A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
The paper develops a theory of discrete Riemann surfaces using quadrilateral cells.
Consider a planar, bounded, -connected region , and let $\bordΩ$ be its boundary. Let be a cellular decomposition of $Ω\cup\bordΩ$, where each 2-cell is either a triangle or a quadrilateral. From these data and a conductance function we construct a canonical pair where is a genus …
This paper gives sharp linear bounds on the genus of a normal surface in a triangulated compact, orientable 3--manifold in terms of the quadrilaterals in its cell decomposition---different bounds arise from varying hypotheses on the surface or triangulation. Two applications of these bounds are given. First, the minima…
Paper introduces quadrilateral labyrinth fractals and their properties.
Any cyclic quadrilateral can fit inside any smooth curve.
Formulae quantify gaps in geodesic quadrilaterals on manifolds.
Spherical quadrilaterals classified based on geometric properties.
New method characterizes surface quadrilateral layouts as special immersions.
Similarity maps cyclic quadrilaterals onto smooth curves.
Paper shows any link can be diagrammed with only triangles and quadrilaterals.
We interpret a normal surface in a (singular) three-manifold in terms of the homology of a chain complex. This allows us to study the relation between normal surfaces and their quadrilateral co-ordinates. Specifically, we give a proof of an (unpublished) observation independently given by Casson and Rubinstein saying t…
The study classifies tilings of the sphere by congruent quadrilaterals.
The paper classifies surfaces formed by quadrilateral gluings.
Investigates stability on toric surfaces with quadrilateral moment polytopes.
Constructs minimal surfaces over Pitot quadrilaterals using harmonic diffeomorphisms.
Constructs constant mean curvature surfaces using geometric flow.
Square-like quadrilaterals inscribed in space curves proven for finite total curvature.
In this paper we continue the study started in part I (posted). We consider a planar, bounded, -connected region , and let $\bordΩ$ be its boundary. Let be a cellular decomposition of $Ω\cup\bordΩ$, where each 2-cell is either a triangle or a quadrilateral. From these data and a conductance function…
We discuss discretization of Koenigs nets (conjugate nets with equal Laplace invariants) and of isothermic surfaces. Our discretization is based on the notion of dual quadrilaterals: two planar quadrilaterals are called dual, if their corresponding sides are parallel, and their non-corresponding diagonals are parallel.…
Let be a compact 3-manifold with a triangulation . We give an inequality relating the Euler characteristic of a surface normally embedded in with the number of normal quadrilaterals in . This gives a relation between a topological invariant of the surface and a quantity derived from its combinatorial …
Normal and almost normal surfaces are essential tools for algorithmic 3-manifold topology, but to use them requires exponentially slow enumeration algorithms in a high-dimensional vector space. The quadrilateral coordinates of Tollefson alleviate this problem considerably for normal surfaces, by reducing the dimension …
Study examines Hilbert area of inscribed polygons in projective geometry.
The enumeration of normal surfaces is a crucial but very slow operation in algorithmic 3-manifold topology. At the heart of this operation is a polytope vertex enumeration in a high-dimensional space (standard coordinates). Tollefson's Q-theory speeds up this operation by using a much smaller space (quadrilateral coord…
New relations for vertex polynomial in graphs of any degree.
Two-dimensional affine A-nets in 3-space are quadrilateral meshes that discretize surfaces parametrized along asymptotic lines. The characterizing property of A-nets is planarity of vertex stars, so for generic A-nets the elementary quadrilaterals are skew. We classify the simply connected affine A-nets that can be ext…
We study geometric consistency relations between angles on 3-dimensional (3D) circular quadrilateral lattices -- lattices whose faces are planar quadrilaterals inscribable into a circle. We show that these relations generate canonical transformations of a remarkable ``ultra-local'' Poisson bracket algebra defined on di…
The B-quadrilateral lattice (BQL) provides geometric interpretation of Miwa's discrete BKP equation within the quadrialteral lattice (QL) theory. After discussing the projective-geometric properties of the lattice we give the algebro-geometric construction of the BQL ephasizing the role of Prym varieties and the corres…
Study on random triangles and quadrilaterals in plane geometry.
We determine all the Q-fundamental surfaces in -lens spaces and -lens spaces with respect to natural triangulations with tetrahedra. For general -lens spaces, we give an upper bound for elements of vectors which represent Q-fundamental surfaces with no quadrilateral normal disks disjoint from t…
Study proves spectral determination of triangles and quadrilaterals, with restrictions on higher-order polygons.
While conformal transformations of the plane preserve Laplace's equation, Lorentz-conformal mappings preserve the wave equation. We discover how simple geometric objects, such as quadrilaterals and pairs of crossing curves, are transformed under nonlinear Lorentz-conformal mappings. Squares are transformed into curvili…
The paper explores discrete isothermic nets using checkerboard patterns in quadrilateral nets.
IC-nets on confocal conics found in grid-like straight lines.
We provide an explicit resolution of the Abreu equation on convex labeled quadrilaterals. This confirms a conjecture of Donaldson in this particular case and implies a complete classification of the explicit toric Kähler-Einstein and toric Sasaki-Einstein metrics constructed in [6,22,14]. As a byproduct, we obtain a we…
Discrete analogues of ellipsoids with preserved circular cross sections.
Characterizes discrete nets with characteristic properties on larger parameter rectangles.
Approximates smooth surfaces using Laguerre geometry meshes.
Approximates surfaces using Laguerre geometry with spherical faces.
Paper finds optimal shapes for minimizing average lengths of billiard trajectories in specific polygons.
The curve shortening flow transforms figure-eight curves into bowties.
We give a parametrization to the asymptotic Teichmuller space of the open unit disk through equivalent classes of shear functions induced by quasisymmetric homeomorphisms on the Farey tesselation of the unit disk. Then using the parametrization, we define a new metric on the asymptotic Teichmuller space. Two other rela…
We show that a topologically minimal disk in a tetrahedron with index is either a normal triangle, a normal quadrilateral, or a normal helicoid with boundary length 4(n+1). This mirrors geometric results of Colding and Minicozzi.
New surgery exact triangles in Heegaard Floer homology for rational slopes.
A new lower bound on the complexity of a 3-manifold is given using the Z2-Thurston norm. This bound is shown to be sharp, and the minimal triangulations realising it are characterised using normal surfaces consisting entirely of quadrilateral discs.
The asymptotic lattices and their transformations are studied within the line geometry approach. It is shown that the discrete asymptotic nets are represented by isotropic congruences in the Plucker quadric. On the basis of the Lelieuvre-type representation of asymptotic lattices and of the discrete analog of the Mouta…
A Margulis spacetime is a complete affine 3-manifold M with nonsolvable fundamental group. Associated to every Margulis spacetime is a noncompact complete hyperbolic surface S. We show that every Margulis spacetime is orientable, even though S may be nonorientable. We classify Margulis spacetimes when S is homeomorphic…
Study classifies graphs with positive curvature without quadrilaterals.
Kernel testing compares cell states in single-cell data.