448 research outputs found
Embedded Eigenvalues and the Nonlinear Schrodinger Equation
A common challenge to proving asymptotic stability of solitary waves is
understanding the spectrum of the operator associated with the linearized flow.
The existence of eigenvalues can inhibit the dispersive estimates key to
proving stability. Following the work of Marzuola & Simpson, we prove the
absence of embedded eigenvalues for a collection of nonlinear Schrodinger
equations, including some one and three dimensional supercritical equations,
and the three dimensional cubic-quintic equation. Our results also rule out
nonzero eigenvalues within the spectral gap and, in 3D, endpoint resonances.
The proof is computer assisted as it depends on the sign of certain inner
products which do not readily admit analytic representations. Our source code
is available for verification at
http://www.math.toronto.edu/simpson/files/spec_prop_asad_simpson_code.zip.Comment: 29 pages, 27 figures: fixed a typo in an equation from the previous
version, and added two equations to clarif
An Efficient Runge-Kutta (4,5) pair
A pair of explicit Runge-Kutta formulas of orders 4 and 5 is derived. It is significantly more efficient than the Fehlberg and Dormand-Prince pairs, and by standard measures it is of at least as high quality. There are two independent estimates of the local error. The local error of the interpolant is, to leading order, a problem-independent function of the local error at the end of the step
Hugoniot data for pyrrhotite and the Earth's core
New shock wave Hugoniot data for pyrrhotite (Fe_(0.9S)) now describe the equation of state to nearly twofold compression at a maximum pressure of 274 GPa. A minor discontinuity on the Hugoniot between 100 and 150 GPa is interpreted as the melting transition. While not tightly constrained, the inferred melting point lies below lower-bound temperature estimates based on the Lindemann criterion. The highest-pressure Hugoniot data (representing the melted phase) are used to model the equation of state for liquid iron sulfide. A density for liquid pyrrhotite of 7.80±0.20 Mg/m^3 under core-mantle boundary conditions (P = 135 GPa, T = 4000 K) is calculated. Assuming that sulfur is the primary alloying element in a predominately iron core, the present data are consistent with a homogeneous outer core containing 10±4 wt % sulfur
Path integrals and symmetry breaking for optimal control theory
This paper considers linear-quadratic control of a non-linear dynamical
system subject to arbitrary cost. I show that for this class of stochastic
control problems the non-linear Hamilton-Jacobi-Bellman equation can be
transformed into a linear equation. The transformation is similar to the
transformation used to relate the classical Hamilton-Jacobi equation to the
Schr\"odinger equation. As a result of the linearity, the usual backward
computation can be replaced by a forward diffusion process, that can be
computed by stochastic integration or by the evaluation of a path integral. It
is shown, how in the deterministic limit the PMP formalism is recovered. The
significance of the path integral approach is that it forms the basis for a
number of efficient computational methods, such as MC sampling, the Laplace
approximation and the variational approximation. We show the effectiveness of
the first two methods in number of examples. Examples are given that show the
qualitative difference between stochastic and deterministic control and the
occurrence of symmetry breaking as a function of the noise.Comment: 21 pages, 6 figures, submitted to JSTA
The structure of the QED-Vacuum and Electron-Positron Pair Production in Super-Intense, pulsed Laser Fields
We discuss electron-positron pair-production by super-intense, short laser
pulses off the physical vacuum state locally deformed by (stripped) nuclei with
large nuclear charges. Consequences of non-perturbative vacuum polarisation
resulting from such a deformation are shortly broached. Production
probabilities per pulse are calculated.Comment: 10 pages, 1 figure, submitted to Journal of Physics
Singularity subtraction for nonlinear weakly singular integral equations of the second kind
The singularity subtraction technique for computing an approximate solution of a linear weakly singular Fredholm integral equation of the second kind is generalized to the case of a nonlinear integral equation of the same kind. Convergence of the sequence of approximate solutions to the exact one is proved under mild standard hypotheses on the nonlinear factor, and on the sequence of quadrature rules used to build an approximate equation. A numerical example is provided with a Hammerstein operator to illustrate some practical aspects of effective computations.The third author was partially supported by CMat (UID/MAT/00013/2013), and the second and fourth authors were partially supported by CMUP (UID/ MAT/ 00144/2013), which are funded by FCT (Portugal) with national funds (MCTES) and European structural funds (FEDER) under the partnership agreement PT2020
A Retrospective Analysis of the AT&T/Time Warner Merger
This article provides a retrospective of a litigated vertical merger: the 2018 AT&T/Time Warner merger, which was challenged by the US Department of Justice, litigated, and permitted to proceed by the court. We describe and evaluate in detail the economic model used by the government’s expert and then focus our empirical work on the accuracy of the predictions made by that model. We also discuss evidence related to the Comcast/NBC Universal merger, which involved the same theory of harm and was allowed to proceed with a remedy similar to the contractual commitment that AT&T/Time Warner unilaterally adopted. We conclude that the evidence from the time of trial showed the theory of harm to be weak and the specific empirical predictions made by the government’s expert to be wrong. Postmerger evidence confirms that conclusion, as does new evidence from the earlier Comcast/NBC Universal merger
Theoretical analysis of the implementation of a quantum phase gate with neutral atoms on atom chips
We present a detailed, realistic analysis of the implementation of a proposal
for a quantum phase gate based on atomic vibrational states, specializing it to
neutral rubidium atoms on atom chips. We show how to create a double--well
potential with static currents on the atom chips, using for all relevant
parameters values that are achieved with present technology. The potential
barrier between the two wells can be modified by varying the currents in order
to realize a quantum phase gate for qubit states encoded in the atomic external
degree of freedom. The gate performance is analyzed through numerical
simulations; the operation time is ~10 ms with a performance fidelity above
99.9%. For storage of the state between the operations the qubit state can be
transferred efficiently via Raman transitions to two hyperfine states, where
its decoherence is strongly inhibited. In addition we discuss the limits
imposed by the proximity of the surface to the gate fidelity.Comment: 9 pages, 5 color figure
Quantum control theory for coupled 2-electron dynamics in quantum dots
We investigate optimal control strategies for state to state transitions in a
model of a quantum dot molecule containing two active strongly interacting
electrons. The Schrodinger equation is solved nonperturbatively in conjunction
with several quantum control strategies. This results in optimized electric
pulses in the THz regime which can populate combinations of states with very
short transition times. The speedup compared to intuitively constructed pulses
is an order of magnitude. We furthermore make use of optimized pulse control in
the simulation of an experimental preparation of the molecular quantum dot
system. It is shown that exclusive population of certain excited states leads
to a complete suppression of spin dephasing, as was indicated in Nepstad et al.
[Phys. Rev. B 77, 125315 (2008)].Comment: 24 pages, 9 figure
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