We consider the Frenet-Serret geometry of null curves in a three and a four-dimensional Minkowski background. We develop a theory of deformations adapted to the Frenet-Serret frame. We exploit it to provide a Lagrangian description of the dynamics of geometric models for null curves.
Improved modeling of chaotic systems using time-delay embeddings and Frenet-Serret frame.
problem Identifying effective coordinate systems for nonlinear dynamical systems.
method Developed a new algorithm to identify more stable and accurate models from less data, leveraging the connection between HAVOK and Frenet-Serret frame.
result The sub- and super-diagonal entries of the linear model correspond to intrinsic curvatures in Frenet-Serret frame.
The Frenet frame generalizes the Park transform for multi-phase circuits.
problem Generalizing the Park transform for multi-phase circuits.
method Using the Frenet frame and Cartan's moving frames.
result The Frenet frame provides a new approach to circuit analysis.
We study variational systems for space curves, for which the Lagrangian or action principle has a Euclidean symmetry, using the Rotation Minimising frame, also known as the Normal, Parallel or Bishop frame. Such systems have previously been studied using the Frenet-Serret frame. The Rotation Minimising frame has many a…
Let γ:I→Rn be a parametric curve of class Cn+1, regular of order n. The Frenet-Serret apparatus of γ at γ(t) consists of a frame e1(t),…,en(t) and generalized curvature values κ1(t),…,κn−1(t). Associated with each point of γ there are also local singular vecto…
Notes on Frenet-Serret formulas for curves in flat pseudo-hermitian manifolds.
problem Analyzing curves in flat pseudo-hermitian manifolds.
method Deriving Frenet-Serret formulas and applying them to specific conditions.
result Characterizations of curves and classification based on their geometric properties.
The submanifold Dirac operator has been studied for this decade, which is closely related to Frenet-Serret and generalized Weierstrass relations. In this article, we will give a submanifold Dirac operator defined over a surface immersed in $\EE^4$ with U(1)-gauge field as torsion in the sense of the Frenet-Serret relat…
The paper explores quaternionic curves using differential geometry.
problem Understanding quaternionic curves.
method Differential geometry applied to quaternionic curves.
result Simpler formulations of quaternionic curves.
A set of equations is developed to describe a curve in space given the curvature κ and the angle of rotation θ of the osculating plane. The set of equations has a solution (in terms of κ and θ) that indirectly solves the Frenet-Serret equations, with a unique value of θ for each specified value of τ. Explic…
Any two equivalent discrete curves must have the same invariants at the corresponding points under an affine transformation. In this paper, we construct the moving frame and invariants for the discrete centroaffine curves, which could be used to discriminate the same discrete curves from different graphics, and estimat…
The study quantifies geometric differences between axonal branches using splines.
problem Neuromorphology's focus on macroscopic features neglects neuron internal geometry.
method Fitting splines to neuron traces, using Frenet-Serret formulas to compute curvature and torsion.
result Parameters of curvature and torsion are distributed differently between axonal branches.
Study constructs closed curves with constant curvature on cylinders and tori.
problem Creating closed curves with constant curvature.
method ODEs and symmetry arguments, starting with cylinders, then tori, and finally Frenet-Serret equations.
result Closed constant curvature space curves constructed on cylinders and tori.
In this work, first, we express some characterizations of helices and ccr curves in the Euclidean 4-space. Thereafter, relations among Frenet-Serret invariants of Bertrand curve of a helix are presented. Moreover, in the same space, some new characterizations of involute of a helix are presented.
This article is one of a series of papers. For this decade, the Dirac operator on a submanifold has been studied as a restriction of the Dirac operator in n-dimensional euclidean space $\EE^n$ to a surface or a space curve as physical models. These Dirac operators are identified with operators of the Frenet-Serret re…
Using the submanifold quantum mechanical scheme, the restricted Dirac operator in a submanifold is defined. Then it is shown that the zero mode of the Dirac operator expresses the local properties of the submanifold, such as the Frenet-Serret and generalized Weierstrass relations. In other words this article gives a re…
This paper presents a new framework for manifold learning based on a sequence of principal polynomials that capture the possibly nonlinear nature of the data. The proposed Principal Polynomial Analysis (PPA) generalizes PCA by modeling the directions of maximal variance by means of curves, instead of straight lines. Co…
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.
For this quarter of century, differential operators in a lower dimensional submanifold embedded or immersed in real n-dimensional euclidean space $\EE^n$ have been studied as quantum mechanical models, which are realized as restriction of the operators in $\EE^n$ to the submanifold. For this decade, the Dirac operato…
Bertrand framed surfaces defined in Euclidean 3-space with applications.
problem Defining and characterizing Bertrand framed surfaces.
method Using moving frames to define Bertrand framed surfaces and analyzing their caustics and involutes.
result Conditions for caustics and involutes to be inverse operations of framed surfaces.
Quaternionic frames' admissibility and homotopy proven.
problem Existence and interpolation of quaternionic frames.
method Interpreting frames as adjoint orbits.
result Spaces of quaternionic frames are path-connected.
Introduces hyperbolic generalized framed surfaces and their properties.
problem None explicitly stated; focuses on introducing new geometric objects.
method Generalization of hyperbolic framed surfaces and curves.
result Established conditions for a surface to be a hyperbolic generalized framed base surface and explored their singularities.
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. The main drawback of the Frenet frame is that it is undefined at those points where the curvature is zero. Further- more, in the case of planar curves, the Frenet frame does not agree with the standard framing of curves in the plane. The main drawback of the Bishop frame is that the principle normal vector N is not in …
New framed moves extend classical knot theory results.
problem Extending classical knot theory to framed braids.
method Introduced framed versions of L-moves, Hilden, Pure Hilden groups, and framed versions of the Birman theorem.
result Proved a framed version of the Birman theorem for framed links in plat representation.
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.
The paper extends BPS invariants for framed knots and links.
problem Investigating BPS invariants for framed knots and links.
method Using the dual A-polynomial and framing change formula, the paper extends the relationship between algebraic curves and BPS invariants to framed knots and links.
result Explicit formulas for extremal A-polynomials and BPS invariants of framed knots, and numerical calculations for framed Whitehead links and Borromean rings.
Simply connected spaces of tight frames identified.
problem Understanding the connectivity of spaces of tight frames.
method Viewing tight frames as elements of Stiefel manifolds and identifying simply connected spaces.
result Spaces of tight frames, including finite unit-norm tight frames, are simply connected.
Gradient descent constructs tight fusion frames.
problem Constructing tight fusion frames from prescribed subspaces.
method Gradient descent and symplectic geometry.
result Gradient descent can be used to construct tight fusion frames.
New invariants defined for framed knots and links.
problem Defining invariants for framed knots and links.
method Introducing birack brackets and categorifying their multiset.
result Quiver-valued invariant defined for framed knots and links.
Study reveals LLM personas have two distinct components: frame-robust aggregated traits and frame-dependent geometric features.
problem Evaluation of LLM personas via psychometric questionnaires discards within-instance correlation structure.
method Constructed within-instance correlation matrices from IPIP-50 responses and analyzed geometry on SPD manifolds under manipulated question orderings.
result Persona expression comprises two dissociable components: aggregated features (Big Five scores) and geometric features (SPD manifold).
Higher-dimensional Milnor frames are characterized and contrasted with 3D Heisenberg and 4D nilpotent Lie algebras.
problem Characterizing higher-dimensional Milnor frames and their properties.
method Definition and classification of higher-dimensional Milnor frames and their relationship to known Lie algebras.
result Higher-dimensional Milnor frames are isomorphic to direct sums of 3D Heisenberg and 4D nilpotent Lie algebras and an abelian Lie algebra.
Canonical framings and stable framings for the tangent bundle of a spin 3-manifold are introduced, and illustrated by a number of familiar examples. Methods for constructing canonical framings, and for comparing them with other naturally defined framings, are discussed.
Study on focal surfaces of lightcone framed surfaces in Lorentz-Minkowski 3-space.
problem Investigate differential geometry properties of focal surfaces of lightcone framed surfaces.
method Introduced lightcone frame to define lightcone framed surfaces, then investigated their differential geometry properties.
result Investigated differential geometry properties of focal surfaces of lightcone framed surfaces.
The paper shows that random frames have full spark with high probability.
problem The probability of a random frame having full spark.
method Relating frame spaces to toric symplectic manifolds to analyze geometric and spectral properties.
result The probability of a random frame having full spark is one.
The paper connects hyperbolic spinors to non-null framed curves in Minkowski 3-space.
problem Understanding geometric properties of non-null framed curves.
method Developed new adapted frames for non-null framed curves and investigated their hyperbolic spinor representations.
result Found geometric results and interpretations for non-null framed curves.
This note is dedicated to the study of a Hopf module structures on the space of framed chord diagrams and framed graphs. We also introduce a framed version of the chromatic polynomial and propose two methods to construct framed weight systems.
Clarifies Einstein-Cartan gravitation with Dirac spinor on generalized frame bundle.
problem Formulating Einstein-Cartan gravitation on a frame bundle.
method Integrates Dirac spinor into the Einstein-Cartan spacetime structure.
result Variational equations imply standard field equations under standard frame bundle condition.
Frame-spun knots are constructed by spinning a knot of lower dimension about a framed submanifold of S^n. We show that all frame-spun knots are slice (null-cobordant).
Introduces formal frames for manifolds and their properties.
problem Understanding and generalizing frames and connections on manifolds.
method Introduces formal frames, canonical forms, and torsions.
result Equivalence of vanishing torsions to realizability of formal frames as ordinary frames.
Generalized Frenet frames for singular space curves
problem Revisiting the Frenet frame for singular space curves
method Introducing a generalized Frenet frame and frame sequence
result Unified framework for Frenet and Bishop frames
Study of lightcone framed surfaces in Lorentz-Minkowski 3-space, focusing on curvature behavior.
problem Investigate differential geometric properties of lightcone framed surfaces.
method Introduced modified frame to study the properties of lightcone framed surfaces.
result Showed behavior of Gaussian and mean curvatures at lightlike and singular points.
Modified knotoids with framing and coframing for quantum invariants.
problem Defining and classifying knotoids with framing.
method Defining framed and biframed knotoids, showing topological correspondence, and constructing quantum invariants.
result Generalized quantum knotoid invariants constructed.
The paper studies circular evolutes and involutes of framed curves in Euclidean space.
problem Investigating properties of framed curves and their evolutes and involutes.
method Definition and analysis of circular evolutes and involutes of framed curves, properties of normal surfaces, and their relations.
result Circular evolutes and involutes of framed curves are opposite operations under suitable assumptions, similar to fronts in the Euclidean plane.
The generic singularities and bifurcations are classified for one-parameter families of curves with frames in a space form, the Euclidean space, the elliptic space or the hyperbolic space via projective geometry. Two kinds of frames are considered, adapted frames and osculating frames, in terms of certain differential …
The paper explores Parseval frames on vector bundles, proving their existence for certain cases.
problem Existence of Parseval frames on vector bundles.
method Using G-bundles and algebraic topology, the authors prove the existence of Parseval frames for orientable vector bundles and provide conditions for smaller size frames. result The existence of Parseval frames for orientable vector bundles and conditions for smaller size frames.
Quantizes neural networks using frame theory for improved accuracy.
problem Improving neural network efficiency and accuracy through quantization.
method Sigma-Delta (ΣΔ) quantization with finite unit-norm tight frames. result Error bound between original and quantized neural networks derived.
In many signal processing problems, it may be fruitful to represent the signal under study in a frame. If a probabilistic approach is adopted, it becomes then necessary to estimate the hyper-parameters characterizing the probability distribution of the frame coefficients. This problem is difficult since in general the …
It is well-known that a knot in a contact manifold (M,C) transverse to a trivialized contact structure possesses the natural framing given by the first of the trivialization vectors along the knot. If the Euler class eC∈H2(M) of C is nonzero, then C is nontrvivializable and the natural framing of transvers…