In this paper we obtain the general solution to the minimal surface equation, namely its local Weierstrass-Enneper representation, by using a system of hodographic coordinates. This is done by using the method of solving the Born-Infeld equations by Whitham. We directly compute conformal coordinates on the minimal surf…
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.
168,695 papers · 148 categories
Trend · papers per month
48 results for “hodographic coordinate system”
The paper classifies affine minimal translation surfaces and finds their properties.
problem Classifying and understanding affine minimal translation surfaces.
method Using Weierstrass-Enneper formula and hodographic coordinate system.
result Classification and properties of affine minimal translation surfaces.
We obtain the Weierstrass-Enneper representation for maximal graphs(whose Gauss map is one-one) in Lorentz-Minkowski space. For this we use the method of Barbishov and Chernikov, which they have used to find the solutions of Born-Infeld equation in hodographic coordinates. We could use their method in our case, because…
The generalised hodograph method for non-diagonalisable integrable systems of hydrodynamic typenlin.SI
Extends method for solving certain hydrodynamic systems.
problem Solving non-diagonalisable integrable systems of hydrodynamic type.
method Generalised hodograph method applied to F-manifolds with compatible connections.
result Provides general solution under certain assumptions.
Local high-degree polynomial integrals of geodesic flows and the generalized hodograph methodmath.DS
Study integrable geodesic flows on 2-surfaces with high-degree polynomial first integrals.
problem Integrable geodesic flows on 2-surfaces with high-degree polynomial first integrals.
method Semi-Hamiltonian systems of PDEs and generalized hodograph method.
result Construction of many local explicit and implicit integrable examples with polynomial first integrals of degrees 3, 4, 5.
The study finds the best elliptical trajectory for planets using a variation of the hodograph theorem.
problem Finding the best elliptical trajectory for planets.
method Using a variation of the circular hodograph theorem, the study finds the best fitting ellipse for planetary trajectories by minimizing the sum of square distances from the points to the plane.
result The study finds that the best fitting ellipse for planetary trajectories minimizes the sum of square distances from the points to the plane.
In this paper we construct multiparametric families of two dimensional metrics with polynomial first integral. Such integrable geodesic flows are described by solutions of some semi-Hamiltonian hydrodynamic type system. We find infinitely many conservation laws and commuting flows for this system. This procedure allows…
Characterizes mappings preserving Pythagorean-hodograph curves.
problem Preserving Pythagorean-hodograph curves in various dimensions.
method Proves conformal functions with square rational dilation are PH-preserving.
result Conformal functions with square rational dilation are the only PH-preserving mappings.
Starting from a homogeneous polynomial in momenta of arbitrary order we extract multi-component hydrodynamic-type systems which describe 2-dimensional geodesic flows admitting the initial polynomial as integral. All these hydrodynamic-type systems are semi-Hamiltonian, thus implying that they are integrable according t…
Hexagonal geodesic 3-websmath.DG
We prove that a surface carries a hexagonal 3-web of geodesics if and only if the geodesic flow on the surface admits a cubic first integral and show that the system of partial differential equations, governing metrics on such surfaces, is integrable by generalized hodograph transform method. We present some new local …
This paper is focused on geometric aspects of two particular types of finite-variable reductions in the dispersionless Toda hierarchy. The reductions are formulated in terms of "Landau-Ginzburg potentials" that play the role of reduced Lax functions. One of them is a generalization of Dubrovin and Zhang's trigonometric…
Paper develops a framework for hyperbolic Monge-Ampère equation on strips, proving well-posedness and stability.
problem Addressing the rigidity-flexibility dichotomy for wrinkled patterns in thin elastic sheets.
method Develops hodograph transformation and parametrix-corrector decomposition to handle corner singularities and prove well-posedness.
result Proves existence and uniqueness of hodograph weak solutions and derives energy estimates for stability.
The paper studies magnetic geodesic flows on 2-surfaces with integrable structures.
problem Analyzing magnetic geodesic flows on 2-surfaces with additional integrals.
method Constructing exact solutions to semi-Hamiltonian systems of PDEs using generalized hodograph method and Legendre transformation.
result Exact solutions constructed for semi-Hamiltonian systems of PDEs.
This paper constructs PH spline curves with prescribed arc lengths.
problem Interpolating points, tangent directions, and curvatures with prescribed arc-length.
method Local construction of planar PH biarc curves of degree 7.
result Prescribed arc-length can be satisfied for any data and any chosen ratio between boundary tangents.
The Dynnikov coordinate system puts global coordinates on the boundary of Teichmüller space of an --punctured disk. We survey the Dynnikov coordinate system, and investigate how we use this coordinate system to study pseudo--Anosov braids making use of results from Thurston's theory on surface homeomorphisms.
Study curvature and torsion in Gaussian distribution's dual coordinate system.
problem Characterize geometric invariants of Gaussian distribution.
method Investigate Riemannian curvature and torsion in a dual coordinate system of Gaussian distribution.
result Explicitly give Amari formulas in the new coordinate system.
Method constructs orthogonal curvilinear coordinates in constant curvature spaces.
problem Creating orthogonal coordinates in spaces of constant curvature.
method Modification of Krichever's method for Euclidean space, applied to constant curvature spaces.
result Examples of orthogonal coordinate systems on the sphere and hyperbolic plane constructed.
We study the limiting case of the Krichever construction of orthogonal curvilinear coordinate systems when the spectral curve becomes singular. We show that the case when the curve is reducible and all its irreducible components are rational curves the construction procedure reduces to solving systems of linear equatio…
In integrable hydrodynamic systems, coordinates exist where generators and symmetries are simple.
problem Existence of Riemannian invariants for integrable systems of hydrodynamic type.
method Finding coordinates where the generator and all symmetries are diagonal.
result In integrable hydrodynamic systems, there exist coordinates where the generator and all symmetries are diagonal.
Study integrable discretizations of cyclic systems with circular coordinate lines.
problem Integrable discretizations of 3D cyclic systems with circular coordinate lines.
method Investigate circle congruences and flat connections in the context of discrete cyclic systems.
result Characterization of circle congruences and existence of certain flat connections.
Orthogonal curvilinear coordinate systems and torsion-free sheaves over reducible spectral curvesmath.DG
Constructs coordinate systems from spectral curve sheaves.
problem Creating coordinate systems from spectral curve sheaves.
method Finite-gap integration methods for orthogonal curvilinear coordinates.
result Constructs coordinate systems over reducible spectral curves.
The study examines the regularity of branched immersions using special coordinate systems.
problem Understanding the regularity of branched immersions and their fundamental elements.
method Development and use of special coordinate systems to express maps with branch points, proving existence and regularity conditions for mean curvature vectors.
result Characterization and existence of special coordinate systems for branch immersions, proving regularity conditions for mean curvature vectors.
Carnot groups can be polarized if they have specific coordinate systems.
problem Understanding when Carnot groups can be polarized.
method Proving Carnot groups with certain coordinate systems are polarizable.
result Carnot groups with suitable horizontal polar coordinates are polarizable.
It is shown that, in four dimensions, it is possible to introduce coordinates so that an analytic metric locally takes block diagonal form. i.e. one can find coordinates such that for where . We call a coordinate system in which the metric takes this for…
Proposes local coordinate frames for improving model performance in complex dynamical systems.
problem Improving model performance in complex, non-linear, and time-dependent dynamical systems.
method Introduces roto-translation invariant local coordinate frames for geometric graphs.
result The approach outperforms state-of-the-art models in various complex scenarios.
Simplifies neural network models by explicitly enforcing constraints in Cartesian coordinates.
problem Learning dynamics of complex systems efficiently and accurately.
method Embedding systems into Cartesian coordinates and using Lagrange multipliers to enforce constraints.
result Explicitly enforcing constraints leads to a 100x improvement in accuracy and data efficiency.
Given a finite collection of vector fields on a manifold which span the tangent space at every point, we consider the question of when there is locally a coordinate system in which these vector fields are real analytic. We give necessary and sufficient, coordinate-free conditions for the existence of such a…
On a discretization of confocal quadrics. II. A geometric approach to general parametrizationsmath.DG
We propose a discretization of classical confocal coordinates. It is based on a novel characterization thereof as factorizable orthogonal coordinate systems. Our geometric discretization leads to factorizable discrete nets with a novel discrete analog of the orthogonality property. A discrete confocal coordinate system…
The study uses isothermic coordinates to analyze space-like surfaces with constant curvature.
problem Global properties of space-like surfaces with constant mean curvature in Lorentz-Minkowski space.
method Isothermic coordinate systems
result Global properties of space-like surfaces with constant mean curvature explored.
Interbank Credit and the Money Manufacturing Process. A Systemic Perspective on Financial Stabilityq-fin.GN
Interbank lending and borrowing occur when financial institutions seek to settle and refinance their mutual positions over time and circumstances. This interactive process involves money creation at the aggregate level. Coordination mismatch on interbank credit may trigger systemic crises. This happened when, since sum…
Extracts intrinsic spatial coordinates for complex agent systems to learn PDEs.
problem Modeling collective dynamics of heterogeneous agents.
method Data-driven extraction of intrinsic spatial coordinates, learning PDEs in emergent space.
result Collective dynamics can be approximated through learned PDEs in emergent coordinates.
Derives new orthogonal coordinates for evolving surfaces and curves.
problem Accounting for geometric effects in boundary layer asymptotics.
method Elementary derivation of orthogonal signed-distance coordinates.
result Provides vector calculus identities for these coordinates.
Bi-Hamiltonian geometry and canonical spectral coordinates for the Rational Calogero-Moser systemmath-ph
We reconsider the (rational) Calogero-Moser system from the point of view of bi-Hamiltonian geometry. By using geometrical tools of the latter, we explicitly construct set(s) of spectral canonical coordinates, that is, complete sets of Darboux coordinates defined by the eigenvalues and the eigenvectors of the Lax matri…
Twisted local systems on surfaces of finite type appear often in geometry and physics. Most of them arise geometrically as local systems of charts for pleated hyperbolic structures. Bonahon and Thurston's "shear-bend coordinates" parameterize these local systems of charts. On a surface …
Discrete conjugate systems are quadrilateral nets with all planar faces. Discrete orthogonal systems are defined by the additional property of all faces being concircular. Their geometric properties allow one to consider them as proper discretization of conjugate, resp. orthogonal coordinate systems of classical differ…
We present explicit formulas for the coordinates in which the Hamiltonians of the Benenti systems with flat metrics take natural form and the metrics in question are represented by constant diagonal matrices.
Discrete-time systems can be characterized by simple flat coordinates and their shifts.
problem Characterizing flatness of discrete-time systems.
method Developed a map from flat coordinates and their shifts to system state and input, fulfilling system equations identically.
result Derived necessary conditions for a system to be flat, without requiring differential geometry methods.
In this paper we attempt to give a systematic account on privileged coordinates and the nilpotent approximation of Carnot manifolds. By a Carnot manifold it is meant a manifold with a distinguished filtration of subbundles of the tangent bundle which is compatible with the Lie bracket of vector fields. This paper lies …
Improved SINDy autoencoder for identifying noisy dynamical systems.
problem Robust identification of noisy dynamical systems from data.
method Incorporates noise-separating neural network structures into SINDy autoencoder architecture.
result Accurately recovers latent dynamics and estimates measurement noise from noisy observations.
AgensFlow learns multi-agent coordination policies from experience.
problem Difficult coordination choices in multi-agent systems built on LLMs.
method Online policy learning from repeated trajectories, treating decisions as learnable.
result Learned routing improves coordination-heavy workflows over static wiring.
Einstein--Weyl spaces and dispersionless Kadomtsev--Petviashvili equation from Painlevé I and IInlin.SI
We present two constructions of new solutions to the dispersionless KP (dKP) equation arising from the first two Painlevé transcendents. The first construction is a hodograph transformation based on Einstein--Weyl geometry, the generalised Nahm's equation and the isomonodromy problem. The second construction, motivated…
Novel deep learning method predicts reaction coordinates and future MD trajectories.
problem Identifying optimal reaction coordinates for chemical reactions.
method Regularized Sparse Autoencoder (RSE) for discovering reaction coordinates and predicting MD trajectory evolution.
result RSE helps in choosing a small but important set of reaction coordinates.
This paper constructs a family of coordinate systems about a point on a quaternionic contact manifold, called quaternionic contact pseudohermitian normal coordinates. Once defined, conformal variations of the quaternionic contact structure induce changes on the coordinates which are studied in an effort to simplify the…
Symmetries of cross capsmath.DG
Paper classifies symmetries of cross caps using invariants.
problem Classifying symmetries of cross caps.
method Used Bruce-West's normal form and associated functions to create invariants.
result Classified possible symmetries on cross caps.
Superintegrable systems on curved manifolds found to have Hessian structures.
problem Characterizing superintegrable systems on curved manifolds.
method Identifying and computing Hessian coordinates for superintegrable systems.
result Examples of superintegrable systems in 2D and 3D have natural Hessian coordinates.
We consider complex Fenchel-Nielsen coordinates on the quasi-Fuchsian space of punctured tori. These coordinates arise from a generalisation of Kra's plumbing construction and are related to earthquakes on Teichmueller space. They also allow us to interpolate between two coordinate systems on Teichmueller space, namely…
New action-angle coordinates found for singular symplectic manifolds.
problem Existence of action-angle coordinates for singular symplectic manifolds.
method Action-angle theorem for folded symplectic integrable systems.
result New topological obstructions found for global existence of action-angle coordinates.
Confocal quadrics lie at the heart of the system of confocal coordinates (also called elliptic coordinates, after Jacobi). We suggest a discretization which respects two crucial properties of confocal coordinates: separability and all two-dimensional coordinate subnets being isothermic surfaces (that is, allowing a con…