All metrics have curvature tensors characterised by its invariants as a limit: the \epsilon-property

We prove a generalisation of the $\epsilon$-property, namely that for any dimension and signature, a metric which is not characterised by its polynomial scalar curvature invariants, there is a frame such that the components of the curvature tensors can be arbitrary close to a certain "background". This "background" is defined by its curvature tensors: it is characterised by its curvature tensors and has the same polynomial curvature invariants as the original metric.Comment: 6 page

Invariant classification and the generalised invariant formalism: conformally flat pure radiation metrics, with zero cosmological constant

Metrics obtained by integrating within the generalised invariant formalism are structured around their intrinsic coordinates, and this considerably simplifies their invariant classification and symmetry analysis. We illustrate this by presenting a simple and transparent complete invariant classification of the conformally flat pure radiation metrics (except plane waves) in such intrinsic coordinates; in particular we confirm that the three apparently non-redundant functions of one variable are genuinely non-redundant, and easily identify the subclasses which admit a Killing and/or a homothetic Killing vector. Most of our results agree with the earlier classification carried out by Skea in the different Koutras-McIntosh coordinates, which required much more involved calculations; but there are some subtle differences. Therefore, we also rework the classification in the Koutras-McIntosh coordinates, and by paying attention to some of the subtleties involving arbitrary functions, we are able to obtain complete agreement with the results obtained in intrinsic coordinates. In particular, we have corrected and completed statements and results by Edgar and Vickers, and by Skea, about the orders of Cartan invariants at which particular information becomes available.Comment: Extended version of GRG publication, with some typos etc correcte

Realizations of Real Low-Dimensional Lie Algebras

Using a new powerful technique based on the notion of megaideal, we construct a complete set of inequivalent realizations of real Lie algebras of dimension no greater than four in vector fields on a space of an arbitrary (finite) number of variables. Our classification amends and essentially generalizes earlier works on the subject. Known results on classification of low-dimensional real Lie algebras, their automorphisms, differentiations, ideals, subalgebras and realizations are reviewed.Comment: LaTeX2e, 39 pages. Essentially exetended version. Misprints in Appendix are correcte

From Navier-Stokes To Einstein

We show by explicit construction that for every solution of the incompressible Navier-Stokes equation in $p+1$ dimensions, there is a uniquely associated "dual" solution of the vacuum Einstein equations in $p+2$ dimensions. The dual geometry has an intrinsically flat timelike boundary segment $\Sigma_c$ whose extrinsic curvature is given by the stress tensor of the Navier-Stokes fluid. We consider a "near-horizon" limit in which $\Sigma_c$ becomes highly accelerated. The near-horizon expansion in gravity is shown to be mathematically equivalent to the hydrodynamic expansion in fluid dynamics, and the Einstein equation reduces to the incompressible Navier-Stokes equation. For $p=2$, we show that the full dual geometry is algebraically special Petrov type II. The construction is a mathematically precise realization of suggestions of a holographic duality relating fluids and horizons which began with the membrane paradigm in the 70's and resurfaced recently in studies of the AdS/CFT correspondence.Comment: 15 pages, 2 figures, typos correcte

Classification of the Weyl Tensor in Higher Dimensions and Applications

We review the theory of alignment in Lorentzian geometry and apply it to the algebraic classification of the Weyl tensor in higher dimensions. This classification reduces to the the well-known Petrov classification of the Weyl tensor in four dimensions. We discuss the algebraic classification of a number of known higher dimensional spacetimes. There are many applications of the Weyl classification scheme, especially in conjunction with the higher dimensional frame formalism that has been developed in order to generalize the four dimensional Newman--Penrose formalism. For example, we discuss higher dimensional generalizations of the Goldberg-Sachs theorem and the Peeling theorem. We also discuss the higher dimensional Lorentzian spacetimes with vanishing scalar curvature invariants and constant scalar curvature invariants, which are of interest since they are solutions of supergravity theory.Comment: Topical Review for Classical and Quantum Gravity. Final published versio

Layer-by-layer deposition of open-pore mesoporous TiO 2- Nafion® film electrodes

The formation of variable thickness TiO2 nanoparticle-Nafion® composite films with open pores is demonstrated via a layer-by-layer deposition process. Films of about 6 nm diameter TiO2 nanoparticles grow in the presence of Nafion® by “clustering” of nanoparticles into bigger aggregates, and the resulting hierarchical structure thickens with about 25 nm per deposition cycle. Film growth is characterized by electron microscopy, atomic force microscopy, and quartz crystal microbalance techniques. Simultaneous small-angle X-ray scattering and wide-angle X-ray scattering measurements for films before and after calcination demonstrate the effect of Nafion® binder causing aggregation. Electrochemical methods are employed to characterize the electrical conductivity and diffusivity of charge through the TiO2-Nafion® composite films. Characteristic electrochemical responses are observed for cationic redox systems (diheptylviologen2+/+, Ru(NH3)3+/2+6, and ferrocenylmethyl-trimethylammonium2+/+) immobilized into the TiO2-Nafion® nanocomposite material. Charge conduction is dependent on the type of redox system and is proposed to occur either via direct conduction through the TiO2 backbone (at sufficiently negative potentials) or via redox-center-based diffusion/electron hopping (at more positive potentials)

Absorbing customer knowledge: how customer involvement enables service design success

Customers are a knowledge resource outside of the firm that can be utilized for new service success by involving them in the design process. However, existing research on the impact of customer involvement (CI) is inconclusive. Knowledge about customers’ needs and on how best to serve these needs (articulated in the service concept) is best obtained from customers themselves. However, codesign runs the risk of losing control of the service concept. This research argues that of the processes of external knowledge, acquisition (via CI), customer knowledge assimilation, and concept transformation form a capability that enables the firm to exploit customer knowledge in the form of a successful new service. Data from a survey of 126 new service projects show that the impact of CI on new service success is fully mediated by customer knowledge assimilation (the deep understanding of customers’ latent needs) and concept transformation (the modification of the service concept due to customer insights). However, its impact is more nuanced. CI exhibits an “∩”-shaped relationship with transformation, indicating there is a limit to the beneficial effect of CI. Its relationship with assimilation is “U” shaped, suggesting a problem with cognitive inertia where initial learnings are ignored. Customer knowledge assimilation directly impacts success, while concept transformation only helps success in the presence of resource slack. An evolving new service design is only beneficial if the firm has the flexibility to adapt to change