We give a proof of the isoperimetric inequality for quermassintegrals of non-convex starshaped domains, using a result of Gerhardt [C. Gerhardt, Flow of nonconvex hypersurfaces into spheres, J. Differential Geometry 32 (1990) 299-314] and Urbas [J. Urbas, On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures, Math. Z. 205 (1990) 355-372] on an expanding geometric curvature flow.

As compared with adjuvant FAC, adjuvant TAC improved the rate of disease-free survival among women with high-risk, node-negative breast cancer. (Funded by GEICAM and Sanofi-Aventis; ClinicalTrials.gov number, NCT00121992.).

Our previous studies of the captioned transition have shown that thermally sensitive poly(Nisopropylacrylamide) (PNIPAM) in water can form stable individual single-chain globules, but not for polystyrene (PS) in cyclohexane. In the current study, using poly(N,N-diethylacrylamide) (PDEAM) (M w ) 1.7 × 10 7 g/mol and M w /M n ) 1.06) with no hydrogen donator site, we intend to find whether the intrachain hydrogen bonding plays a role in stabilizing individual collapsed PNIPAM single-chain globules. We found that PDEAM can also form stable single-chain globules in water even though the transition is less sharp. The resultant individual PDEAM single-chain globules are less compact, reflecting in a lower chain density and a higher ratio of the radius of gyration to hydrodynamic radius, presumably due to the lack of intrachain hydrogen bonding. Our result also shows that, unlike PNIPAM, there is no hysteresis in the transition, indirectly supporting our previous assumption that the hysteresis observed for PNIPAM is due to the formation of some intrachain additional hydrogen bonds formed in the collapsed state.

Formation of stable nanobubbles in aqueous solutions of water-soluble organic molecules is a spontaneous process. Using a combination of laser light scattering (LLS) and zeta-potential measurements, we investigated the effects of salt concentration and pH on their stability in R-cyclodextrin (R-CD) aqueous solutions. Our results reveal that the nanobubbles are unstable in solution with a higher ionic strength, just like colloidal particles in an aqueous dispersion, but become more stable in alkaline solutions. The zeta-potential measurement shows that the nanobubbles are negatively charged with an electric double layer, presumably due to adsorption of negative OH -ions at the gas/water interface. It is this double layer that plays a critical dual role in the formation of stable nanobubbles in aqueous solutions of water-soluble organic molecules, namely, it not only provides a repulsive force to prevent interbubble aggregation and coalescence but also reduces the surface tension at the gas/water interface to decrease the internal pressure inside each bubble.

In this article, we introduce a new type of mean curvature flow (1.3) for bounded star-shaped domains in space forms and prove its longtime existence, exponential convergence without any curvature assumption. Along this flow, the enclosed volume is a constant and the surface area evolves monotonically. Moreover, for a bounded convex domain in R n+1 , the quermassintegrals evolve monotonically along the flow which allows us to prove a class of Alexandrov-Fenchel inequalities of quermassintegrals.

In the past, many laser light-scattering experimental results revealed that besides the fast relaxation mode, there existed an additional slow mode in semidilute solutions. This slow mode has been assigned to a variety of origins, but there has been no clear and well-accepted explanation. As the polymer concentration increases, the slow relaxation mode persists in the concentrated region, in melts and in gels in which polymer chains are crosslinked instead of entangled. The slow relaxation mode has also been reported for charged macromolecules in aqueous and nonaqueous solutions. However, it is generally thought to be different in nature from that observed in semidilute neutral polymer solution. In recent years, armed with novel solution preparation methods and some specially designed polymers, we have reexamined the dynamics of polymer chains, especially the slow mode, in semidilute neutral polymer solutions, dilute polyelectrolyte solutions and gels, which are reviewed here. Our results suggest that the slow mode can be qualitatively considered as hindered motions of interacting chains even though the nature of interaction can be very different; namely, from the weak segment-segment interaction in a less good solvent to strong electrostatic interaction among polyelectrolyte chains, and even to chemical crosslinking inside gel networks.

Abstract. In this note, we construct families of functionals of the type of F-functional and W-functional of Perelman. We prove that these new functionals are nondecreasing under the Ricci flow. As applications, we give a proof of the theorem that compact steady Ricci breathers must be Ricci-flat. Using these new functionals, we also give a new proof of Perelman's no non-trivial expanding breather theorem. Furthermore, we prove that compact expanding Ricci breathers must be Einstein by a direct method. In this note, we also extend Cao's methods of eigenvalues[1] and improve their results.

In the 1970s, de Gennes and Pincus 1,2 used the Rouse model 3 to show that linear polymer chains in a good solvent could undergo a first-order coil-to-stretch transition to pass through a pore much smaller than its coiled size under an elongation flow field with a sufficient hydrodynamic force, independent of both the chain length and the pore size. Namely, a coiled polymer chain can pass through a small pore as long as the flow rate (q) is higher than a threshold [q c = k B T/(3πη)], where k B , T, and η is the Boltzmann constant, the absolutely temperature, and the solvent viscosity, respectively.On the other hand, it has been suggested that the electrostatic blob model for a polyelectrolytes chain would also be applicable for the coil-to-stretch transition of a neutral chain in an ultrafiltration experiment. 4,5 In principle, the stretching of a neutral chain under an elongational flow and the extension of a polyelectrolyte chain under the electrostatic repulsion have a similar physical nature, but not identical. Namely, a neutral chain can be stretched as a string of blobs under an elongational flow. The blob size decreases as the shear rate increases. Finally, each blob can be fully stretched with a sufficiently strong shear force.It is not difficult to realize that the passing through a small pore occurs only when the blob size (ξ b ) of a stretched chain in a flow filed is smaller than the pore size (D). Therefore, the critical flow rate is associated with the relative size of the blob to the pore. Whether a coiled chain can pass through a small pore is related to its local deformability, i.e., the entering of the first blob into the pore, at which the entropic confinement force (k B T/D) is overcame by the hydrodynamic force (3πηq c /D), where it has assumed that each blob is a nondraining ball with a diameter identical to the pore size (D). 1,2 Note that here only the short-range interaction is relevant. In principle, one can reversibly use q c to characterize the local deformability. To our knowledge, the predicted chain-length and pore-size independence of q c has not been confirmed in experiments because such a first-order coil-tostretch transition has not been observed for a long time. Recently, we have observed such a discontinuous first-order transition by ultrafiltrating linear polystyrene chains through specially constructed small pores (20 nm). 6