765 research outputs found
Frequency Precision of Oscillators Based on High-Q Resonators
We present a method for analyzing the phase noise of oscillators based on
feedback driven high quality factor resonators. Our approach is to derive the
phase drift of the oscillator by projecting the stochastic oscillator dynamics
onto a slow time scale corresponding physically to the long relaxation time of
the resonator. We derive general expressions for the phase drift generated by
noise sources in the electronic feedback loop of the oscillator. These are
mixed with the signal through the nonlinear amplifier, which makes them
{cyclostationary}. We also consider noise sources acting directly on the
resonator. The expressions allow us to investigate reducing the oscillator
phase noise thereby improving the frequency precision using resonator
nonlinearity by tuning to special operating points. We illustrate the approach
giving explicit results for a phenomenological amplifier model. We also propose
a scheme for measuring the slow feedback noise generated by the feedback
components in an open-loop driven configuration in experiment or using circuit
simulators, which enables the calculation of the closed-loop oscillator phase
noise in practical systems
Scattering below critical energy for the radial 4D Yang-Mills equation and for the 2D corotational wave map system
We describe the asymptotic behavior as time goes to infinity of solutions of
the 2 dimensional corotational wave map system and of solutions to the 4
dimensional, radially symmetric Yang-Mills equation, in the critical energy
space, with data of energy smaller than or equal to a harmonic map of minimal
energy. An alternative holds: either the data is the harmonic map and the
soltuion is constant in time, or the solution scatters in infinite time
Well-posedness and stability results for the Gardner equation
In this article we present local well-posedness results in the classical
Sobolev space H^s(R) with s > 1/4 for the Cauchy problem of the Gardner
equation, overcoming the problem of the loss of the scaling property of this
equation. We also cover the energy space H^1(R) where global well-posedness
follows from the conservation laws of the system. Moreover, we construct
solitons of the Gardner equation explicitly and prove that, under certain
conditions, this family is orbitally stable in the energy space.Comment: 1 figure. Accepted for publication in Nonlin.Diff Eq.and App
Intrinsic localized modes in parametrically driven arrays of nonlinear resonators
We study intrinsic localized modes (ILMs), or solitons, in arrays of parametrically driven nonlinear resonators with application to microelectromechanical and nanoelectromechanical systems (MEMS and NEMS). The analysis is performed using an amplitude equation in the form of a nonlinear Schrödinger equation with a term corresponding to nonlinear damping (also known as a forced complex Ginzburg-Landau equation), which is derived directly from the underlying equations of motion of the coupled resonators, using the method of multiple scales. We investigate the creation, stability, and interaction of ILMs, show that they can form bound states, and that under certain conditions one ILM can split into two. Our findings are confirmed by simulations of the underlying equations of motion of the resonators, suggesting possible experimental tests of the theory
Weighted maximal regularity estimates and solvability of non-smooth elliptic systems II
We continue the development, by reduction to a first order system for the
conormal gradient, of \textit{a priori} estimates and solvability for
boundary value problems of Dirichlet, regularity, Neumann type for divergence
form second order, complex, elliptic systems. We work here on the unit ball and
more generally its bi-Lipschitz images, assuming a Carleson condition as
introduced by Dahlberg which measures the discrepancy of the coefficients to
their boundary trace near the boundary. We sharpen our estimates by proving a
general result concerning \textit{a priori} almost everywhere non-tangential
convergence at the boundary. Also, compactness of the boundary yields more
solvability results using Fredholm theory. Comparison between classes of
solutions and uniqueness issues are discussed. As a consequence, we are able to
solve a long standing regularity problem for real equations, which may not be
true on the upper half-space, justifying \textit{a posteriori} a separate work
on bounded domains.Comment: 76 pages, new abstract and few typos corrected. The second author has
changed nam
Analycity and smoothing effect for the coupled system of equations of Korteweg - de Vries type with a single point singularity
We study that a solution of the initial value problem associated for the
coupled system of equations of Korteweg - de Vries type which appears as a
model to describe the strong interaction of weakly nonlinear long waves, has
analyticity in time and smoothing effect up to real analyticity if the initial
data only has a single point singularity at $x=0.
Analyticity of layer potentials and solvability of boundary value problems for divergence form elliptic equations with complex coefficients
We consider divergence form elliptic operators of the form L=-\dv
A(x)\nabla, defined in , ,
where the coefficient matrix is , uniformly
elliptic, complex and -independent. We show that for such operators,
boundedness and invertibility of the corresponding layer potential operators on
, is stable under
complex, perturbations of the coefficient matrix. Using a variant
of the Theorem, we also prove that the layer potentials are bounded and
invertible on whenever is real and symmetric (and
thus, by our stability result, also when is complex, is small enough and is real, symmetric,
and elliptic). In particular, we establish solvability of the Dirichlet and
Neumann (and Regularity) problems, with (resp. data, for
small complex perturbations of a real symmetric matrix. Previously,
solvability results for complex (or even real but non-symmetric) coefficients
were known to hold only for perturbations of constant matrices (and then only
for the Dirichlet problem), or in the special case that the coefficients
, , which corresponds to the Kato square
root problem
Eliminating 1/f noise in oscillators
We study 1/f and narrow-bandwidth noise in precision oscillators based on high-quality factor resonators and feedback. The dynamics of such an oscillator are well described by two variables, an amplitude and a phase. In this description we show that low-frequency feedback noise is represented by a single noise vector in phase space. The implication of this is that 1/f and narrow-bandwidth noise can be eliminated by tuning controllable parameters, such as the feedback phase. We present parameter values for which the noise is eliminated and provide specific examples of noise sources for further illustration
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