The paper classifies and studies symplectic and contact properties of circular spherical divisors.
problem Investigating symplectic and contact topology of circular spherical divisors.
method Classification and analysis of concave circular spherical divisors, including embedding, Stein fillability, and rational homology type determination.
result All concave circular spherical divisors up to toric equivalence are realized as symplectic log Calabi-Yau pairs with minimal complements.
Circular nets with spherical parameter lines have geometric properties related to Darboux cyclides and terminating Laplace sequences.
problem Discretizing surfaces with spherical curvature lines.
method Lie-geometric discretisation in terms of principal contact element nets.
result Circular nets with two families of spherical parameter lines are related to Darboux cyclides.
In this article, we investigate Bertrand curves corresponding to the spherical images of the tangent, binormal, principal normal and Darboux indicatrices of a space curve in Euclidean 3-space. As a result, in case of a space curve is a general helix, we show that the curves corresponding to the spherical images of its …
We show how to compute the circular area invariant of planar curves, and the spherical volume invariant of surfaces, in terms of line and surface integrals, respectively. We use the Divergence Theorem to express the area and volume integrals as line and surface integrals, respectively, against particular kernels; our r…
Study finds existence and non-uniqueness of cone spherical metrics on compact Riemann surfaces.
problem Existence and non-uniqueness of cone spherical metrics with prescribed singularities.
method Utilizing polystable extensions of line bundles, the study establishes three primary results concerning these metrics.
result Existence of multiple irreducible and reducible cone spherical metrics for certain effective divisors.
Constructs a map from stable extensions to irreducible metrics on Riemann surfaces.
problem Understanding and constructing cone spherical metrics on Riemann surfaces.
method Using the theory of indigenous bundles, the construction involves developing maps and stable extensions of two line bundles.
result Generically injective map from stable extensions to irreducible metrics, with properties about effective divisors.
Approximates surfaces using Laguerre geometry with spherical faces.
problem Approximating smooth surfaces using Laguerre geometry.
method Using Laguerre conjugate nets and spherical faces to approximate surfaces.
result Laguerre conjugate nets provide a method for surface approximation.
The study examines Riemannian surfaces with simple singularities.
problem Geometry of Riemannian surfaces with discrete singular points.
method Local and global description using divisors, Gauss-Bonnet formula, classifications theorem.
result Classification of flat metrics with simple singularities on compact surfaces.
Approximates smooth surfaces using Laguerre geometry meshes.
problem Approximating smooth surfaces in Laguerre geometry.
method Using Laguerre meshes composed of quadrilaterals, cones, and spherical faces.
result Laguerre conjugate nets and directions for surface approximation.
Characterizes representations for complex projective structures with specific branch data.
problem Understanding representations of surface groups as holonomy of complex projective structures.
method Computing holonomies for spherical metrics and affine structures with prescribed conical angles.
result Computed holonomies for spherical metrics and affine structures with specific conical angles.
New insights into stability of special curves on spheres.
problem Stability of closed p-elastic curves on spheres. method Analytical proof and construction of curves.
result All closed spherical p-elastic curves for p∈(0,1) are unstable. We introduce a compactification of the space of simple positive divisors on a Riemann surface, as well as a compactification of the universal family of punctured surfaces above this space. These are real manifolds with corners. We then study the space of constant curvature metrics on this Riemann surface with prescribe…
We continue our study, initiated in our earlier paper, of Riemann surfaces with constant curvature and isolated conic singularities. Using the machinery developed in that earlier paper of extended configuration families of simple divisors, we study the existence and deformation theory for spherical conic metrics with s…
We show that a capillary surface in a solid cone, that is, a surface that has constant mean curvature and the boundary of surface meets the boundary of the cone with a constant angle, is radially graphical if the mean curvature is non-positive with respect to the Gauss map pointing toward the domain bounded by the surf…
Study circular tractrices and pseudospheres in 3D space.
problem Geometric properties of circular tractrices and pseudospheres.
method Exploration of geometric properties through analysis.
result Characterization of circular tractrices and pseudospheres.
We consider a billiard in the sphere S^2 with circular obstacles, and give a sufficient condition for its flow to be uniformly hyperbolic. We show that the billiard flow in this case is approximated by an Anosov geodesic flow on a surface in the ambiant space S^3. As an application, we show that every orientable surfac…
We consider minimal compact complex surfaces S with Betti numbers b_1=1 and n=b_2>0. A theorem of Donaldson gives n exceptional line bundles. We prove that if in a deformation, these line bundles have sections, S is a degeneration of blown-up Hopf surfaces. Besides, if there exists an integer m>0 and a flat line bundle…
The paper defines circular orderability for quandles and explores their properties.
problem Understanding the structure of quandles through circular orderings.
method Introduced circular orderability for quandles, showed embedding properties, and provided examples.
result Spaces of circular orderings embed in the space of all orderings, and examples of non-circularly orderable quandles are given.
Paper studies geometric and combinatorial properties of circular snakes.
problem Exploring geometric and combinatorial properties of circular snakes.
method Definition and investigation of outer Lipschitz geometry, decomposition of Valette link, construction of combinatorial objects, weakly outer Lipschitz classification.
result Existence of canonical decomposition and necessary/sufficient criteria for removing segments or Hölder triangles.
A curve is rectifying if it lies on a moving hyperplane orthogonal to its curvature vector. In this work, we extend the main result of [Chen 2017, Tamkang J. Math. 48, 209] to any space dimension: we prove that rectifying curves are geodesics on hypercones. We later use this association to characterize rectifying curve…
The study identifies surfaces with Maslovian normal bundles.
problem Characterizing surfaces with specific geometric properties.
method Proving equivalence to round spheres, cylinders, or cones.
result Surfaces with Maslovian normal bundles are limited to specific shapes.
New hexagonal circular 3-webs with reducible curves classified.
problem Classifying hexagonal circular 3-webs with reducible polar curves of degree 3.
method New examples and classifications presented.
result Classification of hexagonal circular 3-webs with reducible polar curves of degree 3.
Paper proposes a method to improve circular coordinate representation for detecting changes in high-dimensional datasets.
problem Detecting changes in high-dimensional datasets with preserved topological structures.
method Adapt circular coordinate framework using a generalized penalty function instead of an L2 penalty.
result Circular coordinates with generalized penalty can detect changes in high-dimensional datasets under different sampling schemes.
The paper studies circular evolutes and involutes of framed curves in Euclidean space.
problem Investigating properties of framed curves and their evolutes and involutes.
method Definition and analysis of circular evolutes and involutes of framed curves, properties of normal surfaces, and their relations.
result Circular evolutes and involutes of framed curves are opposite operations under suitable assumptions, similar to fronts in the Euclidean plane.
CDFD analyzes circularity and directionality in weighted directed networks.
problem Analyzing circularity and directionality in weighted directed networks.
method CDFD framework separates flow into circular and acyclic components.
result CDFD yields a normalized circularity index capturing flow in cycles and directionality.
It was shown by Kaup that every origin-preserving automorphism of quasi-circular domains is a polynomial mapping. In this paper, we study how the weight of quasi-circular domains and the degree of such automorphisms are related. By using the Bergman mapping, we prove that every origin-preserving automorphism of normal …
Classifies hexagonal circular 3-webs with cubic polar curves.
problem Classifying hexagonal circular 3-webs with algebraic polar curves of degree three.
method Analyzes hexagonal circular 3-webs on unit sphere with polar points on a twisted cubic.
result Completes the classification of hexagonal circular 3-webs with algebraic polar curves of degree three.
Study geometrically characterizes piecewise circular curves with decreasing curvature.
problem Characterizing piecewise circular curves with decreasing curvature.
method Introducing moduli spaces and relating them to Legendrian polygons.
result Proves the moduli space contains a connected component homeomorphic to the Fock-Goncharov space of positive flags.
Curves become nearly circular over time without initial assumptions.
problem Understanding the asymptotic behavior of area-preserving flows.
method Proving asymptotic circularity without additional assumptions.
result Immortal solutions become asymptotically circular without initial assumptions.
Study finds a limiting distribution for free path lengths on flat surfaces with circular obstacles.
problem Understanding free path lengths on flat surfaces with circular obstacles.
method Proved the existence of a limiting distribution using radius of obstacles as a parameter.
result Relates the limiting distribution to heights of zippered rectangle decompositions.
We show that circular width is preserved under connected sum of knots for some cases.
Study b-divisors on Kähler manifolds linking them to currents.
problem Intersection theory of b-divisors on Kähler manifolds.
method Established correspondence between closed positive currents and nef b-divisors.
result Intersection theory of nef b-divisors answered.
The paper explores circular orderability in 3-manifold groups, related to the L-space conjecture.
problem Circular orderability of 3-manifold groups and its relation to the L-space conjecture.
method Investigation of finite cyclic covers and Dehn surgeries to establish circular orderability.
result Circularly orderable fundamental groups of compact, connected, P^2-irreducible 3-manifolds are characterized.
Projected random forests improve circular data prediction with adaptive arc length and finite-sample coverage.
problem Regression with circular responses.
method Adapting linear-response models to circular data using projection and random forest out-of-bag mechanism.
result Projected random forest out-of-bag conformal prediction sets are more efficient and shorter than alternative methods.
ANGLE tackles circular data regression, improving predictive performance.
problem Geometrically misleading traditional regression for circular data.
method Generative map optimized via GCES loss for non-parametric distributional regression.
result Unified toolbox for circular statistics challenges.
Derives conformal parameters of curves using inscribed circular polygons.
problem Characterizing conformal invariants of smooth curves in 3D.
method Limiting process with inscribed circular polygons, based on elementary geometry.
result Derives conformal length, curvature, and torsion via a novel method.
Circular variables arise in a multitude of data-modelling contexts ranging from robotics to the social sciences, but they have been largely overlooked by the machine learning community. This paper partially redresses this imbalance by extending some standard probabilistic modelling tools to the circular domain. First w…
We study stable immersed capillary hypersurfaces in a domain B which is either a half-space or a slab in the Euclidean space Rn+1. We prove that such a hypersurface Σ is rotationally symmetric in the following cases: (1) n=2, B is a slab and Σ has genus zero, (2) n≥2, $\mathc…
For a knot K⊂S3, its exterior E(K)=S3\η(K) has a singular foliation by Seifert surfaces of K derived from a circle-valued Morse function f:E(K)→S1. When f is self-indexing and has no critical points of index 0 or 3, the regular levels that separate the index-1 and index-2 critica…
Study contact geometry of symplectic divisors, invariant under specific transformations.
problem Understanding contact structures on symplectic divisors and their boundaries.
method Invariant analysis of contact structures under toric and interior blow-ups/blow-downs, open book decomposition construction.
result Contact structure on divisor boundaries is invariant under specified transformations.
The maximum mean discrepancy (MMD) is a recently proposed test statistic for two-sample test. Its quadratic time complexity, however, greatly hampers its availability to large-scale applications. To accelerate the MMD calculation, in this study we propose an efficient method called FastMMD. The core idea of FastMMD is …
A regular circle-valued Morse function on the knot complement C(K) = S^3\K is a function f from C(K) to S^1 which separates critical points and which behaves nicely in a neighborhood of the knot. Such a function induces a handle decomposition on the knot exterior E(K) = S^3\N (K), with the property that every regular l…
This article provides sufficient conditions for a closed hyperbolic 3-manifold M with non zero first Betti number to fiber over the circle, and to find a fiber in M. Those conditions are formulated in terms of the behavior the circular characteristic in finite regular covers of M. We define the circular character…
Analyzes packing of circles in bounded and unbounded planes using mathematical formulas.
problem Finding optimal radii for packing circles in various plane regions.
method Deterministic analytic formulae and recurrence relations.
result Formulated analytic formulae for 2D circle packing on various plane shapes.
In this paper, we extend the existence and regularity theorems for Kähler-Einstein metrics having conic singularities along a simple normal crossing divisor to the case of normal crossing divisor, i.e. when components of the divisor are allowed to intersect themselves transversely.
We prove that a compact stratied space satises the Riemannian curvature-dimension condition RCD(K, N) if and only if its Ricci tensor is bounded below by K ∈ R on the regular set, the cone angle along the stratum of codimension two is smaller than or equal to 2π and its dimension is at most equal to N. This gives…
The paper extends intersection theory for b-divisors, proving monotonicity and volume inequalities.
problem Intersection theory for b-divisors and monotonicity of intersection products.
method Developed general intersection theory of nef b-divisors, defined restricted volume, proved monotonicity.
result Proved quantitative monotonicity of intersection product and new volume inequalities.
New conditions for circular orderability of direct products, linking to left-orderability of groups.
problem Conditions for circular orderability of direct products GimesZ/nZ. method Cohomological conditions and characterizations for left-orderability.
result New characterization for left-orderability of fundamental groups of rational homology 3-spheres.