Infinite Diameter Research Articles

The diameter of the isomorphism class of a Banach space

We prove that if X is a separable infinite dimensional Banach space then its isomorphism class has infinite diameter with respect to the Banach-Mazur distance. One step in the proof is to show that if X is elastic then X contains an isomorph of c0. We call X elastic if for some K < ?? for every Banach space Y which embeds into X, the space Y is K-isomorphic to a subspace of X. We also prove that if X is a separable Banach space such that for some K <?? every isomorph of X is K-elastic then X is finite dimensional.

Double resonance features in the Raman spectrum of carbon nanotubes

A weak feature is observed in the Raman spectra of single-wall carbon nanotube bundles at a frequency between 1410 and $1430\phantom{\rule{0.3em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$. The band shifts slightly toward lower frequencies as the excitation energy increases from 2.17 to $2.71\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$. An inverse correlation is found between the frequency of the band and that of the average radial-breathing mode, indicating a dependence on the nanotube diameter. The frequency can be extrapolated quadratically to $1487\phantom{\rule{0.3em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$ for infinite diameter. Two different hypotheses are considered, ascribing the band to modes around either $K$ or $M$ points of the graphite dispersion relations. The appearance of a band close to $1480\phantom{\rule{0.3em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$ in defective graphite and its dependence with the laser energy, as well as the strong sensitivity of high-frequency $K$-point modes to structural or impurity defects, support the interpretation of the mode as arising from for the upper phonon branch of graphite around the $K$ point activated by the double-resonance mechanism.

Characterisations of generalized polygons and opposition in rank 2 twin buildings

In this paper, we characterize the natural opposition relation on the set of flags of a generalized polygon. We also investigate when a certain relation on any rank 2 geometry of finite diameter is equivalent to the opposition relation in a generalized polygon. As a consequence we obtain a new definition of generalized polygons. Finally, we also characterize the opposition relation in twin trees, which are the analogues of polygons with infinite diameter.

Wall effects for motion of spheres in power-law fluids

The steady motion of spheres representing particles inside tubes filled with different fluids has been investigated using both a finite-element and a finite-volume method. The rheology of the fluids has been modelled by the power-law able to describe the shear-thinning (pseudoplastic) behaviour of a series of polymer solutions. New results have been obtained for a series of tube/sphere diameter ratios in order to investigate the wall effects on the drag exerted by the fluid on the sphere. The results agree well with previous simulations for an unbounded medium (infinite diameter ratio). Experimental investigations have also been carried out and simulated, and the results compare favourably with the experiments. The present simulations revealed the convergence of the drag coefficient to a constant value independent of tube-to-sphere diameter ratio when the power-law index approaches zero.

Holes in flames, flame isolas, and flame edges

We examine a simple model problem designed to elucidate how easily a hole in a diffusion flame can close and how easily an isolated region of burning in a mixing region (a flame isola) can grow. With the thickness of the mixing layer/diffusion flame defining the characteristic length, we find that a hole of diameter ≈1 closes for all but a tiny interval of Damköhler numbers above the one-dimensional quenching value: a hole of diameter ≈3 closes for Damköhler numbers that exceed the quenching value by more than 30%–40%, depending on the Lewis number: and a hole of infinite diameter closes for Damköhler numbers that exceed the quenching value by more than ∼70%. Even larger Damköhler numbers are required for isolas to grow, and we calculate these values for different radii. We investigate, the speeds with which hole edges or isola edges advance or retreat and provide evidence that for a shrinking hole or isola, a meaningful speed can be defined that depends only on the combustion parameters and the instantaneous hole/isola radius.

Visualization of solute migration in liquid chromatography columns

By matching the refractive indices of the mobile phase, the stationary phase and the material of the bed enclosure, one can render transparent to the eye the chromatographic column which is normally opaque in nature. As a result, band visualization is readily obtained. High-definition on-column detection becomes feasible by using a photographic detector instead of the conventional post-column, on-line (UV–Vis or similar) detector. Quantitative information regarding the concentration distribution in the band is obtained by utilizing optical scanners to obtain a digital image and computer imaging software. The processes of data collection and image analysis are discussed in detail and are illustrated by observing the concept of the infinite diameter column following a central point injection. The performance of the photographic detection method is compared to that of regular detection procedures. The efficiency of the column was determined with both on-column measurements and regular post-column measurements. A minimum reduced plate height ( h) of 1.9 was recorded with post-column detection, in agreement with the average of the results given by on-column detection. On-column analysis allowed the determination of the local plate height which was found to vary across the central region of the column between 2.7 and 0.95.

Hofer's diameter and Lagrangian intersections

We prove that the group of Hamiltonian diffeomorphisms of the 2-sphere has infinite diameter with respect to Hofer's metric. Our approach is based on the theory of Lagrangian intersections.

A mixing rule for the size parameter from Carnahan-Starling equation and Boublik-Mansoori equation

A density-dependent one-fluid mixing rule for the size parameter is proposed from Carnahan-Starling equation and Boublik-Mansoori equation using a numerical method. This mixing rule coincides with the mixing rules usually used at zero pressure and at infinite pressure. It also tends to Carnahan-Starling equations for pure fluids at limiting conditions, such as infinite dilute solution, diameter ratio and equal diameters. Its application to equations of state is also carried out for Lennard-Jones mixtures including binary and ternary mixtures. Compared with the van der Waals one-fluid mixing rule, distinct improvement is obtained, especially for predictions of PVT, chemical potential and vapor-liquid equilibrium data.

Velocity of detonation and charge diameter in some RDX-driven heterogenous explosives: PBXW-115, PBXN-111, composition H-6 and composition B

Abstract : Experimental data on the dependence of velocity of detonation on charge diameter, V(d), of unconfined cylindrical charges of the underwater explosives, PBXW-115 (Aust), PBXN-111 and Composition H-6, and of Composition B, the explosive charge fill offered by Nordic Defence Industries for the Danish Mine Disposal Charge, Damdic, are given. These data are analysed in terms of the previously-reported empirical relationship between the V(d), the detonation velocity at infinite charge diameter (D*) and the reaction zone length parameter, (a*).

Binary classifiers, perceptrons and connectedness in metric spaces and graphs

Minsky and Papert's well-known results about the inability of diameter-limited perceptrons to recognise connectedness in the plane are generalised in three important respects: First, to a large class of metric spaces of arbitrary dimension, not necessarily Euclidean; secondly, to the class of all connected graphs; thirdly, in the case of infinite diameter, including Minsky and Papert's case, it is shown that no diameter-limited binary classifier (including neural nets of arbitrary complexity, etc.) can recognise connectedness.

Bethe ansatz study of 1+1 dimensional Hubbard model

A wave function of Bethe ansatz form is used to obtain approximate solutions for a 1+1 dimensional Hubbard model. In this model a helical structure with infinite diameter is constructed to mimic the square lattice. Betheansatz analysis is performed along the helical line. By including the hopping integrals in perpendicular direction, the solutions show some recovery of two dimensional character. At half-filling, the system undergoes Mott transition, the critical value of on-site repulsion obtained by this scheme is smaller than that of the mean-field theory. For a given particle density, the energy level of corresponding one dimensional model is extended to a “band”. The relation of the results to the properties of two dimensional strong-correlation system is discussed.

Growth of AIGalnP in a high-speed rotating disk OMVPE reactor

In an OMVPE reactor with a high speed rotating disk, growth of a quaternary material AIGalnP is carried out after a basic investigation on the GaAs growth rate uniformity. Experimental growth rate and growth efficiency results are compared with the theoretical results obtained from an infinite diameter rotating disk model. Distributions of the lattice mismatch, PL spectra and carrier concentration were measured to clarify the disk rotation effect on these properties. The main reason for non-uniformities is not the disk rotation but the temperature distribution on a substrate carrier.

Vortex-ring model of the superfluid lambda transition.

An initial model of the superfluid \ensuremath{\lambda} transition is constructed with use of vortex-ring excitations, as originally suggested by Onsager and Feynman. A real-space renormalization technique generates a screened vortex energy and core size, and gives rise to a transition where rings of infinite diameter are excited as the superfluid density approaches zero at ${\mathrm{T}}_{\mathrm{c}}$. Although the model satisfies the Josephson hyperscaling relation, it is not yet a complete theory: The superfluid density exponent is \ensuremath{\nu}=0.53, and does not match the known value \ensuremath{\nu}=0.67.

Vortices and the Super fluid λ-Transition

An initial model of the superfluid λ-transition is constructed using vortex-ring excitations, as originally suggested by Onsager and Feynman. A real-space renormalization technique generates a screened vortex energy and core size, and gives rise to a transition where rings of infinite diameter are excited as the superfluid density approaches zero at Tc. Although the model satisfies the Josephson hyperscaling relation, it is not yet a complete theory: the superfluid density exponent is v=0.53, and does not match the known value v=0.67.

Continuum approximations and particle interactions in concentrated suspensions

Falling‐ball rheometry is used to study hydrodynamic forces in concentrated suspensions. The suspensions consist of large, uniform spheres (diameter ds=0.32 cm) neutrally buoyant in a viscous, Newtonian liquid. Falling‐ball experiments are performed by dropping steel balls of various diameters (df) through suspensions held in cylindrical containers of several diameters. In these optically opaque suspensions, the passage of the falling ball is observed with real‐time radiography and is recorded with digitized, high‐speed video. The average terminal velocities of the falling balls are used to calculate an apparent viscosity of each suspension (ηs). Extrapolation of the data to estimate the velocity which would occur in a cylinder of infinite diameter is used to find the apparent viscosity with zero wall effects (ηs∞). Each ηs∞ agrees with independent measurements throughout the solids concentration range of the data (0.0≤φ≤0.45). In the absence of wall effects, ηs∞ is independent of ds /df. That is, all sizes of falling balls (0.75≤df /ds≤12.0) experience the same mean stress field. In addition, it is seen that in dilute and moderately concentrated suspensions (0.0<φ<0.30), wall effects are identical to those in equivalent Newtonian liquids. At higher solids loadings, we observe additional wall effects that can act over much greater distances than in equivalent Newtonian liquids. The magnitude of additional wall effects increase rapidly as φ increases above 0.30.

Stokes drag on a cylinder in axial motion

Measurements are presented for the drag on a circular cylinder moving along its axis of rotational symmetry at low Reynolds number. The influence of an exterior cylindrical boundary is taken into account by making measurements with boundaries of different diameter and using an empirical correlation to extrapolate to infinite boundary diameter. The results for cylinders with length to diameter ratio (L/D) between 4 and 100 agree well with the calculated values of Youngren and Acrivos. The results for disks (0.019&amp;lt;L/D&amp;lt;0.26) are consistent with a linear interpolation between the value for a disk of zero thickness and the experimental points of Heiss and Coull. Beads-on-a-shell calculations are presented for both axial and transverse motion of cylinders with 0≤L/D≤10. Interpolated values of the ratio of transverse to axial drag are used to obtain approximate values of transverse drag for 10&amp;lt;L/D&amp;lt;100.

Diffusive relaxation processes in liquid3He-4He mixtures. I. Normal phase

When a heat flux is switched on across a fluid binary mixture, steady state conditions for the temperature and mass concentration gradients ∇T and ∇c are reached via a diffusive transient process described by a series of terms “modes” involving characteristic times τn. These are determined by static and transport properties of the mixture, and by the boundary conditions. We present a complete mathematical solution for the relaxation process in a binary normal liquid layer of heightd and infinite diameter, and discuss in particular the role of the parameterA=kT2(∂μ/∂c)T,P/TCP,c coupling the mass and thermal diffusion. HerekT is the thermal diffusion ratio, (∂μ/∂c)T,P−1 is the concentration susceptibility, μ is the chemical potential difference between the components, andCP,c is the specific heat. We present examples of special situations found in relaxation experiments. WhenA is small, the observable times τ(∇T) and τ(∇c) for temperature and concentration equilibration are different, but they tend to the same value asA increases. We present experimental results on four examples of liquid helium of different3He mole fractionX, and discuss these results on the basis of the preceding analysis. In the simple case for pure3He (i.e., in the absence of mass diffusion) we find the observed τ(∇T) to be in good agreement with that calculated from the thermal diffusivity. For all the investigated3He-4He mixtures, we observe τ(∇c) and τ(∇T) to be different whenA is small, a situation occurring at high enough temperatures. AsA increases with decreasingT, they become equal, as predicted. For the mixtures with mole fractionsX(3He)=0.510 and 0.603, we derive the mass diffusionD from the analysis of τ(∇c) and demonstrate that it diverges strongly with an exponent of about 1/3 in the critical region near the superfluid transition. As the tricritical point (Tt,Xt) is approached for the mixtureX=Xt0.675,D tends to zero with an exponent of roughly 0.4. These results are consistent with predictions and also with theD derived from sound attenuation data. We discuss the difficulties of the analysis in the regime close toTλ andTt, with special emphasis on the situation created by the onset of a superfluid film along the wall of the cell forX=0.603 and 0.675.

Comparison of broadening patterns in regular and radially compressed large-diameter columns

Summary The differential broadening pattern of an inert solute injected centrally into a regular (A) and a radially compressed (B) column has been studied. The two columns exhibit identical properties when operated in the infinite diameter mode. A marked difference, to the advantage of column B, develops as the solute becomes appreciably exposed to the wall region. At its fullest extent, the wall effect accounts for a nearly 2.5-fold increase in plate height for column A compared with a 1.5-fold increase for column B. This behavious is rationalized in terms of a semi-theoretical model that takes the radial diffusivity, the radial dependence of the lateral plate height and the peak local velocity into account. It is further shown that the solute radial concentration profile departs significantly from a truncated gaussian shape. The actual profile can be expressed as a Bessel function series.

Detonation Velocity of PETN in Small confined cylindrical charges

AbstractSteady state detonation velocities have been measured in small brass‐confined pressings of PETN. These velocities were found to vary linearly with the reciprocal of the charge diameter and directly with the compact density. For a pressing density of 950 kg/m3 the velocity at infinite diameter was calculated to be 5.27 km/s (50.02 kmk) and the Eyring‐type „reaction‐zone length”︁ 0.54 mm (±0.02 mm).

Interaction of radial and axial dispersion in liquid chromatography in relation to the “infinite diameter effect”

The dynamics of radial dispersion of small solute samples injected centrally into a chromatographic column have been examined using a dual-electrode polarographic detector having one electrode fixed and the other movable radially across the face of the column exit frit. It is shown that the radial dispersion can be characterised by a reduced radial plate height, h r , whose dependence on reduced velocity, ν, obeys the equation h r = 1.4/ν + 0.06 Experiments with trans-column injection using the same equipment show that there is little variation of peak-maximum velocity but that the mean axial plate height is increased up to threefold near the wall. The region of high dispersion extends about 30 particle diameters inwards from the walls. Experiments with conventional high-performance liquid chromatographic columns and equipment, carried out with aromatic hydrocarbon solutes and columns of various lengths and bores packed with 21.5 μm spherical alumina, show that serious loss of efficiency occurs when solute can reach the wall regions. The data are consistent with the view that for efficient chromatography using microparticles the solute should not spread closer to the walls than about 30 particle diameters.