topology   2257

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[1808.06091] Clock theorems for triangulated surfaces
We investigate triangulations of the two-dimensional sphere and torus with the faces properly colored white and black. We focus on matchings between white triangles and incident vertices. On the torus our objects are perfect pairings, whereas on the sphere this is only true after removing one triangle and its vertices. In the latter case, such matchings (first studied by Tutte) extend the notion of state in Kauffman's formal knot theory and we show that his Clock Theorem, in its form due to Gilmer and Litherland, also extends: the set of matchings naturally forms a distributive lattice. Here the role of state transposition is played by a simple local operation about black triangles. By contrast, on the torus, the analogous state transition graph is usually disconnected: some of its components still form distributive lattices with global maxima and minima, while other components contain directed cycles and are without local extrema.
combinatorics  graph-theory  topology  rather-interesting  feature-construction 
7 days ago by Vaguery
Neural Networks, Manifolds, and Topology -- colah's blog
However, there remain a number of concerns about them. One is that it can be quite challenging to understand what a neural network is really doing. If one trains it well, it achieves high quality results, but it is challenging to understand how it is doing so. If the network fails, it is hard to understand what went wrong.
machinelearning  math  topology  visualization  deep.learning  theory 
21 days ago by joecamel
[1812.11592] A Geometric Theory of Higher-Order Automatic Differentiation
First-order automatic differentiation is a ubiquitous tool across statistics, machine learning, and computer science. Higher-order implementations of automatic differentiation, however, have yet to realize the same utility. In this paper I derive a comprehensive, differential geometric treatment of automatic differentiation that naturally identifies the higher-order differential operators amenable to automatic differentiation as well as explicit procedures that provide a scaffolding for high-performance implementations.
gradient-descent  automatic-differentiation  algorithms  topology  algebraic-topology  the-shape-of-data  machine-learning  rather-interesting  to-understand 
23 days ago by Vaguery
JTS Topology Suite download |
The JTS Topology Suite is an API for modelling and manipulating 2-dimensional linear geometry. It provides numerous geometric predicates and functions. JTS conforms to the Simple Features Specification for SQL published by the Open GIS Consortium.
JTS  API  Java  2D  Geometry  Library  Geo  Geospatial  vector  topology  algorithms  tool 
5 weeks ago by searchmeister
[1806.04129] Angels' staircases, Sturmian sequences, and trajectories on homothety surfaces
A homothety surface can be assembled from polygons by identifying their edges in pairs via homotheties, which are compositions of translation and scaling. We consider linear trajectories on a 1-parameter family of genus-2 homothety surfaces. The closure of a trajectory on each of these surfaces always has Hausdorff dimension 1, and contains either a closed loop or a lamination with Cantor cross-section. Trajectories have cutting sequences that are either eventually periodic or eventually Sturmian. Although no two of these surfaces are affinely equivalent, their linear trajectories can be related directly to those on the square torus, and thence to each other, by means of explicit functions. We also briefly examine two related families of surfaces and show that the above behaviors can be mixed; for instance, the closure of a linear trajectory can contain both a closed loop and a lamination.
plane-geometry  geometry  algebra  continued-fractions  fractals  to-understand  topology  several-overlaps 
7 weeks ago by Vaguery

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