Borderline cases and the collapsing principle

John Broome has argued that value incommensurability is vagueness, by appeal to a controversial ‘collapsing principle’ about comparative indeterminacy. I offer a new counterexample to the collapsing principle. That principle allows us to derive an outright contradiction from the claim that some object is a borderline case of some predicate. But if there are no borderline cases, then the principle is empty. The collapsing principle is either false or empty.

Evolution of a collapsing and exploding Bose-Einstein condensate in different trap symmetries

Based on the time-dependent Gross-Pitaevskii equation we study the evolution of a collapsing and exploding Bose-Einstein condensate in different trap symmetries to see the effect of confinement on collapse and subsequent explosion, which can be verified in future experiments. We make a prediction for the evolution of the shape of the condensate and the number of atoms in it for different trap symmetries (cigar to pancake) as well as in the presence of an optical lattice potential. We also make a prediction for the jet formation in different cases when the collapse is suddenly terminated by changing the scattering length to zero via a Feshbach resonance. In addition to the usual global collapse to the center of the condensate, in the presence of an optical-lattice potential one could also have in certain cases independent collapse of parts of the condensate to local centers, which could be verified in experiments.

Dynamics of collapsing and exploding Bose-Einstein condensate

Recently, Donley et al. performed an experiment on the dynamics of collapsing and exploding Bose-Einstein condensates by suddenly changing the scattering length of atomic interaction to a large negative value on a preformed repulsive condensate of Rb-85 atoms in an axially symmetric trap. Consequently, the condensate collapses and ejects atoms via explosions, We show that the accurate numerical solution of the time-dependent Gross-Pitaevskii equation with axial symmetry can explain some aspects of the dynamics of the collapsing condensate. (C) 2002 Published by Elsevier B.V. B.V.

Chaos in collapsing Bose-condensed gas

We reinvestigate the dynamics of the grow and collapse of Bose-Einstein condensates in a system of trapped ultracold atoms with negative scattering lengths, and found a new behavior in the long time scale evolution: the number of atoms can go far beyond the static stability limit. The condensed state is described by the solution of the time-dependent nonlinear Schrödinger equation, in a model that includes atomic feeding and three-body dissipation. Our results for the model show that, by changing the feeding parameter and when a substantial depletion of the ground-state exists, a chaotic behavior is found. We consider a criterion proposed by Deissler and Kaneko [Phys. Lett. A 119, 397 (1987)] to diagnose spatiotemporal chaos. ©2000 The American Physical Society.

Mean-field description of collapsing and exploding Bose-Einstein condensates

Numerical simulations based on the time-dependent mean-field Gross-Pitaevskii equation was performed to explain the dynamics of collapsing and exploding Bose-Einstein condensates (BEC) of 85Rb atoms. The atomic interaction was manipulated by an external magnetic field via a Feshbach resonance. On changing the scattering length of atomic interaction from a positive to a large negative value, the condensate collapsed and ejected atoms via explosion.

Dynamics of a collapsing and exploding Bose-Einstein condensed vortex state

The dynamics of small repulsive Bose-Einstein condensed vortex states of 85Rb atoms in a cylindrical traps with low angular momentum was studied. The time-dependent mean-field Gross-Pitaevskii equation was used for the study. The condensates collapsed and atoms ejected via explosion and a remnant condensate with a smaller number of atoms emerges that survived for a long time.

Mean-field model of jet formation in a collapsing Bose-Einstein condensate

We perform a systematic numerical study, based on the time-dependent Gross-Pitaevskii equation, of jet formation in collapsing and exploding Bose-Einstein condensates as in the experiment by Donley et al (2001 Nature 412 295). In the actual experiment, via a Feshbach resonance, the scattering length of atomic interaction was suddenly changed from positive to negative on a pre-formed condensate. Consequently, the condensate collapsed and ejected atoms via explosion. On disruption of the collapse by suddenly changing the scattering length to zero, a radial jet of atoms was formed in the experiment. We present a satisfactory account of jet formation under the experimental conditions and also make predictions beyond experimental conditions which can be verified in future experiments.

Equation of Motion Governing the Dynamics of Vertically Collapsing Buildings

The present paper aims at contributing to a discussion, opened by several authors, on the proper equation of motion that governs the vertical collapse of buildings. The most striking and tragic example is that of the World Trade Center Twin Towers, in New York City, about 10 years ago. This is a very complex problem and, besides dynamics, the analysis involves several areas of knowledge in mechanics, such as structural engineering, materials sciences, and thermodynamics, among others. Therefore, the goal of this work is far from claiming to deal with the problem in its completeness, leaving aside discussions about the modeling of the resistive load to collapse, for example. However, the following analysis, restricted to the study of motion, shows that the problem in question holds great similarity to the classic falling-chain problem, very much addressed in a number of different versions as the pioneering one, by von Buquoy or the one by Cayley. Following previous works, a simple single-degree-of-freedom model was readdressed and conceptually discussed. The form of Lagrange's equation, which leads to a proper equation of motion for the collapsing building, is a general and extended dissipative form, which is proper for systems with mass varying explicitly with position. The additional dissipative generalized force term, which was present in the extended form of the Lagrange equation, was shown to be derivable from a Rayleigh-like energy function. DOI: 10.1061/(ASCE)EM.1943-7889.0000453. (C) 2012 American Society of Civil Engineers.

Study of the morphologic variants of focal segmental glomerulosclerosis: a Brazilian report

INTRODUCTION: Focal segmental glomerulosclerosis (FSGS) is the most frequent primary glomerulopathy in Brazil and its incidence is increasing worldwide. Pathogenesis is related to podocyte injury, which may be due to several factors including viruses, drugs, genetics and immunological factors. In 2004, the Columbia classification of FSGS identified five histological variants of the disease: collapsing (COL), usual (NOS), tip lesion (TIP), perihilar (PHI) and cellular variant (CEL). The objective of this study was to classify the FSGS biopsies in these morphological variants. METHODS: One hundred thirty-one cases of renal biopsies with primary FSGS diagnosis, which had been performed at a Brazilian reference center from 1996 to 2006, were classified according to the Columbia criteria. RESULTS: FSGS cases were distributed as follows: 38.2% NOS variant, 36.6% COL, 14.5% TIP, 6.9% PHI and 3.8% CEL. CONCLUSION: COL variant of FSGS seems to be more prevalent in Brazil in comparison with other centers worldwide, which may be related to environmental and socioeconomic factors.

The collapse of linear polyelectrolyte chains in a poor solvent: When does a collapsing polyelectrolyte collect its counter ions?

The collapse of linear polyelectrolyte chains in a poor solvent: When does a collapsing polyelectrolyte collect its counter ions? The collapse of polyions in a poor solvent is a complex system and is an active research subject in the theoretical polyelectrolyte community. The complexity is due to the subtle interplay between hydrophobic effects, electrostatic interactions, entropy elasticity, intrinsic excluded volume as well as specific counter-ion and co-ion properties. Long range Coulomb forces can obscure single molecule properties. The here presented approach is to use just a small amount of screening salt in combination with a very high sample dilution in order to screen intermolecular interaction whereas keeping intramolecular interaction as much as possible (polyelectrolyte concentration cp ≤ 12 mg/L, salt concentration; Cs = 10^-5 mol/L). This is so far not described in literature. During collapse, the polyion is subject to a drastic change in size along with strong reduction of free counterions in solution. Therefore light scattering was utilized to obtain the size of the polyion whereas a conductivity setup was developed to monitor the proceeding of counterion collection by the polyion. Partially quaternized PVP’s below and above the Manning limit were investigated and compared to the collapse of their uncharged precursor. The collapses were induced by an isorefractive solvent/non-solvent mixture consisting of 1-propanol and 2-pentanone, with nearly constant dielectric constant. The solvent quality for the uncharged polyion could be quantified which, for the first time, allowed the experimental investigation of the effect of electrostatic interaction prior and during polyion collapse. Given that the Manning parameter M for QPVP4.3 is as low as lB / c = 0.6 (lB the Bjerrum length and c the mean contour distance between two charges), no counterion binding should occur. However the Walden product reduces with first addition of non solvent and accelerates when the structural collapse sets in. Since the dielectric constant of the solvent remains virtually constant during the chain collapse, the counterion binding is entirely caused by the reduction in the polyion chain dimension. The collapse is shifted to lower wns with higher degrees of quaternization as the samples QPVP20 and QPVP35 show (M = 2.8 respectively 4.9). The combination of light scattering and conductivity measurement revealed for the first time that polyion chains already collect their counter ions well above the theta-dimension when the dimensions start to shrink. Due to only small amounts of screening salt, strong electrostatic interactions bias dynamic as well as static light scattering measurements. An extended Zimm formula was derived to account for this interaction and to obtain the real chain dimensions. The effective degree of dissociation g could be obtained semi quantitatively using this extrapolated static in combination with conductivity measurements. One can conclude the expansion factor a and the effective degree of ionization of the polyion to be mutually dependent. In the good solvent regime g of QPVP4.3, QPVP20 and QPVP35 exhibited a decreasing value in the order 1 > g4.3 > g20 > g35. The low values of g for QPVP20 and QPVP35 are assumed to be responsible for the prior collapse of the higher quaternized samples. Collapse theory predicts dipole-dipole attraction to increase accordingly and even predicts a collapse in the good solvent regime. This could be exactly observed for the QPVP35 sample. The experimental results were compared to a newly developed theory of uniform spherical collapse induced by concomitant counterion binding developed by M. Muthukumar and A. Kundagrami. The theory agrees qualitatively with the location of the phase boundary as well as the trend of an increasing expansion with an increase of the degree of quaternization. However experimental determined g for the samples QPVP4.3, QPVP20 and QPVP35 decreases linearly with the degree of quaternization whereas this theory predicts an almost constant value.

Basic research in crashworthiness II -- testing and evaluation of collapsing forward structure modification concept in car-to-car frontal collisions. Interim technical report.

National Highway Traffic Safety Administration, Washington, D.C.

Development of a collapsing ring bridge railing system. Final report.

Federal Highway Administration, Office of Research, Washington, D.C.