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.
Investigates Darboux rectifying curves on smooth surfaces.
problem Characterizing Darboux rectifying curves on smooth surfaces.
method Analyzes the position vector under isometry and finds conformal invariance conditions.
result Identifies sufficient conditions for conformal invariance of Darboux rectifying curves.
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. Recalls and refines the concept of algebraically rectifiable curves.
problem Classical notion of algebraically rectifiable plane curves.
method Provides new criteria, relates to quadratic differentials, and generalizes to higher order differentials.
result Generalization and new criteria for algebraic rectifiability.
The notion of rectifying curve in the Euclidean space is introduced by Chen as a curve whose position vector always lies in its rectifying plane spanned by the tangent and the binormal vector field t and n_2 of the curve. In this study, we have obtained some characterizations of semi-real spatial quaternionic rectifyin…
In this paper, we introduce a new class of curves αcalled a f-rectifying curves, which its f-position vector defined by α_{f}(s)=\int f(s)T(s)ds always lie in the rectifying plane of α, where f is an integrable function and T is the speed curve of α. In particular case, when the function f=0 or constant, the class of f…
In this article, we study rectifying curves in arbitrary dimensional Euclidean space. A curve is said to be a rectifying curve if, in all points of the curve, the orthogonal complement of its normal vector contains a fixed point. We characterize rectifying curves in the n-dimensional Euclidean space in different ways…
A space curve in a Euclidean 3-space E3 is called a rectifying curve if its position vector field always lies in its rectifying plane. This notion of rectifying curves was introduced by the author in [Amer. Math. Monthly {\bf 110} (2003), no. 2, 147-152]. In this present article, we introduce and study the n…
The paper characterizes rectifying curves on smooth surfaces using isometries and Darboux frames.
problem Characterizing rectifying curves on smooth surfaces under isometries.
method Using Darboux frames and isometries to investigate rectifying curves.
result Find deviations of rectifying curves under isometries and analyze their properties.
The paper characterizes timelike rectifying curves in De Sitter 3-space.
problem Characterizing timelike rectifying curves in De Sitter 3-space.
method Defining timelike rectifying curves and conical surfaces, providing characterizations and results.
result Characterizations and results of timelike rectifying curves in De Sitter 3-space.
Rectangular peg problem solved for many curves.
problem Rectangular peg problem for continuous Jordan curves.
method Microlocal sheaf theory and recent work of Greene and Lobb.
result Affirmative answer for a large class of rectifiable curves.
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. In this paper, we define a rectifying spacelike curve in the Minkowski space-time E14 as a curve whose position vector always lies in orthogonal complement N⊥ of its principal normal vector field N. In particular, we study the rectifying spacelike curves in E14 and characterize such curves in terms of…
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…
We defined normal and rectifying curves in Pseudo-Galilean Space G_3^1. Also we obtained some characterizations of this curves in G_3^1.
In this paper, we have first given easily the characterization of special curves with the help of the Rotation minimizing frame (RMF). Also, rectifying-type curves are generalized n-dimensional space Rn.
Study introduces weak elastic energy for curves on Riemannian surfaces.
problem Detecting curvature of curves on Riemannian surfaces.
method Relaxation starting from inscribed geodesic polygonals, defined in normalized isothermal coordinates.
result Relaxed energy detects intrinsic second-order Sobolev regularity and agrees with geodesic curvature.
Legendrian Lavrentiev links are shown to be equivalent to smooth links.
problem Understanding equivalence of Legendrian and smooth links.
method Definition and analysis of Legendrian isotopies.
result Equivalence classes of Legendrian Lavrentiev links coincide with smooth links.
Study calculates the elastic energy of curves on a sphere.
problem Elastic energy of curves on a sphere.
method Introduced p-curvature functional for rectifiable curves in the sphere and proved its finiteness. result The p-curvature functional agrees with the integral of geodesic curvature raised to the power p for curves in W2,p. Study of spatial curves in generalized Minkowski spaces.
problem Characterizing and invariants of spatial curves in non-Euclidean spaces.
method Derive Frenet-type results and invariants for spatial curves in generalized Minkowski spaces.
result Characterization of cylindrical helices and rectifying curves in generalized Minkowski spaces.
New method classifies geodesics on cones.
problem Classifying geodesics on cones in 3D space.
method Using necessary and sufficient conditions for rectifying curves and their traces in spheres.
result Established conditions for geodesics on cones.
Timelike curves in Lorentzian length spaces have a total curvature notion that agrees with smooth curves.
problem Estimating the length of timelike curves in Lorentzian length spaces.
method Introducing a synthetic timelike total curvature notion.
result Proving timelike curves of finite total curvature are rectifiable.
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.
Study extends geodesic curvature formula to higher dimensions.
problem Extending curvature formula to higher-dimensional spheres.
method Using new integral-geometric formulas for Euclidean and geodesic total curvature.
result Explicit formula for geodesic total curvature on higher-dimensional spheres.
Study of flat ribbons constructed along curves in 3D space.
problem Determine the conditions for a ruled structure to form a flat ribbon.
method Investigate the ruled structure of flat ribbons and calculate energy bounds.
result There exists a well-defined flat ribbon only up to an initial condition.
In this paper we study the singular set of Dirichlet-minimizing Q-valued maps from Rm into a smooth compact manifold N without boundary. Similarly to what happens in the case of single valued minimizing harmonic maps, we show that this set is always (m−3)-rectifiable with uniform Minkowski b…
Defines a new family of curves in space with applications.
problem Finding shapes similar to whirls in space.
method Intrinsic equation of curvature and torsion, position vector with arc length parameter.
result Necessary and sufficient conditions for the existence of the family of curves.
Floer homology applied to inscribing rectangles into curves.
problem Determining if a Jordan curve can inscribe a square.
method Constructing Floer homology from inscribed rectangles and using spectral invariants.
result A Jordan curve inscribes a square if its enclosed area exceeds half a circle's area.
Tensor measures chirality for curves, even those with rough edges.
problem Quantifying chirality for complex, possibly irregular curves.
method Developed a tensorial chirality measure for rigid filaments and curves.
result A curve's chirality can be determined by its twist about perpendicular axes.
Two curves in hyperbolic space share a bounded distance, leading to a minimizing surface.
problem Finding a surface minimizing area between two disjoint curves in hyperbolic space.
method Analyzing the asymptotic boundary of hyperbolic 3-space, applying Definition 1.8 for distance bounds, and proving Theorems 1.7 and 1.11.
result Existence of an area-minimizing surface between two disjoint curves with bounded distance.
New rectifiability criteria for finite-perimeter sets in Carnot groups.
problem Rectifiability of finite-perimeter sets in Carnot groups.
method Introducing a new notion of rectifiability based on cone properties and studying semigroups generated by horizontal half-spaces.
result Finite-perimeter subsets in Carnot groups can be covered by countably many subsets with cone properties, leading to countable rectifiability with respect to intrinsic Lipschitz graphs.
Sharp estimates on 2-step nilpotent Lie groups' metrics and cones.
problem Estimating asymptotic metrics in 2-step nilpotent Lie groups.
method Developed a novel technique to perturb rectifiable curves.
result Every 2-step nilpotent Riemannian Lie group is at bounded distance from its asymptotic cone.
If Γ is the range of a Jordan curve that bounds a convex set in R2, then 21(Γ+Γ)=co(Γ), where + is the Minkowski sum and co is the convex hull. Answering a question of V.N. Ushakov, we construct a simple closed curve in R3 with range Γ such that $\frac{1}{2}(…
The abstract proves the existence and regularity of Brakke flows starting from a given set.
problem Existence and regularity of Brakke flows starting from a given set.
method Proves the existence and regularity of Brakke flows using a closed countably 1-rectifiable set in R^2.
result For almost all time, the flow locally consists of a finite number of embedded curves of class W^{2,2} whose endpoints meet at junctions with angles of 0, 60, or 120 degrees.
We prove that 2 dimensional Integral currents (i.e. integer multiplicity 2 dimensional rectifiable currents) which are almost complex cycles in an almost complex manifold admitting locally a compatible symplectic form are smooth surfaces aside from isolated points and therefore are J-holomorphic curves.
Study natural and conjugate mates of Frenet curves in Lie groups.
problem Characterize Frenet curves and their mates in Lie groups.
method Introduced natural and conjugate mates, derived relationships, analyzed specific curves.
result Obtained results for various Frenet curves in Lie groups.
We provide a new proof of the classical result that any closed rectifiable Jordan curve Gamma in space being piecewise of class C^2 bounds at least one immersed minimal surface of disc-type, under the additional assumption that the total curvature of Gamma is smaller than 6*Pi. In contrast to the methods due to Osserma…
I show that every rectifiable simple closed curve in the plane can be continuously deformed into a convex curve in a motion which preserves arc length and does not decrease the Euclidean distance between any pair of points on the curve. This result is obtained by approximating the curve with polygons and invoking the r…
Deep neural networks enforce non-crossing quantile regression curves.
problem Estimating quantile regression curves without crossing.
method Penalized deep ReQU neural networks with a non-crossing penalty.
result Established non-asymptotic risk and error bounds for the estimated QRP.
In this work, we give parallel transport frame of a curve and we introduce the relations between the frame and Frenet frame of the curve in 4-dimensional Euclidean space. The relation which is well known in Euclidean 3-space is generalized for the first time in 4-dimensional Euclidean space. Then we obtain the conditio…
We deal with a notion of weak binormal and weak principal normal for non-smooth curves of the Euclidean space with finite total curvature and total absolute torsion. By means of piecewise linear methods, we first introduce the analogous notation for polygonal curves, where the polarity property is exploited, and then m…
Study of curves and surfaces in Riemannian spaces making a constant angle with a parallel transported direction.
problem Understanding geometric properties of curves and surfaces in Riemannian spaces.
method Developing a theoretical framework to study curves and surfaces by their angle with a parallel transported vector field.
result Surfaces making a constant angle with a parallel transported direction are extrinsically flat ruled surfaces.
In this paper we study the sub-Finsler geometry as a time-optimal control problem. In particular, we consider non-smooth and non-strictly convex sub-Finsler structures associated with the Heisenberg, Grushin, and Martinet distributions. Motivated by problems in geometric group theory, we characterize extremal curves, d…
Study characterizes k-rectifiable sets in homogeneous groups.
problem Characterizing k-rectifiable sets in arbitrary homogeneous groups. method Proves characterizations using (k,G)-approximate tangent groups. result Existence of (k,G)-approximate tangent groups implies k-rectifiability. New elastic energy for irregular curves defined through polygonal approximations.
problem Defining elastic energy for irregular curves in any space dimension.
method Relaxation process with p-rotation of inscribed polygonals, focusing on geometric curvature distribution. result Energy finite if and only if curve's arc-length parameterization has second order summability.
Flow of curves with curvature and forcing vector field exists.
problem Existence of a curve flow with curvature and forcing.
method Proved existence through Brakke motion law.
result Non-trivial flow of curves exists through singularities.
Harmonic maps to Euclidean buildings have rectifiable singular strata.
problem Understanding the structure of singular points for harmonic maps.
method Defining singular strata and proving rectifiability using the rectifiable Reifenberg program.
result Rectifiability of singular strata for harmonic maps into F-connected complexes. We prove (without using Federer's structure theorem) that a finite-mass flat chain over any coefficient group is rectifiable if and only if almost all of its 0-dimensional slices are rectifiable. This implies that every flat chain of finite mass and finite size is rectifiable. It also leads to a simple necessary and su…