Characterizes billiard and quasigeodesic flows in polyhedral convex bodies.
problem Characterizing billiard and quasigeodesic flows in polyhedral convex bodies.
method Alexandrov geometry methods.
result Optimal regularity result for convex bodies: billiard dynamics is continuous if boundary is of class C2,1. Paper constructs two series of Lorentz bi-quotients from polyhedra.
problem Creating fundamental domains for Lorentz bi-quotients.
method Explicit construction of polyhedral fundamental domains.
result Two infinite series of Lorentz bi-quotients constructed.
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.
The paper defines Dirichlet domains for Anosov subgroups in Lie groups.
problem Finding finite-sided Dirichlet domains for Anosov subgroups in Lie groups.
method Introducing a sufficient condition for finite-sided Dirichlet domains in semisimple Lie groups for polyhedral Finsler metrics.
result Finitely generated subgroups of semisimple Lie groups can have finite-sided Dirichlet domains under certain conditions.
We show that the notion of 3-hyperconvexity on oriented flag manifolds defines a partial cyclic order. Using the notion of interval given by this partial cyclic order, we construct Schottky groups and show that they correspond to images of positive representations in the sense of Fock and Goncharov. We construct poly…
Proves smoothness of minimal surfaces near polyhedral boundaries.
problem Smoothness of free-boundary minimal surfaces near polyhedral domains.
method Allard-type regularity theorem for minimal surfaces in convex polyhedra.
result Minimal surfaces are C1,α graphical over a free-boundary plane if close to it. 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…
We investigate subgroups of SL (n,Z) which preserve an open nondegenerate convex cone in real n-space and admit in that cone as fundamental domain a polyhedral cone of which some faces are allowed to lie on the boundary. Examples are arithmetic groups acting on selfdual cones, Weyl groups of certain Kac-Moody algebras …
We classify the 3-dimensional hyperbolic polyhedral orbifolds that contain no embedded essential 2-suborbifolds, up to decomposition along embedded hyperbolic triangle orbifolds (turnovers). We give a necessary condition for a 3-dimensional hyperbolic polyhedral orbifold to contain an immersed (singular) hyperbolic tur…
We prove that, both in the hyperbolic and spherical 3-spaces, there exist nonconvex compact boundary-free polyhedral surfaces without selfintersections which admit nontrivial continuous deformations preserving all dihedral angles and study properties of such polyhedral surfaces. In particular, we prove that the volume …
The paper proves convergence of discrete maps to Riemann mappings for polyhedral surfaces.
problem Discrete conformal geometry of polyhedral surfaces.
method Establishing rigidity for hexagonal triangulations and estimating quasiconformal constants.
result Discrete conformal maps converge to Riemann mappings for Jordan domains.
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.
Computational knot theory and 3-manifold topology have seen significant breakthroughs in recent years, despite the fact that many key algorithms have complexity bounds that are exponential or greater. In this setting, experimentation is essential for understanding the limits of practicality, as well as for gauging the …
New periodic polyhedra found in curved spaces.
problem Existence of periodic polyhedra in curved spaces.
method Using Archimedean solids and transformations, constructing polyhedra with specific properties.
result Existence of compact polyhedral surfaces in spaceforms.
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.
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 …
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 CAT(0) metrics are arborescent.
problem Characterizing locally finite complexes with CAT(0) metrics. method Proving arborescence for complexes with polyhedral CAT(0) metrics. result Locally finite complexes with polyhedral CAT(0) 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.
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.
Polyhedra can mimic constant curvature surfaces, even with self-intersections.
problem Understanding curvature constraints in discrete vs. smooth settings.
method Constructive proof showing any surface can be realized as a polyhedral surface with uniform angular defect.
result Closed surfaces can be realized as polyhedral surfaces with constant angular defect.
We study hyperbolic polyhedral surfaces with faces isometric to regular hyperbolic polygons satisfying that the total angles at vertices are at least 2π. The combinatorial information of these surfaces is shown to be identified with that of Euclidean polyhedral surfaces with negative combinatorial curvature everywher…
Constructs symplectic structures from rational functions on fans.
problem Creating symplectic structures from rational functions on fans.
method Constructs exact symplectic structures and polyhedral Hamiltonians.
result Level sets of polyhedral Hamiltonians are hypersurfaces of contact type.
Finite element method approximates scalar curvature in arbitrary dimensions.
problem Approximating scalar curvature using finite elements in arbitrary dimensions.
method Piecewise polynomial interpolants of a smooth Riemannian metric on a triangulated polyhedral domain.
result Finite element interpolants converge to scalar curvature with rate O(hr+1) in H−2(Ω) norm. 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…
We prove that an open manifold M of dimension at least 5 which admits a complete CAT(0) polyhedral metric is pseudo-collarable, its fundamental group at infinity is strongly perfectly semistable and has vanishing Chapman-Siebenmann obstruction τ∞(M). Moreover, this implies that M is topologically collap…
A polyhedral map is called {p,q}-equivelar if each face has p edges and each vertex belongs to q faces. In 1983, it was shown that there exist infinitely many geometrically realizable {p,q}-equivelar polyhedral maps if q>p=4, p>q=4 or q−3>p=3. 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.
We study Atlas-type models of equity markets with local characteristics that depend on both name and rank, and in ways that induce a stable capital distribution. Ergodic properties and rankings of processes are examined with reference to the theory of reflected Brownian motions in polyhedral domains. In the context of …
Enhances neural network robustness with polyhedral envelope regularization.
problem Improving neural network robustness against adversarial attacks.
method Introduces polyhedral envelope regularization to bound the robustness region.
result Demonstrates improved robustness guarantees with minimal computational overhead.
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.
New insights into currents of Hitchin representations with combinatorial restrictions.
problem Understanding currents associated with Hitchin representations.
method Defining dual spaces and analyzing combinatorial restrictions on self-intersection.
result Dual spaces of discrete boundary currents are polyhedral complexes with dimension at most n-1.
Proposes a mixture of expert architecture for polyhedral classifiers.
problem Learning polyhedral classifiers with high accuracy.
method Uses an expectation maximization algorithm to learn parameters.
result Generalization bounds are derived and the method performs comparably to state-of-the-art approaches.
The paper approximates Einstein tensor using finite elements.
problem Approximating Einstein tensor for piecewise polynomial metrics.
method Finite element method applied to Riemannian metrics.
result Convergence rate of O(hr+1) in H−2(Ω)-norm. 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 develop a framework for consistent polyhedral surrogates in classification and prediction.
problem Designing consistent polyhedral surrogates for classification and prediction problems.
method Formalizing and studying embeddings of predictions as points in R^d, assigning original loss values, and convexifying to create surrogates.
result Established a strong connection between embeddings and polyhedral surrogates, providing constructions and proofs of consistency or inconsistency.
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 …
Non-rigidity degree of a lattice L, nrdL, is dimension of the L-type domain to which L belongs. We complete here the table of nrd's of all root lattices and their duals; namely, the hardest remaining case of Dn∗, and the case of E7∗ are decided. We describe explicitly the L-type domain D(Dn∗)…
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.
We study some basic analytic questions related to differential operators on Lie manifolds, which are manifolds whose large scale geometry can be described by a a Lie algebra of vector fields on a compactification. We extend to Lie manifolds several classical results on Sobolev spaces, elliptic regularity, and mapping p…
Study of Green function and Laplacians on polyhedral surfaces, focusing on genus two with a conical point.
problem Analyzing the behavior of Green function and self-adjoint Laplacians on polyhedral surfaces.
method Explicit construction of a basis in the kernel of the adjoint Laplacian, computation of S-matrix, study of various self-adjoint extensions.
result The behavior of the S-matrix at the zero value of the spectral parameter is sensitive to the geometry of the polyhedron.
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.