In this article we provide a stack-theoretic framework to study the universal tropical Jacobian over moduli space of curves. We develop two approaches process tropicalization compactified curves -- one from logarithmic and other non-Archimedean analytic point view. The central result both points view is that maps in each contexts are compatible with tautological morphisms. sequel will use techn...

This work presents a re-derivation of relative Jacobian matrix for parallel manipulators, expressed in terms of the individual manipulator Jacobians and multiplied by their corresponding transformation matrices. This is particularly useful when the individual manipulator Jacobians are given, such that one would not need to derive an entirely new expression of a relative Jacobian but will only u...

Recent work (Pennington et al., 2017) suggests that controlling the entire distribution of Jacobian singular values is an important design consideration in deep learning. Motivated by this, we study the distribution of singular values of the Jacobian of the generator in Generative Adversarial Networks (GANs). We find that this Jacobian generally becomes ill-conditioned at the beginning of train...

Gray matter atrophy provides important insights into neurodegeneration in multiple sclerosis (MS) and can be used as a marker of neuroprotection in clinical trials. Jacobian integration is a method for measuring volume change that uses integration of the local Jacobian determinants of the nonlinear deformation field registering two images, and is a promising tool for measuring gray matter atrop...

We give upper and lower bounds for the number of points on abelian varieties over finite fields, and lower bounds specific to Jacobian varieties. We also determine exact formulas for the maximum and minimum number of points on Jacobian surfaces.

We give a proof of the two dimensional Jacobian conjecture. We also prove that if (F, G) is a Jacobian pair with deg y F ≥ 1, then F is a monic polynomial of y up to a scalar.

An externalised surface scheme like SURFEX allows computationally cheap offline runs. This is a major advantage for surface assimilation techniques such as the extended Kalman filter (EKF), where the offline runs allow a cheaper numerical estimation of the observation operator Jacobian. In the recent past an EKF has been developed within SURFEX for the initialisation of soil water content and s...