We investigate the mechanical response of jammed packings of repulsive, frictionless spherical particles undergoing isotropic compression. Prior simulations of the soft-particle model, where the repulsive interactions scale as a power law in the interparticle overlap with exponent α, have found that the ensemble-averaged shear modulus 〈G(P)〉 increases with pressure P as ∼P^(α-3/2)/(α-1) at large pressures. 〈G〉 has two key contributions (1) continuous variations as a function of pressure along geometrical families, for which the interparticle contact network does not change, and (2) discontinuous jumps during compression that arise from changes in the contact network. Using numerical simulations, we show that the form of the shear modulus G^f for jammed packings within near-isostatic geometrical families is largely determined by the affine response G^f∼G_a^f, where G_a^f/G_a0=(P/P_0)^(α-2)/(α-1)-P/P_0, P_0∼N^-2(α-1) is the characteristic pressure at which G_a^f=0, G_a0 is a constant that sets the scale of the shear modulus, and N is the number of particles. For near-isostatic geometrical families that persist to large pressures, deviations from this form are caused by significant nonaffine particle motion. We further show that the ensemble-averaged shear modulus 〈G(P)〉 is not simply a sum of two power laws, but 〈G(P)〉∼(P/P_c)^a, where a≈(α-2)/(α-1) in the P→0 limit and 〈G(P)〉∼(P/P_c)^b, where b≳(α-3/2)/(α-1), above a characteristic pressure that scales as P_c∼N^-2(α-1).Molecular expressions for thermodynamic properties and derivatives of the Gibbs energy up to third order in the isobaric-isothermal (NpT) ensemble are systematically derived using the methodology developed by Lustig for the microcanonical and canonical ensembles [J. Chem. Phys. 100, 3048 (1994)10.1063/1.466446; Mol. Phys. 110, 3041 (2012)10.1080/00268976.2012.695032]. They are expressed by phase-space functions, which represent derivatives of the Gibbs energy with respect to temperature and pressure. Additionally, expressions for the phase-space functions for temperature-dependent potentials are provided, which, for example, are required when quantum corrections, e.g., Feynman-Hibbs corrections, are applied in classical simulations. The derived expressions are validated by Monte Carlo simulations for the simple Lennard-Jones model fluid at three selected state points. A unique result is that the phase-space functions contain only ensemble averages of combinations of powers of enthalpy and volume. Thus, the calculation of thermodynamic properties in the NpT ensemble does not require volume derivatives of the potential energy. This is particularly advantageous in Monte Carlo simulations when the interactions between molecules are described by empirical force fields or very accurate ab initio pair and nonadditive three-body potentials.Many sensory pathways in the brain include sparsely active populations of neurons downstream from the input stimuli. The biological purpose of this expanded structure is unclear, but it may be beneficial due to the increased expressive power of the network. In this work, we show that certain ways of expanding a neural network can improve its generalization performance even when the expanded structure is pruned after the learning period. To study this setting, we use a teacher-student framework where a perceptron teacher network generates labels corrupted with small amounts of noise. We then train a student network structurally matched to the teacher. In this scenario, the student can achieve optimal accuracy if given the teacher's synaptic weights. We find that sparse expansion of the input layer of a student perceptron network both increases its capacity and improves the generalization performance of the network when learning a noisy rule from a teacher perceptron when the expansion is pruned after learning. We find similar behavior when the expanded units are stochastic and uncorrelated with the input and analyze this network in the mean-field limit. By solving the mean-field equations, we show that the generalization error of the stochastic expanded student network continues to drop as the size of the network increases. This improvement in generalization performance occurs despite the increased complexity of the student network relative to the teacher it is trying to learn. We show that this effect is closely related to the addition of slack variables in artificial neural networks and suggest possible implications for artificial and biological neural networks.Recent literature indicates that attractive interactions between particles of a dense liquid play a secondary role in determining its bulk mechanical properties.  https://www.selleckchem.com/products/borussertib.html  Here we show that, in contrast with their apparent unimportance to the bulk mechanics of dense liquids, attractive interactions can have a major effect on macro- and microscopic elastic properties of glassy solids. We study several broadly applicable dimensionless measures of stability and mechanical disorder in simple computer glasses, in which the relative strength of attractive interactions-referred to as "glass stickiness"-can be readily tuned. We show that increasing glass stickiness can result in the decrease of various quantifiers of mechanical disorder, on both macro- and microscopic scales, with a pair of intriguing exceptions to this rule. Interestingly, in some cases strong attractions can lead to a reduction of the number density of soft, quasilocalized modes, by up to an order of magnitude, and to a substantial decrease in their core size, similar to the effects of thermal annealing on elasticity observed in recent works. Contrary to the behavior of canonical glass models, we provide compelling evidence indicating that the stabilization mechanism in our sticky-sphere glasses stems predominantly from the self-organized depletion of interactions featuring large, negative stiffnesses. Finally, we establish a fundamental link between macroscopic and microscopic quantifiers of mechanical disorder, which we motivate via scaling arguments. Future research directions are discussed.