The full conjecture is stated over abstract fields, but the counterexample is a concrete 3D function that we can explain and visualize using familiar geometric ideas and a little algebra. The post A Simplified View of the Jacobian Conjecture appeared first on Towards Data Science .
arXiv:2606. 10806v1 Announce Type: new Abstract: Moonshine is an autonomous agent whose central objective is to generate mathematical conjectures.
An OpenAI model solved the 80-year-old unit distance problem, disproving a major conjecture in discrete geometry and marking a milestone in AI-driven mathematics.
arXiv:2606. 08799v1 Announce Type: cross Abstract: We study the generalization of ridge-regularized nonlinear least-squares models via on-average algorithmic stability, deriving error bounds for local minimizers in terms of a data-dependent effective dimension that reflects the geometry of the gradient model at the trained parameters, through the empirical Jacobian Gram matrix and a residual--curvature term.
arXiv:2606. 01596v1 Announce Type: cross Abstract: Learning chaotic dynamical systems from data requires more than short-term predictive accuracy: the learned model must preserve the attractor geometry and its invariant statistics.
arXiv:2209. 15130v3 Announce Type: replace-cross Abstract: We study a general matrix optimization problem with a fixed-rank positive semidefinite (PSD) constraint.
arXiv:2606. 15442v1 Announce Type: cross Abstract: We design a new unconstrained coordinate system where a $p\times p$ symmetric positive definite (SPD) matrix $\Theta$ is represented by a reverse telescoping map $\Theta(x)=\rm{RT}(x)$, with $x=(v,d,r)\in\mathbb{R}\times\mathbb{R}^{(p-1)}\times\mathbb{R}^{p(p-1)/2}$, representing respectively the log volume or log determinant; and the shape, as encoded by log relative diagonal scales and partial covariances among the nodes.
arXiv:2607. 17673v1 Announce Type: cross Abstract: Contrastive learning is increasingly moving toward settings with three or more modalities instead of image-text pairs.
arXiv:2510. 21033v3 Announce Type: replace-cross Abstract: We develop a theory of iso-Riemannian optimization for problems constrained to learned data manifolds, a setting in which classical Riemannian optimization - and Riemannian gradient descent in particular - can be poorly suited.
arXiv:2505. 05168v4 Announce Type: replace-cross Abstract: Under mild conditions, a least-squares local linear Fr\'echet curve predictor is derived for a response and a regressor evaluated in a separable Hilbert space.
arXiv:2410. 00722v3 Announce Type: replace Abstract: We study convolutional neural networks with monomial activation functions.
arXiv:2605. 26234v2 Announce Type: replace-cross Abstract: A recent conjecture by Joel Fine posits a relationship between the coefficients of the HOMFLY polynomial of a knot $K$ in the 3-sphere $S^3$, and the signed count of minimal surfaces in hyperbolic 4-space $\mathrm{H}^4$ meeting the sphere at infinity at $K$, with prescribed genus and self-intersection number.
The clipping bug has lived in every 3D simulation pipeline for three decades. Here is exactly why it happens, how the math breaks, and how swapping one equation fixes it; as well as the python code to see it for yourself!