751 research outputs found
Transmission eigenvalues and thermoacoustic tomography
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
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 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
function displays this crossover explicitly. This behavior is reminiscent
of Zamolodchikov's -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
-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
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
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
. This improves upon the previously known
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~ equations of degree at most in
homogeneous variables with continuation steps. This is a
decisive improvement over previous bounds that prove no better than
continuation steps on the average
QED Fermi-Fields as Operator Valued Distributions and Anomalies
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
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 ) is examplified in a simple way.Comment: soumis le 22/09/200
Stability in Conductivity Imaging from Partial Measurements of One Interior Current
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 ?
The electronic ground state for Lithium is , and not . The
traditional argument for why this is so is based on a screening argument that
claims that the electron is better shielded by the electrons, and
therefore higher in energy then the configuration that includes the
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 repulsion than it is for the repulsion.Comment: 4 page
Lepskii Principle in Supervised Learning
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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