For the class of quasi-periodic solutions of the vortex filament equation, we study connections between the algebro-geometric data used for their explicit construction and the geometry of the evolving curves. We give a complete description of genus one solutions, including geometrically interesting special cases such a…
On-device research index
arXiv research
A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
169,341 papers · 148 categories
Trend · papers per month
48 results for “filament equation”
If a curve in R^3 is closed, then the curvature and the torsion are periodic functions satisfying some additional constraints. We show that these constraints can be naturally formulated in terms of the spectral problem for a 2x2 matrix differential operator. This operator arose in the theory of the self-focusing Nonlin…
We study the topology of quasiperiodic solutions of the vortex filament equation in a neighborhood of multiply covered circles. We construct these solutions by means of a sequence of isoperiodic deformations, at each step of which a real double point is "unpinched" to produce a new pair of branch points and therefore a…
In this paper, the parallel transport frames over non-lightlike curves in Minkowski 3-space are introduced. Evolution equations of these frames with respect to arc length and time are calculated over the space of these curves. Then the equivalence of the non-linear Schrödinger equation and non-linear heat system to the…
We present a Hamiltonian framework for higher-dimensional vortex filaments (or membranes) and vortex sheets as singular 2-forms with support of codimensions 2 and 1, respectively, i.e. singular elements of the dual to the Lie algebra of divergence-free vector fields. It turns out that the localized induction approximat…
Discrete model of curve deformation using discrete nonlinear Schrödinger equation.
problem Deformation of discrete space curves.
method Discrete analogue of the local induction equation using the discrete nonlinear Schrödinger equation.
result Explicit formulas for smooth and discrete curves in terms of τ functions of the two-component KP hierarchy.
Proves local existence and uniqueness of SMCF in Euclidean spaces.
problem Proving existence and uniqueness of SMCF.
method Local existence and uniqueness proof for SMCF.
result Local existence and uniqueness of SMCF proved.
The bicycle model's front end trajectories are integrable in 3D.
problem Integrability of bicycle model trajectories.
method Bicycle kinematics model and Backlund transformation for the filament equation.
result The bicycle correspondence is integrable in 3D.
In this paper, we generalize the famous Hasimoto's transformation by showing that the dynamics of a closed unidimensional vortex filament embedded in a three-dimensional manifold of constant curvature gives rise under Hasimoto's transformation to the non-linear Schrodinger equation. We also give a natural interpretatio…
New flow generates surfaces with constant curvature.
problem Creating surfaces with specific curvature properties.
method Framed curvature flow, analyzing trajectory surfaces.
result Trajectory surfaces of constant mean or Gaussian curvature.
The bicycle model describes curve evolution and is linked to integrable systems.
problem Modeling bicycle motion and its geometric properties.
method Analytical and geometric methods, including differential equations and integrable systems.
result The bicycle model is related to integrable systems and describes curve evolution.
The large-scale structure of the universe is comprised of virialized blob-like clusters, linear filaments, sheet-like walls and huge near empty three-dimensional voids. Characterizing the large scale universe is essential to our understanding of the formation and evolution of galaxies. The density range of clusters, wa…
We define a class of geometric flows on a complete Kähler manifold to unify some physical and mechanical models such as the motion equations of vortex filament, complex-valued mKdV equations, derivative nonlinear Schrödinger equations etc. Furthermore, we consider the existence for these flows from into a complet…
Inhomogeneous plasmas filaments instabilities are investigated by using the techniques of classical differential geometry of curves where Frenet torsion and curvature describe completely the motion of curves. In our case the Frenet frame changes in time and also depends upon the other coordinates taking into account th…
In 2002, D. Hrencecin and L.H. Kauffman defined a filamentation invariant on oriented chord diagrams that may determine whether the corresponding flat virtual knot diagrams are non-trivial. A virtual knot diagram is non-classical if its related flat virtual knot diagram is non-trivial. Hence filamentations can be used …
This paper gives a polynomial invariant for flat virtual links. In the case of one component, the polynomial specializes to Turaev's virtual string polynomial. We show that Turaev's polynomial has the property that it is non-zero precisely when there is no filamentation of the knot, as described by Hrencecin and Kauffm…
In this paper, we discuss filamentations on oriented chord diagrams. When a filamentation cannot be realized on an oriented chord diagram, then the corresponding flat virtual knot is non-trivial. If a flat knot diagram is non-trivial, then any virtual diagram whose shadow is the flat diagram must also be non-trivial. W…
The Backlund transformation for pseudospherical surfaces, which is equivalent to that of the sine-Gordon equation, can be restricted to give a transformation on space curves that preserves constant torsion. We study its effects on closed curves (in particular, elastic rods) that generate multiphase solutions for the vo…
Gauss map of skew mean curvature flow satisfies Schrödinger flow.
problem Understanding the geometry of skew mean curvature flow.
method Exploring the oriented Grassmannian manifold and embedding it into exterior product space.
result Gauss map of SMCF satisfies a Schrödinger flow equation.
3D filament plots visualize curves in datasets, avoiding visual clutter.
problem Visualizing high-dimensional data relationships in scatterplots of points.
method Construct 3D filament plots using linear isometries and Frenet-Serret systems.
result 3D filament plots preserve Euclidean distances and avoid visual clutter.
The article extends vortex filament theory to Hermitian reductive Lie algebras.
problem Investigating vortex filaments in Hermitian reductive Lie algebras.
method Develops vortex models in a purely geometric way.
result Vortex models revert to known models in Hermitian symmetric Lie algebras.
Stable knots and links can exist in electromagnetic fields.
problem Stability of knots and links in electromagnetic fields.
method Proving the existence of electromagnetic fields preserving link topology.
result Every link can be realized as stable field lines in electromagnetic fields.
The theory of the vortex filament in three-dimensional fluid dynamics, consisting mainly of the models up to the third-order approximation, is an attractive subject in both physics and mathematics. Many efforts have been devoted to the extension of the theory to higher-dimensional symmetric Lie algebras. However, such …
A Lie algebra structure on variation vector fields along an immersed curve in a -dimensional real space form is investigated. This Lie algebra particularized to plane curves is the cornerstone in order to define a Hamiltonian structure for plane curve motions. The Hamiltonian form and the integrability of the planar…
Coupling between Phase Separation and Geometry on a Closed Elastic Curve: Free Energy Minimization and Dynamicscond-mat.soft
Study of phase separation and geometry on a closed elastic curve, including dynamics and free energy minimization.
problem Free energy and dynamics of a closed elastic filament coupled to a scalar concentration field.
method Analytical and numerical simulations of coupled Willmore flow and Cahn--Hilliard gradient flow on differential geometry.
result Qualitative changes in free energy landscape due to closure constraint, leading to metastable and stable multi-domain morphologies.
Study on elastic curves with variable stiffness, derived from bending energy.
problem Modeling elastic wires with varying thickness.
method Derive Euler-Lagrange equations for curves with variable bending stiffness.
result Characterizations of elastic curves with variable stiffness.
Counterexamples show no simple generalization of Hasimoto transform for higher-dimensional Euler fluids.
problem No straightforward generalization of Hasimoto transform for higher-dimensional Euler fluids.
method Derivation of evolution equations for mean curvature and torsion form for membranes.
result Existence of counterexamples implies no simple generalization of Hasimoto transform.
For a given manifold we consider the non-linear Grassmann manifold of -dimensional submanifolds in . A closed -form on gives rise to a closed 2-form on . If the original form was integral, the 2-form will be the curvature of a principal -bundle over . Using this $S^…
We discuss a short-time existence theorem of solutions to the initial value problem for a third order dispersive flow for closed curves into a compact almost Hermitian manifold. Our equations geometrically generalize a physical model describing the motion of vortex filament. The classical energy method cannot work for …
Chirality for crooked curvescond-mat.soft
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.
Topologically Directed Simulations Reveal the Impact of Geometric Constraints on Knotted Proteinsmath.GT
Method optimizes knotting pathways in constrained polymers.
problem Understanding how geometric constraints affect knot formation in polymers.
method Topological steering using knotoid spectrum and mean unravelling number.
result Geometric constraints increase the frequency of twist knots in polymers.
Generalizes Hasimoto transformation to arbitrary flows on space curves.
problem Modeling and analyzing wave motions on space curves.
method Develops a mapping between curve evolution and scalar equations, transforming general vector fields in the Frenet frame.
result Binormal flows are generally length preserving but bending energy is fragile.
Bayesian method classifies actin cytoskeleton networks using topological data.
problem Classifying the structure of biological networks, especially actin cytoskeleton networks.
method Transform actin cytoskeleton networks into persistence diagrams, quantify variability with Bayesian framework, estimate posterior distributions.
result Bayesian framework successfully classifies actin filament networks, outperforming state-of-the-art methods.
Euler-Plateau, with a twistcond-mat.soft
A generalization of the Euler-Plateau problem to account for the energy contribution due to twisting of the bounding loop is proposed. Euler-Lagrange equations are derived in a parameterized setting and a bifurcation analysis is performed. A pair of dimensionless parameters govern bifurcations from a flat, circular gro…
New method identifies knots in protein chains using virtual knots.
problem Identifying knots in open protein chains.
method Introducing virtual knots to analyze open curves without closure.
result Recovering and extending previous knotting results in proteins.
Study investigates multicomponent bilayers in Cahn-Hilliard models.
problem Stability and evolution of multicomponent bilayer interfaces.
method Developed mFCH model to study gradient flows of bilayer interfaces, derived sharp interface energy.
result Identified codimensional bifurcation mechanism leading to layer-by-layer pearling.
Two new algorithms recover ridge lines from point clouds with convergence guarantees.
problem Extracting filamentary structure from point clouds.
method Proposes two novel algorithms with convergence guarantees.
result The algorithms can asymptotically recover the full ridge set.
Study cosmic structures using Topological Data Analysis and Persistence Energy.
problem Investigate cosmic web evolution in CDM cosmologies.
method Apply LITE method to embed persistence diagrams into vector spaces and analyze cosmic structures.
result Discover a correlation between Persistence Energy and redshift values.
New material groupoid theory subdivides non-uniform bodies into smoothly uniform parts and isolated points.
problem Lack of differentiability in material bodies leads to non-uniformity.
method Introducing material groupoid and material distribution to study non-uniform bodies rigorously.
result Material bodies can be subdivided into smoothly uniform parts and isolated points.
Two-sample tests improve on existing methods for microtubule data.
problem Testing differences between two groups of filament data.
method Optimal lifts and manifold stability theorem applied to microtubule data.
result New tests outperform existing methods on simulated and real data.
In this mostly pedagogical tutorial article a brief introduction to modern geometrical treatment of fluid dynamics and electrodynamics is provided. The main technical tool is standard theory of differential forms. In fluid dynamics, the approach is based on general theory of integral invariants (due to Poincare and Car…
Paper defines and analyzes mathematical equivalence of periodic tangles.
problem Understanding the equivalence of periodic tangles.
method Established mathematical framework, characterized isotopies, generalized results.
result Characterization of DP tangle equivalence based on motifs.
In mathematics, the classical Plateau problem consists of finding the surface of least area that spans a given rigid boundary curve. A physical realization of the problem is obtained by dipping a stiff wire frame of some given shape in soapy water and then removing it; the shape of the spanning soap film is a solution …
Parallel algorithm speeds up Jones polynomial computation.
problem Efficient computation of knot complexity measures.
method First parallel algorithm for exact Jones polynomial computation.
result Reduces computational time by an exponential factor.
New geometric approach to curve flows in space forms.
problem Understanding geometric flows on space curves.
method Directly describe flows as vector fields on the manifold of space curves, using symplectic geometry.
result Characterization of elastic curves as solutions to an isoperimetric problem.
New geometric mechanics approach to elastic curves.
problem Understanding elastic curves in mechanics and geometry.
method Developed a new geometric mechanics perspective on elastic curves.
result Elastic curves are critical points of length under fixed area and volume constraints.
New methods assess topological entanglement in periodic systems.
problem Assessing topological entanglement in systems with periodic boundary conditions.
method Introducing Periodic Jones polynomial and Cell Jones polynomial.
result Periodic Jones polynomial is a recurring factor of Jones polynomial of finite cutoffs.
GANs generate realistic cosmic web models in seconds, overcoming computational limits.
problem Resource-intensive N-body simulations limit upcoming cosmology experiments.
method Generative Adversarial Networks (GANs) trained on 2D snapshots of N-body simulations.
result GAN-generated cosmic web models are qualitatively and quantitatively similar to originals.