2,346 research outputs found
On the one-dimensional cubic nonlinear Schrodinger equation below L^2
In this paper, we review several recent results concerning well-posedness of
the one-dimensional, cubic Nonlinear Schrodinger equation (NLS) on the real
line R and on the circle T for solutions below the L^2-threshold. We point out
common results for NLS on R and the so-called "Wick ordered NLS" (WNLS) on T,
suggesting that WNLS may be an appropriate model for the study of solutions
below L^2(T). In particular, in contrast with a recent result of Molinet who
proved that the solution map for the periodic cubic NLS equation is not weakly
continuous from L^2(T) to the space of distributions, we show that this is not
the case for WNLS.Comment: 14 pages, additional reference
Dynamic robust duality in utility maximization
A celebrated financial application of convex duality theory gives an explicit
relation between the following two quantities:
(i) The optimal terminal wealth of the problem
to maximize the expected -utility of the terminal wealth
generated by admissible portfolios in a market
with the risky asset price process modeled as a semimartingale;
(ii) The optimal scenario of the dual problem to minimize
the expected -value of over a family of equivalent local
martingale measures , where is the convex conjugate function of the
concave function .
In this paper we consider markets modeled by It\^o-L\'evy processes. In the
first part we use the maximum principle in stochastic control theory to extend
the above relation to a \emph{dynamic} relation, valid for all .
We prove in particular that the optimal adjoint process for the primal problem
coincides with the optimal density process, and that the optimal adjoint
process for the dual problem coincides with the optimal wealth process, . In the terminal time case we recover the classical duality
connection above. We get moreover an explicit relation between the optimal
portfolio and the optimal measure . We also obtain that the
existence of an optimal scenario is equivalent to the replicability of a
related -claim.
In the second part we present robust (model uncertainty) versions of the
optimization problems in (i) and (ii), and we prove a similar dynamic relation
between them. In particular, we show how to get from the solution of one of the
problems to the other. We illustrate the results with explicit examples
Temperature induced pore fluid pressurization in geomaterials
The theoretical basis of the thermal response of the fluid-saturated porous
materials in undrained condition is presented. It has been demonstrated that
the thermal pressurization phenomenon is controlled by the discrepancy between
the thermal expansion of the pore fluid and of the solid phase, the
stress-dependency of the compressibility and the non-elastic volume changes of
the porous material. For evaluation of the undrained thermo-poro-elastic
properties of saturated porous materials in conventional triaxial cells, it is
important to take into account the effect of the dead volume of the drainage
system. A simple correction method is presented to correct the measured pore
pressure change and also the measured volumetric strain during an undrained
heating test. It is shown that the porosity of the tested material, its drained
compressibility and the ratio of the volume of the drainage system to the one
of the tested sample, are the key parameters which influence the most the error
induced on the measurements by the drainage system. An example of the
experimental evaluation of undrained thermoelastic parameters is presented for
an undrained heating test performed on a fluid-saturated granular rock
Experimental artefacts in undrained triaxial testing
For evaluation of the undrained thermo-poro-elastic properties of saturated
porous materials in conventional triaxial cells, it is important to take into
account the effect of the dead volume of the drainage system. The
compressibility and the thermal expansion of the drainage system along with the
dead volume of the fluid filling this system, influence the measured pore
pressure and volumetric strain during undrained thermal or mechanical loading
in a triaxial cell. A correction method is presented in this paper to correct
these effects during an undrained isotropic compression test or an undrained
heating test. A parametric study has demonstrated that the porosity and the
drained compressibility of the tested material and the ratio of the vol-ume of
the drainage system to the one of the tested sample are the key parameters
which influence the most the error induced on the measurements by the drainage
system
HMM-based Offline Recognition of Handwritten Words Crossed Out with Different Kinds of Strokes
In this work, we investigate the recognition of words that have been crossed-out by the writers and are thus degraded. The degradation consists of one or more ink strokes that span the whole word length and simulate the signs that writers use to cross out the words. The simulated strokes are superimposed to the original clean word images. We considered two types of strokes: wave-trajectory strokes created with splines curves and line-trajectory strokes generated with the delta-lognormal model of rapid line movements. The experiments have been performed using a recognition system based on hidden Markov models and the results show that the performance decrease is moderate for single writer data and light strokes, but severe for multiple writer data
Convolutions of singular measures and applications to the Zakharov system
Uniform L^2-estimates for the convolution of singular measures with respect
to transversal submanifolds are proved in arbitrary space dimension. The
results of Bennett-Bez are used to extend previous work of
Bejenaru-Herr-Tataru. As an application, it is shown that the 3D Zakharov
system is locally well-posed in the full subcritical regime
Focusing Singularity in a Derivative Nonlinear Schr\"odinger Equation
We present a numerical study of a derivative nonlinear Schr\"odinger equation
with a general power nonlinearity, . In the
-supercritical regime, , our simulations indicate that there is
a finite time singularity. We obtain a precise description of the local
structure of the solution in terms of blowup rate and asymptotic profile, in a
form similar to that of the nonlinear Schr\"odinger equation with supercritical
power law nonlinearity.Comment: 24 pages, 17 figure
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