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We consider a class of deterministic mean field games, where the state associated with each player evolves according to an ODE which is linear w.r.t. the control. Existence, uniqueness, and stability of solutions are studied from the point of viewof generic theory. Within a suitable topological space of dynamics and cost functionals, we prove that, for “nearly all” mean field games (in the Baire category sense) the best reply map is single-valued for a.e. player. As a consequence, the mean field game admits a strong (not randomized) solution. Examples are given of open sets of games admitting a single solution, and other open sets admitting multiple solutions. Further examples show the existence of an open set of MFG having a unique solution which is asymptotically stable w.r.t. the best reply map, and another open set of MFG having a unique solution which is unstable. We conclude with an example of a MFG with terminal constraints which does not have any solution, not even in the mild sense with randomized strategies. 

The paper discusses various regularity properties for solutions to a scalar, 1-dimensional conservation law with strictly convex flux and integrable source. In turn, these yield compactness estimates on the solution set. Similar properties are expected to hold for 2x2 genuinely nonlinear systems. 

One surprising trait of neural networks is the extent to which their connections can be pruned with little to no effect on accuracy. But when we cross a critical level of parameter sparsity, pruning any further leads to a sudden drop in accuracy. This drop plausibly reflects a loss in model complexity, which we aim to avoid. In this work, we explore how sparsity also affects the geometry of the linear regions defined by a neural network, and consequently reduces the expected maximum number of linear regions based on the architecture. We observe that pruning affects accuracy similarly to how sparsity affects the number of linear regions and our proposed bound for the maximum number. Conversely, we find out that selecting the sparsity across layers to maximize our bound very often improves accuracy in comparison to pruning as much with the same sparsity in all layers, thereby providing us guidance on where to prune. 

Abstract Two common hemoglobinopathies, sickle cell disease (SCD) and β-thalassemia, arise from genetic mutations within the β-globin gene. In this work, we identified a 500-bp motif (Fetal Chromatin Domain, FCD) upstream of human ϒ-globin locus and showed that the removal of this motif using CRISPR technology reactivates the expression of ϒ-globin. Next, we present two different cell morphology-based machine learning approaches that can be used identify human blood cells (KU-812) that harbor CRISPR-mediated FCD genetic modifications. Three candidate models from the first approach, which uses multilayer perceptron algorithm (MLP 20-26, MLP26-18, and MLP 30-26) and flow cytometry-derived cellular data, yielded 0.83 precision, 0.80 recall, 0.82 accuracy, and 0.90 area under the ROC (receiver operating characteristic) curve when predicting the edited cells. In comparison, the candidate model from the second approach, which uses deep learning (T2D5) and DIC microscopy-derived imaging data, performed with less accuracy (0.80) and ROC AUC (0.87). We envision that equivalent machine learning-based models can complement currently available genotyping protocols for specific genetic modifications which result in morphological changes in human cells.