1,312 research outputs found
Emergence of Periodic Structure from Maximizing the Lifetime of a Bound State Coupled to Radiation
Consider a system governed by the time-dependent Schr\"odinger equation in
its ground state. When subjected to weak (size ) parametric forcing
by an "ionizing field" (time-varying), the state decays with advancing time due
to coupling of the bound state to radiation modes. The decay-rate of this
metastable state is governed by {\it Fermi's Golden Rule}, , which
depends on the potential and the details of the forcing. We pose the
potential design problem: find which minimizes (maximizes
the lifetime of the state) over an admissible class of potentials with fixed
spatial support. We formulate this problem as a constrained optimization
problem and prove that an admissible optimal solution exists. Then, using
quasi-Newton methods, we compute locally optimal potentials. These have the
structure of a truncated periodic potential with a localized defect. In
contrast to optimal structures for other spectral optimization problems, our
optimizing potentials appear to be interior points of the constraint set and to
be smooth. The multi-scale structures that emerge incorporate the physical
mechanisms of energy confinement via material contrast and interference
effects.
An analysis of locally optimal potentials reveals local optimality is
attained via two mechanisms: (i) decreasing the density of states near a
resonant frequency in the continuum and (ii) tuning the oscillations of
extended states to make , an oscillatory integral, small. Our
approach achieves lifetimes, , for locally
optimal potentials with as compared with
for a typical potential. Finally, we
explore the performance of optimal potentials via simulations of the
time-evolution.Comment: 33 pages, 6 figure
Sharp spectral stability estimates via the Lebesgue measure of domains for higher order elliptic operators
We prove sharp stability estimates for the variation of the eigenvalues of
non-negative self-adjoint elliptic operators of arbitrary even order upon
variation of the open sets on which they are defined. These estimates are
expressed in terms of the Lebesgue measure of the symmetric difference of the
open sets. Both Dirichlet and Neumann boundary conditions are considered
On thin plate spline interpolation
We present a simple, PDE-based proof of the result [M. Johnson, 2001] that
the error estimates of [J. Duchon, 1978] for thin plate spline interpolation
can be improved by . We illustrate that -matrix
techniques can successfully be employed to solve very large thin plate spline
interpolation problem
On the existence of effective potentials in time-dependent density functional theory
We investigate the existence and properties of effective potentials in
time-dependent density functional theory. We outline conditions for a general
solution of the corresponding Sturm-Liouville boundary value problems. We
define the set of potentials and v-representable densities, give a proof of
existence of the effective potentials under certain restrictions, and show the
set of v-representable densities to be independent of the interaction.Comment: 13 page
Conforming finite element methods for the clamped plate problem
Finite element methods for solving biharmonic boundary value problems are considered. The particular problem discussed is that of a clamped thin plate. This problem is reformulated in a weak, form in the Sobolev space Techniques for setting up conforming trial
Functions are utilized in a Galerkin technique to produce finite element solutions. The shortcomings of various trial function formulations are discussed, and a macroâelement approach to local mesh refinement using rectangular elements is given
Control theory for principled heap sizing
We propose a new, principled approach to adaptive heap sizing based on control theory. We review current state-of-the-art heap sizing mechanisms, as deployed in Jikes RVM and HotSpot. We then formulate heap sizing as a control problem, apply and tune a standard controller algorithm, and evaluate its performance on a set of well-known benchmarks. We find our controller adapts the heap size more responsively than existing mechanisms. This responsiveness allows tighter virtual machine memory footprints while preserving target application throughput, which is ideal for both embedded and utility computing domains. In short, we argue that formal, systematic approaches to memory management should be replacing ad-hoc heuristics as the discipline matures. Control-theoretic heap sizing is one such systematic approach
Sobolev Inequalities for Differential Forms and -cohomology
We study the relation between Sobolev inequalities for differential forms on
a Riemannian manifold and the -cohomology of that manifold.
The -cohomology of is defined to be the quotient of the space
of closed differential forms in modulo the exact forms which are
exterior differentials of forms in .Comment: This paper has appeared in the Journal of Geometric Analysis, (only
minor changes have been made since verion 1
On the existence of initial data containing isolated black holes
We present a general construction of initial data for Einstein's equations
containing an arbitrary number of black holes, each of which is instantaneously
in equilibrium. Each black hole is taken to be a marginally trapped surface and
plays the role of the inner boundary of the Cauchy surface. The black hole is
taken to be instantaneously isolated if its outgoing null rays are shear-free.
Starting from the choice of a conformal metric and the freely specifiable part
of the extrinsic curvature in the bulk, we give a prescription for choosing the
shape of the inner boundaries and the boundary conditions that must be imposed
there. We show rigorously that with these choices, the resulting non-linear
elliptic system always admits solutions.Comment: 11 pages, 2 figures, RevTeX
Rendezvous of Two Robots with Constant Memory
We study the impact that persistent memory has on the classical rendezvous
problem of two mobile computational entities, called robots, in the plane. It
is well known that, without additional assumptions, rendezvous is impossible if
the entities are oblivious (i.e., have no persistent memory) even if the system
is semi-synchronous (SSynch). It has been recently shown that rendezvous is
possible even if the system is asynchronous (ASynch) if each robot is endowed
with O(1) bits of persistent memory, can transmit O(1) bits in each cycle, and
can remember (i.e., can persistently store) the last received transmission.
This setting is overly powerful.
In this paper we weaken that setting in two different ways: (1) by
maintaining the O(1) bits of persistent memory but removing the communication
capabilities; and (2) by maintaining the O(1) transmission capability and the
ability to remember the last received transmission, but removing the ability of
an agent to remember its previous activities. We call the former setting
finite-state (FState) and the latter finite-communication (FComm). Note that,
even though its use is very different, in both settings, the amount of
persistent memory of a robot is constant.
We investigate the rendezvous problem in these two weaker settings. We model
both settings as a system of robots endowed with visible lights: in FState, a
robot can only see its own light, while in FComm a robot can only see the other
robot's light. We prove, among other things, that finite-state robots can
rendezvous in SSynch, and that finite-communication robots are able to
rendezvous even in ASynch. All proofs are constructive: in each setting, we
present a protocol that allows the two robots to rendezvous in finite time.Comment: 18 pages, 3 figure
Stability Of contact discontinuity for steady Euler System in infinite duct
In this paper, we prove structural stability of contact discontinuities for
full Euler system
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