Formula derived for spherical growth series of specific groups.
problem Calculating the spherical growth series of specific groups.
method Developed a formula and derived a rational function expression for the spherical growth series.
result Explicit rational function expressions for the spherical growth series of specific groups.
Survey on 4-manifolds with specific curvature properties.
problem Understanding the structure of 4-manifolds with nonnegative Ricci curvature and Euclidean volume growth.
method Analysis of blow-downs and cone-like structures at infinity.
result Manifolds look like cones over spherical space forms at infinity.
Model predicts growth competition on curved surfaces.
problem Growth dynamics of two subsets on Riemannian manifolds.
method Modeling growth rates on spherically symmetric Riemannian manifolds.
result Conditions for bounded or unbounded growth on different manifolds.
We prove that the exponential growth rate of the regular language of penetration sequences is smaller than the growth rate of the regular language of normal form words, if the acceptor of the regular language of normal form words is strongly connected. Moreover, we show that the latter property is satisfied for all irr…
We give sufficient conditions for a noncompact Riemannian manifold, which has quadratic curvature decay, to have finite topological type with ends that are cones over spherical space forms.
Ancient convex solutions to flow equations are limited to simple shapes.
problem Characterizing ancient convex solutions to flow equations.
method Analyzing mean curvature flow and curvature functions of convex hypersurfaces.
result Ancient convex solutions to flow equations are limited to spherical, cylindrical, or planar shapes.
The paper proves the existence and uniqueness of Calabi-Yau metrics on affine spherical varieties.
problem Existence and uniqueness of Calabi-Yau metrics on affine spherical varieties.
method Explicit K-stability condition, degeneration, and asymptotic cone analysis.
result Uniqueness of K-invariant Calabi-Yau metrics on affine spherical manifolds. Paper characterizes DLN distribution, its properties, and estimation methods.
problem No specific problem stated, focuses on DLN distribution properties.
method Characterization of PDF, CDF, moments; generalization to N-dimensions; methods to handle double-exponential nature.
result Characterization of DLN distribution and its properties, including estimation methods.
The paper proves properties of fundamental groups of 4-manifolds with nonnegative Ricci curvature and Euclidean volume growth.
problem Properties of fundamental groups of 4-manifolds with nonnegative Ricci curvature and Euclidean volume growth.
method Proves properties of fundamental groups of 4-manifolds with nonnegative Ricci curvature and Euclidean volume growth.
result Fundamental groups of 4-manifolds with nonnegative Ricci curvature and Euclidean volume growth are finitely generated.
We study minimal hypersurfaces in manifolds of non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay at infinity. By comparison with capped spherical cones, we identify a precise borderline for the Ricci curvature decay. Above this value, no complete area-minimizing hypersurfaces exist. Be…
For spherically symmetric distributions, efficient quantisation can be achieved with moderate sample sizes.
problem Optimal quantisation in high dimensions requires large sample sizes, making it impractical.
method Uniformly distributed random quantisers on a sphere of suitable radius achieve exceptional performance.
result For moderate sample sizes, quantisation error can be efficiently computed and approximated.
We present a new procedure to determine the growth function of a homogeneous Garside monoid, with respect to the finite generating set formed by the atoms. In particular, we present a formula for the growth function of each Artin--Tits monoid of spherical type (hence of each braid monoid) with respect to the standard g…
The study examines the growth of reciprocal classes in Hecke groups, proving an asymptotic formula.
problem Analyzing the growth of reciprocal classes in Hecke groups.
method Utilizes the free product structure of Hecke groups, combinatorial counting, and recurrence relations.
result Proves an asymptotic formula for the number of reciprocal classes in Hecke groups.
The paper proposes a method to determine the geometric priors of relational data.
problem Identifying geometric structure in heterogeneous, high-dimensional data.
method Combinatorial approach analyzing nearest-neighbor structures and local neighborhood growth rates.
result The method can identify the geometric priors of suitable embedding spaces for relational data.
Stability of extremal Reissner-Nordström black holes proven in spherical symmetry.
problem Stability of extremal Reissner-Nordström black holes in spherical symmetry.
method Proved nonlinear asymptotic stability through spherically symmetric characteristic data.
result Existence of a submanifold Mstab leading to stable solutions. Study on regularity of exceptional actions and moduli of continuity for circle diffeomorphisms.
problem Regularity of exceptional actions of groups on the circle.
method Analysis of C1,α diffeomorphisms with free orbits and bounded derivatives. result Existence of exceptional C1,α diffeomorphisms under certain conditions on α. Harmonic functions on compact symmetric spaces exhibit strong convexity properties.
problem Understanding the convexity of harmonic functions on compact symmetric spaces.
method Analyzing the nonnegativity of the Laplacian powers of harmonic functions.
result Harmonic functions on compact symmetric spaces have nonnegative Laplacian powers, demonstrating strong convexity.
Paper introduces spherical knot mosaics for knot and link invariants.
problem Representing knots on a sphere with tiles.
method Tiling a 2-sphere with 11 knot mosaic tiles to define new invariants.
result New knot invariants derived from spherical mosaic tiling.
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.
The abstract proves spherical surface decompositions with conical singularities.
problem Decomposing surfaces with spherical metrics and conical singularities.
method Geometric triangulations and irreducible components of standard shapes.
result Spherical polygons, including half-spherical concave polygons, can be arbitrarily complicated.
For the n-dimensional spherical pedal curve pedγ,P with respect to an n-dimensional spherical unit speed curve γ and a given point P∈Sn, we define the spherical orthotomic curve of γ relative to the point P, and classify singularities of spherical orthotomic curves.
Study spherical curves with curvature dependent on distance to a great circle.
problem Understanding spherical curves with curvature dependent on distance to a great circle.
method Introducing spherical angular momentum, characterizing known curves, finding new families, and obtaining arc length parametrizations.
result New families of spherical curves with intrinsic equations in elementary or Jacobi elliptic functions.
Study calculates eigenvalues and eigenfunctions for spherical triangles and finds fundamental gap behavior.
problem Understanding eigenvalues and gaps in spherical triangles.
method Explicit computation of Dirichlet eigenvalues and eigenfunctions for spherical lunes and triangles.
result Fundamental gap of spherical triangles increases as the angle of the lune decreases.
New findings show fundamental group is not audible in spherical space forms.
problem Isospectral spherical space forms with non-cyclic fundamental groups.
method Revisited and found new examples of spherical space forms.
result Fundamental group is not audible among spherical space forms.
Existence and uniqueness of spherical helicoidal surfaces in 3-sphere via spherical curves.
problem Existence and uniqueness of spherical helicoidal surfaces in 3-sphere.
method Continuous function of distance to axis, spherical angular momentum of spherical curves.
result Existence and uniqueness theorem for spherical helicoidal surfaces in 3-sphere.
New solutions for soft materials' nonlinear behavior in growing spheroidal inclusions.
problem Understanding soft materials' behavior under microstructure effects.
method Presented an accurate semi-inverse solution for isotropically growing spheroidal inclusions.
result Existence of a non-spherical asymptotic shape and associated asymptotic pressure (isomorphic pressure).
The reductivity of a spherical curve represents how reduced the spherical curve is. It is unknown if there exists a spherical curve whose reductivity is four. In this paper we give an unavoidable set for spherical curves with reductivity four by considering 4-gons.
The paper improves inequalities for nearly spherical sets using quermassintegrals.
problem Improving inequalities for nearly spherical sets.
method Establishing quantitative Alexandrov-Fenchel inequalities for quermassintegrals.
result Lower bounds on the (k,m)-isoperimetric deficit found using spherical deviation and asymmetry. The reductivity of a spherical curve is the minimal number of a local transformation called an inverse-half-twisted splice required to obtain a reducible spherical curve from the spherical curve. It is unknown if there exists a spherical curve whose reductivity is four. In this paper, an unavoidable set of configuratio…
We show that we can obtain a reducible spherical curve from any non-trivial spherical curve by four or less inverse-half-twisted splices, i.e., the reductivity, which represents how reduced a spherical curve is, is four or less. We also discuss unavoidable sets of tangles for spherical curves.
The paper classifies Landsberg spherically symmetric Finsler metrics in various dimensions.
problem Investigating compatibility conditions on spherically symmetric Finsler metrics.
method Using the inverse problem of calculus of variations, the paper focuses on Landsberg and Berwald types.
result All Landsberg spherically symmetric manifolds in higher dimensions are either Riemannian or have specific geodesic spray formulas.
For a convex body on the Euclidean unit sphere the spherical convex floating body is introduced. The asymptotic behavior of the volume difference of a spherical convex body and its spherical floating body is investigated. This gives rise to a new spherical area measure, the floating area. Remarkably, this floating area…
Study on spherical CR manifolds with non-trivial Chern classes.
problem Understanding Chern classes in spherical CR manifolds.
method Construction and proof of constraints on Chern classes.
result Topological obstruction to spherical CR structures on contact manifolds.
PGF kernels analyze spherical data using generalized RBF kernels.
problem Analysis of spherical data.
method Introduced PGF kernels and a semi-parametric learning algorithm.
result PGF kernels generalize RBF kernels for spherical data.
The paper examines properties of spherical Finsler metrics, proving their semi-C-reducibility and conditions for vanishing mean stretch curvature.
problem Analyzing curvature properties of spherical Finsler metrics.
method Proving semi-C-reducibility and finding conditions for vanishing mean stretch curvature.
result Conditions for a general spherically symmetric Finsler metric to have vanishing mean stretch curvature.
Classifies involutions on spherical 3-manifolds.
problem Classifying involutions on spherical 3-manifolds.
method Geometric approach to conjugacy classification.
result Insights into topological properties of involutions.
The paper classifies spherically symmetric Finsler metrics as Berwaldian or Riemannian.
problem Classifying spherically symmetric Finsler metrics.
method Proving all spherically symmetric Landsberg surfaces are Berwaldian and modifying the classification of spherically symmetric Finsler metrics.
result All Berwald spherically symmetric metrics of dimension n≥3 are Riemannian or given by a specific formula. Paper develops invariants for spherical curves using chord diagrams.
problem Developing invariants for spherical curves under local moves.
method Using based chord diagrams and local moves from Reidemeister moves.
result Invariants include both classical and new spherical curve invariants.
In this paper we consider the spherical slant helices in R3. More- over, we show how could be obtained to a spherical slant helix and we give some spherical slant helix examples in Euclidean 3-space.
The paper characterizes spherically symmetric metrics with scalar curvature.
problem Characterizing spherically symmetric metrics with scalar curvature.
method Established a curvature compatibility condition on spherically symmetric Finsler metrics and constructed a Berwald frame.
result Characterized spherically symmetric metrics with scalar curvature.
The study proves spherical polygon analogs of curve theorems, finding bounds on intersections and inflections.
problem Finding bounds on intersections and inflections for spherical polygons.
method Adapting smooth curve theorems to spherical polygons using discrete tools.
result Proves discrete analogs of four-vertex theorems for spherical polygons.
New formula for spherical polygon area via prequantization.
problem Traditional area formula for spherical polygons requires measuring angles.
method Uses prequantization to create a new formula that doesn't require angle measurement.
result New formula applicable to a wider range of degenerate curves and polygons.
Euler explored spherical geometry using trigonometric formulae and solid geometry methods.
problem Developing trigonometric formulae for spherical geometry.
method Used calculus of variations and classical solid geometry methods.
result Established trigonometric formulae and computed spherical triangle areas.
A spherical polyhedron surface is a triangulated surface obtained by isometric gluing of spherical triangles. For instance, the boundary of a generic convex polytope in the 3-sphere is a spherical polyhedron surface. This paper investigates these surfaces from the point of view of inner angles. A rigidity result is obt…
Recently we initiated the study of spherical T-duality for spacetimes that are principal SU(2)-bundles. In this paper, we extend spherical T-duality to spacetimes that are oriented non-principal SU(2)-bundles. There are several interesting new examples in this case and a new phenomenon appearing in the non-principal ca…
Kähler-Ricci flow on spherical Fano manifolds converges to a soliton.
problem Analyzing the behavior of Kähler-Ricci flow on spherical Fano manifolds.
method Gromov-Hausdorff limit and torus degeneration.
result The limit of Kähler-Ricci flow on spherical Fano manifolds is a spherical Fano variety with a Kähler-Ricci soliton.
Spherical Plateau problem studies minimal surfaces in quotients of spheres.
problem Minimal surfaces in quotients of spheres.
method Metric currents, barycenter map method.
result Intrinsic uniqueness of solutions for negatively curved manifolds.
The class of spherically symmetric Finsler metrics is studied and locally dually flat and projectively flat spherically symmetric Finsler metrics is classified.