Related papers: South Pointing Chariot: An Invitation to Different…
This paper presents relevant modern mathematical formulations for (classical) gauge field theories, namely, ordinary differential geometry, noncommutative geometry, and transitive Lie algebroids. They provide rigorous frameworks to describe…
We solve the problem of prescribing different types of curvatures (principal, mean or Gaussian) on rotational surfaces in terms of arbitrary continuous functions depending on the distance from the surface to the axis of revolution. In this…
In a variety of scientific applications we wish to characterize a physical system using measurements or observations. This often requires us to solve an inverse problem, which usually has non-unique solutions so uncertainty must be…
The main aim of this work is to present the interpretation of the Ising type models as a kind of field theory in the framework of noncommutative geometry. We present the method and construct sample models of field theory on discrete spaces…
Navigation is an essential ability for mobile agents to be completely autonomous and able to perform complex actions. However, the problem of navigation for agents with limited (or no) perception of the world, or devoid of a fully defined…
Differentiable physics is a powerful tool in computer vision and robotics for scene understanding and reasoning about interactions. Existing approaches have frequently been limited to objects with simple shape or shapes that are known in…
This is an introduction to geometric algebra, an alternative to traditional vector algebra that expands on it in two ways: 1. In addition to scalars and vectors, it defines new objects representing subspaces of any dimension. 2. It defines…
In this paper we make an overview of results relating the recent "discoveries" in differential geometry, such as higher structures and differential graded manifolds with some natural problems coming from mechanics. We explain that a lot of…
This is an expository paper in which we define projective GIT quotients and introduce toric varieties from this perspective. It is intended primarily for readers who are learning either invariant theory or toric geometry for the first time.
We introduce differentiable indirection -- a novel learned primitive that employs differentiable multi-scale lookup tables as an effective substitute for traditional compute and data operations across the graphics pipeline. We demonstrate…
Geometric predicates are a basic ingredient to implement a vast range of algorithms in computational geometry. Modern implementations employ floating point filtering techniques to combine efficiency and robustness, and state-of-the-art…
This essay, an excerpt of the author's Ph.D. in Philosophy of mathematics (2012) thought of as being a companion to recent discoveries of new explicit Cartan geometry curvatures, analyzes how Gauss, after having devised the isometrically…
Many learning problems require predicting sets of objects when the number of objects is not known beforehand. Examples include object detection, molecular modeling, and scientific inference tasks such as astrophysical source detection.…
Synthesizing extrapolated views from recorded driving logs is critical for simulating driving scenes for autonomous driving vehicles, yet it remains a challenging task. Recent methods leverage generative priors as pseudo ground truth, but…
In this paper, we introduce some known map projections from a model of the Earth to a flat sheet of paper or map and derive the plotting equations for these projections. The first fundamental form and the Gaussian fundamental quantities are…
This is an attempt to present axioms for Euclidean geometry, aiming at the following goals: to work with geometric notions (thus not merely identify points with pairs of numbers, giving a special status to a particular coordinate system);…
Classical geometric mechanics, including the study of symmetries, Lagrangian and Hamiltonian mechanics, and the Hamilton-Jacobi theory, are founded on geometric structures such as jets, symplectic and contact ones. In this paper, we shall…
Shape inference is classically ill-posed, because it involves a map from the (2D) image domain to the (3D) world. Standard approaches regularize this problem by either assuming a prior on lighting and rendering or restricting the domain,…
Mathematics is a mountain, but students need more than descriptions of the view: they need a trail they can actually walk. This paper presents the Math Teaching Atlas, a framework for mathematical exposition built around route units (single…
This paper aims to systematically and comprehensively initiate a foundation for using concepts from computational differential geometry as instruments for power flow computing and research. At this point we focus our discussion on the…