The recovery of a spherically-symmetric wave speed v is considered in a bounded spherical region of radius b from the set of the corresponding transmission eigenvalues for which the corresponding eigenfunctions are also spherically symmetric. If the integral of 1/v on the interval [0, b] is less than b, assuming that there exists at least one v corresponding to the data, it is shown that v is uniquely determined by the data consisting of such transmission eigenvalues and their "multiplicities," where the "multiplicity" is defined as the multiplicity of the transmission eigenvalue as a zero of a key quantity.When that integral is equal to b, the unique recovery is obtained when the data contains one additional piece of information. Some similar results are presented for the unique determination of the potential from the transmission eigenvalues with "multiplicities" for a related Schrödinger equation.
Mathematics Subject Classification (2010): 34B07 34B24 47E05Short title: Inverse problem for transmission eigenvalues
The unique reconstruction of a spherically-symmetric wave speed v is considered in a bounded spherical region of radius b from the set of corresponding transmission eigenvalues for which the corresponding eigenfunctions are also spherically symmetric. If the integral of 1/v on the interval [0, b] is less than b, assuming that there exists at least one v corresponding to the data, v is uniquely reconstructed from the data consisting of such transmission eigenvalues and their "multiplicities," where the multiplicity is defined as the multiplicity of the transmission eigenvalue as a zero of a key quantity. When that integral is equal to b, the unique reconstruction is presented when the data set contains one additional piece of information. Some similar results are presented for the unique reconstruction of the potential from the transmission eigenvalues with multiplicities for a related Schrödinger equation.
Mathematics Subject Classification (2010): 34B07 34B24 47E05Short title: Inverse problem for transmission eigenvalues
We study the spectral theory of the fourth-order eigenvalue problemwhere the functions a and p are periodic and strictly positive. This equation models the transverse vibrations of a thin straight (periodic) beam whose physical characteristics are described by a and p.We examine the structure of the spectrum establishing the fact that the periodic and antiperiodic eigenvalues are the endpoints of the spectral bands. We also introduce an entire function, which we denote by E(λ), connected to the spectral theory, whose zeros (at least the ones of odd multiplicity) are shown to lie on the negative real axis, where they define a collection of "pseudogaps." Next we prove some inverse results in the spirit of two old theorems of Borg for the Hill's equation. We finish with a "determinant formula" (i.e. a multiplicative trace formula) and some comments on its role in the formulation of the general inverse problem.
We develop techniques for computing the asymptotics of the first and second moments of the number T
N
of coupons that a collector has to buy in order to find all N existing different coupons as N → ∞. The probabilities (occurring frequencies) of the coupons can be quite arbitrary. From these asymptotics we obtain the leading behavior of the variance V[T
N
] of T
N
(see Theorems 3.1 and 4.4). Then, we combine our results with the general limit theorems of Neal in order to derive the limit distribution of T
N
(appropriately normalized), which, for a large class of probabilities, turns out to be the standard Gumbel distribution. We also give various illustrative examples.
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