Image Reconstruction with Analytical Point Spread Functions

The image degradation produced by atmospheric turbulence and optical aberrations is usually alleviated using post-facto image reconstruction techniques, even when observing with adaptive optics systems. These techniques rely on the development of the wavefront using Zernike functions and the non-linear optimization of a certain metric. The resulting optimization procedure is computationally heavy. Our aim is to alleviate this computationally burden. To this aim, we generalize the recently developed extended Zernike-Nijboer theory to carry out the analytical integration of the Fresnel integral and present a natural basis set for the development of the point spread function in case the wavefront is described using Zernike functions. We present a linear expansion of the point spread function in terms of analytic functions which, additionally, takes defocusing into account in a natural way. This expansion is used to develop a very fast phase-diversity reconstruction technique which is demonstrated through some applications. This suggest that the linear expansion of the point spread function can be applied to accelerate other reconstruction techniques in use presently and based on blind deconvolution.Comment: 10 pages, 4 figures, accepted for publication in Astronomy & Astrophysic

A non-linear observer for unsteady three-dimensional flows

A method is proposed to estimate the velocity field of an unsteady flow using a limited number of flow measurements. The method is based on a non-linear low-dimensional model of the flow and on expanding the velocity field in terms of empirical basis functions. The main idea is to impose that the coefficients of the modal expansion of the velocity field give the best approximation to the available measurements and that at the same time they satisfy as close as possible the non-linear low-order model. The practical use may range from feedback flow control to monitoring of the flow in non-accessible regions. The proposed technique is applied to the flow around a confined square cylinder, both in two- and three-dimensional laminar flow regimes. Comparisons are provided. with existing linear and non-linear estimation techniques

Large-N expansions applied to gravitational clustering

We develop a path-integral formalism to study the formation of large-scale structures in the universe. Starting from the equations of motion of hydrodynamics (single-stream approximation) we derive the action which describes the statistical properties of the density and velocity fields for Gaussian initial conditions. Then, we present large-N expansions (associated with a generalization to N fields or with a semi-classical expansion) of the path-integral defined by this action. This provides a systematic expansion for two-point functions such as the response function and the usual two-point correlation. We present the results of two such expansions (and related variants) at one-loop order for a SCDM and a LCDM cosmology. We find that the response function exhibits fast oscillations in the non-linear regime with an amplitude which either follows the linear prediction (for the direct steepest-descent scheme) or decays (for the 2PI effective action scheme). On the other hand, the correlation function agrees with the standard one-loop result in the quasi-linear regime and remains well-behaved in the highly non-linear regime. This suggests that these large-N expansions could provide a good framework to study the dynamics of gravitational clustering in the non-linear regime. Moreover, the use of various expansion schemes allows one to estimate their range of validity without the need of N-body simulations and could provide a better accuracy in the weakly non-linear regime.Comment: 27 pages, published in A&

Recurrence relations for the number of solutions of a class of Diophantine equations

Recursive formulas are derived for the number of solutions of linear and quadratic Diophantine equations with positive coefficients. This result is further extended to general non-linear additive Diophantine equations. It is shown that all three types of the recursion admit an explicit solution in the form of complete Bell polynomial, depending on the coefficients of the power series expansion of the logarithm of the generating functions for the sequences of individual terms in the Diophantine equations.Comment: 11 pages, Latex. Dedicated to the 70-th anniversary of Apolodor Radut

A Parameterized Post-Friedmann Framework for Modified Gravity

We develop a parameterized post-Friedmann (PPF) framework which describes three regimes of modified gravity models that accelerate the expansion without dark energy. On large scales, the evolution of scalar metric and density perturbations must be compatible with the expansion history defined by distance measures. On intermediate scales in the linear regime, they form a scalar-tensor theory with a modified Poisson equation. On small scales in dark matter halos such as our own galaxy, modifications must be suppressed in order to satisfy stringent local tests of general relativity. We describe these regimes with three free functions and two parameters: the relationship between the two metric fluctuations, the large and intermediate scale relationships to density fluctuations and the two scales of the transitions between the regimes. We also clarify the formal equivalence of modified gravity and generalized dark energy. The PPF description of linear fluctuation in f(R) modified action and the Dvali-Gabadadze-Porrati braneworld models show excellent agreement with explicit calculations. Lacking cosmological simulations of these models, our non-linear halo-model description remains an ansatz but one that enables well-motivated consistency tests of general relativity. The required suppression of modifications within dark matter halos suggests that the linear and weakly non-linear regimes are better suited for making complementary test of general relativity than the deeply non-linear regime.Comment: 12 pages, 9 figures, additional references reflect PRD published versio

Geometrically non-linear vibration and meshless discretization of the composite laminated shallow shells with complex shape

To study the geometrically non-linear vibrations of the composite laminated shallow shells with complex plan form the approach, based on meshless discretization, is proposed. Non-linear equations of motion for shallow shells based on the first order shear deformation shell theories are considered. The discretization of the motion equations is carried out by method based on expansion of the unknown functions in series for which eigenvectors of the linear vibration obtained by RFM (R-functions method) are employed as basic functions. The factors of these series are functions (generalizing coordinates) depending on time. Due to applying the basic variational principle in mechanics by Ostrogradsky-Hamilton the corresponding system of the ordinary differential equations by Euler is obtained The non-linear ordinary differential equations are derived in terms of amplitudes of the vibration modes. The offered method is expounded for multi-modal approximation of unknown functions. Backbone curves of the spherical shallow shell with complex plan form are obtained using only the first vibration mode by the Bubnov-Galerkin method. The effects of lamination schemes on the behavior are discussed

A redshift distortion free correlation function at third order in the nonlinear regime

The zeroth-order component of the cosine expansion of the projected three-point correlation function is proposed for clustering analysis of cosmic large scale structure. These functions are third order statistics but can be measured similarly to the projected two-point correlations. Numerical experiments with N-body simulations indicate that the advocated statistics are redshift distortion free within 10% in the non-linear regime on scales ~0.2-10Mpc/h. Halo model prediction of the zeroth-order component of the projected three-point correlation function agrees with simulations within ~10%. This lays the ground work for using these functions to perform joint analyses with the projected two-point correlation functions, exploring galaxy clustering properties in the framework of the halo model and relevant extensions.Comment: 10 pages, 6 figs; MNRAS accepte

A CMOS-based Analog Function Generator: HSPICE Modeling and Simulation

In many Engineering applications, analog circuits present many advantagesover their digital counterparts and have recently been particularly used in awide range of signal processor circuits. In this paper, an analog non-linearfunction synthesizer is presented based on a polynomial expansion model.The proposed function synthesizer model is based on a 10th orderpolynomial approximation of any of the required non-linear functions. Thepolynomial approximations of these functions can then be implemented usingbasic CMOS circuit blocks. The proposed circuit model can simultaneouslysynthesize and generate many different mathematical functions. The circuitmodel is designed and simulated with HSPICE and its performance isdemonstrated through the simulation of a number of non-linear functions.DOI:http://dx.doi.org/10.11591/ijece.v4i4.598