We study incidences between points and (constant-degree algebraic) curves in three dimensions, taken from a family C of curves that have almost two degrees of freedom, meaning that (i) every pair of curves of C intersect in O(1) points, (ii) for any pair of points p, q, there are only O(1) curves of C that pass through both points, and (iii) a pair p, q of points admit a curve of C that passes through both of them if and only if F (p, q) = 0 for some polynomial F of constant degree associated with the problem. (As an example, the family of unit circles in R 3 that pass through some fixed point is such a family.)We begin by studying two specific instances of this scenario. The first instance deals with the case of unit circles in R 3 that pass through some fixed point (so called anchored unit circles). In the second case we consider tangencies between directed points and circles in the plane, where a directed point is a pair (p, u), where p is a point in the plane and u is a direction, and (p, u) is tangent to a circle γ if p ∈ γ and u is the direction of the tangent to γ at p. A lifting transformation due to Ellenberg et al. [4] maps these tangencies to incidences between points and curves ('lifted circles') in three dimensions. In both instances we have a family of curves in R 3 with almost two degrees of freedom.We show that the number of incidences between m points and n anchored unit circles in R 3 , as well as the number of tangencies between m directed points and n arbitrary circles in the plane, is O(m 3/5 n 3/5 + m + n) in both cases.We then derive a similar incidence bound, with a few additional terms, for more general families of curves in R 3 with almost two degrees of freedom, under a few additional natural assumptions.The proofs follow standard techniques, based on polynomial partitioning, but they face a critical novel issue involving the analysis of surfaces that are infinitely ruled by the respective family of curves, as well as of surfaces in a dual three-dimensional space that are infinitely ruled by the respective family of suitably defined dual curves. We either show that no such surfaces exist, or develop and adapt techniques for handling incidences on such surfaces.The general bound that we obtain is O(m 3/5 n 3/5 + m + n) plus additional terms that depend on how many curves or dual curves can lie on an infinitely-ruled surface.

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<p>Standard approaches to earthquake forecasting - both statistics-based models, e.g. the epidemic type aftershock (ETAS), and physics-based models, e.g. models based on the Coulomb failure stress (CFS) criteria, estimate the probability of an earthquake occurring at a certain time and location. In both modeling approaches the time and location of an earthquake are commonly assumed to be distributed independently of their magnitude. That is, the magnitude of a given earthquake is taken to be the marginal magnitude distribution, the Gutenberg-Richter (GR) distribution, typically constant in time,or fitted to recent seismic history. Such model construction implies an assumption that the underlying process determining where and when an earthquake occurs is decoupled from the process that determines its magnitude.</p> <p>In this work we address the question of magnitude independence directly. We build a machine learning model that predicts earthquake magnitudes based on their location, region history, and other geophysical properties. We use neural networks to encode these properties and output a  conditional magnitude probability distribution, maximizing on the log-likelihood of the model’s prediction. We discuss the model architecture, performance, and evaluate this model against the GR distribution.</p>