This study finds the shape of centrally symmetric octahedra with specific angles.
problem Finding the shape of centrally symmetric octahedra with prescribed cone-deficits.
method Inspired by Thurston's work, the paper studies the set of shapes of centrally symmetric octahedra with prescribed cone-deficits, showing it forms a real hyperbolic ideal tetrahedron.
result The set of shapes of centrally symmetric octahedra with prescribed cone-deficits forms a real hyperbolic ideal tetrahedron with dihedral angles half of the prescribed cone-deficits.
A knot complement admits a pseudo-hyperbolic structure by solving Thurston's gluing equations for an octahedral decomposition. It is known that a solution to these equations can be described in terms of region variables, also called w-variables. In this paper, we consider the case when pinched octahedra appear as a b…
We show existence of centrally symmetric maps on surfaces all of whose faces are quadrangles and pentagons for each orientable genus g≥0. We also show existence of centrally symmetric maps on surfaces all of whose faces are hexagons for each orientable genus g=2k−1, k∈N. We enumerate centrally …
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.
We give a rigorous geometric proof of the Murakami-Yano formula for the volume of a hyperbolic tetrahedron. In doing so, we are led to consider generalized hyperbolic tetrahedra, which are allowed to be non-convex, and have vertices `beyond infinity'; and we uncover a group, which we call 22.5K, of 23040 scissors-class…
We present constructions inspired by the Ma-Schlenker example of~\cite{Ma:2012hl} that show the non-rigidity of spherical inversive distance circle packings. In contrast to the use in~\cite{Ma:2012hl} of an infinitesimally flexible Euclidean polyhedron, embeddings in de Sitter space, and Pogorelov maps, our elementary …
Paper computes link determinants using Fourier-Hadamard transforms.
problem Computing determinants of complex link structures.
method Fourier-Hadamard transforms of Boolean functions.
result Determinant of centrally symmetric links with even components equals zero.
Geometric proof confirms link volume conjecture.
problem Volume conjecture for hyperbolic 3-manifolds.
method Topological and skein theoretic techniques.
result Complements of octahedral fully augmented links are isometric to fundamental shadow links.
The article considers the problem of existence and uniqueness of centrally symmetrical convex body for which the projection curvature radius function coincides with a given flag function. A necessary and sufficient condition is found that ensures a positive answer. An algorithm for construction the body in question is …
Polyhedral surfaces can be broken down into parallelograms.
problem Decomposing polyhedral surfaces into simpler shapes.
method Analyzing moduli spaces and using geometric properties.
result Polyhedral surfaces with 8 vertices can be decomposed into at most 20 parallelograms.
The paper studies billiards in symmetric tables and finds a measure bound for maximizing orbits.
problem Understanding the measure of maximizing orbits in symmetric billiard tables.
method Introduced a closed invariant set of locally maximizing orbits and gave an effective bound on its measure.
result An effective bound on the measure of the invariant set in terms of the isoperimetric defect of the curve.
We prove that every complete finite-volume hyperbolic 3-manifold M that is tessellated into (embedded) right-angled regular polyhedra (dodecahedra or ideal octahedra) embeds geodesically in a complete finite-volume connected orientable hyperbolic 4-manifold W, which is also tessellated into right-angled regular pol…
The paper establishes a majorization result for symmetric matrices.
problem Majorization of symmetric matrices under certain conditions.
method Using Garding-Dirichlet operator and properties of I-central operators. result A definitive inequality for symmetric matrices.
Makeev proved that among centrally symmetric four-dimensional polytopes, with more than twenty facets and circumscribed about the Euclidean ball of diameter one, there is no universal cover for the family of unit diameter sets. In this paper we examine the converse problem, and prove that each centrally symmetric polyt…
The conformal Laplacian's algebraic structure is explored in 2D, revealing a central charge.
problem Exploring the algebraic structure of the conformal Laplacian in 2D.
method Using prefactorization algebras and Green functions.
result In 2D, the conformal Laplacian's algebraic structure is revealed through a central charge.
New central elements found in a quantum algebra related to knot theory.
problem Understanding the algebraic structure of SLd-skein algebras. method Threaded polynomials from symmetric functions.
result Extraction of central elements in SLd-skein algebra. Proves Birkhoff-Poritsky conjecture for centrally-symmetric billiards.
problem Proves conjecture for centrally-symmetric billiards.
method Uses non-standard generating function, invariant curve structure, and integral-geometry approach.
result Billiard curve is an ellipse under given conditions.
New random walk results on rank one symmetric spaces.
problem Analyzing random walks on noncompact rank one symmetric spaces.
method Unified algebraic framework using Möbius addition and harmonic analysis of spherical functions.
result Renormalized walk converges to heat kernel on Laplace-Beltrami operator.
Constructs minimal submanifolds in symmetric spaces using eigenfunctions.
problem Finding minimal submanifolds in symmetric spaces.
method Employing recent results from S. Gudmundsson and T.J. Munn, constructing submanifolds using eigenfunctions.
result Constructs minimal submanifolds of classical compact Riemannian symmetric spaces.
We prove: If a complete connected smooth surface M in euclidean 3-space has general position, intersects some plane along a clean figure-8 (a loop with total curvature zero) and all compact intersections with planes have central symmetry, then M is a (geometric) cylinder over some central figure-8. On the way, we estab…
An algebra A with identity (a∘b)∘c−a∘(b∘c)=(a∘c)∘b−a∘(c∘b), is called right-symmetric. Cohomology and deformation theory for right-symmetric algebras are developed. Cohomologies of gln and half-Witt algebras Wnrsym,p=0, Wnrsym(m),p>0, are calculated. In p…
Linearizes Virasoro symmetries for semisimple Frobenius manifolds.
problem Linearizing Virasoro symmetries for semisimple Frobenius manifolds.
method Proving the existence of an infinite family of linearizable Virasoro symmetries under specific conditions.
result The Dubrovin-Zhang hierarchy associated with semisimple Frobenius manifolds has a bihamiltonian structure that can be represented by differential polynomials.
Explains the Borromean rings, icosahedron, and Poincaré homology sphere.
problem Exploring the relationship between Borromean rings, icosahedron, and Poincaré homology sphere.
method Introduction of topological concepts and geometric construction of icosahedral compound of octahedra.
result Proofs about the orientation-preserving symmetry group of an icosahedron and the linked nature of Borromean rings.
We consider two types of p-centro affine flows on smooth, centrally symmetric, closed convex planar curves, p-contracting, respectively, p-expanding. Here p is an arbitrary real number greater than 1. We show that, under any p-contracting flow, the evolving curves shrink to a point in finite time and the only…
Formula establishes determinant majorization for symmetric matrices.
problem Determining determinant majorization for symmetric matrices.
method Establishes determinant majorization formula for symmetric matrices using invariant Garding-Dirichlet polynomials.
result Formula F(A)N1≥det(A)n1 for symmetric matrices. Let G/H be a compact 4-symmetric space of inner type such that the dimension of the center Z(H) of H is at most one. In this paper we shall classify involutions of G preserving H for the case where dim Z(H)=0, or H is a centralizer of a toral subgroup of G.
The paper proves a convex polytope conjecture with specific symmetry conditions.
problem Proving Kalai's 3^d conjecture for centrally symmetric polytopes.
method Using hyperplanes with orthogonal normal vectors to prove the conjecture.
result The conjecture is proven if the polytope is symmetric about multiple hyperplanes.
This paper studies the large time existence for the motion of closed hypersurfaces in a radially symmetric potential. In physical, this surface can be considered as an electrically charged membrane with a constant charge per area in a radially symmetric potential. The evolution of such surface has been investigated by …
Develops symmetric Cartan calculus linking to Patterson-Walker metric.
problem No specific problem stated; focuses on developing a new calculus.
method Symmetric Cartan calculus, using torsion-free affine connections.
result Symmetric Cartan calculus is a complete analogue of classical Cartan calculus.
We describe complex twistor spaces over inner 3-symmetric spaces G/H, such that H acts transitively on the fibre. Like in the symmetric case, these are flag manifolds G/K where K is the centralizer of a torus in G. Moreover, they carry an almost complex structure defined using the horizontal distribution of t…
Study surjections between braid groups on surfaces, focusing on lower central series.
problem Determine conditions for surjections between braid groups on surfaces.
method Utilizes properties of lower central series and combinatorial methods.
result Identifies specific values of m and n for which surjections exist between braid groups.
A Minkowski class is a closed subset of the space of convex bodies in Euclidean space Rn which is closed under Minkowski addition and non-negative dilatations. A convex body in Rn is universal if the expansion of its support function in spherical harmonics contains non-zero harmonics of all orders. If K is universal, t…
The paper explores symmetric representations of links and conditions for amphichirality.
problem Investigating symmetric representations of links and conditions for amphichirality.
method Using antipodally self-dual and antipodally symmetric maps, the authors provide sufficient combinatorial conditions for amphichirality.
result A link is amphichiral if its self-dual pairing is not one of 6 specific ones.
Cyclohedra are a well-known infinite familiy of finite-dimensional polytopes that can be constructed from centrally symmetric triangulations of even-sided polygons. In this article we introduce an infinite-dimensional analogue and prove that the group of symmetries of our construction is a semidirect product of a degre…
It is shown that the multiplicative monoids of Brauer's centralizer algebras generated out of the basis are isomorphic to monoids of endomorphisms in categories where an endofunctor is adjoint to itself, and where, moreover, a kind of symmetry involving the self-adjoint functor is satisfied. As in a previous paper, of …
The article explores causal structures in symmetric spaces and their relation to AQFT.
problem Understanding causal structures in symmetric spaces and their applications in AQFT.
method Classification of reductive causal symmetric spaces using Euler elements and 3-grading.
result Extraction of real Matsuki crowns and description of stabilizer groups of Euler elements.
Extended current algebra on S^3 with new bilinear form and 2-cocycle.
problem Investigate the current algebra on S^3 of complex Lie algebras.
method Defined a new quaternion-valued 2-cocycle and symmetric invariant bilinear form.
result Extended affine Kac-Moody algebra to Lie algebra of smooth mappings.
Study shows a central limit theorem for random coverings of manifolds with nilpotent groups.
problem Understanding the distribution of connected components in random coverings of manifolds with nilpotent fundamental groups.
method Used sampling homomorphisms from the fundamental group into the symmetric group and subgroup growth zeta functions of nilpotent groups.
result Proved a central limit theorem for the number of connected components of these random coverings.
New findings on magnetic geodesic flows and periodic motions.
problem Characterizing superintegrable systems in magnetic geodesic flows.
method Analyzing rotationally symmetric magnetic geodesic flows.
result All sufficiently slow motions in a central magnetic field are periodic under specific curvature and homogeneity conditions.
This paper studies Ptolemy coordinates for knot complements, providing explicit formulas and algorithms.
problem Computing obstructions and holonomy representations for knot complements.
method Explicit computation of Ptolemy coordinates and gluing equations, providing a diagrammatic algorithm.
result Explicit formula for lifting representations and a diagrammatic algorithm for holonomy representations.
Study shows conjugacy of torsion in genus 2 surfaces.
problem Torsion elements in mapping class groups of surfaces.
method Proved congruence subgroup property for centralizers of finite subgroups.
result Torsion elements in surfaces of genus ≤ 2 are conjugacy distinguished.
The signed volume function for polyhedra can be generalized to a mean volume function for volume elements by averaging over the triangulations of the underlying polyhedron. If we consider these up to translation and scaling, the resulting quotient space is diffeomorphic to a sphere. The mean volume function restricted …
We discuss the concept of the shadow boundary of a centrally symmetric convex ball K (actually being the unit ball of a Minkowski normed space) with respect to a direction x of the Euclidean n-space Rn. We introduce the concept of general parameter spheres of K corresponding to this direction and prove t…
Kähler-Einstein metrics on toric submanifolds cannot be induced by projective immersions.
problem Characterizing Kähler-Einstein metrics induced by projective immersions.
method Analyzing four families of symmetric and non-symmetric toric Fano manifolds.
result Kähler-Einstein metrics on toric submanifolds cannot be induced by projective immersions.
Paper introduces a diagnostic for approximate inference methods.
problem Estimating errors in probabilistic inference algorithms, especially for approximate methods.
method Repeatedly simulate datasets from the prior and perform inference on each, estimating a symmetric KL-divergence.
result A diagnostic for approximate inference methods can be estimated using symmetric KL-divergence.
The paper studies Stein-Weiss operators on symmetric tensors, extending previous work.
problem Understanding Stein-Weiss operators on symmetric tensors of arbitrary rank.
method Analyzing the decomposition of tensor spaces into irreducible components and computing Weitzenbock formulas.
result Unified framework for second-order Stein-Weiss operators and tools for geometric analysis.
We relate the distribution of eigenvalues of a random symmetric matrix in the Gaussian Orthogonal Ensemble to the distribution of critical values of a random linear combination of eigenfunctions of the Laplacian on a compact Riemann manifold. We then prove a central limit theorem describing what happens when the dimens…
Quantum approach to volume computation from colored Jones polynomials.
problem Computing volumes of hyperbolic 3-manifolds from knot polynomials.
method Categorification of Jones polynomials and asymptotic analysis of skein elements.
result Asymptotic growth rate of Kauffman bracket relates to volumes of ideal octahedra.