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Purpose: To investigate the effect of dry coating the amount and type of silica on powder flowability enhancement using a comprehensive set of 19 pharmaceutical powders having different sizes, surface roughness, morphology, and aspect ratios, as well as assess flow predictability via Bond number estimated using a mechanistic multi-asperity particle contact model. Method: Particle size, shape, density, surface energy and area, SEM-based morphology, and FFC were assessed for all powders. Hydrophobic (R972P) or hydrophilic (A200) nano-silica were dry coated for each powder at 25%, 50%, and 100% surface area coverage (SAC). Flow predictability was assessed via particle size and Bond number. Results: Nearly maximal flow enhancement, one or more flow category, was observed for all powders at 50% SAC of either type of silica, equivalent to 1 wt% or less for both the hydrophobic R972P or hydrophilic A200, while R972P generally performed slightly better. Silica amount as SAC better helped understand the relative performance. The power-law relation between FFC and Bond number was observed. Conclusion: Significant flow enhancements were achieved at 50% SAC, validating previous models. Most uncoated very cohesive powders improved by two flow categories, attaining easy flow. Flowability could not be predicted for both the uncoated and dry coated powders via particle size alone. Prediction was significantly better using Bond number computed via the mechanistic multi-asperity particle contact model accounting for the particle size, surface energy, roughness, and the amount and type of silica. The widely accepted 200 nm surface roughness was not valid for most pharmaceutical powders. 

Structural supercapacitors reach high performance with a gradient electrolyte and redox polymer electrodes. 

Boolean functions play an important role in many different areas of computer science. The _local sensitivity_ of a Boolean function $f:\{0,1\}^n\to \{0,1\}$ on an input $x\in\{0,1\}^n$ is the number of coordinates whose flip changes the value of $f(x)$, i.e., the number of i's such that $f(x)\not=f(x+e_i)$, where $e_i$ is the $i$-th unit vector. The _sensitivity_ of a Boolean function is its maximum local sensitivity. In other words, the sensitivity measures the robustness of a Boolean function with respect to a perturbation of its input. Another notion that measures the robustness is block sensitivity. The _local block sensitivity_ of a Boolean function $f:\{0,1\}^n\to \{0,1\}$ on an input $x\in\{0,1\}^n$ is the number of disjoint subsets $I$ of $\{1,..,n\}$ such that flipping the coordinates indexed by $I$ changes the value of $f(x)$, and the _block sensitivity_ of $f$ is its maximum local block sensitivity. Since the local block sensitivity is at least the local sensitivity for any input $x$, the block sensitivity of $f$ is at least the sensitivity of $f$.The next example demonstrates that the block sensitivity of a Boolean function is not linearly bounded by its sensitivity. Fix an integer $k\ge 2$ and define a Boolean function $f:\{0,1\}^{2k^2}\to\{0,1\}$ as follows: the coordinates of $x\in\{0,1\}^{2k^2}$ are split into $k$ blocks of size $2k$ each and $f(x)=1$ if and only if at least one of the blocks contains exactly two entries equal to one and these entries are consecutive. While the sensitivity of the function $f$ is $2k$, its block sensitivity is $k^2$. The Sensitivity Conjecture, made by Nisan and Szegedy in 1992, asserts that the block sensitivity of a Boolean function is polynomially bounded by its sensivity. The example above shows that the degree of such a polynomial must be at least two.The Sensitivity Conjecture has been recently proven by Huang in [Annals of Mathematics 190 (2019), 949-955](https://doi.org/10.4007/annals.2019.190.3.6). He proved the following combinatorial statement that implies the conjecture (with the degree of the polynomial equal to four): any subset of more than half of the vertices of the $n$-dimensional cube $\{0,1\}^n$ induces a subgraph that contains a vertex with degree at least $\sqrt{n}$. The present article extends this result as follows: every Cayley graph with the vertex set $\{0,1\}^n$ and any generating set of size $d$ (the vertex set is viewed as a vector space over the binary field) satisfies that any subset of more than half of its vertices induces a subgraph that contains a vertex of degree at least $\sqrt{d}$. In particular, when the generating set consists of the $n$ unit vectors, the Cayley graph is the $n$-dimensional hypercube.