The exfoliation of acid-exchanged K4Nb6O17 with tetra(n-butyl)ammonium hydroxide in water produces a colloidal suspension of individual sheets, which roll into loosely bound tubular structures. The tubule shape can be made permanent via precipitation of the colloid with alkali cations. Atomic force microscopy and transmission electron micrographs reveal that the tubules have outer diameters ranging from 15 to 30 nm and that they are 0.1 to 1 μm in length. The observed curling tendency, preferential folding, and cleavage angles of the individual sheets are interpreted in terms of the crystal structure of the parent solid, K4Nb6O17. The driving force for tubule formation appears to be relief of strain that is inherent in the asymmetric single sheets. This driving force is absent in bilayer colloids formed early in the exfoliation process, which are found only as flat sheets. Tubules in colloidal suspensions that have been subjected to turbulence have a tendency to unroll into flat sheets on surfaces, indicating that the forces controlling rolling and unrolling are closely balanced.

Location-aware networks are of great importance and interest in both civil and military applications. This paper determines the localization accuracy of an agent, which is equipped with an antenna array and localizes itself using wireless measurements with anchor nodes, in a far-field environment. In view of the Cram\'er-Rao bound, we first derive the localization information for static scenarios and demonstrate that such information is a weighed sum of Fisher information matrices from each anchor-antenna measurement pair. Each matrix can be further decomposed into two parts: a distance part with intensity proportional to the squared baseband effective bandwidth of the transmitted signal and a direction part with intensity associated with the normalized anchor-antenna visual angle. Moreover, in dynamic scenarios, we show that the Doppler shift contributes additional direction information, with intensity determined by the agent velocity and the root mean squared time duration of the transmitted signal. In addition, two measures are proposed to evaluate the localization performance of wireless networks with different anchor-agent and array-antenna geometries, and both formulae and simulations are provided for typical anchor deployments and antenna arrays.Comment: to appear in IEEE Transactions on Information Theor

We consider the problem of estimating functionals of discrete distributions, and focus on tight nonasymptotic analysis of the worst case squared error risk of widely used estimators. We apply concentration inequalities to analyze the random fluctuation of these estimators around their expectations, and the theory of approximation using positive linear operators to analyze the deviation of their expectations from the true functional, namely their \emph{bias}. We characterize the worst case squared error risk incurred by the Maximum Likelihood Estimator (MLE) in estimating the Shannon entropy $H(P) = \sum_{i = 1}^S -p_i \ln p_i$, and $F_\alpha(P) = \sum_{i = 1}^S p_i^\alpha,\alpha>0$, up to multiplicative constants, for any alphabet size $S\leq \infty$ and sample size $n$ for which the risk may vanish. As a corollary, for Shannon entropy estimation, we show that it is necessary and sufficient to have $n \gg S$ observations for the MLE to be consistent. In addition, we establish that it is necessary and sufficient to consider $n \gg S^{1/\alpha}$ samples for the MLE to consistently estimate $F_\alpha(P), 0<\alpha<1$. The minimax rate-optimal estimators for both problems require $S/\ln S$ and $S^{1/\alpha}/\ln S$ samples, which implies that the MLE has a strictly sub-optimal sample complexity. When $1<\alpha<3/2$, we show that the worst-case squared error rate of convergence for the MLE is $n^{-2(\alpha-1)}$ for infinite alphabet size, while the minimax squared error rate is $(n\ln n)^{-2(\alpha-1)}$. When $\alpha\geq 3/2$, the MLE achieves the minimax optimal rate $n^{-1}$ regardless of the alphabet size. As an application of the general theory, we analyze the Dirichlet prior smoothing techniques for Shannon entropy estimation. We show that no matter how we tune the parameters in the Dirichlet prior, this technique cannot achieve the minimax rates in entropy estimation.Comment: 27 pages, 1 figure, published in IEEE Transactions on Information Theor

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