1,720,987 research outputs found
THE NORM OF TIME-FREQUENCY AND WAVELET LOCALIZATION OPERATORS
Full text linkTime-frequency localization operators (with Gaussian window) L_F : L^2(R^d) → L^2(R^d), where F is a weight in R^2d, were introduced in signal processing by I. Daubechies [IEEE Trans. Inform. Theory 34 (1988), pp. 605–612], inaugurating a new, geometric, phase-space perspective. Sharp upper bounds for the norm (and the singular values) of such operators turn out to be a challenging issue with deep applications in signal recovery, quantum physics and the study of uncertainty principles. In this note we provide optimal upper bounds for the operator norm ||L_F||_{L^2→L^2}, assuming F ∈ L^p(R^2d), 1 < p < ∞ or F ∈ L^p(R^2d) ∩ L^∞(R^2d), 1 ≤ p < ∞. It turns out that two regimes arise, depending on whether the quantity ||F||_{L^p}/||F||_{L^∞} is less or greater than a certain critical value. In the first regime the extremal weights F, for which equality occurs in the estimates, are certain Gaussians, whereas in the second regime they are proved to be Gaussians truncated above, degenerating into a multiple of a characteristic function of a ball for p = 1. This phase transition through Gaussians truncated above appears to be a new phenomenon in time-frequency concentration problems. For the analogous problem for wavelet localization operators—where the Cauchy wavelet plays the role of the above Gaussian window—a complete solution is also provided
NLS ground states on metric trees: existence results and open questions
Full text linkWe consider the minimization of the NLS energy on a metric tree, either rooted or unrooted, subject to a mass constraint. With respect to the same problem on other types of metric graphs, several new features appear, such as the existence of minimizers with positive energy, and the emergence of unexpected threshold phenomena. We also study the problem with a radial symmetry constraint that is in principle different from the free problem due to the failure of the Pólya–Szegő inequality for radial rearrangements. A key role is played by a new Poincaré inequality with remainder
The low-frequency limiting behavior of ambipolar diffusive models of impedance spectroscopy
No full textThe Poisson–Nernst–Planck (PNP) diffusional model is a successful theoretical framework to investigate the electrochemical impedance response of insulators containing ionic impurities to an external ac stimulus. Apparent deviations of the experimental spectra from the predictions of the PNP model in the low frequency region are usually interpreted as an interfacial property. Here, we provide a rigorous mathematical analysis of the low-frequency limiting behavior of the model, analyzing the possible origin of these deviation related to bulk properties. The analysis points toward the necessity to consider a bulk effect connected with the difference in the diffusion coefficients of cations and anions (ambipolar diffusion). The ambipolar model does not continuously reach the behavior of the one mobile ion diffusion model when the difference in the mobility of the species vanishes, for a fixed frequency, in the cases of ohmic and adsorption–desorption boundary conditions. The analysis is devoted to the low frequency region, where the electrodes play a fundamental role in the response of the cell; thus, different boundary conditions, charged to mimic the non-blocking character of the electrodes, are considered. The new version of the boundary conditions in the limit in which one of the mobility is tending to zero is deduced. According to the analysis in the dc limit, the phenomenological parameters related to the electrodes are frequency dependent, indicating that the exchange of electric charge from the bulk to the external circuit, in the ohmic model, is related to a surface impedance, and not simply to an electric resistance
Uniqueness and non–uniqueness of prescribed mass NLS ground states on metric graphs
Full text linkWe consider the problem of uniqueness of ground states of prescribed mass for the Nonlinear Schrödinger Energy with power nonlinearity on noncompact metric graphs. We first establish that the Lagrange multiplier appearing in the NLS equation is constant on the set of ground states of mass μ, up to an at most countable set of masses. Then we apply this result to obtain uniqueness of ground states on two specific noncompact graphs. Finally we construct a graph that admits at least two ground states with the same mass having different Lagrange multipliers. Our proofs are based on careful variational arguments and rearrangement techniques, and hold both for the subcritical range p∈(2,6) and in the critical case p=6
The Faber-Krahn inequality for the Short-time Fourier transform
Full text linkIn this paper we solve an open problem concerning the characterization of
those measurable sets that, among all sets
having a prescribed Lebesgue measure, can trap the largest possible energy
fraction in time-frequency space, where the energy density of a generic
function is defined in terms of its Short-time Fourier
transform (STFT) , with Gaussian window. More
precisely, given a measurable set having measure
, we prove that the quantity is largest possible if and only if
is equivalent, up to a negligible set, to a ball of measure , and
in this case we characterize all functions that achieve equality. This
result leads to a sharp uncertainty principle for the "essential support" of
the STFT (when , this can be summarized by the optimal bound
, with equality if and only if is a
ball). Our approach, using techniques from measure theory after suitably
rephrasing the problem in the Fock space, also leads to a local version of
Lieb's uncertainty inequality for the STFT in when , as
well as to -concentration estimates when , thus proving a
related conjecture. In all cases we identify the corresponding extremals.Comment: 23 page
A minimization procedure to the existence of segregated solutions to parabolic reaction-diffusion systems
Full text linkA variational approach to the Hele-Shaw flow with injection
No full textWe investigate a variational approach to the Hele–Shaw flow \partial_t\chi = \Delta u + f\chi, f ≥ 0 in R^n, where \chi is the characteristic function of an open set \Omega(t)\in\R^n and u(\cdot, t)\in H^1_0(\Omega(t)) solves −\Delta u(\cdot, t)= f in \Omega(t).
By iteratively solving a variational problem in R^n, we construct a staircase family of opens sets
and a corresponding family of functions: both sets and functions converge increasingly, at fixed time, to a weak solution of the problem. When the latter is not unique, the solution thus obtained is characterized by a minimality property, with respect to set inclusion, at fixed time.
We also prove several monotonicity results of the solutions thus obtained, with respect to both the initial set and the forcing term f
Topics on analysis in metric spaces
No full textBased on lecture notes from the Scuola Normale this book presents the main mathematical prerequisites for analysis in metric spaces. Supplemented with exercises of varying difficulty it is ideal for a graduate-level short course for applied mathematicians and engineers
On the Dirichlet problem with several volume constraints on the level sets
No full textWe consider minimization problems involving the Dirichlet integral under an arbitrary number of volume constraints on the level sets and a generalized boundary condition. More precisely, given a bounded open domain Ω ⊂ Rn with smooth boundary, we study the problem of minimizing fΩ |∇u|2 among all those functions u ∈ H1 that simultaneously satisfy n-dimensional measure constraints on the level sets of the kind |{u = li}| = α1, i = 1,..., k, and a generalized boundary condition u ∈. Here, Κ is a closed convex subset of H1 such that Κ + H01 = Κ; the invariance of Κ under H01 provides that the condition u ∈ Κ actually depends only on the trace of u along δΩ. By a penalization approach, we prove the existence of minimizers and their Hölder continuity, generalizing previous results that are not applicable when a boundary condition is prescribed. Finally, in the case of just two volume constraints, we investigate the Γ-convergence of the above (rescaled) functionals when the total measure of the two prescribed level sets tends to saturate the whole domain Ω. It turns out that the resulting Γ-limit functional can be split into two distinct parts: the perimeter of the interface due to the Dirichlet energy that concentrates along the jump, and a boundary integral term due to the constraint u ∈ Κ. In the particular case where Κ = H1 (i.e. when no boundary condition is prescribed), the boundary term vanishes and we recover a previous result due to Ambrosio et al
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