Levy and set theory

Azriel Levy (1934–) did fundamental work in set theory when it was transmuting into a modern, sophisticated field of mathematics, a formative period of over a decade straddling Cohen’s 1963 founding of forcing. The terms “Levy collapse”, “Levy hierarchy”, and “Levy absoluteness” will live on in set theory, and his technique of relative constructibility and connections established between forcing and definability will continue to be basic to the subject. What follows is a detailed account and analysis of Levy’s work and contributions to set theory.     

Dual entropic and enthalpic processes in the lower critical solution temperature phase separation of poly(vinyl methyl ether) aqueous solutions

In the lower critical solution temperature phase separation of poly(vinyl methyl ether) aqueous solutions, the process corresponding to the weakening of the hydrogen bond interaction with increasing temperature is dominant and occurs over the entire concentration region of solutions and over a broad temperature range from 30 to 41°C, giving rise to the energetic enthalpic effect during phase separation, while the conformational change, that is, collapse of the swollen polymer coils, occurs only in the swelling polymer solution when the water concentration is above 38.3 wt %, giving rise to the entropic effect during phase separation. In addition, the entropic process corresponding to the collapse of the polymer coils occurs in a much narrow theta temperature range from 35.5 to 37°C. If the solution is held at a constant temperature for a sufficiently long time, 90% collapse of the polymer coils occurs in only the 0.5 °C temperature region between 35.5 and 36°C. Accordingly, in the enthalpic process, the most dramatic blueshift of the νC‐O bond peak occurs in the temperature range between 35 and 41°C, while this blueshift is only approximately 2 cm^?1 in the temperature range from 30 to 35°C, prior to the collapse of the polymer coils due to the entropic effect. © 2019 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2019 , 57, 323–330     

Numerical study of real gas effects during bubble collapse using a disequilibrium multiphase model

An explicit density-based solver of the Euler equations for inviscid and immiscible gas–liquid flow media is coupled with real-fluid thermodynamic equations of state supporting mild dissociation and calibrated with shock tube data up to 5000 K and 28 GPa. The present work expands the original 6-equation disequilibrium method by generalising the numerical approach required for estimating the equilibrium pressure in computational cells where both gas and liquid phases co-exist while enforcing energy conservation for all media. An iterative numerical procedure is suggested for taking into account the properties of the gas content as derived from highly non-linear real gas equations of state and implemented in a tabulated form during the numerical solution. The developed method is subsequently used to investigate gaseous bubble collapse cases considering both spherical and 2D asymmetric arrangements as induced by the presence of a rigid wall. It is demonstrated that the predicted maximum temperatures are strongly influenced by the equations of state used; the real gas model predicts a temperature reduction in the bubble interior up to $41\%$ space-averaged and $50\%$ locally during the collapse phase compared to the predictions obtained with the aid of the widely used ideal gas approximation.     

Effects of medium viscoelasticity on bubble collapse strength of interacting polydisperse bubbles

Due to its physical and/or chemical effects, acoustic cavitation plays a crucial role in various emerging applications ranging from advanced materials to biomedicine. The cavitation bubbles usually undergo oscillatory dynamics and violent collapse within a viscoelastic medium, which are closely related to the cavitation-associated effects. However, the role of medium viscoelasticity on the cavitation dynamics has received little attention, especially for the bubble collapse strength during multi-bubble cavitation with the complex interactions between size polydisperse bubbles. In this study, modified Gilmore equations accounting for inter-bubble interactions were coupled with the Zener viscoelastic model to simulate the dynamics of multi-bubble cavitation in viscoelastic media. Results showed that the cavitation dynamics (e.g., acoustic resonant response, nonlinear oscillation behavior and bubble collapse strength) of differently-sized bubbles depend differently on the medium viscoelasticity and each bubble is affected by its neighboring bubbles to a different degree. More specifically, increasing medium viscosity drastically dampens the bubble dynamics and weakens the bubble collapse strength, while medium elasticity mainly affects the bubble resonance at which the bubble collapse strength is maximum. Differently-sized bubbles can achieve resonances and even subharmonic resonances at high driving acoustic pressures as the elasticity changes to certain values, and the resonance frequency of each bubble increases with the elasticity increasing. For the interactions between the size polydisperse bubbles, it indicated that the largest bubble generally has a dominant effect on the dynamics of smaller ones while in turn it is almost unaffected, exhibiting a pattern of destructive and constructive interactions. This study provides a valuable insight into the acoustic cavitation dynamics of multiple interacting polydisperse bubbles in viscoelastic media, which may offer a potential of controlling the medium viscoelasticity to appropriately manipulate the dynamics of multi-bubble cavitation for achieving proper cavitation effects according to the desired application.     

Comparisons between sine-Gordon and perturbed nonlinear Schrödinger equations for modeling light bullets beyond critical collapse

The sine-Gordon (SG) equation and perturbed nonlinear Schrödinger (NLS) equations are studied numerically for modeling the propagation of two space dimensional (2D) localized pulses (the so-called light bullets) in nonlinear dispersive optical media. We begin with the (2 + 1) SG equation obtained as an asymptotic reduction in the two level dissipationless Maxwell-Bloch system, followed by the review on the perturbed NLS equation in 2D for SG pulse envelopes, which is globally well posed and has all the relevant higher order terms to regularize the collapse of standard critical (cubic focusing) NLS. The perturbed NLS is approximated by truncating the nonlinearity into finite higher order terms undergoing focusing-defocusing cycles. Efficient semi-implicit sine pseudospectral discretizations for SG and perturbed NLS are proposed with rigorous error estimates. Numerical comparison results between light bullet solutions of SG and perturbed NLS as well as critical NLS are reported, which validate that the solution of the perturbed NLS as well as its finite-term truncations are in qualitative and quantitative agreement with the solution of SG for the light bullets propagation even after the critical collapse of cubic focusing NLS. In contrast, standard critical NLS is in qualitative agreement with SG only before its collapse. As a benefit of such observations, pulse propagations are studied via solving the perturbed NLS truncated by reasonably many nonlinear terms, which is a much cheaper task than solving SG equation directly.     

Sweeping and new on-line sample preconcentration techniques in capillary electrophoresis

Sweeping is a powerful on-line sample preconcentration technique that improves the concentration sensitivity of capillary electrophoresis (CE). This approach is designed to focus the analyte into narrow bands within the capillary, thereby increasing the sample volume that can be injected, without any loss of CE efficiency. It utilizes the interactions between an additive [i.e., a pseudostationary phase (PS) or complexing agent] in the separation buffer and the sample in a matrix that is devoid of the additive used. The accumulation occurs due to chromatographic partitioning, complexation or any interaction between analytes and the additive through electrophoresis. The extent of the preconcentration is dependent on the strength of interaction involved. Both charged and neutral analytes can be preconcentrated. Remarkable improvements—up to several thousandfold—in detection sensitivity have been achieved. This suggests that sweeping is a superior and general approach to on-line sample preconcentration in CE. The focusing mechanism of sweeping under different experimental conditions and its combination with other on-line preconcentration techniques are discussed in this review. The recently introduced techniques of transient trapping (tr-trapping) and analyte focusing by micelle collapse (AFMC) as well as other novel approaches to on-line sample preconcentration are also described.

Hydrophobic collapse in (in silico) protein folding

A model of hydrophobic collapse, which is treated as the driving force for protein folding, is presented. This model is the superposition of three models commonly used in protein structure prediction: (1) 'oil-drop' model introduced by Kauzmann, (2) a lattice model introduced to decrease the number of degrees of freedom for structural changes and (3) a model of the formation of hydrophobic core as a key feature in driving the folding of proteins. These three models together helped to develop the idea of a fuzzy-oil-drop as a model for an external force field of hydrophobic character mimicking the hydrophobicity-differentiated environment for hydrophobic collapse. All amino acids in the polypeptide interact pair-wise during the folding process (energy minimization procedure) and interact with the external hydrophobic force field defined by a three-dimensional Gaussian function. The value of the Gaussian function usually interpreted as a probability distribution is treated as a normalized hydrophobicity distribution, with its maximum in the center of the ellipsoid and decreasing proportionally with the distance versus the center. The fuzzy-oil-drop is elastic and changes its shape and size during the simulated folding procedure.     

Quantum Phase Dynamics of Interaction between Photon Field and Magnetic System: Effects of Magnetic Quantum Tunnelling

We investigate the dynamics of probability distributions of an initially one-mode coherent field interacting with a four-state molecular system, which is a single magnet with a tunneling across an anisotropic barrier, using a numerically exact approach. The population for each state, the phase properties of and , and ), the entropy are calculated for a model system. The model predicts that the molecule and field become asymptotically disentangled at half of the revival time, and that optical Schrödinger-cat and magnetic Schrödinger-cat states are generated.This paper was originally presented at the 5th International Conference on NEAR FIELD OPTICS and RELATED TECHNOLOGIES (NFO-5), which was held on December 6–10, 1998 at Coganoi Bay Hotel, Shirahama, Japan, in cooperation with the Japan Society of Applied Physics and Mombusho Grant-in Aid for Scientific Research on Priority Areas “Near-field Nano-optics” Project, sponsored by Japan Society for the Promotion of Science.     

Cluster Structure of Collapsing Polymers

In order to better understand the geometry of the polymer collapse transition, we study the distribution of geometric clusters made up of the nearest neighbor interactions of an interacting self-avoiding walk. We argue for this new correlated percolation problem that in two dimensions, and possibly also in three dimensions, a percolation transition takes place at a temperature lower than the collapse transition. Hence this novel transition should be governed by exponents unrelated to the -point exponents. This also implies that there is a temperature range in which the polymer has collapsed, but has no long-range cluster structure. We use Monte Carlo to study the distribution of clusters on the simple cubic and Manhattan lattices. On the Manhattan lattice, where the data are most convincing, we find that the percolation transition occurs at _p =1.461(3), while the collapse transition is known to occur exactly at =1.414.... We propose a finite-size scaling form for the cluster distribution and estimate several of the critical exponents. Regardless of the value of _p , this percolation problem sheds new light on polymer collapse.     

Swelling behaviors of polyelectrolyte hydrogels containing sulfonate groups

Series of maleate monoester and diester monomers based on poly(ethylene glycol) monomethyl ether (MPEG) were copolymerized using the ionizable 2‐acrylamido‐2–methyl propane sulfonic acid (AMPS) via different dose rate of electron‐beam irradiation (40–150 kGy). The crosslinking of the copolymers were carried out in aqueous acidic solutions at pH 1 or in the presence of 1% N,N‐methylene bisacrylamide (MBA) as crosslinking agent. The final equilibrium water content and swelling capacities for the prepared hydrogels were determined in aqueous solutions at pH 1, 6.8, and 12 and in aqueous salt solutions at 298 K. Swelling equilibria for prepared hydrogels were determined in different molar salt solutions of NaCl, KCl, CaCl₂, Na₂SO₄, K₂SO₄, and CaSO₄. The swelling ratios of gels in pure water and in the salt solutions were found to depend on the counterion species in the increasing sequence of Ca²⁺, Na⁺ and K⁺. Copyright © 2002 John Wiley & Sons, Ltd.