Hyers-Ulam stability of elliptic M\"obius difference equation

The linear fractional map $f(z) = \frac{az+ b}{cz + d}$ on the Riemann sphere with complex coefficients $ad-bc \neq 0$ is called M\"obius map. If $f$ satisfies $ad-bc=1$ and $-2<a+d<2$, then $f$ is called $\textit{elliptic}$ M\"obius map. Let $\{ b_n \}_{n \in \mathbb{N}_0}$ be the solution of the elliptic M\"obius difference equation $b_{n+1} = f(b_n)$ for every $n \in \mathbb{N}_0$. Then the sequence $\{ b_n \}_{n \in \mathbb{N}_0}$ has no Hyers-Ulam stability.Comment: 12 page

Design of Composite Double-Slab Radar Absorbing Structures Using Forward, Inverse, and Tandem Neural Networks

The survivability and mission of a military aircraft is often designed with minimum radar cross section (RCS) to ensure its long-term operation and maintainability. To reduce aircraft’s RCS, a specially formulated Radar Absorbing Structures (RAS) is primarily applied to its external skins. A Ni-coated glass/epoxy composite is a recent RAS material system designed for decreasing the RCS for the X-band (8.2 – 12.4 GHz), while maintaining efficient and reliable structural performance to function as the skin of an aircraft. Experimentally measured and computationally predicted radar responses (i.e., return loss responses in specific frequency ranges) of multi-layered RASs are expensive and labor-intensive. Solving their inverse problems for optimal RAS design is also challenging due to their complex configuration and physical phenomena. An artificial neural network (ANN) is a machine learning method that uses existing data from experimental results and validated models (i.e., transfer learning) to predict complex behavior. Training an ANN can be computationally expensive; however, training is a one-time cost. In this work, three different Three ANN models are presented for designing dual slab Ni-coated glass/epoxy composite RASs: (1) the feedforward neural network (FNN) model, (2) the inverse neural network (INN) model – an inverse network, which maintains a parallel structure to the FNN model, and (3) the tandem neural network (TNN) model – an alternative to the INN model which uses a pre-trained FNN in the training process. The FNN model takes the thicknesses of dual slab RASs to predict their returns loss in the X-band range. The INN model solves the inverse problem for the FNN model. The TNN model is established with a pre-trained FNN to train an INN that exactly reverses the operation done in the FNN rather than solving the inverse problem independently. These ANN models will assist in reducing the time and cost for designing dual slab (and further extension to multi-layered) RASs