852 research outputs found
Carleman estimate for an adjoint of a damped beam equation and an application to null controllability
In this article we consider a control problem of a linear Euler-Bernoulli
damped beam equation with potential in dimension one with periodic boundary
conditions. We derive a new Carleman estimate for an adjoint of the equation
under consideration. Then using a well known duality argument we obtain
explicitly the control function which can be used to drive the solution
trajectory of the control problem to zero state
Reionization constraints using Principal Component Analysis
Using a semi-analytical model developed by Choudhury & Ferrara (2005) we
study the observational constraints on reionization via a principal component
analysis (PCA). Assuming that reionization at z>6 is primarily driven by
stellar sources, we decompose the unknown function N_{ion}(z), representing the
number of photons in the IGM per baryon in collapsed objects, into its
principal components and constrain the latter using the photoionization rate
obtained from Ly-alpha forest Gunn-Peterson optical depth, the WMAP7 electron
scattering optical depth and the redshift distribution of Lyman-limit systems
at z \sim 3.5. The main findings of our analysis are: (i) It is sufficient to
model N_{ion}(z) over the redshift range 2<z<14 using 5 parameters to extract
the maximum information contained within the data. (ii) All quantities related
to reionization can be severely constrained for z<6 because of a large number
of data points whereas constraints at z>6 are relatively loose. (iii) The weak
constraints on N_{ion}(z) at z>6 do not allow to disentangle different feedback
models with present data. There is a clear indication that N_{ion}(z) must
increase at z>6, thus ruling out reionization by a single stellar population
with non-evolving IMF, and/or star-forming efficiency, and/or photon escape
fraction. The data allows for non-monotonic N_{ion}(z) which may contain sharp
features around z \sim 7. (iv) The PCA implies that reionization must be 99%
completed between 5.8<z<10.3 (95% confidence level) and is expected to be 50%
complete at z \approx 9.5-12. With future data sets, like those obtained by
Planck, the z>6 constraints will be significantly improved.Comment: Accepted in MNRAS. Revised to match the accepted versio
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