751 research outputs found

    Transmission eigenvalues and thermoacoustic tomography

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    The spectrum of the interior transmission problem is related to the unique determination of the acoustic properties of a body in thermoacoustic imaging. Under a non-trapping hypothesis, we show that sparsity of the interior transmission spectrum implies a range separation condition for the thermoacoustic operator. In odd dimension greater than or equal to three, we prove that the transmission spectrum for a pair of radially symmetric non-trapping sound speeds is countable, and conclude that the ranges of the associated thermoacoustic maps have only trivial intersection

    The Thermodynamics of Quantum Systems and Generalizations of Zamolodchikov's C-theorem

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    In this paper we examine the behavior in temperature of the free energy on quantum systems in an arbitrary number of dimensions. We define from the free energy a function CC of the coupling constants and the temperature, which in the regimes where quantum fluctuations dominate, is a monotonically increasing function of the temperature. We show that at very low temperatures the system is controlled by the zero-temperature infrared stable fixed point while at intermediate temperatures the behavior is that of the unstable fixed point. The CC function displays this crossover explicitly. This behavior is reminiscent of Zamolodchikov's CC-theorem of field theories in 1+1 dimensions. Our results are obtained through a thermodynamic renormalization group approach. We find restrictions on the behavior of the entropy of the system for a CC-theorem-type behavior to hold. We illustrate our ideas in the context of a free massive scalar field theory, the one-dimensional quantum Ising Model and the quantum Non-linear Sigma Model in two space dimensions. In regimes in which the classical fluctuations are important the monotonic behavior is absent.Comment: 25 pages, LateX, P-92-10-12

    Fate of Quasiparticle at Mott Transition and Interplay with Lifshitz Transition Studied by Correlator Projection Method

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    Filling-control metal-insulator transition on the two-dimensional Hubbard model is investigated by using the correlator projection method, which takes into account momentum dependence of the free energy beyond the dynamical mean-field theory. The phase diagram of metals and Mott insulators is analyzed. Lifshitz transitions occur simultaneously with metal-insulator transitions at large Coulomb repulsion. On the other hand, they are separated each other for lower Coulomb repulsion, where the phase sandwiched by the Lifshitz and metal-insulator transitions appears to show violation of the Luttinger sum rule. Through the metal-insulator transition, quasiparticles retain nonzero renormalization factor and finite quasi-particle weight in the both sides of the transition. This supports that the metal-insulator transition is caused not by the vanishing renormalization factor but by the relative shift of the Fermi level into the Mott gap away from the quasiparticle band, in sharp contrast with the original dynamical mean-field theory. Charge compressibility diverges at the critical end point of the first-order Lifshitz transition at finite temperatures. The origin of the divergence is ascribed to singular momentum dependence of the quasiparticle dispersion.Comment: 24 pages including 10 figure

    Rigid continuation paths I. Quasilinear average complexity for solving polynomial systems

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    How many operations do we need on the average to compute an approximate root of a random Gaussian polynomial system? Beyond Smale's 17th problem that asked whether a polynomial bound is possible, we prove a quasi-optimal bound (input size)1+o(1)\text{(input size)}^{1+o(1)}. This improves upon the previously known (input size)32+o(1)\text{(input size)}^{\frac32 +o(1)} bound. The new algorithm relies on numerical continuation along \emph{rigid continuation paths}. The central idea is to consider rigid motions of the equations rather than line segments in the linear space of all polynomial systems. This leads to a better average condition number and allows for bigger steps. We show that on the average, we can compute one approximate root of a random Gaussian polynomial system of~nn equations of degree at most DD in n+1n+1 homogeneous variables with O(n5D2)O(n^5 D^2) continuation steps. This is a decisive improvement over previous bounds that prove no better than 2min(n,D)\sqrt{2}^{\min(n, D)} continuation steps on the average

    QED Fermi-Fields as Operator Valued Distributions and Anomalies

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    The treatment of fields as operator valued distributions (OPVD) is recalled with the emphasis on the importance of causality following the work of Epstein and Glaser. Gauge invariant theories demand the extension of the usual translation operation on OPVD, thereby leading to a generalized QEDQED formulation. At D=2 the solvability of the Schwinger model is totally preserved. At D=4 the paracompactness property of the Euclidean manifold permits using test functions which are decomposition of unity and thereby provides a natural justification and extension of the non perturbative heat kernel method (Fujikawa) for abelian anomalies. On the Minkowski manifold the specific role of causality in the restauration of gauge invariance (and mass generation for QED2QED_{2}) is examplified in a simple way.Comment: soumis le 22/09/200

    Stability in Conductivity Imaging from Partial Measurements of One Interior Current

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    We prove a stability result in the hybrid inverse problem of recovering the electrical conductivity from partial knowledge of one current density field generated inside a body by an imposed boundary voltage. The region where interior data stably reconstructs the conductivity is well defined by a combination of the exact and perturbed data

    Why is the ground state electron configuration for Lithium 1s22s1s^22s ?

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    The electronic ground state for Lithium is 1s22s1s^22s, and not 1s22p1s^22p. The traditional argument for why this is so is based on a screening argument that claims that the 2p2p electron is better shielded by the 1s1s electrons, and therefore higher in energy then the configuration that includes the 2s2s electron. We show that this argument is flawed, and in fact the actual reason for the ordering is because the electron-electron interaction energy is higher for the 2p1s2p-1s repulsion than it is for the 2s1s2s-1s repulsion.Comment: 4 page

    Lepskii Principle in Supervised Learning

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    In the setting of supervised learning using reproducing kernel methods, we propose a data-dependent regularization parameter selection rule that is adaptive to the unknown regularity of the target function and is optimal both for the least-square (prediction) error and for the reproducing kernel Hilbert space (reconstruction) norm error. It is based on a modified Lepskii balancing principle using a varying family of norms
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