The authors consider Sidon sets of first kind. By comparing them with the Steinhaus sequence, they prove a local Khintchine-Kahane inequality on compact sets. As consequences, they prove the following results for Sido...The authors consider Sidon sets of first kind. By comparing them with the Steinhaus sequence, they prove a local Khintchine-Kahane inequality on compact sets. As consequences, they prove the following results for Sidon series taking values in a Banach space: the summability on a set of positive measure implies the almost everywhere convergence; the contraction principle of Billard-Kahane remains true for Sidon series. As applications, they extend a uniqueness theorem of Zygmund concerning lacunary Fourier series and an analytic continuation theorem of Hadamard concerning lacunary Taylor series. Some of their results still hold for Sidon sets of second kind.展开更多
This work deals with the numerical localization of small electromagnetic inhomogeneities. The underlying inverse problem considers, in a three-dimensional bounded domain, the time-harmonic Maxwell equations formulated...This work deals with the numerical localization of small electromagnetic inhomogeneities. The underlying inverse problem considers, in a three-dimensional bounded domain, the time-harmonic Maxwell equations formulated in electric field. Typically, the domain contains a finite number of unknown inhomogeneities of small volume and the inverse problem attempts to localize these inhomogeneities from a finite number of boundary measurements. Our localization approach is based on a recent framework that uses an asymptotic expansion for the perturbations in the tangential boundary trace of the curl of the electric field. We present three numerical localization procedures resulting from the combination of this asymptotic expansion with each of the following inversion algorithms： the Current Projection method, the MUltiple Signal Classification （MUSIC） algorithm, and an Inverse Fourier method. We perform a numerical study of the asymptotic expansion and compare the numerical results obtained from the three localization procedures in different settings.展开更多
From a limit model in electric field obtained by letting the frequency vanish in the time-harmonic Maxwell equations, we consider a limit perturbation model in the tangential boundary trace of the curl of the electric...From a limit model in electric field obtained by letting the frequency vanish in the time-harmonic Maxwell equations, we consider a limit perturbation model in the tangential boundary trace of the curl of the electric field for localizing numerically certain small electromagnetic inhomogeneities, in a three-dimensional bounded domain. We introduce here two localization procedures resulting from the combination of this limit perturbation model with each of the following inversion processes： the Current Projection method and an Inverse Fourier method. Each localization procedure uses, as data, a finite number of boundary measurements, and is employed in the single inhomogeneity case; only the one based on an Inverse Fourier method is required in the multiple inhomogeneities case. Our localization approach is numerically suitable for the context of inhomogeneities that are not purely electric. We compare the numerical results obtained from the two localization procedures in the single inhomogeneity configuration, and describe, in various settings of multiple inhomogeneities, the results provided by the procedure based on an Inverse Fourier method.展开更多
Let(■_n)≥0 be the Markov chain of two states with respect to the probability measure of the maximal entropy on the subshift spaceΣ_A defined by Fibonacci incident matrix A.We consider the measureμ_λof the probabi...Let(■_n)≥0 be the Markov chain of two states with respect to the probability measure of the maximal entropy on the subshift spaceΣ_A defined by Fibonacci incident matrix A.We consider the measureμ_λof the probability distribution of the random seriesΣ_(n=0)~∞■_nλ~n(0<λ<1).It is proved thatμ_λis singular ifλ∈(0,(5^(1/2)-1)/2)and thatμ_λis absolutely continuous for almost allλ∈((5^(1/2)-1)/2,0.739).展开更多
The aim of this paper is to solve numerically the inverse problem of reconstructing small amplitude perturbations in the magnetic permeability of a dielectric material from partial or total dynamic boundary measuremen...The aim of this paper is to solve numerically the inverse problem of reconstructing small amplitude perturbations in the magnetic permeability of a dielectric material from partial or total dynamic boundary measurements. Our numerical algorithm is based on the resolution of the time-dependent Maxwell equations, an exact controllability method and Fourier inversion for localizing the perturbations. Two-dimensional numerical experiments illustrate the performance of the reconstruction method for different configurations even in the case of limited-view data.展开更多
The random trigonometric series∑∞n=1ρn cos(nt+ωn)on the circle T are studied under the conditions∑|ρn|^(2)=∞andρn→0,where{ωn}are independent and uniformly distributed random variables on T.They are almost su...The random trigonometric series∑∞n=1ρn cos(nt+ωn)on the circle T are studied under the conditions∑|ρn|^(2)=∞andρn→0,where{ωn}are independent and uniformly distributed random variables on T.They are almost surely not Fourier-Stieltjes series but determine pseudo-functions.This leads us to develop the theory of trigonometric multiplicative chaos,which produces a class of random measures.The kernel and the image of chaotic operators are fully studied and the dimensions of chaotic measures are exactly computed.The behavior of the partial sums of the above series is proved to be multifractal.Our theory holds on the torus Tdof dimension d≥1.展开更多
文摘The authors consider Sidon sets of first kind. By comparing them with the Steinhaus sequence, they prove a local Khintchine-Kahane inequality on compact sets. As consequences, they prove the following results for Sidon series taking values in a Banach space: the summability on a set of positive measure implies the almost everywhere convergence; the contraction principle of Billard-Kahane remains true for Sidon series. As applications, they extend a uniqueness theorem of Zygmund concerning lacunary Fourier series and an analytic continuation theorem of Hadamard concerning lacunary Taylor series. Some of their results still hold for Sidon sets of second kind.
基金supported by ACI NIM (171) from the French Ministry of Education and Scientific Research
文摘This work deals with the numerical localization of small electromagnetic inhomogeneities. The underlying inverse problem considers, in a three-dimensional bounded domain, the time-harmonic Maxwell equations formulated in electric field. Typically, the domain contains a finite number of unknown inhomogeneities of small volume and the inverse problem attempts to localize these inhomogeneities from a finite number of boundary measurements. Our localization approach is based on a recent framework that uses an asymptotic expansion for the perturbations in the tangential boundary trace of the curl of the electric field. We present three numerical localization procedures resulting from the combination of this asymptotic expansion with each of the following inversion algorithms： the Current Projection method, the MUltiple Signal Classification （MUSIC） algorithm, and an Inverse Fourier method. We perform a numerical study of the asymptotic expansion and compare the numerical results obtained from the three localization procedures in different settings.
基金supported by ACI NIM(171)from the French Ministry of Education and Scientific Research
文摘From a limit model in electric field obtained by letting the frequency vanish in the time-harmonic Maxwell equations, we consider a limit perturbation model in the tangential boundary trace of the curl of the electric field for localizing numerically certain small electromagnetic inhomogeneities, in a three-dimensional bounded domain. We introduce here two localization procedures resulting from the combination of this limit perturbation model with each of the following inversion processes： the Current Projection method and an Inverse Fourier method. Each localization procedure uses, as data, a finite number of boundary measurements, and is employed in the single inhomogeneity case; only the one based on an Inverse Fourier method is required in the multiple inhomogeneities case. Our localization approach is numerically suitable for the context of inhomogeneities that are not purely electric. We compare the numerical results obtained from the two localization procedures in the single inhomogeneity configuration, and describe, in various settings of multiple inhomogeneities, the results provided by the procedure based on an Inverse Fourier method.
文摘Let(■_n)≥0 be the Markov chain of two states with respect to the probability measure of the maximal entropy on the subshift spaceΣ_A defined by Fibonacci incident matrix A.We consider the measureμ_λof the probability distribution of the random seriesΣ_(n=0)~∞■_nλ~n(0<λ<1).It is proved thatμ_λis singular ifλ∈(0,(5^(1/2)-1)/2)and thatμ_λis absolutely continuous for almost allλ∈((5^(1/2)-1)/2,0.739).
文摘The aim of this paper is to solve numerically the inverse problem of reconstructing small amplitude perturbations in the magnetic permeability of a dielectric material from partial or total dynamic boundary measurements. Our numerical algorithm is based on the resolution of the time-dependent Maxwell equations, an exact controllability method and Fourier inversion for localizing the perturbations. Two-dimensional numerical experiments illustrate the performance of the reconstruction method for different configurations even in the case of limited-view data.
基金supported by National Natural Science Foundation of China (Grant No.11971192)。
文摘The random trigonometric series∑∞n=1ρn cos(nt+ωn)on the circle T are studied under the conditions∑|ρn|^(2)=∞andρn→0,where{ωn}are independent and uniformly distributed random variables on T.They are almost surely not Fourier-Stieltjes series but determine pseudo-functions.This leads us to develop the theory of trigonometric multiplicative chaos,which produces a class of random measures.The kernel and the image of chaotic operators are fully studied and the dimensions of chaotic measures are exactly computed.The behavior of the partial sums of the above series is proved to be multifractal.Our theory holds on the torus Tdof dimension d≥1.
