost planar centrosymmetric inorganic Sn₂Te₂O₈C₂ core and hypercoordinated Sn and Te atoms. The structure features four secondary intramolecular Te...O contacts.

The determination of all syzygies for the dependent polynomial invariants of the Riemann Tensor. II. mixed invariants of Eeven degree in the Ricci Spinor

We continue our analysis of the polynomial invariants of the Riemann tensor in a four-dimensional Lorentzian space. We concentrate on the mixed invariants of even degree in the Ricci spinor Φ_ABȦḂ and show how, using constructive graph-theoretic methods, arbitrary scalar contractions between copies of the Weyl spinor ψ_ABCD, its conjugate ψ_ȦḂĊḊ and an even number of Ricci spinors can be expressed in terms of paired contractions between these spinors. This leads to an algorithm for the explicit expression of dependent invariants as polynomials of members of the complete set. Finally, we rigorously prove that the complete set as given by Sneddon [J. Math. Phys. 39, 1659-1679 (1998)] for this case is both complete and minimal.

Old buildings, new landscape : Redeveloping historic neighbourhoods - A case study of Xin Tian Di, Shanghai, China

This paper explores the use of tradition by the redevelopment project ofXin Tian Di (New Heaven and Earth) in Shanghai. The paper also identifies the implications of this urban redevelopment project to historic quarters in Chinese cities. An enormous commercial success and new Shanghai icon, Xin Tian Di project has transformed a former residential neighbourhood into a high end entertainment precinct. While the redevelopment ofXin Tian Di has retained the architectural features of the place, the use and interpretation of tradition by the project is problematic. The project shows a tendency where the established idea of tradition is destabilised and challenged. It also highlights issues of cultural authenticity in the redevelopment of historic quarters.

Determination of all syzygies for the dependent polynomial invariants of the Riemann Tensor. III. mixed invariants of arbitrary degree in the Ricci Spinor

In this paper, we rigorously prove that the complete set of Riemann tensor invariants given by Sneddon [J. Math. Phys. 40, 5905 (1999)] is both minimal and complete. Furthermore, we provide a two-stage algorithm for the explicit construction of polynomial syzygies relating any dependent Riemann tensor invariant to members of the complete set.