2,772 research outputs found
Sharing storage using dirty vectors
Consider a computation F with n inputs (independent variables) and m outputs (dependent variables) and suppose that we wish to evaluate the Jacobian of F. Automatic differentiation commonly performs this evaluation by associating vector storage either with the program variables (in the case of forward-mode automatic differentiation) or with the adjoint variables (in the case of reverse). Each vector component contains a partial derivative with respect to an independent variable, or a partial derivative of a dependent variable, respectively. The vectors may be full vectors, or they may be dynamically managed sparse data structures. In either case, many of these vectors will be scalar multiples of one another. For example, any intermediate variable produced by a unary operation in the forward mode will have a derivative vector that is a multiple of the derivative for the argument. Any computational graph node that is read just once during its lifetime will have an adjoint vector that is a multiple of the adjoint of the node that reads it. It is frequently wasteful to perform component multiplications explicitly. A scalar multiple of another vector can be replaced by a single multiplicative "scale factor" together with a pointer to the other vector. Automated use of this "dirty vector" technique can save considerable memory management overhead and dramatically reduce the number of floating-point operations required. In particular, dirty vectors often allow shared threads of computation to be reverse-accumulated cheaply. The mechanism permits a number of generalizations, some of which give efficient techniques for preaccumulation
Inelastic neutron scattering study on the resonance mode in an optimally doped superconductor LaFeAsOF
An optimally doped iron-based superconductor LaFeAsOF with
K has been studied by inelastic powder neutron scattering. The
magnetic excitation at \AA is enhanced below , leading to
a peak at meV as the resonance mode, in addition to the
formation of a gap at low energy below the crossover energy . The peak energy at \AA corresponds to in
good agreement with the other values of resonance mode observed in the various
iron-based superconductors, even in the high- cuprates. Although the
phonon density of states has a peak at the same energy as the resonance mode in
the present superconductor, the -dependence is consistent with the resonance
being of predominately magnetic origin.Comment: 4 pages, 5 Postscript figure
Quantum ergodicity for restrictions to hypersurfaces
Quantum ergodicity theorem states that for quantum systems with ergodic
classical flows, eigenstates are, in average, uniformly distributed on energy
surfaces. We show that if N is a hypersurface in the position space satisfying
a simple dynamical condition, the restrictions of eigenstates to N are also
quantum ergodic.Comment: 22 pages, 1 figure; revised according to referee's comments. To
appear in Nonlinearit
Near Sharp Strichartz estimates with loss in the presence of degenerate hyperbolic trapping
We consider an -dimensional spherically symmetric, asymptotically
Euclidean manifold with two ends and a codimension 1 trapped set which is
degenerately hyperbolic. By separating variables and constructing a
semiclassical parametrix for a time scale polynomially beyond Ehrenfest time,
we show that solutions to the linear Schr\"odiner equation with initial
conditions localized on a spherical harmonic satisfy Strichartz estimates with
a loss depending only on the dimension and independent of the degeneracy.
The Strichartz estimates are sharp up to an arbitrary loss. This is
in contrast to \cite{ChWu-lsm}, where it is shown that solutions satisfy a
sharp local smoothing estimate with loss depending only on the degeneracy of
the trapped set, independent of the dimension
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