arXiv subjects

Tuhin Ghosh

Publications and source records attributed to Tuhin Ghosh.

71 records · Page 4Linked to original sources

arXiv2017

The Calderón problem for variable coefficients nonlocal elliptic operators

In this paper, we introduce an inverse problem of a Schrödinger type variable nonlocal elliptic operator $(-\nabla\cdot(A(x)\nabla))^{s}+q)$, for $0<s<1$. We determine the unknown bounded potential $q$ from the exterior partial measurements associated with the nonlocal Dirichlet-to-Neumann map for any dimension $n\geq2$. Our results generalize the recent initiative [16] of introducing and solving inverse problem for fractional Schrödinger operator $((-Δ)^{s}+q)$ for $0<s<1$. We also prove some regularity results of the direct problem corresponding to the variable coefficients fractional differential operator and the associated degenerate elliptic operator.

math.AP↗

arXiv2017

Approximate Isotropic Cloak for the Maxwell equations

We construct a regular isotropic approximate cloak for the Maxwell system of equations. The method of transformation optics has enabled the design of electromagnetic parameters that cloak a region from external observation. However, these constructions are singular and anisotropic, making practical implementation difficult. Thus, regular approximations to these cloaks have been constructed that cloak a given region to any desired degree of accuracy. In this paper, we show how to construct isotropic approximations to these regularized cloaks using homogenization techniques, so that one obtains cloaking of arbitrary accuracy with regular and isotropic parameters.

math.AP↗

arXiv2017

Statistical simulations of the dust foreground to CMB polarization

The characterization of the dust polarization foreground to the Cosmic Microwave Background (CMB) is a necessary step towards the detection of the B-mode signal associated with primordial gravitational waves. We present a method to simulate maps of polarized dust emission on the sphere, similarly to what is done for the CMB anisotropies. This method builds on the understanding of Galactic polarization stemming from the analysis of Planck data. It relates the dust polarization sky to the structure of the Galactic magnetic field and its coupling with interstellar matter and turbulence. The Galactic magnetic field is modelled as a superposition of a mean uniform field and a random component with a power-law power spectrum of exponent $α_{\rm M}$. The model parameters are constrained to fit the power spectra of dust polarization EE, BB and TE measured using Planck data. We find that the slopes of the E and B power spectra of dust polarization are matched for $α_{\rm M} = -2.5$. The model allows us to compute multiple realizations of the Stokes Q and U maps for different realizations of the random component of the magnetic field, and to quantify the variance of dust polarization spectra for any given sky area outside of the Galactic plane. The simulations reproduce the scaling relation between the dust polarization power and the mean total dust intensity including the observed dispersion around the mean relation. We also propose a method to carry out multi-frequency simulations including the decorrelation measured recently by Planck, using a given covariance matrix of the polarization maps. These simulations are well suited to optimize component separation methods and to quantify the confidence with which the dust and CMB B-modes can be separated in present and future experiments. We also provide an astrophysical perspective on our modeling of the dust polarization spectra.

astro-ph.CO↗

arXiv2017

Cloaking for a quasi-linear elliptic partial differential equation

In this article we consider cloaking for a quasi-linear elliptic partial differential equation of divergence type defined on a bounded domain in $\mathbb{R}^N$ for $N=2,3$. We show that a perfect cloak can be obtained via a singular change of variables scheme and an approximate cloak can be achieved via a regular change of variables scheme. These approximate cloaks though non-degenerate are anisotropic. We also show, within the framework of homogenization, that it is possible to get isotropic regular approximate cloaks. This work generalizes to quasi-linear settings previous work on cloaking in the context of Electrical Impedance Tomography for the conductivity equation.

math.AP↗

arXiv2017

The Calderón problem for the fractional Schrödinger equation

We show global uniqueness in an inverse problem for the fractional Schrödinger equation: an unknown potential in a bounded domain is uniquely determined by exterior measurements of solutions. We also show global uniqueness in the partial data problem where the measurements are taken in arbitrary open, possibly disjoint, subsets of the exterior. The results apply in any dimension $\geq 2$ and are based on a strong approximation property of the fractional equation that extends earlier work. This special feature of the nonlocal equation renders the analysis of related inverse problems radically different from the traditional Calderón problem.

math.AP↗

arXiv2017

Modelling and simulation of large-scale polarized dust emission over the southern Galactic cap using the GASS HI data

The Planck survey has quantified polarized Galactic foregrounds and established that they are a main limiting factor in the quest for the cosmic microwave background (CMB) B-mode signal induced by primordial gravitational waves during cosmic inflation. Accurate separation of the Galactic foregrounds therefore binds this quest to our understanding of the magnetized interstellar medium (ISM). The two most relevant empirical results from analysis of Planck data are line of sight depolarization arising from fluctuations of the Galactic magnetic field orientation and alignment of filamentary dust structures with the magnetic field at high Galactic latitude. Furthermore, Planck and HI emission data in combination indicate that most of the filamentary dust structures are in the cold neutral medium. The goal of this paper is to test whether these salient observational results, taken together, can account fully for the statistical properties of the dust polarization over a selected low column density region comprising 34 % of the southern Galactic cap ($b \le -30°$). To do this, we construct a dust model that incorporates HI column density maps as tracers of the dust intensity structures and a phenomenological description of the Galactic magnetic field. By adjusting the parameters of the dust model, we were able to reproduce the Planck dust observations at 353 GHz in the selected region. Realistic simulations of the polarized dust emission enabled by such a dust model are useful for testing the accuracy of component separation methods, studying non-Gaussianity, and constraining the amount of decorrelation with frequency.

astro-ph.GA↗

arXiv2016

Convergence Relative to a Microstructure : Properties, Optimal Bounds and Application

In this work, we study a new notion involving convergence of microstructures represented by matrices $B^ε$ related to the classical $H$-convergence of $A^ε$. It incorporates the interaction between the two microstructures. This work is about its effects on various aspects : existence, examples, optimal bounds on emerging macro quantities, application etc. Five among them are highlighted below : $(1)$ The usual arguments based on translated inequality, $H$-measures, Compensated Compactness etc for obtaining optimal bounds are not enough. Additional compactness properties are needed. $(2)$ Assuming two-phase microstructures, the bounds define naturally four optimal regions in the phase space of macro quantities. The classically known single region in the self-interacting case , namely $B^ε= A^ε$ can be recovered from them, a result that indicates we are dealing with a true extension of the $\mathcal{G}$-closure problem. $(3)$ Optimality of the bounds is not immediate because of (a priori) non-commutativity of macro-matrices, an issue not present in the self-interacting case. Somewhat surprisingly though, commutativity follows a posteriori. $(4)$ From the application to "Optimal Oscillation-Dissipation Problems", it emerges that oscillations and dissipation can co-exist optimally and the microstructures behind them need not be the same though they are closely linked. Furthermore, optimizers are found among $N$-rank laminates with interfaces. This is a new feature. $(5)$ Explicit computations in the case of canonical microstructures are performed, in which we make use of $H$-measure in a novel way.

math.AP↗

arXiv2016

Homogenization of Stokes System using Bloch Waves

In this work, we study the Bloch wave homogenization for the Stokes system with periodic viscosity coefficient. In particular, we obtain the spectral interpretation of the homogenized tensor. The presence of the incompressibility constraint in the model raises new issues linking the homogenized tensor and the Bloch spectral data. The main difficulty is a lack of smoothness for the bottom of the Bloch spectrum, a phenomenon which is not present in the case of the elasticity system. This issue is solved in the present work, completing the homogenization process of the Stokes system via the Bloch wave method.

math.AP↗

arXiv2016

Inverse Diffusivity Problem via Homogenization Theory

Polarization tensor corresponding to near zero volume inhomogeneities was introduced in the pioneering work by Capdeboscq-Vogelius \cite{CV1,CV2}. A beautiful application of the polarization tensor to an inverse problem involving inhomogeneities was also given by them. In this article, we take an approach toward polarization tensor via homogenized tensor. Accordingly, we introduce polarization tensor corresponding to inhomogeneities with positive volume fraction.A relation between this tensor and the homogenized tensor is found. Next, we proceed to examine the sense in which this tensor is continuous as the volume fraction tends to zero. Our approach has its own advantages, as we will see. In particular, it provides another method to deduce optimal estimates on polarization tensors in any dimension from those on homogenized tensors, along with the information on underlying microstructures.

math.AP↗

arXiv2016

Bloch wave spectral analysis in the class of generalized Hashin-Shtrikman micro-structures

In this paper, we use spectral methods by introducing the Bloch waves to study the homogenization process in the non-periodic class of generalized Hashin-Shtrikman micro-structures \cite[page no. 281]{T}, which incorporates both translation and dilation with a family of scales, including one subclass of laminates. We establish the classical homogenization result with providing the spectral representation of the homogenized coefficients. It offers a new lead towards extending the Bloch spectral analysis in the non-periodic, non-commutative class of micro-structures.

math.AP↗

arXiv2016

Dispersion tensor and its unique minimizer in Hashin-Shtrikman micro-structures

In this paper, we introduce the macroscopic quantity, namely the dispersion tensor or the \textit{Burnett coefficient}s in the class of generalized Hashin-Shtrikman micro-structures \cite[page no. 281]{T}. In the case of two-phase materials associated with the periodic Hashin-Shtrikman structures, we settle the issue that the dispersion tensor has an unique minimizer, which is so called Apollonian-Hashin-Shtrikman micro-structure.

math.AP↗

arXiv2016

Bloch Wave Homogenization Relative to a Microstructure

In this work, we study the aspect of Bloch wave homogenization of the new notion of convergence of microstructures represented by matrices $B^ε$ related to the classical $H$-convergence of $A^ε$ introduced in \cite{TG-MV-1}. The new macro quantity $B^{#}$ appears to incorporate the interaction between the two microstructures $A^ε,B^ε$. Here we present its Bloch spectral representation along with the homogenization result.

math.AP↗

arXiv2011

Foreground Analysis Using Cross-Correlations of External Templates on the 7-year WMAP data

WMAP data when combined with ancillary data on free-free, synchrotron and dust allow an improved understanding of the spectrum of emission from each of these components. Here we examine the sky variation at intermediate and high latitudes using a cross-correlation technique. In particular, we compare the observed emission in several large partitions of the sky plus 33 selected sky regions to three "standard" templates. The regions are selected using a criterion based on the morphology of these template maps. The synchrotron emission shows evidence of steepening between GHz frequencies and the \emph{WMAP} bands. There are indications of spectral index variations across the sky but the current data are not precise enough to accurately quantify this from region-to-region. The emission correlated with the $H_α$ template shows clear evidence of deviation from a free-free spectrum. The emission can be decomposed into a contribution from both free-free and spinning dust in the warm ionised medium of the Galaxy. The derived free-free emissivity corresponds to a mean electron temperature of $\sim 6000$ K, although the value depends critically on the impact of dust absorption on the $H_α$ intensity. The WIM spinning dust emission has a peak emission in intensity in the range 40--50 GHz. The anomalous microwave emission associated with dust is detected at high significance in most of the 33 fields studied. The anomalous emission correlates well with the Finkbeiner et al. (1999) model 8 predictions (FDS8) at 94 GHz, and is well described globally by a power-law emission model with an effective spectral index between 20 and 60 GHz of $β\approx -2.7$. It is clear that attempts to explain the emission by spinning dust models require multiple components, which presumably relates to a complex mix of emission regions along a given line-of-sight.

astro-ph.CO↗

arXiv2010

Foreground Maps in WMAP frequency bands

This paper provides full sky maps of foreground emission in all WMAP channels, with very low residual contamination from the Cosmic Microwave Background (CMB) anisotropies and controlled level of instrumental noise. Foreground maps are obtained by subtraction of a properly filtered CMB map, obtained from linear combinations of needlet-based representations of all WMAP observations and of a 100-micron map. The error in the reconstructed foreground maps on large scales is significantly lower than the original error due to CMB contamination, while remaining of the order of the original WMAP noise on small scales. The level of the noise is estimated, which permits to implement local filters for maximising the local signal to noise ratio. An example of such filtering, which reduces the small scale noise using latitude dependent filters is implemented. This enhances significantly the contrast of galactic emission, in particular on intermediate angular scales and at intermediate galactic latitude. The clean WMAP foreground maps can be used to study the galactic interstellar medium, in particular for the highest frequency channels for which the proper subtraction of CMB contamination is mandatory. The foregrounds maps can be downloaded from a dedicated web site.

astro-ph.CO↗

arXiv2009

Model Independent Foreground Power Spectrum Estimation using WMAP 5-year Data

In this paper, we propose & implement on WMAP 5-year data, a model independent approach of foreground power spectrum estimation for multifrequency observations of CMB experiments. Recently a model independent approach of CMB power spectrum estimation was proposed by Saha et al. 2006. This methodology demonstrates that CMB power spectrum can be reliably estimated solely from WMAP data without assuming any template models for the foreground components. In the current paper, we extend this work to estimate the galactic foreground power spectrum using the WMAP 5 year maps following a self contained analysis. We apply the model independent method in harmonic basis to estimate the foreground power spectrum and frequency dependence of combined foregrounds. We also study the behaviour of synchrotron spectral index variation over different regions of the sky. We compare our results with those obtained from MEM foreground maps which are formed in pixel space. We find that relative to our model independent estimates MEM maps overestimates the foreground power close to galactic plane and underestimates it at high latitudes.

astro-ph.CO↗

arXiv2007

Unveiling Hidden Patterns in CMB Anisotropy Maps

Bianchi VII_h models have been recently proposed to explain potential anomalies in the CMB anisotropy as observed by WMAP. We investigate the violation of statistical isotropy due to an embedded Bianchi VII_h templates in the CMB anisotropy maps to determine whether the existence of a hidden Bianchi template in the WMAP data is consistent with the previous null detection of the bipolar power spectrum in the WMAP first year maps. We argue that although correcting the WMAP maps for the Bianchi template may explain some features in the WMAP data it may cause other anomalies such as preferred directions leading to detectable levels of violation of statistical isotropy in the Bianchi corrected maps. We compute the bipolar power spectrum for the low density Bianchi VII_h models embedded in the background CMB anisotropy maps with the power spectrum that have been shown in recent literature to best fit the first year WMAP data. By examining statistical isotropy of these maps, we put a limit of {σ/H}_0 < 2.77E-10 (99% CL) on the shear parameter in Bianchi VII_h models.

astro-ph↗

arXiv2007

Effects of lengthscales and attractions on the collapse of hydrophobic polymers in water

We present results from extensive molecular dynamics simulations of collapse transitions of hydrophobic polymers in explicit water focused on understanding effects of lengthscale of the hydrophobic surface and of attractive interactions on folding. Hydrophobic polymers display parabolic, protein-like, temperature-dependent free energy of unfolding. Folded states of small attractive polymers are marginally stable at 300 K, and can be unfolded by heating or cooling. Increasing the lengthscale or decreasing the polymer-water attractions stabilizes folded states significantly, the former dominated by the hydration contribution. That hydration contribution can be described by the surface tension model, $ΔG=γ(T)ΔA$, where the surface tension, $γ$, is lengthscale dependent and decreases monotonically with temperature. The resulting variation of the hydration entropy with polymer lengthscale is consistent with theoretical predictions of Huang and Chandler (Proc. Natl. Acad. Sci.,97, 8324-8327, 2000) that explain the blurring of entropy convergence observed in protein folding thermodynamics. Analysis of water structure shows that the polymer-water hydrophobic interface is soft and weakly dewetted, and is characterized by enhanced interfacial density fluctuations. Formation of this interface, which induces polymer folding, is strongly opposed by enthalpy and favored by entropy, similar to the vapor-liquid interface.

cond-mat.stat-mech↗