Multi-normed spaces

We modify the very well known theory of normed spaces (E, \norm) within functional analysis by considering a sequence (\norm_n : n\in\N) of norms, where \norm_n is defined on the product space $E^n$ for each $n\in\N$. Our theory is analogous to, but distinct from, an existing theory of `operator spaces'; it is designed to relate to general spaces $L^p$ for $p\in [1,\infty]$, and in particular to $L^1$-spaces, rather than to $L^2$-spaces. After recalling in Chapter 1 some results in functional analysis, especially in Banach space, Hilbert space, Banach algebra, and Banach lattice theory that we shall use, we shall present in Chapter 2 our axiomatic definition of a `multi-normed space' ((E^n, \norm_n) : n\in \N), where (E, \norm) is a normed space. Several different, equivalent, characterizations of multi-normed spaces are given, some involving the theory of tensor products; key examples of multi-norms are the minimum and maximum multi-norm based on a given space. Multi-norms measure `geometrical features' of normed spaces, in particular by considering their `rate of growth'. There is a strong connection between multi-normed spaces and the theory of absolutely summing operators. A substantial number of examples of multi-norms will be presented. Following the pattern of standard presentations of the foundations of functional analysis, we consider generalizations to `multi-topological linear spaces' through `multi-null sequences', and to `multi-bounded' linear operators, which are exactly the `multi-continuous' operators. We define a new Banach space ${\mathcal M}(E,F)$ of multi-bounded operators, and show that it generalizes well-known spaces, especially in the theory of Banach lattices. We conclude with a theory of `orthogonal decompositions' of a normed space with respect to a multi-norm, and apply this to construct a `multi-dual' space.Comment: Many update

Synthesis of perbromates

Salts of heptavalent bromine were synthesized by a hot atom process, the beta decay of radioactive selenium-83 incorporated into a selenate. Formation of an unreactive perbromate ion led to preparation of macro amounts of perborate. A rubidium salt was isolated

A Fluid Dynamics Calculation of Sputtering from a Cylindrical Thermal Spike

The sputtering yield, Y, from a cylindrical thermal spike is calculated using a two dimensional fluid dynamics model which includes the transport of energy, momentum and mass. The results show that the high pressure built-up within the spike causes the hot core to perform a rapid expansion both laterally and upwards. This expansion appears to play a significant role in the sputtering process. It is responsible for the ejection of mass from the surface and causes fast cooling of the cascade. The competition between these effects accounts for the nearly linear dependence of $Y$ with the deposited energy per unit depth that was observed in recent Molecular Dynamics simulations. Based on this we describe the conditions for attaining a linear yield at high excitation densities and give a simple model for this yield.Comment: 10 pages, 9 pages (including 9 figures), submitted to PR

Wear-resistant ball bearings for space applications

Ball bearings for hostile environments were developed. They consist of normal ball bearing steel parts of which the rings are coated with hard, wear-resistant, chemical vapor deposited (C.V.D) TiC. Experiments in ultrahigh vacuum, using cages of various materials with self-lubricating properties, have shown that such bearings are suitable for space applications

A continuum model for entangled fibres

Motivated by the study of fibre dynamics in the carding machine, a continuum model for the motion of a medium composed of fibres is derived under the assumption that the dominant forces are due to fibre-fibre interactions and that the material is in tension. To characterise the material we include the averaged values of density and velocity and introduce variables to describe the mean direction, alignment and entanglement. We assume that the bulk stress of the material depends on the density, entanglement, degree of alignment, average direction and shear-rates. A kinematic equation for the average direction and two proposed heuristic laws for the evolution of entanglement and degree of alignment are given to close the system. Extensional and shearing simulations are in good qualitative agreement with experimental results

Attaching of strain gages to substrates

A method and apparatus for attaching strain gages to substrates is described. A strain gage having a backing plate is attached to a substrate by using a foil of brazing material between the backing plate and substrate. A pair of electrodes that are connected to a current source, are applied to opposite sides of the backing plate, so that heating of the structure occurs primarily along the relatively highly conductive foil of brazing material. Field installations are facilitated by utilizing a backing plate with wings extending at an upward incline from either side of the backing plate, by attaching the electrodes to the wings to perform the brazing operation, and by breaking off the wings after the brazing is completed

Multiphase modelling of vascular tumour growth in two spatial dimensions

In this paper we present a continuum mathematical model of vascular tumour growth which is based on a multiphase framework in which the tissue is decomposed into four distinct phases and the principles of conservation of mass and momentum are applied to the normal/healthy cells, tumour cells, blood vessels and extracellular material. The inclusion of a diffusible nutrient, supplied by the blood vessels, allows the vasculature to have a nonlocal influence on the other phases. Two-dimensional computational simulations are carried out on unstructured, triangular meshes to allow a natural treatment of irregular geometries, and the tumour boundary is captured as a diffuse interface on this mesh, thereby obviating the need to explicitly track the (potentially highly irregular and ill-defined) tumour boundary. A hybrid finite volume/finite element algorithm is used to discretise the continuum model: the application of a conservative, upwind, finite volume scheme to the hyperbolic mass balance equations and a finite element scheme with a stable element pair to the generalised Stokes equations derived from momentum balance, leads to a robust algorithm which does not use any form of artificial stabilisation. The use of a matrix-free Newton iteration with a finite element scheme for the nutrient reaction-diffusion equations allows full nonlinearity in the source terms of the mathematical model. Numerical simulations reveal that this four-phase model reproduces the characteristic pattern of tumour growth in which a necrotic core forms behind an expanding rim of well-vascularised proliferating tumour cells. The simulations consistently predict linear tumour growth rates. The dependence of both the speed with which the tumour grows and the irregularity of the invading tumour front on the model parameters are investigated

Critical State in Thin Anisotropic Superconductors of Arbitrary Shape

A thin flat superconductor of arbitrary shape and with arbitrary in-plane and out-of-plane anisotropy of flux-line pinning is considered, in an external magnetic field normal to its plane. It is shown that the general three-dimensional critical state problem for this superconductor reduces to the two-dimensional problem of an infinitely thin sample of the same shape but with a modified induction dependence of the critical sheet current. The methods of solving the latter problem are well known. This finding thus enables one to study the critical states in realistic samples of high-Tc superconductors with various types of anisotropic flux-line pinning. As examples, we investigate the critical states of long strips and rectangular platelets of high-Tc superconductors with pinning either by the ab-planes or by extended defects aligned with the c-axis.Comment: 13 pages including 13 figure files in the tex

Thin superconductors and SQUIDs in perpendicular magnetic field

It is shown how the static and dynamic electromagnetic properties can be calculated for thin flat superconducting films of any shape and size, also multiply connected as used for SQUIDs, and for any value of the effective magnetic London penetration depth Lambda. As examples, the distributions of sheet current and magnetic field are obtained for rectangular and circular films without and with slits and holes, in response to an applied perpendicular magnetic field and to magnetic vortices moving in the film. The self energy and interaction of vortices with each other and with an applied magnetic field and/or transport current are given. Due to the long ranging magnetic stray field, these energies depend on the size and shape of the film and on the vortex position even in large films, in contrast to the situation in large bulk superconductors. The focussing of magnetic flux into the central hole of square films without and with a radial slit is compared.Comment: 12 pages, 10 figure