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Charged lipids violate this criterion unless there is sufficient salt in the solution to provide good screening. Nonlocal interactions induced by backflow effects play a role in the hydrodynamics but do not affect the static energetics described here.
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It is worth pointing out that many irregular shapes are seen in the laboratory, which do not correspond to the simple shape classes described here. Some of these are related to multilamellar structures; others are doubtless associated with different kinds of defects. Some probably appear in the phase diagram at lower reduced volumes than those so-far studied. There are many open questions!
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Such tubes cannot, it turns out, be stabilized in equilibrium in the absence of an applied axial force. Applying such a force pulls out a tubular "tether" from an otherwise compact vesicle. See, e.g., R.M. Hochmuth, H.C. Wiles, E.A. Evans, and J.T. McCowen, Biophys J. 39, 83 (1982) or R.E. Waugh, J. Song, S.Svetina, and B. Žekš, Biophys. J. 61, 974 (1992).
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Of course, V and a both change in general with temperature; however, these dependences are typically weaker.
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l,m) in each subsector of given |m| and all ℓ's of given parity. The diagonalization scheme truncates this infinite set when addition of more coefficients fails to change the low-lying levels and eigenmodes.
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