The paper extends Gelfand-Kapranov-Zelevinsky construction to hyperbolic Riemann surfaces with punctures.
problem Stratifying the space of weight vectors for hyperbolic Riemann surfaces with punctures.
method Analogous to Gelfand-Kapranov-Zelevinsky construction, associates polyhedral fans to hyperbolic Riemann surfaces with punctures.
result The secondary fan of a hyperbolic Riemann surface with punctures is the normal fan of a convex polyhedron, the secondary polyhedron.
We are generalizing to higher dimensions the Bavard-Ghys construction of the hyperbolic metric on the space of polygons with fixed directions of edges. The space of convex d-dimensional polyhedra with fixed directions of facet normals has a decomposition into type cones that correspond to different combinatorial types …
Moduli spaces of doubly periodic monopoles, also called monopole walls or monowalls, are hyperkähler; thus, when four-dimensional, they are self-dual gravitational instantons. We find all monowalls with lowest number of moduli. Their moduli spaces can be identified, on the one hand, with Coulomb branches of five-dimens…
Löbell polyhedra have small systoles and are quasi-arithmetic.
problem Finding compact hyperbolic polyhedra with small systoles.
method Elementary and conceptual means to observe systole behavior, number theoretic invariants to refine results.
result Löbell polyhedra give examples of closed hyperbolic 3-manifolds with arbitrarily small systole and are quasi-arithmetic.
Introduces Coxeter polyhedra in various geometries.
problem None explicitly stated, focuses on introducing concepts.
method Introduction of Coxeter polyhedra in spherical, Euclidean, and hyperbolic geometries.
result Fundamental theorems and descriptions of polyhedra and tessellations.
New hyperbolic polyhedra with π/3 angles and volumes calculated.
problem Finding new hyperbolic polyhedra with specific dihedral angles.
method Constructed a new sequence of hyperbolic polyhedra with π/3 angles and determined their volumes. result Volumes of some constructed polyhedra determined.
The paper sets new limits on hyperbolic polyhedra volumes.
problem Finding upper bounds on volumes of hyperbolic polyhedra.
method Analyzes three types of polyhedra: ideal, compact with finite vertices, and finite volume with mixed vertices.
result Establishes new upper bounds for polyhedra volumes in hyperbolic space.
Study of polyhedra on a sphere in projective 3-space.
problem Characterizing polyhedra with vertices on a sphere.
method Purely combinatorial and linear programming approaches.
result Characterization of dihedral angles and hyperbolic-de Sitter structure.
Study mutant pairs of hyperbolic polyhedra, focusing on commensurability.
problem Determine commensurability of mutant pairs of hyperbolic polyhedra.
method Introduce mutation concept, develop new techniques for non-cusped polyhedra.
result New techniques needed for studying mutant pairs of polyhedra.
The study sets limits on dihedral angles of large hyperbolic polyhedra.
problem Establishing bounds on dihedral angles of hyperbolic Coxeter polyhedra.
method Developed a constructive procedure for Coxeter polyhedra with prescribed dihedral angles.
result Classification of ADEG-polyhedra with specific dihedral angles and no disjoint facets.
Easy condition for local k-connectedness in inverse limits of polyhedra.
problem Local k-connectedness of inverse limits of polyhedra.
method Easy condition for local k-connectedness.
result Provided an easily verifiable condition.
The study finds upper bounds and computes volumes of ideal right-angled polyhedra in Lobachevsky space.
problem Finding upper bounds and computing volumes of ideal right-angled polyhedra in Lobachevsky space.
method Analyzing a class of right-angled polyhedra with vertices on the absolute, obtaining upper bounds on volumes, computing volumes for polyhedra with up to 23 faces, and introducing the class of polyhedra with isolated triangles.
result Minimum volumes are realized on antiprisms and twisted antiprisms, and the first 248 values of volumes are presented.
Improved volume estimates for right-angled polyhedra in hyperbolic space.
problem Estimating volumes of right-angled polyhedra in hyperbolic space.
method Combining Andreev theorem and Atkinson's results, improved upper volume estimates.
result Upper volume estimates for both compact and ideal right-angled polyhedra improved.
We review several results related to the characterization of polyhedra in hyperbolic 3-space. In particular we present Rivin's theorem that gives a characterization of compact convex hyperbolic polyhedra, and Hodgson's proof of the Adreev's theorem. We also review the analogous characterization of ideal polyhedra, and …
Improved bounds on ideal vertices in right-angled hyperbolic polyhedra.
problem Finding bounds on ideal vertices in hyperbolic polyhedra.
method Improved Nikulin's inequality and Nonaka's lower bound.
result Shorter proofs and improved bounds on ideal vertices.
Study calculates mass of special polyhedra in hyperbolic space.
problem Evaluating mass in hyperbolic geometry.
method Used upper half space model and special polyhedra.
result Evaluated mass functional on polyhedra.
Mass in relativity linked to polyhedra geometry.
problem Mass in general relativity.
method Riemannian polyhedra geometry.
result Mass connected to polyhedra geometry.
Flexible models of non-flexible polyhedra explained.
problem Non-flexible Siamese dipyramids behave like flexible ones.
method Simple mathematical method to explain model flexibility.
result Physical models of Siamese dipyramids are flexible.
Polyhedra volume conjecture supports Stoker conjecture weakly.
problem Proving the Stoker conjecture for polyhedra.
method Using an extension of Montcouquiol and Weiss' result on polyhedra angles as local coordinates.
result Volume Conjecture for polyhedra implies a weak version of the Stoker conjecture.
This article defines a pair of combinatorial operations on the combinatorial structure of compact right-angled hyperbolic polyhedra in dimension three called decomposition and edge surgery. It is shown that these operations simplify the combinatorics of such a polyhedron, while keeping it within the class of right-angl…
Intermediate logic of all convex polyhedra is axiomatized.
problem Defining and axiomatizing intermediate logic for convex polyhedra.
method Using Jankov-Fine formulas, classical polyhedral geometry, and p-morphic images to establish completeness.
result A finite axiomatisation of PL for all convex polyhedra.
Study approximates Riemannian manifolds using polyhedra.
problem Understanding Tullio Regge's approximation theorem.
method Proof of Regge theorem using polyhedra approximation.
result Integral of scalar curvature approximated by polyhedral curvature.
Analyzes 2D polyhedra decompositions into PL collapsible subpolyhedra.
problem Characterizing 2D polyhedra that can be split into two PL collapsible parts.
method Examines the minimum size of PL collapsible covers and uses topological and geometric analysis.
result Special class of 2D polyhedra appears naturally, including all closed surfaces and complexes from one-relator presentations.
New families of hyperbolic polyhedra yield infinitely many unique reflection groups.
problem Understanding commensurability classes of compact Coxeter polyhedra in hyperbolic spaces.
method Analyzing families of compact Coxeter polyhedra constructed by Makarov.
result Proves infinitely many commensurability classes in 4- and 5-dimensional hyperbolic spaces.
The paper solves three problems related to monostable polyhedra.
problem Three problems related to monostable polyhedra posed by Conway and Goldberg.
method General theorem describing approximations of smooth convex bodies by convex polyhedra in terms of static equilibrium points.
result Existence of a convex polyhedron with only one stable and one unstable point.
New invariant from links to polyhedra volumes.
problem Computing hyperbolic volumes of link complements.
method Geometric, topological, and combinatorial methods to decompose link complements into ideal polyhedra.
result A new geometric link invariant, the right-angled volume, is a lower bound for hyperbolic volume.
Improved algorithm learns convex polyhedra with margin.
problem Learning convex polyhedra with margin in the realizable PAC setting.
method Constructs a consistent polyhedron as an intersection of halfspaces with constant-size margins.
result Polyhedra can be learned efficiently with margin constraints.
New framework for folding polyhedra into two planes.
problem Creating polyhedra that can be folded into two planes.
method General framework for bifoldable polyhedra construction.
result Examples of infinite triply periodic and fractal bifoldable polyhedra.
In this article we establish the relation between the spines of 3-manifolds and the polyhedra with identified faces. We do this by showing that the spines of the closed, connected, orientable 3-manifolds can be presented through polyhedra with identified faces in a very natural way. We also prove the equivalence betwee…
Machine learning improves RNA secondary structure prediction.
problem Stagnant performance of RNA secondary structure prediction methods.
method Machine learning, especially deep learning, is used to predict RNA secondary structures.
result Machine learning methods have improved the prediction of RNA secondary structures.
Paper solves long-standing problem of infinite ideal polyhedra in hyperbolic space.
problem Characterize infinite ideal polyhedra in hyperbolic 3-space.
method Introduced combinatorial Ricci flow for infinite ideal circle patterns.
result Proved characterization of infinite ideal circle patterns under specific conditions.
Mass in relativity linked to polyhedra geometry.
problem Understanding ADM mass in general relativity.
method Relating ADM mass to the total mean curvature and defect of dihedral angles of Riemannian polyhedra.
result Expressed n-dimensional mass as an integral of geometric quantities. We consider a natural question: "Is it true that each homotopy domination of a polyhedron over itself is a homotopy equivalence?" and a strongly related problem of K. Borsuk (1967): "Is it true that two ANR's homotopy dominating each other have the same homotopy type?" The answer was earlier known to be positive for ma…
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.
An algorithm for determining the list of smallest volume right-angled hyperbolic polyhedra in dimension 3 is described. This algorithm has been implemented on computer using the program Orb to compute volumes, and the first 825 polyhedra in the list have been determined.
Study geodesics on spherical polyhedra, estimating their number.
problem Counting simple closed geodesics on spherical polyhedra.
method Examined regular spherical octahedra, cubes, and tetrahedra.
result Estimated the number of simple closed geodesics on spherical polyhedra.
Study of hyperideal polyhedra in anti-de Sitter space.
problem Characterizing hyperideal polyhedra in anti-de Sitter space.
method Defined hyperideal polyhedra as intersections with convex polyhedra in projective model of anti-de Sitter space.
result Hyperideal polyhedra uniquely determined by combinatorics, dihedral angles, and induced metrics on boundary.
Paper finds knots with vanishing Upsilon but non-trivial secondary Upsilon.
problem Understanding the Upsilon invariant and its secondary version.
method Constructing infinite families of knots and proving conjectures.
result Secondary Upsilon invariant can be non-trivial even if Upsilon is zero.
Software finds ideal polyhedra with rational dihedral angles and volume maxima.
problem Finding ideal convex polyhedra with maximal volume in hyperbolic 3-space.
method Rivin's variational characterization and combinatorial optimization algorithms.
result Maximal volume ideal polyhedra have dihedral angles that are rational multiples of π.
The paper proves rigidity and uniformization theorems for infinite circle patterns and convex polyhedra in hyperbolic 3-space.
problem Characterize infinite circle patterns and convex polyhedra in hyperbolic 3-space.
method Extends techniques from previous work to prove rigidity and uniformization theorems for infinite circle patterns and convex polyhedra.
result Establishes existence and rigidity of infinite regular circle patterns and convex trivalent polyhedra.
In [6], Kellerhals and Perren conjectured that the growth rates of the reflection groups given by hyperbolic Coxeter polyhedra are always Perron numbers. We prove that this conjecture is always true for the case of ideal Coxeter polyhedra in H3. We also find out the ideal Coxeter polyhedron in $\mathbb{H}^3…
Polyhedra rigidity theorem in hyperbolic space proved.
problem Dihedral rigidity conjecture in hyperbolic 3-space.
method Comparison theorem for polyhedra in a 3-manifold with scalar curvature bounded below.
result Confirms Gromov dihedral rigidity conjecture in hyperbolic 3-space.
New bounds found for vertices of hyperbolic polyhedra in dimensions 5 to 12.
problem Determining minimum number of ideal and finite vertices in hyperbolic polyhedra.
method Geometric method of orthogonal gluings combined with double counting and recurrence relations.
result Improved lower bounds for vertices in all dimensions up to 12.
We prove that there are thirteen Archimedean/semiregular polyhedra by using Euler's polyhedral formula.
We define the injectivity radius of a Coxeter polyhedron in H^3 to be half the shortest translation length among hyperbolic/loxodromic elements in the orientation-preserving reflection group. We show that, for finite-volume polyhedra, this number is always less than 2.6339..., and for compact polyhedra it is always les…
The paper classifies 10 antipodal pairings of self-dual maps.
problem Understanding the antipodal pairings of strongly involutive polyhedra.
method Classification of self-dual pairings and construction of polyhedra.
result Determination of 10 antipodal pairings among 24 self-dual pairings.
In this paper, we prove the existence of energy minimizers in each free homotopy class of maps between polyhedra with target space without focal points. Our proof involves a careful study of some geometric properties of riemannian polhyedra without focal points. Among other things, we show that on the relevant polyhedr…
Classical H.Minkowski theorems on existence and uniqueness of convex polyhedra with prescribed directions and areas of faces as well as the well-known generalization of H.Minkowski uniqueness theorem due to A.D.Alexandrov are extended to a class of nonconvex polyhedra which are called polyhedral herissons and may be de…