22,879 research outputs found
The Novikov conjecture on Cheeger spaces
We prove the Novikov conjecture on oriented Cheeger spaces whose fundamental
group satisfies the strong Novikov conjecture. A Cheeger space is a stratified
pseudomanifold admitting, through a choice of ideal boundary conditions, an
L2-de Rham cohomology theory satisfying Poincare duality. We prove that this
cohomology theory is invariant under stratified homotopy equivalences and that
its signature is invariant under Cheeger space cobordism. Analogous results,
after coupling with a Mishchenko bundle associated to any Galois covering,
allow us to carry out the analytic approach to the Novikov conjecture: we
define higher analytic signatures of a Cheeger space and prove that they are
stratified homotopy invariants whenever the assembly map is rationally
injective. Finally we show that the analytic signature of a Cheeger space
coincides with its topological signature as defined by Banagl.Comment: To appear in JNC
High-order, Dispersionless "Fast-Hybrid" Wave Equation Solver. Part I: Sampling Cost via Incident-Field Windowing and Recentering
This paper proposes a frequency/time hybrid integral-equation method for the
time dependent wave equation in two and three-dimensional spatial domains.
Relying on Fourier Transformation in time, the method utilizes a fixed
(time-independent) number of frequency-domain integral-equation solutions to
evaluate, with superalgebraically-small errors, time domain solutions for
arbitrarily long times. The approach relies on two main elements, namely, 1) A
smooth time-windowing methodology that enables accurate band-limited
representations for arbitrarily-long time signals, and 2) A novel Fourier
transform approach which, in a time-parallel manner and without causing
spurious periodicity effects, delivers numerically dispersionless
spectrally-accurate solutions. A similar hybrid technique can be obtained on
the basis of Laplace transforms instead of Fourier transforms, but we do not
consider the Laplace-based method in the present contribution. The algorithm
can handle dispersive media, it can tackle complex physical structures, it
enables parallelization in time in a straightforward manner, and it allows for
time leaping---that is, solution sampling at any given time at
-bounded sampling cost, for arbitrarily large values of ,
and without requirement of evaluation of the solution at intermediate times.
The proposed frequency-time hybridization strategy, which generalizes to any
linear partial differential equation in the time domain for which
frequency-domain solutions can be obtained (including e.g. the time-domain
Maxwell equations), and which is applicable in a wide range of scientific and
engineering contexts, provides significant advantages over other available
alternatives such as volumetric discretization, time-domain integral equations,
and convolution-quadrature approaches.Comment: 33 pages, 8 figures, revised and extended manuscript (and now
including direct comparisons to existing CQ and TDIE solver implementations)
(Part I of II
Model Reduction Near Periodic Orbits of Hybrid Dynamical Systems
We show that, near periodic orbits, a class of hybrid models can be reduced
to or approximated by smooth continuous-time dynamical systems. Specifically,
near an exponentially stable periodic orbit undergoing isolated transitions in
a hybrid dynamical system, nearby executions generically contract
superexponentially to a constant-dimensional subsystem. Under a non-degeneracy
condition on the rank deficiency of the associated Poincare map, the
contraction occurs in finite time regardless of the stability properties of the
orbit. Hybrid transitions may be removed from the resulting subsystem via a
topological quotient that admits a smooth structure to yield an equivalent
smooth dynamical system. We demonstrate reduction of a high-dimensional
underactuated mechanical model for terrestrial locomotion, assess structural
stability of deadbeat controllers for rhythmic locomotion and manipulation, and
derive a normal form for the stability basin of a hybrid oscillator. These
applications illustrate the utility of our theoretical results for synthesis
and analysis of feedback control laws for rhythmic hybrid behavior
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