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.
Study evolutes of curves with varying smoothness.
problem Understanding evolutes of curves with low smoothness.
method Analyzing the relationship between curve smoothness and evolute regularity.
result Evolutes have one less order of smoothness than the parent curve in generic cases.
Curve shortening flow's regularity depends on initial conditions after a certain time.
problem Understanding the regularity of evolving curves under curve shortening flow.
method Proposing and proving principles of controllable regularity based on initial conditions.
result No regularity estimate holds before a specific time, A/π. Study of generalized Bishop frames on curves in 4D space.
problem Understanding frames on curves in 4D space.
method Introducing and studying four types of generalized Bishop frames on curves in E4. result Every regular curve in E4 admits all four types of generalized Bishop frames. Characterizes minimizing curves in Riemannian manifolds.
problem Finding optimal paths in curved spaces.
method Characterization of prox-regular sets and tangent cones.
result Necessary condition for minimizing curves in prox-regular sets.
We define winding numbers of regular closed curves on surfaces with a nice euclidean or hyperbolic geometry. We prove that two regular closed curves are regularly homotopic if and only if they are freely homotopic and have the same winding number.
New estimate for Curve Shortening Flow improves graphical solutions.
problem Improving regularity estimates for Curve Shortening Flow.
method Generalizing delayed parabolic regularity for Curve Shortening Flow.
result Interior graphical estimate for Curve Shortening Flow.
Mathematical formulas for elliptic curve integrals solve anomaly equations.
problem Mathematical formulation of contact term singularities on elliptic curves.
method Residue formulas and holomorphic anomaly equations.
result Regularized integrals on elliptic curves satisfy holomorphic anomaly equations.
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 paper proves properties of curves in Riemannian manifolds.
problem Characterizing curves in Riemannian manifolds.
method Analyzing locally minimizing and weak geodesics.
result Locally minimizing curves are weak geodesics under certain conditions.
We prove that any two irreducible cuspidal Hurwitz curves C0 and C1 (or more generally, curves with A-type singularities) in the Hirzebruch surface FN with coinciding homology classes and sets of singularities are regular homotopic; and symplectically regular homotopic if C0 and C1 are symplectic with re…
The paper studies a flow of Legendre curves, generalizing the inverse curvature flow of regular curves.
problem Analyzing the inverse curvature flow of Legendre curves.
method Investigates the unique existence, monotonicity, and asymptotic behavior of the flow.
result The flow asymptotically converges to a self-similar solution, categorized by initial curve.
Groups with specific curvature have a regular language of geodesics.
problem Understanding the language of geodesics in non-positively curved triangle groups.
method Proving finitely many cone types and regularity of geodesic languages.
result The language of lexicographically first geodesics is regular and satisfies the fellow traveller property.
We study the horizontally regular curves in the Heisenberg groups Hn. We show the fundamental theorem of curves in Hn (n≥2) and define the concept of the orders for horizontally regular curves. We also show that the curve γ is of order k if and only if γ lies in Hk but not in Hk−1 up to a Heis…
In an earlier paper, I defined a new winding number of regular closed curves on complete euclidean/hyperbolic surfaces and showed that this winding number, together with the free homotopy class, determines the regular homotopy class. In this paper, I give a Whitney-type formula for the winding number of non-null-homoto…
We study pairs of curves with Poncelet's porism properties and compute their vertex curves.
problem Understanding pairs of curves with Poncelet's porism properties.
method Developed formulas to compute vertex curves for given envelope curves and vice versa, for all sufficiently regular pairs of Poncelet curves.
result Formulas produce all possible sufficiently regular pairs of Poncelet curves, including sets of curves analogous to pencils of conic sections.
Study of generalized Bishop frames on time-like curves in 4D Lorentz space.
problem Characterize frames for time-like curves in 4D Lorentz space.
method Introduced and studied generalized Bishop frames for regular time-like curves in 4D Lorentz space.
result Hierarchy of frames exists for time-like curves in 4D Lorentz space, similar to Euclidean case.
Study on Brownian motion on discrete curve spaces, proving stochastic completeness.
problem Analyzing Brownian motion on spaces of discrete curves.
method Introduced and studied Brownian motion on spaces of discrete regular curves with Sobolev-type metrics.
result All geodesically complete spaces of discrete regular curves are stochastically complete.
In this paper we study the general affine geometry of curves in affine space A2. For a regular plane curves we define two kinds of moving frames. The first is of minimal order in all moving frames.The second is the Frenet moving frame. We get the moving equations of these moving frames. And we prove that curvature a…
We define a computable topological invariant μ(γ) for generic closed planar regular curves γ, which gives an effective lower bound for the number of inflection points on a given generic closed planar curve. Using it, we classify the topological types of locally convex curves (i.e. closed planar regular curves witho…
Study reveals learning curves and benign overfitting in spectral algorithms for large dimensions.
problem Understanding learning curves and benign overfitting in spectral algorithms for large-dimensional data.
method Analysis of learning curves and benign overfitting in spectral algorithms for inner-product kernels on the sphere and general domains.
result Characterization of three distinct regimes: over-regularized, under-regularized, and interpolation regimes, revealing benign overfitting across both under-regularized and interpolation regimes.
Every curve can fit countless rhombuses.
problem Finding many rhombi within any curve.
method No curve regularity assumed.
result Uncountably many rhombi fit every curve.
Using the moving frame and invariants, any discrete curve in R3 could be uniquely identified by its centroaffine curvatures and torsions. In this paper, depending on the affine curvatures of the fractal curves, such as Koch curve and Hilbert curve, we can clearly describe their iterative regularities. Interestingly…
It is well-known that the class of piecewise smooth curves together with a smooth Riemannian metric induces a metric space structure on a manifold. However, little is known about the minimal regularity needed to analyze curves and particularly to study length-minimizing curves where neither classical techniques such as…
Survey on geodesics on tetrahedra in curved spaces.
problem Understanding geodesics on tetrahedra in curved spaces.
method Analyzing geodesics on regular tetrahedra in spaces of constant curvature.
result Results on the behavior of simple closed geodesics.
Study Lipschitz regularity for manifold-constrained ROF model on curved surfaces.
problem Lipschitz regularity for manifold-constrained ROF model on curved surfaces.
method Generalization of ROF model, existence and uniqueness of minimizers, regularity results on PDE system.
result Lipschitz regularity of minimizers without convexity requirements.
We analyze learning curves of RF models with convex regularization and derive precise asymptotic expressions.
problem Understanding the learning curves of RF models with general convex regularization.
method Novel multi-level application of the convex Gaussian min max theorem (CGMT) to compute precise asymptotic expressions.
result Precise asymptotic expressions for learning curves of RF models with separable strongly convex regularization or ℓ1 regularization. This paper proves geodesic curvature measures are bounded for curves near cross cap singularities.
problem Boundedness of geodesic curvature measures near cross cap singularities.
method Analyzes intrinsic cross cap singularities and extends Gauss-Bonnet formula.
result Proves boundedness of geodesic curvature measures for curves near cross cap singularities.
Study on isoperimetric inequalities and regularity of A-harmonic functions on surfaces.
problem Investigating isoperimetric inequalities and regularity of A-harmonic functions on smooth surfaces. method Logarithmic and power-type convexity of the length of level curves, higher Sobolev regularity properties, and estimates for derivatives.
result Higher Sobolev regularity properties of solutions, including W2,2 regularity. Regular Lie groups are infinite dimensional Lie groups with the property that smooth curves in the Lie algebra integrate to smooth curves in the group in a smooth way (an `evolution operator' exists). Up to now all known smooth Lie groups are regular. We show in this paper that regular Lie groups allow to push surprisi…
New metrics on curve spaces improve shape analysis.
problem Discretization of curve spaces and metric completeness.
method Sobolev metrics on discrete regular curves, completeness analysis.
result The finite-dimensional Riemannian manifolds are complete.
We demonstrate the breakdown of several fundamentals of Lorentzian causality theory in low regularity. Most notably, chronological futures (defined naturally using locally Lipschitz curves) may be non-open, and may differ from the corresponding sets defined via piecewise C1-curves. By refining the notion of a causal…
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.
We prove short-time existence of φ-regular solutions to the anisotropic and crystalline curvature flow of immersed planar curves.
New concept of regular separation for ODEs leads to improved Hardy field results.
problem Understanding solutions of definable ODEs with specific properties.
method Introducing regular separation and proving its implications for ODEs and vector fields.
result The regular separation property leads to improved Hardy field results and non-empty sets of trajectories.
Proves Gannon-Lee theorem for C1 spacetimes.
problem Classical singularity theorems for C1 spacetimes. method Proves theorem for C1 spacetimes, shows geodesic properties. result Gannon-Lee theorem holds for C1 spacetimes. We show short-time existence for curves driven by curve diffusion flow with a prescribed contact angle α∈(0,π): The evolving curve has free boundary points, which are supported on a line and it satisfies a no-flux condition. The initial data are suitable curves of class W2γ with γ∈(23,2]. For …
Square-like quadrilaterals inscribed in space curves proven for finite total curvature.
problem Square-peg problem in space curves.
method Local curvature analysis and limiting argument on approximating curves.
result Every embedded curve with finite total curvature has an inscribed square-like quadrilateral.
We establish an area-type formula for the intrinsic spherical Hausdorff measure of every regular curve embedded in an arbitrary graded group.
In this paper, we consider a regular curve on an oriented surface in Euclidean 3-space with the Darboux frame {T,V,U} along the curve, where T is the unit tangent vector field of the curve, U is the surface normal restricted to the curve and $\mathsf{V}=\mathsf{U}\ti…
Soft diamond regularizers improve deep learning performance and sparsity.
problem Improving deep learning performance and sparsity of trained weights.
method New soft diamond synaptic weight priors based on thick-tailed symmetric alpha stable probability curves.
result Soft diamond regularizers outperform state-of-the-art methods in deep learning tasks.
We consider the problem of minimizing the bending or elastic energy among Jordan curves confined in a given open set Ω. We prove existence, regularity and some structural properties of minimizers. In particular, when Ω is convex we show that a minimizer is necessarily a convex curve. We also provide an example of a…
The study counts closed elliptic curves and Reeb orbits on Vaisman and Sasakian manifolds.
problem Counting closed orbits and elliptic curves on Vaisman and Sasakian manifolds.
method Analyzes the structure of Vaisman and Sasakian manifolds, uses quasi-regular and S1-quotients, and counts closed orbits and curves. result The number of closed elliptic curves and Reeb orbits is either infinite or equal to the sum of all Betti numbers of a Kähler orbifold.
Sharp proof of sub-Riemannian length-minimizing curves being at least C2
problem Smoothness of sub-Riemannian length-minimizing curves
method Study of a class of sub-Riemannian structures, proving C2 regularity result Theorem 1.1 in [6] is sharp
Suppose S is a closed orientable surface and S~ is a finite sheeted regular cover of S. The following question was posed by Julién Marché in Mathoverflow: Do the lifts of simple curves from S generate H1(S~,Z)? A family of examples is given for which the answer is "no".
Low regularity spacetimes split into simpler structures.
problem Proving splitting theorem for C1 metrics and weights. method Combining elliptic techniques and line-adapted curves.
result Extends Lorentzian splitting theorem to C1 settings. In this paper, we study the computation of curvatures at the singular points of algebraic curves and surfaces. The idea is to convert the problem to compute the curvatures of the corresponding regular parametric curves and surfaces, which have intersections with the original curves and surfaces at the singular points. …
Hopf's Umlaufsatz relates the total curvature of a closed immersed plane curve to its rotation number. While the curvature of a curve changes under local deformations, its integral over a closed curve is invariant under regular homotopies. A natural question is whether one can find some non-trivial densities on a curve…