Isospectral theory of Euler equations

Isospectral problem of both 2D and 3D Euler equations of inviscid fluids, is investigated. Connections with the Clay problem are described. Spectral theorem of the Lax pair is studied.     

Effects of the H-bond donor structure on the properties of supramolecular side chain liquid crystalline polymers based on poly(3-carboxypropylmethylsiloxane-co-dimethylsiloxane)s and stilbazoles

Supramolecular side chain liquid crystalline polymers (SCLCPs) based on poly(3-carboxypropylmethylsiloxane-co-dimethylsiloxane) (PSIX, X=100, 76, 60, 41 or 23, denoting the mole percentage of 3-carboxypropylmethylsiloxane unit in the polymer) and stilbazole derivatives have been obtained through intermolecular hydrogen bonding (H-bonding) interactions between the carboxylic acid and the pyridyl moieties. The formation of H-bonding and self-assembly results in the formation of new mesogenic units, in which H-bonds function as molecular connectors. FTIR shows the existence of H-bonding in the complexes. The polymeric complexes behave as single component liquid crystalline polymers and exhibit stable and enantiotropic mesophases. The liquid crystalline properties of the supramolecular SCLCPs were studied using differential scanning calorimetry, polarizing optical microscopy and X-ray diffraction, and were found to exhibit smectic A phases with focal-conic textures. The thermal stability of the SCLCP increases on increasing the carboxylic acid content in the polysiloxane and the concentration of the stilbazole derivative in the complex. However, the thermal stability decreases on increasing the chain length of the stilbazole derivative. The crystal phase was not formed even on cooling to the glass transition temperature of the polymeric complex.     

The Spectrum of a Linearized 2D Euler Operator

We study the spectral properties of the linearized Euler operator obtained by linearizing the equations of incompressible two-dimensional fluid at a steady state with the vorticity that contains only two nonzero complex conjugate Fourier modes. We prove that the essential spectrum coincides with the imaginary axis, and give an estimate from above for the number of isolated nonimaginary eigenvalues. In addition, we prove that the spectral mapping theorem holds for the group generated by the linearized 2D Euler operator.     

A minimal pair joining to a plus cupping Turing degree

A computably enumerable (c.e.) degree a is called nonbounding, if it bounds no minimal pair, and plus cupping, if every nonzero c.e. degree x below a is cuppable. Let NB and PC be the sets of all nonbounding and plus cupping c.e. degrees, respectively. Both NB and PC are well understood, but it has not been possible so far to distinguish between the two classes. In the present paper, we investigate the relationship between the classes NB and PC , and show that there exists a minimal pair which join to a plus cupping degree, so that PC ? NB . This gives a first known difference between NB and PC . (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear rheological behavior of silica filled solution‐polymerized styrene butadiene rubber

The reinforcement and nonlinear viscoelastic behavior have been investigated for silica (SiO₂) filled solution‐polymerized styrene butadiene rubber (SSBR). Experimental results reveal that the nonlinear viscoelastic behavior of the filled rubber is similar to that of unfilled SSBR, which is inconsistent with the general concept that this characteristic comes from the breakdown and reformation of the filler network. It is interesting that the curves of either dynamic storage modulus (G′) or loss tangent (tan δ) versus strain amplitude (γ) for the filled rubber can be superposed, respectively, on those for the unfilled one, suggesting that the primary mechanism for the Payne effect is mainly involved in the nature of the entanglement network in rubbery matrix. It is believed there exists a cooperation between the breakdown and reformation of the filler network and the molecular disentanglement, resulting in enhancing the Payne effect and improving the mechanical hysteresis at high strain amplitudes. Moreover, the vertical and the horizontal shift factors for constructing the master curves could be well understood on the basis of the reinforcement factor f(φ) and the strain amplification factor A(φ), respectively. The surface modification of SiO₂ causes a decrease in f(φ), which is ascribed to weakeness of the filler–filler interaction and improvement of the filler dispersion. However, the surface nature of SiO₂ hardly affects A(φ). © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 2594‐2602, 2007     

Role of Na(+)-Ca2+ exchange system in the pathogenesis of myocardial ischemia-reperfusion damage

Results from a systematic experiment on isolated perfused rat heart and isolated myocytes of adult rat showed that the mechanism of calcium influx during myocardial ischemia-reperfusion is due to the development of intracellular sodium overload during ischemic period; on reperfusion, the high intracellular Na+ content activated the reverse direction of Na(+)-Ca2+ exchange over myocardial sarcolemma (SL), thus a large quantity of extracellular Ca2+ fluxed over the SL to the intracellular space, forming a condition of intracellular Ca2+ overload, which leads to irreversible damage of the myocardium.     

Improvement of least-squares finite difference method on simulating slider air bearings in head disk systems

Least-squares finite difference (LSFD) method, one of mesh-free methods, is used to solve slider air bearings problem through discritizing the generalized Reynolds equation into nonlinear systems of algebraic equations. Two approximation schemes for the linearization of these equations are presented and compared. And, some new techniques to search supporting points for the reference node in the mesh-free method were proposed and explored. Therefore, these improvements eliminate some potential limitation of the LSFD method previously published and further facilitate its employment in complex slider models. Advanced step slider as an example of negative pressure sliders is simulated and verified using the improved LSFD mesh-free method in head disk systems.     

A comparison of spin relaxation and local motion between symmetrically and asymmetrically ring-substituted bisphenol units in dissolved polycarbonates

Carbon-13 and proton spin-lattice relaxation times were measured at two field strengths on solutions 10% by weight of two polycarbonates in C₂D₂Cl₄ from ?20 to +120°C. The first polycarbonate is an asymmetrically substituted form with one chlorine on one of the two phenylene aromatic rings of the bisphenol unit, whereas the second polycarbonate is symmetrically substituted with two chlorines on each of the two rings. The nuclear spin relaxation data are interpreted in terms of several local motions likely in these polymers. Segmental motion was described by the Hall–Helfand correlation function. Segmental motion in the monosubstituted polycarbonate is somewhat slower than in unsubstituted polycarbonate, whereas segmental motion in the tetrasubstituted polycarbonate is considerably slower. Phenylene ring rotation is observed in unsubstituted polycarbonate and in the monosubstituted polycarbonate above 40°C. Below 40°C in the monosubstituted species, and at all temperatures in the tetrasubstituted species, ring rotation is replaced by ring libration as the predominant motion contributing to spin lattice relaxation. In addition, the rotational motion of the two types of rings in the asymmetric monosubstituted form are very similar although not identical. The substituted ring is slightly less mobile than the rings of unsubstituted polycarbonate. This indicates a strong coupling of ring motion, although the coupling leads to less than synchronous motion. Methyl group rotation is present in both polymers and is little affected by the various structural modifications.     

Some results on the unramified principal series ofp-adic groups

Supported in part by NSF grant No. DMS-9003999