Given a finite simplicial complex L and a collection of pairs of spaces indexed by its vertex set, one can define their polyhedral product. We record a simple formula for its Euler characteristic. In special cases the formula simplifies further to one involving the h-polynomial of L.
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48 results for “polyhedral product”
Golod and tight 3-manifoldsmath.CO
3-manifold triangulations are Golod and tight, proven through a topological characterization.
problem Understanding Golodness and tightness in 3-manifold triangulations.
method Topological characterization of a polyhedral product for a tight-neighborly manifold triangulation.
result Golodness and tightness are equivalent for 3-manifold triangulations.
The Cartesian subgroup in graph products of groups is studied with bounds and algorithms.
problem Understanding the structure of Cartesian subgroups in graph products of groups.
method Theory of polyhedral products, lower and upper bounds, algorithm for small presentations.
result Bounds on the number of relations and deficiency in presentations of Cartesian groups.
The "polyhedral product functor" produces a space from a simplicial complex L and a collection of pairs of spaces, {(A(i),B(i))}, where i ranges over the vertex set of L. We give necessary and sufficient conditions for the resulting space to be aspherical. There are two similar constructions, each of which starts with …
The Polyhedral Tree Complexmath.CO
Tree complex linked to polyhedral shapes like associahedra and cyclohedra.
problem Understanding the structure of mapping class groups and complex dynamics.
method Characterizing associahedra and cyclohedra using planar tree embeddings and barycentric subdivision.
result Tree complex is a barycentric subdivision of a polyhedral cell complex made of associahedra and cyclohedra.
New oriented matroids from simplex triangulations.
problem Constructing oriented matroids from triangulations.
method Using polyhedral matching fields and matroids over hyperfields.
result Generalized construction of matroids over hyperfields.
Paper proposes a method to estimate consumer valuations from bundle sales data.
problem Estimating consumer valuations from bundle sales data using classical methods is challenging.
method Proposes an approach using EM algorithm and Monte Carlo simulation to estimate consumer valuations from bundle sales data.
result The approach can recover the distribution of consumers' valuations and is robust to unobserved no-purchases and clustered market segments.
Study on discrete Gaussian curvature for polyhedral surfaces.
problem Discretization of Gaussian curvature for polyhedral surfaces.
method Generalization of discrete conformal equivalence to define discrete Gaussian curvature and classify polyhedral surfaces.
result Existence of polyhedral surfaces with constant discrete Gaussian curvature in every discrete conformal class.
Smooth polyhedral surfacesmath.MG
Polyhedral surfaces are fundamental objects in architectural geometry and industrial design. Whereas closeness of a given mesh to a smooth reference surface and its suitability for numerical simulations were already studied extensively, the aim of our work is to find and to discuss suitable assessments of smoothness of…
A discrete conformality for hyperbolic polyhedral surfaces is introduced in this paper. This discrete conformality is shown to be computable. It is proved that each hyperbolic polyhedral metric on a closed surface is discrete conformal to a unique hyperbolic polyhedral metric with a given discrete curvature satisfying …
The paper proves rigidity for submanifolds in warped product manifolds.
problem Rigidity of submanifolds in warped product manifolds.
method Dihedral extremality and rigidity theorem for submanifolds with polyhedral boundary.
result Dihedral rigidity results for hyperbolic polyhedra in flat warped product spaces.
Polyhedral semantics for intermediate logics; Nerve Criterion ensures completeness.
problem Characterize polyhedrally-complete intermediate logics.
method Developed Nerve Criterion to characterize polyhedrally-complete logics combinatorially.
result Nerve Criterion provides a necessary and sufficient condition for polyhedrally-completeness.
Locally finite complexes with polyhedral metrics are arborescent.
problem Characterizing locally finite complexes with metrics.
method Proving arborescence for complexes with polyhedral metrics.
result Locally finite complexes with polyhedral metrics are arborescent.
The paper proves rigidity of bordered polyhedral surfaces using variational principles.
problem Determining the rigidity of bordered polyhedral surfaces.
method Using the variational principle, the paper shows that bordered polyhedral surfaces are determined by boundary values and discrete curvatures on interior edges.
result The paper re-proves the classical result that two Euclidean or hyperbolic cyclic polygons are congruent if their side lengths are equal.
Differential chains are a proper subspace of de Rham currents given as an inductive limit of Banach spaces endowed with a geometrically defined strong topology. Boundary is a continuous operator, as are operators that dualize to Hodge star, Lie derivative, pullback and interior product. Partitions of unity exist in thi…
We develop a method to find a set of diminimal polyhedral maps on the torus from which all other polyhedral maps on the torus may be generated by face splitting and vertex splitting. We employ this method, though not to its completion, to find 53 diminimal polyhedral maps on the Torus.
Characterizes metrics on triangulated surfaces using glued Euclidean triangles.
problem Describing metrics on triangulated surfaces constructed from glued Euclidean triangles.
method Carefully constructing polyhedral metrics and proving their uniqueness.
result Polyhedral metrics are the only intrinsic metrics preserving Euclidean triangle lengths.
We study hyperbolic polyhedral surfaces with faces isometric to regular hyperbolic polygons satisfying that the total angles at vertices are at least The combinatorial information of these surfaces is shown to be identified with that of Euclidean polyhedral surfaces with negative combinatorial curvature everywher…
A discrete conformality for polyhedral metrics on surfaces is introduced in this paper which generalizes earlier work on the subject. It is shown that each polyhedral metric on a surface is discrete conformal to a constant curvature polyhedral metric which is unique up to scaling. Furthermore, the constant curvature me…
A polyhedral map is called -equivelar if each face has edges and each vertex belongs to faces. In 1983, it was shown that there exist infinitely many geometrically realizable -equivelar polyhedral maps if , or . It was shown in 2001 that there exist infi…
Smooth 2-tori in R^4 can be approximated by polyhedral Lagrangian or isotropic tori.
problem Approximating smooth 2-tori in high-dimensional spaces.
method Polyhedral approximation using Lagrangian and isotropic tori.
result Smooth 2-tori can be approximated by polyhedral Lagrangian or isotropic tori in C0 or C1 sense.
New research limits how deep neural networks can be for certain functions.
problem Understanding the depth required for neural networks to represent specific functions.
method Mixed-integer optimization, polyhedral theory, tropical geometry.
result Neural networks with more than one layer are necessary to represent certain functions.
Discrete curvaturemath.DG
Survey on discrete curvature concepts for polygons and polyhedral surfaces.
problem Defining curvature for discrete structures like polygons and polyhedral surfaces.
method Explains curvature notions for polygons, polyhedral surfaces, and abstract polyhedral manifolds.
result Discrete curvature theorems parallel classical theorems in differential geometry.
We show that area minimizing polyhedral surfaces are saddle.
The study provides conditions for approximating Riemannian manifolds with polyhedral metrics.
problem Approximating Riemannian manifolds with polyhedral metrics.
method Conditions on curvature tensors for Lipschitz and local polyhedral approximations.
result Conditions are sufficient for local polyhedral approximations, conjectured to be sufficient for global approximations.
Study calculates Floer homology for binary polyhedral spaces.
problem Calculating Floer homology for specific polyhedral spaces.
method Equivariant instanton Floer homology, modified algebraic construction.
result Equivariant instanton Floer homology values for binary polyhedral spaces.
We study rigidity of polyhedral surfaces and the moduli space of polyhedral surfaces using variational principles. Curvature like quantities for polyhedral surfaces are introduced. Many of them are shown to determine the polyhedral metric up to isometry. The action functionals in the variational approaches are derived …
This paper investigates several global rigidity issues for polyhedral surfaces including inversive distance circle packings. Inversive distance circle packings are polyhedral surfaces introduced by P. Bowers and K. Stephenson as a generalization of Andreev-Thurston's circle packing. They conjectured that inversive dist…
New bounds show polyhedral surrogates are optimal for generalization.
problem Proving generalization rates for polyhedral loss functions.
method Developed two general results for polyhedral surrogates.
result Polyhedral surrogates provide linear surrogate regret bounds, translating directly to target rates.
New index theory proves Gromov's dihedral conjectures.
problem Comparisons and rigidity of scalar curvatures, mean curvatures, and dihedral angles.
method Developed a new index theory for manifolds with polyhedral boundary.
result Proved Gromov's dihedral extremality and rigidity conjectures.
Constructs a moment map flow for isotropic maps on surfaces.
problem Understanding isotropic maps on surfaces and their properties.
method Develops a Kähler moment map geometry and a modified moment map flow.
result Polyhedral modified moment map flow induces a strong deformation retraction.
This work describes compactifications of metric spaces and vector spaces using asymmetric norms.
problem Compactifying metric spaces and vector spaces using asymmetric norms.
method Nonstandard methods, ultrapowers of the spaces at hand.
result Polyhedral compactifications of vector spaces with stratified structure.
We investigate the rigidity of hyperbolic cone metrics on -manifolds which are isometric gluing of ideal and hyper-ideal tetrahedra in hyperbolic spaces. These metrics will be called ideal and hyper-ideal hyperbolic polyhedral metrics. It is shown that a hyper-ideal hyperbolic polyhedral metric is determined up to i…
Researchers decompose hyperbolic n-manifolds with totally geodesic boundaries into polyhedral cells.
problem Decomposing hyperbolic n-manifolds with totally geodesic boundaries into polyhedral cells.
method Two different approaches to demonstrate the existence of polyhedral decompositions.
result The number of polyhedral decompositions of is finite.
Polyhedral Kahler Manifoldsmath.DG
In this article we introduce the notion of Polyhedral Kahler manifolds, even dimensional polyhedral manifolds with unitary holonomy. We concentrate on the 4-dimensional case, prove that such manifolds are smooth complex surfaces, and classify the singularities of the metric. The singularities form a divisor and the res…
The study connects polyhedral manifolds to Riemannian ones with geometric bounds.
problem Connecting polyhedral manifolds to Riemannian manifolds with geometric constraints.
method Using a theorem by C. Lange and B. Bowditch, the study bounds the curvature and injectivity radius of Riemannian manifolds.
result Polyhedral manifolds with bounded geometry are bi-Lipschitz homeomorphic to Riemannian manifolds with controlled curvature and injectivity radius.
We show that a compact length space is polyhedral if a small spherical neighborhood of any point is conic.
Soft tilingsmath.MG
Softens tilings in 3D space, proving conjectures about polyhedral tilings.
problem Proving that all locally polyhedral tilings in 3D space can be softened.
method Developed a new edge-bending algorithm to prove the statement.
result Proved conjectures about polyhedral tilings in 3D space and the plane.
Study shows non-polyhedral structure in moduli spaces for n≥8.
problem Identifying non-polyhedral structure in moduli spaces of pointed stable curves.
method Constructing an extremal non-polyhedral ray via maps on meromorphic strata of differentials.
result Moduli spaces are not Mori Dream Spaces for n≥8.
Polyhedral Kähler metrics on math.DG
Conditions for polyhedral Kähler metrics on CP^n with specific singularities.
problem Existence of polyhedral Kähler metrics on complex projective space with specified singularities.
method Parabolic Kobayashi-Hitchin correspondence, linear and quadratic constraints on cone angles.
result Necessary and sufficient conditions for the existence of polyhedral Kähler metrics on CP^n.
Study circle patterns and polyhedral surfaces in hyperbolic ends, proving manifold properties.
problem Understanding the space of complex projective structures on surfaces with circle patterns.
method Analyzing ideal polyhedral surfaces in hyperbolic ends, proving manifold properties and Lagrangian immersions.
result The space of complex projective structures on surfaces with circle patterns is a manifold of dimension 6g-6.
Polyhedral surfaces in flat (2+1)-spacetimes and balanced cellulations on hyperbolic surfacesmath.MG
Flat metrics on hyperbolic surfaces embed as polyhedral surfaces in (2+1)-spacetimes.
problem Embedding flat metrics on hyperbolic surfaces into (2+1)-spacetimes.
method Using convex polyhedral Cauchy surfaces and Teichmüller space properties.
result Existence and uniqueness of flat metrics embedding in (2+1)-spacetimes.
Study shows Bergman metric is non-Einstein for certain domains.
problem Characterizing the Bergman metric of specific domains.
method Analyzing pseudoconvex domains with strongly pseudoconvex polyhedral boundaries.
result Bergman metric is not Einstein for the studied domains.
We study the rigidity of polyhedral surfaces using variational principle. The action functionals are derived from the cosine laws. The main focus of this paper is on the cosine law for a non-triangular region bounded by three possibly disjoint geodesics. Several of these cosine laws were first discovered and used by Fe…
We present and apply a method for disproving the existence of polyhedral immersions in of certain triangulations on non-orientable surfaces. In particular, it is proved that neither of the two vertex-minimal, neighborly 9-vertex triangulations of the non-orientable surface of genus 5 are realizable as im…
We show that 3-dimensional polyhedral manifolds with nonnegative curvature in the sense of Alexandrov can be approximated by nonnegatively curved 3-dimensional Riemannian manifolds.
A hyperbolic conjugacy class in the modular group PSL(2,Z) corresponds to a closed geodesic in the modular orbifold. Some of these geodesics virtually bound immersed surfaces, and some do not; the distinction is related to the polyhedral structure in the unit ball of the stable commutator length norm. We prove the foll…
The paper introduces combinatorial curvature and flow for polyhedral surfaces, proving rigidity and solving the Yamabe problem.
problem Discrete conformal structures on polyhedral surfaces and their rigidity.
method Parameterized combinatorial curvature, combinatorial α-Ricci flow, and flow extension through singularities.
result Existence and convergence of combinatorial α-Ricci flow for solving the Yamabe problem.