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We consider the Ising perceptron model with N spins and M = N*alpha patterns, with a general activation function U that is bounded above. For U bounded away from zero, or U a one-sided threshold function, it was shown by Talagrand (2000, 2011) that for small densities alpha, the free energy of the model converges in the large-N limit to the replica symmetric formula conjectured in the physics literature (Krauth–Mezard 1989, see also Gardner–Derrida 1988). We give a new proof of this result, which covers the more general class of all functions U that are bounded above and satisfy a certain variance bound. The proof uses the (first and second) moment method conditional on the approximate message passing iterates of the model. In order to deduce our main theorem, we also prove a new concentration result for the perceptron model in the case where U is not bounded away from zero.

In this poster we describe Make with Data, a two-year project that invites teachers and students from public high schools to work with professional data scientists and open-source data to explore issues important to their local community. While the negotiation of the personal and the quantitative resulted in tensions, Make with Data students found their personal experiences a useful tool for adding context and complexity to the phenomena being studied.

This chapter describes two mechanical expansion microscopy methods with accompanying step-by-step protocols. The first method, mechanically resolved expansion microscopy, uses non-uniform expansion of partially digested samples to provide the imaging contrast that resolves local mechanical properties. Examining bacterial cell wall with this method, we are able to distinguish bacterial species in mixed populations based on their distinct cell wall rigidity and detect cell wall damage caused by various physiological and chemical perturbations. The second method is mechanically locked expansion microscopy, in which we use a mechanically stable gel network to prevent the original polyacrylate network from shrinking in ionic buffers. This method allows us to use anti-photobleaching buffers in expansion microscopy, enabling detection of novel ultra-structures under the optical diffraction limit through super-resolution single molecule localization microscopy on bacterial cells and whole-mount immunofluorescence imaging in thick animal tissues. We also discuss potential applications and assess future directions.