On the mod-p distribution of discriminants of G-extensions

This paper was motivated by a recent paper by Krumm and Pollack ([Twists of hyperelliptic curves by integers in progressions modulo [Formula: see text], preprint (2018); https://arXiv.org/abs/1807.00972] ) investigating modulo-[Formula: see text] behavior of quadratic twists with rational points of a given hyperelliptic curve, conditional on the abc-conjecture. We extend those results to twisted Galois covers with arbitrary Galois groups. The main point of this generalization is to interpret those results as statements about the sets of specializations of a given Galois cover under restrictions on the discriminant. In particular, we make a connection with existing heuristics about the distribution of discriminants of Galois extensions such as the Malle conjecture: our results show in a precise sense the non-existence of “local obstructions” to such heuristics, in many cases essentially only under the assumption that [Formula: see text] occurs as the Galois group of a Galois cover defined over [Formula: see text]. This complements and generalizes a similar result in the direction of the Malle conjecture by Dèbes ([On the Malle conjecture and the self-twisted cover, Israel J. Math. 218(1) (2017) 101–131]).

SYSTEM IDENTIFICATION WITH ARTIFICIAL NEURAL NETWORKS

System identification is the term scientists and engineers use to refer to the process of building mathematical models of dynamical systems based on observed data. This paper approaches system identification as a pattern recognition problem. We use computers to simulate the system response for a variety of different mathematical models. For each distinct system model, simulated system responses tend to remain segregated in one or more amorphous regions of system response space despite (1) large variations in system parameters, (2) experimental errors, and (3) noise. The actual system response is classified with the model corresponding to the region of system response space where it is found. The classifier is an Artificial Neural Network (ANN) which implements a Generalized Vector Quantizer (GVQ). A small number of simple but powerful discriminant functions facilitate the correct classification of most of the responses in any given region. The required distribution of discriminants among the regions evolves automatically as they learn their respective functions.