7,434 research outputs found
The squares of the Laplacian-Dirichlet eigenfunctions are generically linearly independent
The paper deals with the genericity of domain-dependent spectral properties
of the Laplacian-Dirichlet operator. In particular we prove that, generically,
the squares of the eigenfunctions form a free family. We also show that the
spectrum is generically non-resonant. The results are obtained by applying
global perturbations of the domains and exploiting analytic perturbation
properties. The work is motivated by two applications: an existence result for
the problem of maximizing the rate of exponential decay of a damped membrane
and an approximate controllability result for the bilinear Schr\"odinger
equation
How to play a disc brake
We consider a gyroscopic system under the action of small dissipative and
non-conservative positional forces, which has its origin in the models of
rotating bodies of revolution being in frictional contact. The spectrum of the
unperturbed gyroscopic system forms a "spectral mesh" in the plane "frequency
-gyroscopic parameter" with double semi-simple purely imaginary eigenvalues at
zero value of the gyroscopic parameter. It is shown that dissipative forces
lead to the splitting of the semi-simple eigenvalue with the creation of the
so-called "bubble of instability" - a ring in the three-dimensional space of
the gyroscopic parameter and real and imaginary parts of eigenvalues, which
corresponds to complex eigenvalues. In case of full dissipation with a
positive-definite damping matrix the eigenvalues of the ring have negative real
parts making the bubble a latent source of instability because it can "emerge"
to the region of eigenvalues with positive real parts due to action of both
indefinite damping and non-conservative positional forces. In the paper, the
instability mechanism is analytically described with the use of the
perturbation theory of multiple eigenvalues. As an example stability of a
rotating circular string constrained by a stationary load system is studied in
detail. The theory developed seems to give a first clear explanation of the
mechanism of self-excited vibrations in the rotating structures in frictional
contact, that is responsible for such well-known phenomena of acoustics of
friction as the squealing disc brake and the singing wine glass.Comment: 25 pages, 9 figures, Presented at BIRS 07w5068 Workshop "Geometric
Mechanics: Continuous and discrete, finite and infinite dimensional", August
12-17, 2007, Banff, Canad
On the behavior of clamped plates under large compression
We determine the asymptotic behavior of eigenvalues of clamped plates under large compression by relating this problem to eigenvalues of the Laplacian with Robin boundary conditions. Using the method of fundamental solutions, we then carry out a numerical study of the extremal domains for the first eigenvalue, from which we see that these depend on the value of the compression, and start developing a boundary structure as this parameter is increased. The corresponding number of nodal domains of the first eigenfunction of the extremal domain also increases with the compression.This work was partially supported by the Funda ̧c ̃ao para a Ciˆencia e a Tecnologia(Portugal) through the program “Investigador FCT” with reference IF/00177/2013 and the projectExtremal spectral quantities and related problems(PTDC/MAT-CAL/4334/2014).info:eu-repo/semantics/publishedVersio
Local integration by parts and Pohozaev identities for higher order fractional Laplacians
We establish an integration by parts formula in bounded domains for the
higher order fractional Laplacian with . We also obtain the
Pohozaev identity for this operator. Both identities involve local boundary
terms, and they extend the identities obtained by the authors in the case
.
As an immediate consequence of these results, we obtain a unique continuation
property for the eigenfunctions in ,
in .Comment: The sign of the boundary term in Theorem 1.5 has been correcte
Spatial hole burning in thin-disk lasers and twisted-mode operation
Spatial hole burning prevents single-frequency operation of thin-disk lasers
when the thin disk is used as a folding mirror. We present an evaluation of the
saturation effects in the disk for disks acting as end-mirrors and as
folding-mirrors explaining one of the main obstacles towards single-frequency
operation. It is shown that a twisted-mode scheme based on a multi-order
quarter-wave plate combined with a polarizer provides an almost complete
suppression of spatial hole burning and creates an additional wavelength
selectivity that enforces efficient single-frequency operation.Comment: 14 pages, 16 figure
Drifting Oscillations in Axion Monodromy
We study the pattern of oscillations in the primordial power spectrum in
axion monodromy inflation, accounting for drifts in the oscillation period that
can be important for comparing to cosmological data. In these models the
potential energy has a monomial form over a super-Planckian field range, with
superimposed modulations whose size is model-dependent. The amplitude and
frequency of the modulations are set by the expectation values of moduli
fields. We show that during the course of inflation, the diminishing energy
density can induce slow adjustments of the moduli, changing the modulations. We
provide templates capturing the effects of drifting moduli, as well as drifts
arising in effective field theory models based on softly broken discrete shift
symmetries, and we estimate the precision required to detect a drifting period.
A non-drifting template suffices over a wide range of parameters, but for the
highest frequencies of interest, or for sufficiently strong drift, it is
necessary to include parameters characterizing the change in frequency over the
e-folds visible in the CMB. We use these templates to perform a preliminary
search for drifting oscillations in a part of the parameter space in the Planck
nominal mission data.Comment: 48 pages, 5 figure
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