Smooth solutions to a class of free boundary parabolic problems

We establish existence, uniqueness, and regularity results for solutions to a class of free boundary parabolic problems, including the free boundary heat equation which arises in the so-called ``focusing problem' in the mathematical theory of combustion. Such solutions are proved to be smooth with respect to time for positive , if the data are smooth.

     

Free Wreath Product by the Quantum Permutation Group

Let A be a compact quantum group, let nN ^* and let A _aut(X _n ) be the quantum permutation group on n letters. A free wreath product construction A*_w A _aut(X _n ) is introduced. This construction provides new examples of quantum groups, and is useful to describe the quantum automorphism group of the n-times disjoint union of a finite connected graph.     

On the dynamics of generic non-Abelian free actions

We investigate some global generic properties of the dynamics associated to non-Abelian free actions in certain special cases. The main properties considered in this paper are related to the existence of dense orbits, to ergodicity and to topological rigidity. We first deal with them in the case of conservative homeomorphisms of a manifold and C ¹-diffeomorphisms of a surface. Groups of analytic diffeomorphisms of a manifold which, in addition, contain a Morse-Smale element and possess a generating set close to the identity are considered as well. From our discussion we also derive the existence of a rigidity phenomenon for groups of skew-products which is opposed to the phenomenon present in Furstenbergs celebrated example of a minimal diffeomorphism that is not ergodic (cf. [Ma]).     

Equivalence Principle Measurements

Over the centuries there have been many experimental tests of the universality of free-fall. To date, these measurements have established the equivalence between inertial and gravitational mass to high precision, justifying its use as a foundation stone of general relativity by Einstein. There is a surprising richness in the variety of techniques and choice of test bodies which have been used so far, and a brief review is presented. However, future space experiments promise much better precision in this measurement and STEP is presented in some detail as one of a number of such missions. Using pairs of concentric free-falling proof-masses, STEP will be able to test the Equivalence Principle (EP) to a sensitivity at least five orders of magnitude better than currently achievable on ground. The EP is a founding principle of general relativity and STEP is the most sensitive experiment of this type planned so far, aiming at 1 part in 10¹⁸.     

Thermosensitive poly(methyl methacrylate) emulsion prepared in the presence of poly(vinyl alcohol) with a cloud point as a protective colloid

An emulsion of poly(methyl methacrylate) (PMMA) was prepared using poly(vinyl alcohol) (PVA) of low degree of hydrolysis with a cloud point as a protective colloid. The behaviour of an aqueous solution of PVA with 80% degree of hydrolysis was first investigated in terms of the Huggins constant in viscometry. MMA was polymerized using the PVA at 20 °C, where no abnormality in the aqueous PVA was observed. The change in transmittance of the emulsion observed with a UV–vis photometer revealed that in the case of UV light of wavelength 370 nm, the transmittance decreased markedly at around 30 °C with an increase in temperature, and then increased with a decrease in temperature. The thermosensitive property resulted from PVA with a low degree of hydrolysis with a cloud point, at a higher temperature of which the PVA loses solubility in water owing to weakening of the hydrogen bond between PVA molecules and water.     

Monosized cationic nanoparticles prepared by emulsifer-free emulsion polymerization

Monosized poly(styrene/N-[3-(dimethylamino)propyl]methacrylamide/poly(ethylene glycol) ethyl ether methacrylate) [poly(St/PEG-EEM/DMAPM)] cationic nanoparticles were synthesized by emulsifier-free emulsion polymerization conducted in the presence of a cationic initiator, 2,2-azobis(2-methylpropionamidine) dihydrochloride (APDH or V-50). Particle sizes and surface charge densities were measured with a Zeta Sizer. The structure of the terpolymers was determined by Fourier transform IR and ¹H NMR spectroscopies. The amounts of the main monomer (St), cationic comonomer (DMAPM), stabilizer (PEG-EEM), and initiator (APDH), and the water-to-monomer phase ratio were all effective on both the average size and the surface charge of the nanoparicles. The average particle size was in the range 75–400 nm depending on the recipe applied; it decreased on increasing the amount of DMAP or PEG-EEM or the water-to-monomer phase ratio in the feed, while it increased with increasing St or APDH content. These nanoparticles were quite monodisperse with a polydispersity index of 1.008–1.14.     

Interaction of Alkali Blue 6B with Cetyltrimethylammonium Bromide

The microsurface adsorption–spectral correction (MSASC) technique has been applied to investigate the interaction of alkali blue 6B (AB6B) with cetyltrimethylammonium bromide (CTAB). The aggregation of AB6B on CTAB obeys the Langmuir isothermal adsorption. The aggregate was characterized by MSASC. For the monomer aggregate AB6B₂CTAB formed, its binding constant is calculated to be K = 3.01·10⁵ and its molar absorptivity to be 2.45·10⁶ liters·cm^–1 at = 690 nm. The adsorption has been used to carry out quantitative detection of cationic surfactant in samples.     

External Fields, Density Functionals, and the Gibbs Inequality

By combining the upper and lower bounds to the free energy as given by the Gibbs inequality for two systems with the same intermolecular interactions but with external fields differing from each other only in a finite region of space , we show that the corresponding equilibrium densities must also differ from each other somewhere in . We note that the basic equations of density functional theory arise naturally from a simple rearrangement and reinterpretation of the terms in the upper bound Gibbs inequality for such systems and briefly discuss some of the complications that occur when the intermolecular interactions of the two systems also differ.     

The Riemann Surface of the Chiral Potts Model Free Energy Function

In a recent paper we derived the free energy or partition function of the N-state chiral Potts model by using the infinite lattice inversion relation method, together with a non-obvious extra symmetry. This gave us three recursion relations for the partition function per site T _pq of the infinite lattice. Here we use these recursion relations to obtain the full Riemann surface of T _pq . In terms of the t _p ,t _q variables, it consists of an infinite number of Riemann sheets, each sheet corresponding to a point on a (2N–1)-dimensional lattice (for N>2). The function T _pq is meromorphic on this surface: we obtain the orders of all the zeros and poles. For N odd, we show that these orders are determined by the usual inversion and rotation relations (without the extra symmetry), together with a simple linearity ansatz. For N even, this method does not give the orders uniquely, but leaves only [(N+4)/4] parameters to be determined.     

NMR-microscopy with TrueFISP at 11.75T

The purpose of this paper is to demonstrate that a fully balanced gradient echo technique (TrueFISP) can be used for microscopic experiments at high static magnetic field strengths. TrueFISP experiments were successfully performed on homogeneous and inhomogeneous objects at 11.75T. High-resolution TrueFISP images were obtained from phantoms, plants, formalin-fixed samples, and from an isolated beating rat heart with an in-plane resolution of 78 micro m and a slice thickness of 500 micro m. The signal-to-noise ratio (SNR) gain of TrueFISP compared to conventional gradient echo or spin echo sequences will allow faster acquisition times or an improvement in spatial resolution for microscopic experiments.