115 research outputs found
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A Transformational Approach to Japanese Traditional Music of the Edo Period
Analysis of sĆkyoku jiuta, Japanese traditional music of the Edo period for koto and shamisen, has in the past relied primarily on static tetrachordal or hexachordal models. The present study takes a transformational approach to traditional Japanese music. Specifically, it develops a framework for six-pitch hexachordal space inspired by Steven Ringsâs transformational approach to tonal music. This novel voice-leading space yields insights into intervallic structure, trichordal transposition and hexachordal voice leading and transformations of this music at both its surface and large-scale levels. A side-by-side comparison with Ringsâs approach highlights differences between the hexachordal and diatonic systems
Variational formulation of problems involving fractional order differential operators
In this work, we consider boundary value problems involving either Caputo or Riemann-Liouville fractional derivatives of order α â (1, 2) on the unit interval (0, 1). These fractional derivatives lead to nonsymmetric
boundary value problems, which are investigated from a variational point of view. The variational problem for the Riemann-Liouville case is coercive on the space Hα/2
0 (0, 1) but the solutions are less regular, whereas that for the Caputo case involves different test and trial spaces. The numerical analysis of these problems requires the so-called shift theorems which show that the
solutions of the variational problem are more regular. The regularity pickup enables one to establish convergence rates of the finite element approximations.
The analytical theory is then applied to the Sturm-Liouville problem involving a fractional derivative in the leading term. Finally, extensive numerical
results are presented to illustrate the error estimates for the source problem and eigenvalue problem
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Exact de Rham Sequences of Spaces Defined on Macro-elements in Two and Three Spatial Dimensions
This paper proposes new finite element spaces that can be constructed for agglomerates of standard elements that have certain regular structure. The main requirement is that the agglomerates share faces that have closed boundaries composed of 1-d edges. The spaces resulting from the agglomerated elements are subspaces of the original de Rham sequence of H{sup 1}-conforming, H(curl) conforming, H(div) conforming and piecewise constant spaces associated with an unstructured 'fine' mesh. The procedure can be recursively applied so that a sequence of nested de Rham complexes can be constructed. As an illustration we generate coarser spaces from the sequence corresponding to the lowest order Nedelec spaces, lowest order Raviart-Thomas spaces, and for piecewise linear H{sup 1}-conforming spaces, all in three-dimensions. The resulting V-cycle multigrid methods used in preconditioned conjugate gradient iterations appear to perform similar to those of the geometrically refined case
Characteristics of the polymer transport in ratchet systems
Molecules with complex internal structure in time-dependent periodic
potentials are studied by using short Rubinstein-Duke model polymers as an
example. We extend our earlier work on transport in stochastically varying
potentials to cover also deterministic potential switching mechanisms,
energetic efficiency and non-uniform charge distributions. We also use currents
in the non-equilibrium steady state to identify the dominating mechanisms that
lead to polymer transportation and analyze the evolution of the macroscopic
state (e.g., total and head-to-head lengths) of the polymers. Several numerical
methods are used to solve the master equations and nonlinear optimization
problems. The dominating transport mechanisms are found via graph optimization
methods. The results show that small changes in the molecule structure and the
environment variables can lead to large increases of the drift. The drift and
the coherence can be amplified by using deterministic flashing potentials and
customized polymer charge distributions. Identifying the dominating transport
mechanism by graph analysis tools is found to give insight in how the molecule
is transported by the ratchet effect.Comment: 35 pages, 17 figures, to appear in Phys. Rev.
Exact de Rham sequences of spaces defined on macro-elements in two and three spatial dimensions
Abstract. This paper proposes new finite element spaces that can be constructed for agglomerates of standard elements that have certain regular structure. The main requirement is that the agglomerates share faces that have closed boundaries composed of 1-d edges. The spaces resulting from the agglomerated elements are subspaces of the original de Rham sequence of H 1 -conforming, H(curl) conforming, H(div) conforming and piecewise constant spaces associated with an unstructured "fine" mesh. The procedure can be recursively applied so that a sequence of nested de Rham complexes can be constructed. As an illustration we generate coarser spaces from the sequence corresponding to the lowest order Nédélec spaces, lowest order Raviart-Thomas spaces, and for piecewise linear H 1 -conforming spaces, all in three-dimensions. The resulting V -cycle multigrid methods used in preconditioned conjugate gradient iterations appear to perform similar to those of the geometrically refined case
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Mesh independent convergence of the modified inexact Newton method for a second order nonlinear problem
In this paper, we consider an inexact Newton method applied to a second order nonlinear problem with higher order nonlinearities. We provide conditions under which the method has a mesh-independent rate of convergence. To do this, we are required to first, set up the problem on a scale of Hilbert spaces and second, to devise a special iterative technique which converges in a higher than first order Sobolev norm. We show that the linear (Jacobian) system solved in Newton's method can be replaced with one iterative step provided that the initial nonlinear iterate is accurate enough. The closeness criteria can be taken independent of the mesh size. Finally, the results of numerical experiments are given to support the theory
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H(curl) Auxiliary Mesh Preconditioning
This paper analyzes a two-level preconditioning scheme for H(curl) bilinear forms. The scheme utilizes an auxiliary problem on a related mesh that is more amenable for constructing optimal order multigrid methods. More specifically, we analyze the case when the auxiliary mesh only approximately covers the original domain. The latter assumption is important since it allows for easy construction of nested multilevel spaces on regular auxiliary meshes. Numerical experiments in both two and three space dimensions illustrate the optimal performance of the method
Space-Time PetrovâGalerkin FEM for Fractional Diffusion Problems
We present and analyze a space-time Petrov-Galerkin finite element
method for a time-fractional diffusion equation involving a Riemann-Liouville
fractional derivative of order α â (0, 1) in time and zero initial data. We derive a
proper weak formulation involving different solution and test spaces and show
the inf-sup condition for the bilinear form and thus its well-posedness. Further,
we develop a novel finite element formulation, show the well-posedness of the
discrete problem, and derive error bounds in both energy and L
2 norms for the
finite element solution. In the proof of the discrete inf-sup condition, a certain
nonstandard L
2
stability property of the L
2 projection operator plays a key role.
We provide extensive numerical examples to verify the convergence analysis
Single-layer metal-on-metal islands driven by strong time-dependent forces
Non-linear transport properties of single-layer metal-on-metal islands driven
with strong static and time-dependent forces are studied. We apply a
semi-empirical lattice model and use master equation and kinetic Monte Carlo
simulation methods to compute observables such as the velocity and the
diffusion coefficient. Two types of time-dependent driving are considered: a
pulsed rotated field and an alternating field with a zero net force
(electrophoretic ratchet). Small islands up to 12 atoms were studied in detail
with the master equation method and larger ones with simulations. Results are
presented mainly for a parametrization of Cu on Cu(001) surface, which has been
the main system of interest in several previous studies. The main results are
that the pulsed field can increase the current in both diagonal and axis
direction when compared to static field, and there exists a current inversion
in the electrophoretic ratchet. Both of these phenomena are a consequence of
the coupling of the internal dynamics of the island with its transport. In
addition to the previously discovered "magic size" effect for islands in
equilibrium, a strong odd-even effect was found for islands driven far out of
equilibrium. Master equation computations revealed non-monotonous behavior for
the leading relaxation constant and effective Arrhenius parameters. Using cycle
optimization methods, typical island transport mechanisms are identified for
small islands.Comment: 39 pages, 20 figures, to appear in Phys. Rev. E [corrected typo of
the x-axis label in Fig. 6
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