On propagation through long step tapers

In integrated optics, tapered waveguides usually possess step changes in waveguide width or other dimensions, due to lithographic limitations. We present a theoretical asymptotic analysis of the effect of these steps when compared with the ideal smooth taper. We present some figures for the size of the effect, conclude that the effect is significant in certain situation and that making the taper too long will reduce its performance, or in the worst instance cause the taper to function as badly as a butt join.     

Efficient Quantum Algorithms for Simulating Sparse Hamiltonians

We present an efficient quantum algorithm for simulating the evolution of a quantum state for a sparse Hamiltonian H over a given time t in terms of a procedure for computing the matrix entries of H. In particular, when H acts on n qubits, has at most a constant number of nonzero entries in each row/column, and ||H|| is bounded by a constant, we may select any positive integer k such that the simulation requires O((log^* n)t ^1+1/2k ) accesses to matrix entries of H. We also show that the temporal scaling cannot be significantly improved beyond this, because sublinear time scaling is not possible.     

Combinatorics of Maass-Shimura operators

Maass-Shimura operators on holomorphic modular forms preserve the modularity of modular forms but not holomorphy, whereas the derivative preserves holomorphy but not modularity. Rankin-Cohen brackets are bilinear operators that preserve both and are expressed in terms of the derivatives of modular forms. We give identities relating Maass-Shimura operators and Rankin-Cohen brackets on modular forms and obtain a natural expression of the Rankin-Cohen brackets in terms of Maass-Shimura operators. We also give applications to values of L-functions and Fourier coefficients of modular forms.     

Superparamagnetism in Heterogeneous AgFe Thin Films – A Low Energy μSR Study

This paper describes a study of superparamagnetism in iron nanoclusters using low energy μSR, including for the first time zero field measurements, which yield an intrinsic nanocluster superparamagnetic relaxation rate of ν₀=67±15 MHz and an energy barrier of ΔE=37±4 K. TEM micrographs of the embedded clusters show there to be a range of shapes and also sizes, which is bourne out by SQUID magnetometry measurements. This revised version was published online in September 2006 with corrections to the Cover Date.     

Sharp Conditions for Korn Inequalities in Orlicz Spaces

We show that Korn’s inequality in Orlicz spaces holds if and only if the Orlicz function satisfies the Δ₂- and the ${\nabla_2}$ -condition. This result applies to several types of Korn’s inequality. In particular we show that Korn’s inequality is false in L ¹, in L log L, in Exp and in L ^∞.     

Breakdown of Layering in Frustrated Smectic‐A Elastomers

Smectic‐A elastomers combine the positional long‐range order of mesogenic molecules in one dimension with the rubber elasticity of a polymer network. Upon mechanical deformation, completely different responses of the phase structure have been reported. We present a highly distorted system which shows a breakdown of smectic layering but no reorientation under deformation along the layer normal, while the phase structure stays unaffected under uniaxial stress in the perpendicular direction. The thermoelastic properties, macroscopic dimensions and stress–strain behaviour are investigated parallel and perpendicular to the layer normal. SAXS measurements supply evidence for a breakdown of the macroscopically ordered layer structure indicated by the small angle intensities and correlation length, whereas the orientational order is preserved. We propose defects in the smectic layer structure to be the origin of the different responses of different smectic elastomers.