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 study classifies tilings of the sphere by congruent quadrilaterals.
Paper introduces quadrilateral labyrinth fractals and their properties.
Formulae quantify gaps in geodesic quadrilaterals on manifolds.
Spherical quadrilaterals classified based on geometric properties.
New method characterizes surface quadrilateral layouts as special immersions.
We prove that any cyclic quadrilateral can be inscribed in any closed convex -curve. The smoothness condition is not required if the quadrilateral is a rectangle.
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 Specifically, we give a proof of an (unpublished) observation independently given by Casson and Rubinstein saying t…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 paper classifies surfaces formed by quadrilateral gluings.
We investigate the relationship between stability and the existence of extremal Kähler metrics on certain toric surfaces. In particular, we consider how log stability depends on weights for toric surfaces whose moment polytope is a quadrilateral. We introduce a space of symplectic potentials for toric manifolds, which …
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.
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…
We give two proofs that the 3-torus is not weakly d-congruent to the connected sum of three S^1xS^2's, if d>2. We study how cohomology ring structure relates to weak congruence. We give an example of three 3--manifolds which are weakly 5-congruent but are not 5-congruent.
The tilings of the 2-dimensional sphere by congruent triangles have been extensively studied, and the edge-to-edge tilings have been completely classified. However, not much is known about the tilings by other congruent polygons. In this paper, we classify the simplest case, which is the edge-to-edge tilings of the 2-d…
Our aim in this paper is to provide a theory of discrete Riemann surfaces based on quadrilateral cellular decompositions of Riemann surfaces together with their complex structure encoded by complex weights. Previous work, in particular of Mercat, mainly focused on real weights corresponding to quadrilateral cells havin…
Study finds finitely many non-congruent polygonal domains with same Steklov spectrum.
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…
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…
In this paper, we study strictly convex affine hypersurfaces centroaffinely congruent to their centre map, in the case when the shape operator has two distinct eigenvalues: one of multiplicity 1, and one nonzero of multiplicity n-1. We show how to construct them from (n-1)-dimensional affine hyperspheres.
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…
New findings on non-congruent curves with identical signatures.
The paper explores discrete isothermic nets using checkerboard patterns in quadrilateral nets.
Colored HOMFLY-PT invariant, the generalization of the colored Jones polynomial, is one of the most important quantum invariants of links. This paper is devoted to investigating the basic structures of the colored HOMFLY-PT invariants of links. By using the HOMFLY-PT skein theory, firstly, we show that the (reformulate…
ccc-Autoevolutes are closed curves congruent to their evolutes, constructed via symmetry.
We investigate the common underlying discrete structures for various smooth and discrete nets. The main idea is to impose the characteristic properties of the nets not only on elementary quadrilaterals but also on larger parameter rectangles. For discrete planar quadrilateral nets, circular nets, -nets and conical…
We investigate the duality between minimal surfaces in Euclidean space and maximal surfaces in Lorentz-Minkowski space in the family of rotational surfaces. We study if the dual surfaces of two congruent rotational minimal (or maximal) surfaces are congruent. We show that in the duality process by means of a one-parame…
Let K be a knot in the 3-sphere with 2-fold branched covering space M. If for some prime p congruent to 3 mod 4 the p-torsion in the first homology of M is cyclic with odd exponent, then K is of infinite order in the knot concordance group. As one application, recall that the n-twisted double of an arbitrary knot has o…
Uniformly elliptic Weingarten spheres in S2xR are congruent to a canonical example.
We consider the problem of finding the probability that a random triangle is obtuse, which was first raised by Lewis Caroll. Our investigation leads us to a natural correspondence between plane polygons and the Grassmann manifold of 2-planes in real -space proposed by Allen Knutson and Jean-Claude Hausmann. This cor…
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.
4-ball can be tiled with knotted surfaces.
Approximates smooth surfaces using Laguerre geometry meshes.
Approximates surfaces using Laguerre geometry with spherical faces.
Virtual knots can be transformed by -moves, affecting their writhes.
Conway-Gordon proved that for every spatial complete graph on 6 vertices, the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2, and for every spatial complete graph on 7 vertices, the sum of the Arf invariants over all of the Hamiltonian knots is also congruent to 1 mo…
Paper finds optimal shapes for minimizing average lengths of billiard trajectories in specific polygons.
Basket links are shown to be isotopic to .