Study infinite superelliptic curves and their Veech groups, providing geometric and algebraic insights.
problem Characterize Veech groups of infinite superelliptic curves.
method Analyzing geometric properties, differential equations, and group theory.
result Veech groups of infinite superelliptic curves are all matrices permuting branched points.
Study of superelliptic curves and complex braid groups, computing homology.
problem Understanding the homology of superelliptic curves and their associated braid groups.
method Analysis of the universal family of superelliptic curves, computation of homology groups using complex braid groups.
result Complete calculation of integral homology groups of End over finite fields. The reduced Burau representation Vn of the braid group Bn is obtained from the action of Bn on the homology of an infinite cyclic cover of the disc with n punctures. The group homology H∗(Bn;Vn) of braid groups with coefficients in the complexified reduced Burau representation is calculated. Our topolog…
Finite presentation for a specific group in 3D handlebody topology.
problem Finding a finite presentation for a specific group in 3D handlebody topology.
method Constructed a finite presentation for the liftable Hilden group to derive the presentation for the balanced superelliptic handlebody group.
result A finite presentation was given for the balanced superelliptic handlebody group.
Positive factorization found for a specific map on surfaces.
problem Balanced superelliptic rotation on surfaces.
method Positive factorization approach.
result Positive factorization for balanced superelliptic rotation.
Three elements generate balanced superelliptic mapping class groups.
problem Generating balanced superelliptic mapping class groups.
method Proving groups are generated by three elements through normalizers and liftable mapping class groups.
result Balanced superelliptic mapping class groups are generated by three elements.
Study of liftable mapping class group for superelliptic covers.
problem Understanding mapping class groups of superelliptic covers.
method Computational and algebraic methods to study the liftable mapping class group.
result The liftable mapping class group is independent of the degree of the cover and has finite abelianization.
Finite presentations for mapping class groups of surfaces and surfaces with points/boundaries.
problem Finding finite presentations for balanced superelliptic mapping class groups.
method Construct finite presentations for corresponding liftable mapping class groups in a different generating set.
result Finite presentations for balanced superelliptic mapping class groups of various surfaces.
Proves a minimal generating set for a specific group of mapping classes.
problem Finding a minimal generating set for a specific group of mapping classes.
method Proved the group is generated by four elements, with minimal exceptions.
result Minimal generating set for the balanced superelliptic mapping class group.
The Loch Ness Monster admits many regular dessins d'enfants and different holomorphic structures.
problem Classical theory of dessins d'enfants on compact surfaces extended to non-compact surfaces.
method Study of infinite genus surfaces and their connections to Riemann surfaces.
result The Loch Ness monster admits infinitely many regular dessins d'enfants.
Market maker handles negative prices with unique asset swapping.
problem Handling negative prices in financial markets.
method Unique market mechanism with numeraire currency, liquidity extensions.
result Liquidity fingerprint and payoff compared to established models.
The paper constructs braiding structures for a specific subfactor.
problem The challenge is to understand the braiding structures of a Jones-Wassermann subfactor.
method The approach involves constructing braiding structures on the multi-interval Jones-Wassermann subfactor planar algebra.
result The braiding structures induce a projective unitary representation of the balanced superelliptic mapping class group.
In this paper we propose and investigate a novel nonlinear unit, called Lp unit, for deep neural networks. The proposed Lp unit receives signals from several projections of a subset of units in the layer below and computes a normalized Lp norm. We notice two interesting interpretations of the Lp unit. First…
The paper studies liftable mapping class groups of cyclic covers of spheres.
problem Understanding liftable mapping class groups of cyclic covers of spheres.
method Derived finite generating sets, provided algorithms, determined isomorphism classes, derived presentations, and calculated normalizers and centralizers.
result Presentations and isomorphism classes of liftable mapping class groups for various covers.
New methods to construct curve pairs and their applications.
problem Constructing curve pairs and their properties.
method Using integral curves to study direction and donor curves.
result New methods to construct partner curves of unit speed curves.
New method to construct partner curves of non-lightlike curves.
problem Constructing partner curves for non-lightlike curves.
method Using integral curves in Minkowski 3-space, direction curve, and donor curve.
result New methods to construct partner curves of a unit speed non-lightlike curve.
The study explores Bertrand and Mannheim curves in 4D Euclidean space for framed curves.
problem Exploring Bertrand and Mannheim curves in 4D Euclidean space for framed curves.
method Defining and investigating Bertrand and Mannheim curves of framed curves in 4D Euclidean space.
result Bertrand and Mannheim curves exist even for framed curves in 4D Euclidean space, contrary to regular curves.
The study examines Bertrand Legendre curves in the unit tangent bundle over Euclidean plane.
problem Investigating properties of Legendre curves and their associated curves.
method Analyzing Bertrand Legendre curves and their associated curves, including parallel, evolute, and involute curves.
result Existence conditions and inverse operation for Bertrand Legendre curves are provided.
Method for generating new curves from plane curves on cylinders.
problem Generating new space curves from given plane curves.
method Defining a non-planar space curve on a right generalized cylinder and examining its focal curve.
result Parametric representation of the focal curve of a cylindrical curve.
The paper characterizes curves in pseudo-Galilean 4-space.
problem Characterizing curves in the pseudo-Galilean 4-space G14. method Investigation and characterisation of admissible curves in terms of curvature functions.
result Necessary and sufficient conditions for admissible rectifying curves in G14. The paper explores Bertrand and framed curves in 3D space.
problem Characterizing Bertrand and framed curves in Euclidean 3-space.
method Analyzing curves where tangent, normal, or binormal lines match another curve's lines.
result Conditions for the existence of Bertrand and framed curves are clarified.
Study introduces D-type curves on surface pencils and their properties.
problem Characterizing curves on surface pencils.
method Using Frenet vectors and parametric representation, investigate necessary and sufficient conditions for curves to be D-type curves.
result D-type curves are geodesic or asymptotic in some cases.
The paper examines how closed curves on surfaces intersect and how this intersection determines the curves.
problem Determining closed curves on surfaces based on their intersections.
method Constructing and studying k-equivalent curves, analyzing intersections with other curves. result Curves are determined by their intersections with all other curves, but non-simple curves require infinitely many intersections to distinguish.
Approximating complex curves with simple parametric curves is widely used in CAGD, CG, and CNC. This paper presents an algorithm to compute a certified approximation to a given parametric space curve with cubic B-spline curves. By certified, we mean that the approximation can approximate the given curve to any given pr…
The study classifies singularities of spherical orthotomic curves.
problem Classifying singularities of spherical orthotomic curves.
method Defining spherical orthotomic curves and classifying their singularities.
result Singularities of spherical orthotomic curves are classified.
Flow deforms locally convex curves to curves of constant k-order width.
problem Evolve locally convex curves to curves of constant k-order width.
method Introduced a nonlocal curvature flow to evolve locally convex curves in the plane.
result The flow converges to a smooth, locally convex curve of constant k-order width as time goes to infinity.
Modified curve shortening flow constructs λ-Angenent curve.
problem Constructing λ-Angenent curve. method Modified curve shortening flow
result Constructs λ-Angenent curve. Study on CR curves in 3-sphere, focusing on critical curves integration and existence.
problem Addressing the integration and existence of critical curves in the CR 3-sphere.
method Provided a procedure for the explicit integration of general critical curves and characterized closed curves.
result Existence of infinite countably many closed critical curves.
The paper quantifies fractal curves using centroaffine curvatures.
problem Quantifying the irregularities of fractal curves.
method Using moving frame and centroaffine curvatures.
result Fractal curves can be described by a sequence of affine curvatures.
Study rectifying curves in 3D multiplicative Euclidean space.
problem Investigate rectifying curves in a non-Newtonian geometry setting.
method Apply multiplicative differential-geometric concepts to rectifying curves.
result Classify multiplicative rectifying curves using spherical curves.
In classical curve theory, the geometry of a curve in three dimensions is essentially characterized by their invariants, curvature and torsion. When they are given, the problem of finding a corresponding curve is known as 'solving natural equations'. Explicit solutions are known only for a handful of curve classes, inc…
The paper characterizes pedal curves of quadratic curves.
problem Understanding pedal curves of quadratic curves.
method Analyzing the inverse construction of pedal curves.
result Characterization of pedal curves of quadratic curves.
Unified description of aesthetic curves through self-affinities.
problem Characterizing log-aesthetic curves and their properties.
method Reformulating and proving self-affinities of planar curves, integrating equiaffine geometry.
result Unified characterization of constant curvature curves in similarity and equiaffine geometries.
Primitive curves in handlebodies form a connected complex.
problem Understanding the structure of curves in handlebodies.
method Defining and analyzing primitive curves and constructing sequences between them.
result The primitive curve complex for a handlebody is connected.
Study on Bertrand lightcone framed curves in Lorentz-Minkowski 3-space.
problem Analyzing mixed types of curves with singular points in Lorentz-Minkowski 3-space.
method Using lightcone frame to consider Bertrand types for lightcone framed curves.
result Existence conditions of Bertrand lightcone framed curves in all cases.
Study isotopy of rational cuspidal curves in 4-manifolds.
problem Isotopy of rational cuspidal curves in 4-manifolds.
method Tame symplectic curves, pseudo-holomorphic curves, log pairs, 4-dimensional topology.
result Every rational cuspidal curve is isotopic to a complex curve in degrees up to 5.
Study uses isotropic projection to link plane and null curves in Minkowski space.
problem Understanding the geometry of null curves in Minkowski 3-space.
method Utilizes isotropic projection of Laguerre geometry to establish a correspondence.
result Alternative description of plane curves congruent to a given one.
Defines new curves from tangent indicatrix of curves, linking them to helices and slant helices.
problem Understanding and constructing helices and slant helices from spherical curves.
method Defining integral curves of Frenet vectors and using their curvatures.
result Established relationships and methods to create helices and slant helices from specific spherical curves.
Study on triharmonic curves in f-Kenmotsu manifolds.
problem Characterizing triharmonic curves in f-Kenmotsu manifolds.
method Investigation of necessary and sufficient conditions for Frenet curves, slant, and Legendre curves to be triharmonic. Proof of specific properties of triharmonic Frenet curves.
result Triharmonic Frenet curves with constant curvature are Frenet helices in three dimensional f-Kenmotsu manifolds.
The paper generalizes rectifying and normal curves in Lorentzian n-space.
problem Characterizing and classifying g−rectifying and g−normal curves in Lorentzian n-space. method Introducing a g−position vector field and defining g−rectifying and g−normal curves based on this field. result Comprehensive characterization and classification of g−rectifying and g−normal curves. New findings on hyperbolicity of fine curve graphs and their subgraphs.
problem Investigating hyperbolicity of fine curve graphs and their subgraphs.
method Analyzing large subgraphs of fine curve graphs and computing distances in specific cases.
result Large subgraphs of fine curve graphs contain flats of every finite dimension, indicating they are not hyperbolic.
Study of p-biharmonic curves and their properties.
problem Generalizing biharmonic curves to p-biharmonic curves. method Classification and analysis of p-biharmonic curves on surfaces and space forms. result Existence and stability of p-biharmonic curves on closed surfaces. In this paper we study null Bertrand curves in R14 under the assumption the curve has a Cartan frame. We show that if the derivative vectors of the null Cartan curve in R14 is linearly independent, then this curve is not a Bertrand curve. Since then the already known notion of null Bertrand curves in $R…
Compact curve solution emerges from non-compact curve.
problem Constructing solutions from non-compact curves.
method Slingshot solution to curve shortening flow.
result Compact embedded solution exists for a finite time.
Homotopy types of curve and arc complexes are studied.
problem Understanding the homotopy types of curve and arc complexes.
method Proving homotopy equivalence and contractibility of complexes.
result Fine curve complex is homotopy equivalent to curve complex, fine arc complex is contractible.
New curves generalize helix and rectifying curves.
problem Generalizing helix and rectifying curves.
method Introducing f-rectifying curves with f-position vector in rectifying plane.
result Classification and characterization of f-rectifying curves.
We classify curves in the moduli space of curves that are both Shimura- and Teichmueller curves: Except for the moduli space of genus one curves there is only a single such curve. We start with a Hodge-theoretic description of Shimura curves and of Teichmueller curves that reveals similarities and differences of the tw…
In this study, we define a new type of direction curves in the Euclidean 3-space such as osculating-direction curve. We give the characterizations for these curves. Moreover, we obtain the relationships between osculating direction curves and some special curves such as helix, slant helix or rectifying curves.