The paper examines the behavior of Weierstrass measures on stable curves as they approach a nodal stable curve.
problem Understanding the behavior of Weierstrass measures on stable curves as they approach a nodal stable curve.
method Analyzing the limiting behavior of Weierstrass measures on a smooth curve of genus g⩾2 as it approaches a nodal stable curve in the Deligne-Mumford compactification. result The Weierstrass measures on a stable rational curve at the boundary of Mg are completely determined. Characterizes rotational solitons for curve shortening flow on revolution surfaces.
problem Understanding the behavior of curves under curve shortening flow on revolution surfaces.
method Characterization and asymptotic behavior analysis.
result Asymptotic behavior of rotational solitons to parallel geodesics.
In this paper we prove a universal inequality describing the asymptotic behavior of support points for planar continuous curves. As corollaries we get an analogous result for tangent points of differentiable planar curves and some (partially known) assertions on the asymptotic of the mean value points for various class…
In this paper, we analyze the asymptotic behavior of κ-noncollapsed and positively curved steady Ricci solitons and prove that any n-dimensional κ-noncollapsed steady Kähler-Ricci soliton with non-negative sectional curvature must be flat.
In this paper, we introduce the pseudo-torsion functions along spacelike curves whose curvature vector field has isolated lightlike points in Lorentz-Minkowski 3-space, and prove the fundamental theorem. Moreover, we analyze the behavior of the torsion function at such points. As a corollary, we obtain a necessary and …
We describe the curves of constant (geodesic) curvature and torsion in the three-dimensional round sphere. These curves are the trajectory of a point whose motion is the superposition of two circular motions in orthogonal planes. The global behavior may be periodic or the curve may be dense in a Clifford torus embedded…
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.
Study of Moncrief lines' behavior in curved space-times.
problem Understanding the asymptotic behavior of Moncrief lines in curved space-times.
method Analysis of geodesic laminations and convergence to Thurston boundary.
result Moncrief lines converge to a unique point in the Thurston boundary.
In this paper we study the convergence behavior of grafting rays to the Thurston boundary of Teichmuller space. When the grafting is done along a weighted system of simple closed curves or along a maximal uniquely ergodic lamination this behavior is the same as for Teichmuller geodesics and lines of minima. We also sho…
The paper studies geometric flows of spacelike curves in Lorentz-Minkowski plane and proves their long-term behavior.
problem Investigating geometric flows of spacelike curves in Lorentz-Minkowski plane.
method Examining the evolution of spacelike curves along prescribed geometric flows, including curve shortening and mean curvature flows.
result The geometric flows of spacelike curves in Lorentz-Minkowski plane exist for all time and converge to specific curves as time tends to infinity.
We determine the asymptotic behavior of the optimal Lipschitz constant for the systole map from Teichmuller space to the curve complex.
We show that when the genus and punctures of a surface are directly proportional by some rational number the minimal asymptotic translation length in the curve complex has behavior inverse to the square of the Euler characteristic. We also show that when the genus is fixed and the number of punctures varies the behavio…
The paper studies curve shortening flows on non-convex surfaces.
problem Behavior of curve shortening flows on non-convex surfaces.
method Defined a graph property and proved its preservation under curve shortening flow.
result The curve becomes a graph after a finite time under the curve shortening flow.
Study predicts success of crypto-tokens on Pump.fun platform.
problem Identify factors affecting the success of new crypto-tokens.
method Build predictive models using bonding curve mechanism and structural/behavioral variables.
result Conditional variables significantly improve the predictive power of token success.
The paper studies the free elastic flow of closed curves and finds their asymptotic shape converges to a circle.
problem Challenges in studying the asymptotic behavior of the free elastic flow for closed curves.
method Analysis of the free elastic flow as an L2-gradient flow for Euler's elastic energy. result An appropriate rescaling of initial curves geometrically close to circles converges to a unique round circle.
Method constructs WP geodesics to study Teichmüller space behavior.
problem Understanding behavior of WP geodesics in Teichmüller space.
method Constructs WP geodesics with specific properties.
result Demonstrates disparity between geodesics and hyperbolic 3-manifold curves.
We investigate the evolution of open curves with fixed endpoints under the curve shortening flow, which evolves curves in proportion to their curvature. Using a distance comparison of Huisken, we determine the long-term behavior of open curves with fixed endpoints evolving in certain convex domains on surfaces of const…
It is shown that the curvature function satisfies a nonlinear evolution equation under the general curve shortening flow and a detailed asymptotic behavior of the closed curves is presented when they contract to a point in finite time.
In this paper is studied the behavior of principal curvature lines near a curve of umbilic points of a smooth surface.
Curved metrics on Wallach spaces bounded by curves under flow.
problem Properties of positively curved Riemannian metrics on Wallach spaces.
method Normalized Ricci flow analysis on specific Wallach spaces.
result Set of metrics forms bounded curves asymptotically.
Learning curves show more data doesn't always improve performance.
problem The surprising finding that more data doesn't always lead to better generalization performance.
method A survey of learning curves, focusing on those that indicate more data doesn't necessarily improve performance.
result Learning curves can show that more data doesn't always lead to better generalization performance.
Investigates polar tangential angles of curves and their monotonicity.
problem Behavior of polar tangential angles of plane curves.
method Proof of monotonicity for certain curves of monotone curvature.
result Nonexistence results for an obstacle problem involving free elasticae.
For a closed real algebraic plane affine curve dividing its complexification and equipped with a complex orientation, the Whitney number is expressed in terms of behavior of its complexification at infinity.
Study ruled surfaces with finite multiplicity, focusing on their curves and singularities.
problem Understanding ruled surfaces with finite multiplicity.
method Analyzing striction curves and singularities of ruled surfaces.
result Geometric meanings of invariants related to ruled surfaces.
In this paper, we systemally study the long time behavior of the curve shortening flow in a closed or non-compact complete locally Riemannian symmetric manifold. Assume that we have a global flow. Then we can exhibit a a limit for the global behavior of the flow. In particular, we show the following results. 1). Let $\…
We study the role of co-jumps in the interest rate futures markets. To disentangle continuous part of quadratic covariation from co-jumps, we localize the co-jumps precisely through wavelet coefficients and identify statistically significant ones. Using high frequency data about U.S. and European yield curves we quanti…
The paper studies Möbius energy gradient of helix pairs and finds limiting behavior as coiling ratio increases.
problem Characterizing the limiting behavior of Möbius energy gradient for symmetric helix pairs.
method Complex asymptotics
result The gradient diverges in opposing directions based on radius, approaching 1/2 as coiling ratio increases.
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.
We consider an evolving plane curve with two endpoints that can move freely on the x-axis with generating constant contact angles. We discuss the asymptotic behavior of global-in-time solutions when the evolution of this plane curve is governed by area-preserving curvature flow equation. The main result shows that an…
In 1872 G. Darboux defined a family of curves on surfaces of R^3 which are preserved by the action of the Mobius group and share many properties with geodesics. Here we characterize these curves under the view point of Lorentz geometry and prove some general properties and make them explicit them on simple surfaces, re…
The paper studies heat behavior on curved spaces without radiality assumption.
problem Analyzing heat behavior on curved spaces.
method Examining heat equation solutions on specific Riemannian manifolds.
result Long-time convergence results hold on more general manifolds.
Anisotropic curvature flow of networks shows unique solutions and behavior under finite time.
problem Existence and behavior of networks under anisotropic curvature flow.
method Existence, uniqueness, and regularity of maximal geometric solutions proven.
result Existence of maximal geometric solutions and behavior under finite time.
Paper disproves Wright's periodic map conjecture.
problem Existence of periodic maps on closed hyperbolic surfaces.
method Analyzes automorphisms and curve behavior on surfaces.
result No periodic map can fill every curve with its image.
We prove that the space of complete, finite volume, pinched negatively curved Riemannian metrics on a smooth high-dimensional manifold is either empty or it is highly non-connected, provided their behavior at infinity is similar.
The paper extends curve deformation methods in Minkowski plane.
problem Studying deformations of curves in the Minkowski plane considering their geometry and singularities.
method Extends methods from [17, 18] to analyze 2-parameter families of curves in Minkowski plane.
result Obtains geometry of deformed curves, including inflections, vertices, and lightlike points.
This paper concerns with the asymptotic behavior of complete non-compact convex curves embedded in R2 under the α-curve shortening flow for exponents α>21. We show that any such curve having in addition its two ends asymptotic to two parallel lines, converges under α-curve shortening flow to the …
This paper corrects an error in [Keller-Ressel, M. and Steiner T. "Yield curve shapes and the asymptotic short rate distribution in affine one-factor models." Finance and Stochastics 12.2 (2008): 149-172]. The error concerns the correct expression for the boundary between normal and humped yield curve behavior in affin…
Study complex lines in symplectic geometry, generalizing previous results.
problem Understanding symplectic aspects of complex lines and their associated currents.
method Systematic study using Ahlfors currents, generalizing previous results.
result Ahlfors currents control the asymptotic behavior of pseudoholomorphic curves, showing convexity of the space of currents.
Proves identities linking curve lengths and orthogeodesics on hyperbolic surfaces.
problem Exploring relationships between curve lengths and orthogeodesics on hyperbolic surfaces.
method Partitioned orthogeodesics into sets based on dynamical behavior, relating them to geodesics on orbifold surfaces.
result Extends a result to surfaces with cusps, showing how to extend a previous result.
There is a natural duality between line congruences in R3 and surfaces in R4 that sends principal lines into asymptotic lines. The same correspondence takes the discriminant curve of a line congruence into the parabolic curve of the dual surface. Moreover, it takes the ridge curves to the flat r…
We show that for many strata of Abelian differentials in low genus the sum of Lyapunov exponents for the Teichmueller geodesic flow is the same for all Teichmueller curves in that stratum, hence equal to the sum of Lyapunov exponents for the whole stratum. This behavior is due to the disjointness property of Teichmuell…
Study on curve shortening flow in 3D space curves, showing convexity preservation and avoidance principle.
problem Analyzing the behavior of space curves under curve shortening flow in R3. method Analysis of properties of space curves evolved by the curve shortening flow, including convexity preservation and avoidance principle.
result Orthogonal projections of space curves remain convex, and the Avoidance principle is shown for spherical curves.
In the first part of the paper we survey some nonlocal flows of convex plane curves ever studied so far and discuss properties of the flows related to enclosed area and length, especially the isoperimetric ratio and the isoperimetric difference. We also study a new nonlocal flow of convex plane curves and discuss its e…
The paper proves an inequality and describes a curve flow in centro-affine geometry.
problem Proving the isoperimetric inequality in centro-affine plane geometry.
method Investigating a curve flow with centro-affine curvature, expressed as a nonlinear parabolic equation.
result Closed convex curves may converge to ellipses under the described flow.
New method shortens and straightens curves, proving convergence and well-posedness.
problem Shortening and straightening of curves.
method Conceptual shift in curve shortening to tangent aligning, variational study of geometric flows.
result Proves convergence to a straight line and global well-posedness for various geometric flows.
Plotting a learner's average performance against the number of training samples results in a learning curve. Studying such curves on one or more data sets is a way to get to a better understanding of the generalization properties of this learner. The behavior of learning curves is, however, not very well understood and…
Study variational properties of curves in half-plane with area constraints.
problem Characterize critical points of inverse mean curvature.
method Variational analysis of curves with boundary constraints.
result Existence and stability of critical points with prescribed area.
We consider the motion by curvature of a network of curves in the plane and we discuss existence, uniqueness, singularity formation and asymptotic behavior of the flow.