arXiv subjects
Publications and source records attributed to Prasenjit Ghosh.
Generalized fusion frame and some of their properties in tensor product of Hilbert spaces are described. Also, the canonical dual g-fusion frame in tensor product of Hilbert spaces is considered. Finally, the frame operator for a pair of g-fusion Bessel sequences in tensor product of Hilbert spaces is presented.
The concept of b-linear functional and its different types of continuity in linear n-normed space are presented and some of their properties are being established. We derive the Uniform Boundedness Principle and Hahn-Banach extension Theorem with the help of bounded b-linear functionals in the case of linear n-normed spaces and discuss some examples and applications. Finally, we present the concept of weak*convergence for the sequence of bounded b-linear functionals in linear n-normed space.
In this paper, our aim is to introduce the concept of a frame in n-Hilbert space and describe some of their properties. We further discuss tight frame relative to n-Hilbert space. At the end, we study the relationship between frame and bounded linear operator in n-Hilbert space.
In this paper, we will see that the Cartesian product of two 2-Banach spaces is also 2-Banach space and discuss some properties of closed linear operator in linear 2-normed space. We also describe the concept of different types of continuity of b-linear functional and derive the Uniform Boundedness Principle and Hahn-Banach extension theorem for b-linear functionals in the case of linear 2-normed spaces. We also introduce the notion of weak*convergence for the sequence of bounded b-linear functionals relative to linear 2-normed space.
MXenes, a new class of two dimensional materials with several novel properties, are usually prepared from their MAX phases by etching out the A element using strong chemical reagents. This results in passivation of the surfaces of MXene with different functional groups like O, -OH, -F, etc., which in many cases tend to degrade their properties. In this work, using first principle density functional theory based calculations, we propose a novel method to synthesize pristine Ti$_2$C and Ti$_3$C$_2$ MXenes from the bulk titanium carbides with corresponding stoichiometry. Based on the values of cleavage energy obtained from our calculations, we envisage that pristine Ti$_2$C and Ti$_3$C$_2$ MXenes can be prepared, using mechanical or sonification-assisted liquid-phase exfoliation techniques, from their bulk phases.
In this article, we consider the problem of simultaneous testing of hypotheses when the individual test statistics are not necessarily independent. Specifically, we consider the problem of simultaneous testing of point null hypotheses against two-sided alternatives for the mean parameters of normally distributed random variables. We assume that conditionally given the vector of means, these random variables jointly follow a multivariate normal distribution with a known but arbitrary covariance matrix. We consider a Bayesian framework where each unknown mean parameter is modeled through a two-component "spike and slab" mixture prior. This way, unconditionally the test statistics jointly have a mixture of multivariate normal distributions. A new testing procedure is developed that uses the dependence among the test statistics and works in a "step-down" manner. This procedure is general enough to be applied for non-normal data. A decision theoretic justification in favor of the proposed testing procedure has been provided by showing that unlike many traditional p-value based stepwise procedures, this new method possesses a certain "convexity property" which makes it admissible with respect to a vector risk function that captures the risks for the individual testing problems. An alternative representation of the proposed test statistics has also been established resulting in great simplification in the computational complexity. It is demonstrated through extensive simulations that for various forms of dependence and a wide range of sparsity levels, the proposed testing procedure compares quite favorably with several existing multiple testing procedures available in the literature in terms of overall misclassification probability.
In this paper, we consider Bayesian variable selection problem of linear regression model with global-local shrinkage priors on the regression coefficients. We propose a variable selection procedure that select a variable if the ratio of the posterior mean to the ordinary least square estimate of the corresponding coefficient is greater than $1/2$. Under the assumption of orthogonal designs, we show that if the local parameters have polynomial-tailed priors, our proposed method enjoys the oracle property in the sense that it can achieve variable selection consistency and optimal estimation rate at the same time. However, if, instead, an exponential-tailed prior is used for the local parameters, the proposed method does not have the oracle property.
In this article, we investigate certain asymptotic optimality properties of a very broad class of one-group continuous shrinkage priors for simultaneous estimation and testing of a sparse normal mean vector. Asymptotic optimality of Bayes estimates and posterior concentration properties corresponding to the general class of one-group priors under consideration are studied where the data is assumed to be generated according to a multivariate normal distribution with a fixed unknown mean vector. Under the assumption that the number of non-zero means is known, we show that Bayes estimators arising out of this general class of shrinkage priors under study, attain the minimax risk, up to some multiplicative constant, under the $l_2$ norm. In particular, it is shown that for the horseshoe-type priors such as the three parameter beta normal mixtures with parameters $a=0.5, b>0$ and the generalized double Pareto prior with shape parameter $α=1$, the corresponding Bayes estimates become asymptotically minimax. Moreover, posterior distributions arising out of this general class of one-group priors are shown to contract around the true mean vector at the minimax $l_2$ rate for a wide range of values of the global shrinkage parameter depending on the proportion of non-zero components of the underlying mean vector. An important and remarkable fact that emerges as a consequence of one key result essential for proving the aforesaid minimaxity result is that, within the asymptotic framework of Bogdan et al. (2011), the natural thresholding rules due to Carvalho et al. (2010) based on the horseshoe-type priors, asymptotically attain the optimal Bayes risk w.r.t. a $0-1$ loss, up to the correct multiplicative constant and are thus, asymptotically Bayes optimal under sparsity (ABOS).
Suppose we have data generated according to a multivariate normal distribution with a fixed unknown mean vector that is sparse in the sense of being nearly black. Optimality of Bayes estimates and posterior concentration properties in terms of the minimax risk in the $l_2$ norm corresponding to a very general class of continuous shrinkage priors are studied in this work. The class of priors considered is rich enough to include a great variety of heavy tailed prior distributions, such as, the three parameter beta normal mixtures (including the horseshoe), the generalized double Pareto, the inverse gamma and the normal-exponential-gamma priors. Assuming that the number of non-zero components of the mean vector is known, we show that the Bayes estimators corresponding to this general class of priors attain the minimax risk in the $l_2$ norm (possibly up to a multiplicative constant) and the corresponding posterior distributions contract around the true mean vector at the minimax optimal rate for appropriate choice of the global shrinkage parameter. Moreover, we provide conditions for which these posterior distributions contract around the corresponding Bayes estimates at least as fast as the minimax risk in the $l_2$ norm. We also provide a lower bound to the total posterior variance for an important subclass of this general class of shrinkage priors that includes the generalized double Pareto priors with shape parameter $α=0.5$ and the three parameter beta normal mixtures with parameters $a=0.5$ and $b>0$ (including the horseshoe) in particular. The present work is inspired by the recent work of van der Pas et al. (2014) on the posterior contraction properties of the horseshoe prior under the present set-up. We extend their results for this general class of priors and come up with novel unifying proofs which work for a very broad class of one-group continuous shrinkage priors.
Consider the problem of simultaneous testing for the means of independent normal observations. In this paper, we study some asymptotic optimality properties of certain multiple testing rules induced by a general class of one-group shrinkage priors in a Bayesian decision theoretic framework, where the overall loss is taken as the number of misclassified hypotheses. We assume a two-groups normal mixture model for the data and consider the asymptotic framework adopted in Bogdan et al. (2011) who introduced the notion of asymptotic Bayes optimality under sparsity in the context of multiple testing. The general class of one-group priors under study is rich enough to include, among others, the families of three parameter beta, generalized double Pareto priors, and in particular the horseshoe, the normal-exponential-gamma and the Strawderman-Berger priors. We establish that within our chosen asymptotic framework, the multiple testing rules under study asymptotically attain the risk of the Bayes Oracle up to a multiplicative factor, with the constant in the risk close to the constant in the Oracle risk. This is similar to a result obtained in Datta and Ghosh (2013) for the multiple testing rule based on the horseshoe estimator introduced in Carvalho et al. (2009, 2010). We further show that under very mild assumption on the underlying sparsity parameter, the induced decision rules based on an empirical Bayes estimate of the corresponding global shrinkage parameter proposed by van der Pas et al. (2014), attain the optimal Bayes risk up to the same multiplicative factor asymptotically. We provide a unifying argument applicable for the general class of priors under study. In the process, we settle a conjecture regarding optimality property of the generalized double Pareto priors made in Datta and Ghosh (2013). Our work also shows that the result in Datta and Ghosh (2013) can be improved further.
We have studied the adsorption of NO, and the coadsorption of N and O, on four physical and hypothetical systems: unstrained and strained Rh(100) surfaces and monolayers of Rh atoms on strained and unstrained MgO(100) surfaces. We find that as we go from Rh(100) to Rh/Mg0(100), via the other two hypothetical systems, the effective coordination progressively decreases, the $d$-band narrows and its center shifts closer to the Fermi level, and the strength of adsorption and co-adsorption increases. Both strain and the presence of the oxide substrate contribute significantly to this. However, charge-transfer is found to play a negligible role, due to a cancelling out between donation and back-donation processes. Our results suggest that lowering effective coordination of Rh catalysts by strain, roughening or the use of inert substrates might improve reaction rates for the reduction of NO to N$_2$.
We have studied the adsorption of NO on small Rh clusters, containing one to five atoms, using density functional theory in both spin-polarized and non-spin-polarized forms. We find that NO bonds more strongly to Rh clusters than it does to Rh(100) or Rh(111); however, it also quenches the magnetism of the clusters. This (local) effect results in reducing the magnitude of the adsorption energy, and also washes out the clear size-dependent trend observed in the non-magnetic case. Our results illustrate the competition present between the tendencies to bond and to magnetize, in small clusters.
The adsorption of CO on unreconstructed and reconstructed Ir{100} has been studied, using a combination of density functional theory and thermodynamics, to determine the relative stability of the two phases as a function of CO coverage, temperature and pressure. We obtain good agreement with experimentaldata. At zero temperature, the (1X5) reconstruction becomes less stable than the unreconstructed (1X1) surface when the CO coverage exceeds a critical value of 0.09 ML. The interaction between CO molecules is found to be repulsive on the reconstructed surface, but attractive on the unreconstructed, explaining the experimental observation of high CO coverage on growing (1X1) islands. At all temperatures and pressures, we find only two possible stable states: 0.05 ML CO c(2X2) overlayer on the (1X1) substrate, and the clean (1$\times$5) reconstructed surface.
Motivated by the unusual soft mode anomalies in the relaxor ferroelectrics, we present a numerical study of vibrational excitations of a disordered lattice model with chemical order on the nano-scale. We find an Ioffe-Regel crossover in acoustic modes separating propagating low-frequency modes from the intermediate frequency diffusons, and localized modes at the high-frequency end of the spectrum. At a fixed degree of disorder, we find increasing uncertainty in the momentum of long wave-length optical modes with decrease in the size of nano-ordered regions. We determine the cause for this to be a strong mixing between acoustic and optical modes, believed to be central to the model of waterfall phonon anomalies.