We show that for a class of C 2 quasiperiodic potentials and for any Diophantine frequency, the Lyapunov exponents of the corresponding Schrödinger cocycles are uniformly positive and weak Hölder continuous as function of energies. As a corollary, we also obtain that the corresponding integrated density of states (IDS) is weak Hölder continous. Our approach is of purely dynamical systems, which depends on a detailed analysis of asymptotic stable and unstable directions. We also apply it to more general SL(2, R) cocycles, which in turn can be applied to get uniform positivity and continuity of Lyapuonv exponents around unique nondegenerate extremal points of any smooth potential, and to a certain class of C 2 Szegő cocycles.

Aggregation-induced quenching (AIQ) of photo-

We provide a complete and self-contained proof of spectral and dynamical localization for the one-dimensional Anderson model, starting from the positivity of the Lyapunov exponent provided by Fürstenberg's theorem. That is, a Schrödinger operator in ℓ 2 (Z) whose potential is given by independent identically distributed (i.i.d.) random variables almost surely has pure point spectrum with exponentially decaying eigenfunctions and its unitary group exhibits exponential off-diagonal decay, uniformly in time. This is achieved by way of a new result: for the Anderson model, one typically has Lyapunov behavior for all generalized eigenfunctions. We also explain how to obtain analogous statements for extended CMV matrices whose Verblunsky coefficients are i.i.d., as well as for half-line analogs of these models.V.B., D.D., V.G., T.

Abstract. Simon's subshift conjecture states that for every aperiodic minimal subshift of Verblunsky coefficients, the common essential support of the associated measures has zero Lebesgue measure. We disprove this conjecture in this paper, both in the form stated and in the analogous formulation of it for discrete Schrödinger operators. In addition we prove a weak version of the conjecture in the Schrödinger setting. Namely, under some additional assumptions on the subshift, we show that the density of states measure, a natural measure associated with the operator family and whose topological support is equal to the spectrum, is singular. We also consider one-frequency quasi-periodic Schrödinger operators with continuous sampling functions and show that generically, the density of states measure is singular as well.

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In this paper we first obtain a formula of averaged Lyapunov exponents for ergodic Szegő cocycles via the Herman-Avila-Bochi formula. Then using acceleration, we construct a class of analytic quasi-periodic Szegő cocycles with uniformly positive Lyapunov exponents. Finally, a simple application of the main theorem in [Y] allows us to estimate the Lebesgue measure of support of the measure associated to certain class of C 1 quasiperiodic 2sided Verblunsky coefficients. Using the same method, we also recover the [S-S] results for Schrödinger cocycles with nonconstant real analytic potentials and obtain some nonuniform hyperbolicity results for arbitrarily fixed Brjuno frequency and for certain C 1 potentials.

Abstract. We show that for a class of C 2 quasiperiodic potentials and for any Diophantine frequency, the spectrum is Cantor. Our approach is of purely dynamical systems, which depends on a detailed analysis of asymptotic stable and unstable directions. We also apply it to general SL(2, R) cocycles, and obtain that uniform hyperbolic systems form a open and dense set in some one-parameter family.

A novel POSS‐containing methacrylate monomer (HEMAPOSS) was fabricated by extending the side chain between polyhedral oligomeric silsesquioxane (POSS) unit and methacrylate group, which can efficiently decrease the steric hindrance in free‐radical polymerization of POSS‐methacrylate monomer. POSS‐containing homopolymers (PHEMAPOSS) with a higher degree of polymerization (DP) can be prepared using HEMAPOSS monomer via reversible addition–fragmentation chain transfer (RAFT) polymerization. PHEMAPOSS was further used as the macro‐RAFT agent to construct a series of amphiphilic POSS‐containing poly(N, N‐dimethylaminoethyl methacrylate) diblock copolymers, PHEMAPOSS‐b‐PDMAEMA. PHEMAPOSS‐b‐PDMAEMA block copolymers can self‐assemble into a plethora of morphologies ranging from irregular assembled aggregates to core‐shell spheres and further from complex spheres (pearl‐necklace‐liked structure) to large compound vesicles. The thermo‐ and pH‐responsive behaviors of the micelles were also investigated by dynamic laser scattering, UV spectroscopy, SEM, and TEM. The results reveal the reversible transition of the assembled morphologies from spherical micelles to complex micelles was realized through acid‐base control. © 2014 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2014, 52, 2669‐2683