In this paper, we prove that the small energy harmonic maps from H 2 to H 2 are asymptotically stable under the wave map equation in the subcritical perturbation class. This result may be seen as an example supporting the soliton resolution conjecture for geometric wave equations without equivariant assumptions on the initial data. In this paper, we construct Tao's caloric gauge in the case when nontrivial harmonic map occurs. With the "dynamic separation" the master equation of the heat tension field appears as a semilinear magnetic wave equation. By the endpoint and weighted Strichartz estimates for magnetic wave equations obtained by the first author [38], the asymptotic stability follows by a bootstrap argument.

The ability to add synthetic channels to polymersome (polymer vesicle) membranes could lead to novel membrane composites with unique selectivity and permeability. Proton transport through two different synthetic pores, self‐assembled from either a dendritic dipeptide, (6Nf‐3,4‐3,5)12G2‐CH2‐Boc‐L‐Tyr‐L‐Ala‐OMe, or a dendritic ester, (R)‐4Bp‐3,4‐dm8G1‐COOMe, incorporated into polymersome membranes are studied. Polymersomes provide an excellent platform for studying such transport processes due to their robustness and mechanical and chemical stability compared to liposomes. It is found that the incorporated dendritic dipeptide and dendritic ester assemble into stable helical pores in the poly(ethylene oxide)‐polybutadiene (PEO‐PBD) polymersomes but not in the poly(2‐methyloxazoline)‐poly(dimethylsiloxane)‐poly(2‐methyl oxazoline) (PMOX‐PDMS‐PMOX) polymersomes. The incorporation is confirmed by circular dichroism (CD), changes in purely synthetic mechanical strength (e.g., areal expansion modulus) as assessed by micropipette aspiration, and cryo‐TEM. In addition to the structural analyses, a transport measurement shows the incorporated dendritic helical pores allow facile transport of protons across the polymersome membranes after up to one month of storage. This integration of synthetic porous channels with polymersome substrates could provide a valuable tool for studying active transport processes in a composite membrane. These composites will ultimately expand the family of biologically inspired porous‐membrane mimics.

A quantum form of the Hamiltonian for a double-exchange ͑DE͒ system, based on the results of Anderson and Hasegawa's semiclassical treatment, is presented together with several predictions. The magnetization decreases mainly in 3/2 and 5/2 powers of temperature at low temperature but includes a T 3 term. The Curie temperature, in a rescaled DE coupling energy (ͱ2b/2S 2 )J d unit, is dependent on hole concentration x and has highest value that is about 1/3 that of ordinary Heisenberg magnet at xϭ0.5. The susceptibility at high temperatures lacks a 1/T 2 term and has a characteristic temperature in rescaled DE coupling units that is about 3/4 that of Heisenberg magnet in molecular-field theory.

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