This letter deals with neural networks as dynamical systems governed by finite difference equations. It shows that the introduction of [Formula: see text]-many skip connections into network architectures, such as residual networks and additive dense networks, defines [Formula: see text]th order dynamical equations on the layer-wise transformations. Closed-form solutions for the state-space representations of general [Formula: see text]th order additive dense networks, where the concatenation operation is replaced by addition, as well as [Formula: see text]th order smooth networks, are found. The developed provision endows deep neural networks with an algebraic structure. Furthermore, it is shown that imposing [Formula: see text]th order smoothness on network architectures with [Formula: see text]-many nodes per layer increases the state-space dimension by a multiple of [Formula: see text], and so the effective embedding dimension of the data manifold by the neural network is [Formula: see text]-many dimensions. It follows that network architectures of these types reduce the number of parameters needed to maintain the same embedding dimension by a factor of [Formula: see text] when compared to an equivalent first-order, residual network. Numerical simulations and experiments on CIFAR10, SVHN, and MNIST have been conducted to help understand the developed theory and efficacy of the proposed concepts.

It is unclear what triggers the behavior of active shooters, but their consequences are severe. There is opportunity to implement an automated response system capable of delivery guidance to evacuees' to aide people to safety. Optimizing guidance delivery is challenging because active shooter incidents evolve quickly and are unpredictable. In this paper, we develop a new problem formulation specific for active shooter events by utilizing the structure of each room. To effectively solve this problem, a divide-and-conquer approach is deployed to split evacuees into groups. The egress routes are decomposed and coordinated for each group are optimized using stochastic dynamic programming. Numerical testing and simulation show our solution with a safe room the solution fast relevant to shooting events and effectively.