539 research outputs found
Upper semi-continuity of the Royden-Kobayashi pseudo-norm, a counterexample for H\"olderian almost complex structures
If is an almost complex manifold, with an almost complex structure of
class \CC^\alpha, for some , for every point and every
tangent vector at , there exists a germ of -holomorphic disc through
with this prescribed tangent vector. This existence result goes back to
Nijenhuis-Woolf. All the holomorphic curves are of class \CC^{1,\alpha}
in this case.
Then, exactly as for complex manifolds one can define the Royden-Kobayashi
pseudo-norm of tangent vectors. The question arises whether this pseudo-norm is
an upper semi-continuous function on the tangent bundle. For complex manifolds
it is the crucial point in Royden's proof of the equivalence of the two
standard definitions of the Kobayashi pseudo-metric. The upper semi-continuity
of the Royden-Kobayashi pseudo-norm has been established by Kruglikov for
structures that are smooth enough. In [I-R], it is shown that \CC^{1,\alpha}
regularity of is enough.
Here we show the following:
Theorem. There exists an almost complex structure of class \CC^{1\over
2} on the unit bidisc \D^2\subset \C^2, such that the Royden-Kobayashi
seudo-norm is not an upper semi-continuous function on the tangent bundle.Comment: 5 page
Tungsten resonance integrals and Doppler coefficients Third quarterly report, Jan. - Mar. 1966
Reactivities, Doppler coefficients, and resonance integrals for tungsten isotope
Quantum graphs as holonomic constraints
We consider the dynamics on a quantum graph as the limit of the dynamics
generated by a one-particle Hamiltonian in R^2 with a potential having a deep
strict minimum on the graph, when the width of the well shrinks to zero. For a
generic graph we prove convergence outside the vertices to the free dynamics on
the edges. For a simple model of a graph with two edges and one vertex, we
prove convergence of the dynamics to the one generated by the Laplacian with
Dirichlet boundary conditions in the vertex.Comment: 28 pages, 3 figure
New Classes of Potentials for which the Radial Schrodinger Equation can be solved at Zero Energy
Given two spherically symmetric and short range potentials and V_1 for
which the radial Schrodinger equation can be solved explicitely at zero energy,
we show how to construct a new potential for which the radial equation can
again be solved explicitely at zero energy. The new potential and its
corresponding wave function are given explicitely in terms of V_0 and V_1, and
their corresponding wave functions \phi_0 and \phi_1. V_0 must be such that it
sustains no bound states (either repulsive, or attractive but weak). However,
V_1 can sustain any (finite) number of bound states. The new potential V has
the same number of bound states, by construction, but the corresponding
(negative) energies are, of course, different. Once this is achieved, one can
start then from V_0 and V, and construct a new potential \bar{V} for which the
radial equation is again solvable explicitely. And the process can be repeated
indefinitely. We exhibit first the construction, and the proof of its validity,
for regular short range potentials, i.e. those for which rV_0(r) and rV_1(r)
are L^1 at the origin. It is then seen that the construction extends
automatically to potentials which are singular at r= 0. It can also be extended
to V_0 long range (Coulomb, etc.). We give finally several explicit examples.Comment: 26 pages, 3 figure
Some homogenization and corrector results for nonlinear monotone operators
This paper deals with the limit behaviour of the solutions of quasi-linear
equations of the form \ \ds -\limfunc{div}\left(a\left(x, x/{\varepsilon
_h},Du_h\right)\right)=f_h on with Dirichlet boundary conditions.
The sequence tends to and the map is
periodic in , monotone in and satisfies suitable continuity
conditions. It is proved that weakly in , where is the solution of a homogenized problem \
-\limfunc{div}(b(x,Du))=f on . We also prove some corrector results,
i.e. we find such that in
Constraints on unroofing rates in the high Himalaya, eastern Nepal
Thermobarometric data for samples across the Main Central thrust zone in eastern Nepal show an inversion in temperature but not in pressure. These data have been interpreted to represent a portion of the paleogeotherm at the time of Main Central thrust deformation. A 40Ar/39Ar age on hornblende (closure temperature (Tc)=500±50°C) constrains the timing of this deformation to be âŒ21±0.2 Ma. The 40Ar/39Ar ages of other minerals (muscovite, Tc=350°C, age (t)=12.0±0.2 Ma; K-feldspar, Tc=220°C, t=8.0±0.2 Ma) from the same location further constrain the cooling history of this region. Together the geochronologic and thermobarometric data yield an average unroofing rate of 1.2±0.6 mm/yr for the High Himalaya of eastern Nepal.
Simple thermal models show that these geochronologic and thermobarometric data are consistent with a wide range of different initial geotherms, applied boundary conditions and magnitude of radiogenic heat production. The variation through time of the unroofing rates can only be poorly constrained, however. The unroofing histories were found to be largely insensitive to the details of the assumed initial geotherm, fairly sensitive to the magnitude of radiogenic heat production, and extremely sensitive to the nature of the boundary conditions applied below the fault zone. This study underscores the difficulty in constraining uplift histories on the basis of cooling rates even when thermobarometric data are available to supplement geochronologic constraints on the cooling history of the region
The Absence of Positive Energy Bound States for a Class of Nonlocal Potentials
We generalize in this paper a theorem of Titchmarsh for the positivity of
Fourier sine integrals. We apply then the theorem to derive simple conditions
for the absence of positive energy bound states (bound states embedded in the
continuum) for the radial Schr\"odinger equation with nonlocal potentials which
are superposition of a local potential and separable potentials.Comment: 23 page
Towards absorbing outer boundaries in General Relativity
We construct exact solutions to the Bianchi equations on a flat spacetime
background. When the constraints are satisfied, these solutions represent in-
and outgoing linearized gravitational radiation. We then consider the Bianchi
equations on a subset of flat spacetime of the form [0,T] x B_R, where B_R is a
ball of radius R, and analyze different kinds of boundary conditions on
\partial B_R. Our main results are: i) We give an explicit analytic example
showing that boundary conditions obtained from freezing the incoming
characteristic fields to their initial values are not compatible with the
constraints. ii) With the help of the exact solutions constructed, we determine
the amount of artificial reflection of gravitational radiation from
constraint-preserving boundary conditions which freeze the Weyl scalar Psi_0 to
its initial value. For monochromatic radiation with wave number k and arbitrary
angular momentum number l >= 2, the amount of reflection decays as 1/(kR)^4 for
large kR. iii) For each L >= 2, we construct new local constraint-preserving
boundary conditions which perfectly absorb linearized radiation with l <= L.
(iv) We generalize our analysis to a weakly curved background of mass M, and
compute first order corrections in M/R to the reflection coefficients for
quadrupolar odd-parity radiation. For our new boundary condition with L=2, the
reflection coefficient is smaller than the one for the freezing Psi_0 boundary
condition by a factor of M/R for kR > 1.04. Implications of these results for
numerical simulations of binary black holes on finite domains are discussed.Comment: minor revisions, 30 pages, 6 figure
Barycentric decomposition of quantum measurements in finite dimensions
We analyze the convex structure of the set of positive operator valued
measures (POVMs) representing quantum measurements on a given finite
dimensional quantum system, with outcomes in a given locally compact Hausdorff
space. The extreme points of the convex set are operator valued measures
concentrated on a finite set of k \le d^2 points of the outcome space, d<
\infty being the dimension of the Hilbert space. We prove that for second
countable outcome spaces any POVM admits a Choquet representation as the
barycenter of the set of extreme points with respect to a suitable probability
measure. In the general case, Krein-Milman theorem is invoked to represent
POVMs as barycenters of a certain set of POVMs concentrated on k \le d^2 points
of the outcome space.Comment: !5 pages, no figure
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