New example shows multisymplectic forms without Darboux charts.
problem Existence of multisymplectic forms without Darboux charts.
method Construction of multisymplectic three-forms on R6. result Provided an explicit counterexample to Darboux theorem in multisymplectic framework.
The study offers conditions for Darboux charts on specific types of manifolds.
problem Existence of Darboux charts on weakly symplectic manifolds.
method Using Moser's trick to find sufficient conditions.
result Sufficient conditions for Darboux charts on weakly symplectic manifolds.
The paper finds charts for Legendrian curves in complex contact manifolds.
problem Finding charts for Legendrian curves in complex contact manifolds.
method Holomorphic Darboux charts and approximations of Legendrian immersions.
result Holomorphic Legendrian immersions can be approximated by embeddings.
Suppose M be the projective limit of weak symplectic Banach manifolds \{(M_i,φ_{ij})\}_{i,j\in\mathbb N}, where M_i are modeled over reflexive Banach space and σis compatible with the inverse system(defined in the article). We associate to each point x\in M, a Fréchet space H_x(defined in section 3). We prove that if H…
For a finite dimensional symplectic manifold (M,ω) with a symplectic form ω, corresponding loop space (LM=C∞(S1,M)) admits a weak symplectic form Ωω. We prove that the loop space over $\mbr^n$ admits Darboux chart for the weak symplectic structure Ωω. Further, we show that inclusion map from the symp…
We prove the vanishing of the first Chern class of a codimension 2 closed contact submanifold of a cooriented contact manifold with trivial integral 2-dimensional cohomology group. Hence the first Chern class is an obstruction for the existence of codimension 2 contact embeddings in a Darboux chart. For the existence o…
We study the number of Darboux charts needed to cover a closed connected symplectic manifold (M,ω), and effectively estimate this number from below and from above in terms of the Lusternik--Schnirelmann category of M and the Gromov width of (M,ω).
The study of multisymplectic structures using Spencer cohomology.
problem Integrability of multisymplectic structures.
method Applying Spencer cohomology to identify multisymplectic structures as G-structures and giving conditions for integrability. result Conditions for a multisymplectic form to admit a chart with constant coefficients.
In this paper we compute the minimal number of Darboux chart needed to cover a Hermitian symmetric space of compact type in terms of the degree of their embeddings in CPN. The proof is based on the recent work of Y. B. Rudyak and F. Schlenk [18] and on the symplectic geometry tool developed by the first au…
Classifies different types of Darboux transformations for multidimensional operators.
problem Classifying Darboux transformations for multidimensional operators.
method Analyzes all known types of Darboux transformations and introduces new types.
result Full classification of first-order Darboux transformations and a description of higher-order transformations.
Defines and studies minimal charts with theorems.
problem Characterizing minimal charts with theorems.
method Definitions and fundamental theorems.
result Established fundamental theorems for minimal charts.
No minimal charts with exactly seven white vertices found.
problem Finding minimal charts with specific vertex counts.
method Investigating charts representing embedded surfaces in 4-space.
result No minimal chart with exactly seven white vertices exists.
This paper studies minimal charts of a specific type to understand embedded surfaces in 4-space.
problem Investigating minimal charts of a specific type to understand embedded surfaces in 4-space.
method Analyzing charts of type (5,2) to find a minimal chart.
result Identified a minimal chart of type (5,2) representing an embedded surface in 4-space.
The paper proves conditions for Darboux integrability in diagonal hydrodynamic systems.
problem Conditions for Darboux integrability in diagonal hydrodynamic systems.
method Proof of conditions using Laplace transformation sequences and geometric interpretations.
result Diagonal systems of hydrodynamic type are Darboux integrable if and only if the corresponding systems for commuting flows are Darboux integrable.
No minimal chart of type (7) exists.
problem Characterizing minimal charts of specific types.
method Analyzing charts based on edge labels and white vertex counts.
result There is no minimal chart of type (7).
Paper defines conditions for chart equivalence and composition.
problem Conditions for chart equivalence and composition.
method Defined conditions for charts to be equivalent and composed.
result Charts can be composed if they meet certain conditions.
No minimal chart of type (4,3) exists in 4-space.
problem Existence of minimal charts of specific type in 4-space.
method Investigation of charts and their properties in 4-space.
result No minimal chart of type (4,3) exists.
Investigates Darboux rectifying curves on smooth surfaces.
problem Characterizing Darboux rectifying curves on smooth surfaces.
method Analyzes the position vector under isometry and finds conformal invariance conditions.
result Identifies sufficient conditions for conformal invariance of Darboux rectifying curves.
We study an analogue of the classical Bianchi-Darboux transformation for L-isothermic surfaces in Laguerre geometry, the Bianchi-Darboux transformation. We show how to construct the Bianchi-Darboux transforms of an L-isothermic surface by solving an integrable linear differential system. We then establish a permutabili…
Paper classifies surface-links using charts with specific properties.
problem Classifying surface-links with specific charts.
method Using quandle colorings to differentiate charts representing different surface-links.
result Charts in the second class represent different surface-links.
Introduces Darboux-Lie derivative for fiber bundles.
problem None explicitly stated; focuses on introducing a new derivative.
method Study of Darboux-Lie derivative for fiber-bundle maps.
result Properties of Darboux-Lie derivative for fiber bundles.
The paper studies 4-charts with three crossings and their equivalence to a specific knot.
problem Investigating the structure and equivalence of 4-charts with three crossings.
method Examining charts as oriented labeled graphs in a disk, focusing on acyclic components and equivalence through label-orientation-reflection.
result Any linear minimal 4-chart with three crossings is equivalent to a 2-twist spun trefoil knot.
Researchers create coordinates for hyperbolic surfaces, proving a magic formula.
problem Constructing coordinates for hyperbolic structures on genus-2 surfaces.
method Developed Fenchel-Nielsen coordinates and Wolpert's magic formula analogues.
result Found Darboux charts for the Goldman symplectic form on branched hyperbolic structures.
Minimal surfaces proof via Darboux transformations.
problem Minimal surface properties and transformations.
method Christoffel, Goursat, and Darboux transformations of isothermic surfaces.
result Simple proof of Darboux pairs relation.
No minimal chart of type (2,3,2) exists.
problem Characterizing minimal charts of specific types.
method Analyzing charts with given edge label constraints.
result No minimal chart of type (2,3,2) exists.
No minimal chart of type (3,2,2) exists.
problem Characterizing minimal charts of specific types.
method Analyzing charts of type (3,2,2) and proving non-existence.
result There is no minimal chart of type (3,2,2).
The paper explores unique properties of Darboux transformations of spacelike curves in the Lorentz-Minkowski plane.
problem Exploring Darboux transformations of spacelike curves in the Lorentz-Minkowski plane.
method Using the Penrose diagram for conformal compactification, the paper investigates unique properties of Darboux transformations of spacelike curves.
result The paper identifies unique properties of Darboux transformations of spacelike curves in the Lorentz-Minkowski plane, especially regarding singularities and blowup.
Darboux inverses explain Kepler orbits on curved surfaces.
problem Understanding orbits on curved surfaces.
method Analyzing the Darboux inverses of the Kepler problem.
result Kepler orbits are periodic on open sets of phase space.
In this study, we introduce Darboux slant ruled surfaces in the Euclidean 3-space which is defined by the property that the Darboux vector of orthonormel frame of ruled surface makes a constant angle with a fixed, non-zero direction. We obtain the characterizations of Darboux slant ruled surfaces regarding the conical …
A new SVDD-based control chart for high-frequency multivariate data.
problem Challenges in interpreting kernel distance plots for high-frequency multivariate data.
method Proposes a new SVDD-based control chart, KT chart, to track process variation and central tendency. result Demonstrates successful use of KT chart on the Tennessee Eastman process data. Minimal charts of specific type contain unique subgraphs.
problem Characterizing minimal charts of type (m;2,3,2). method Analyzing the structure of charts and their subgraphs.
result Each of Γm+1 and Γm+2 contains one of three specific subgraphs. We give a full description of Darboux transformations of any order for arbitrary (nondegenerate) differential operators on the superline. We show that every Darboux transformation of such operators factorizes into elementary Darboux transformations of order one. Similar statement holds for operators on the ordinary lin…
The paper discusses a new method for constructing two-step Darboux transforms of isothermic surfaces.
problem Constructing two-step Darboux transforms of isothermic surfaces.
method Sym-type construction using parallel sections of the associated family.
result All two-step Darboux transforms of an isothermic surface are given without further integration.
Minimal charts with loops have at most 6 white vertices.
problem Properties of minimal charts with loops.
method Analyzing minimal charts with exactly one white vertex per loop.
result There is no loop in a minimal chart with 7 white vertices.
Simplifies surface-knots using chart moves involving black vertices.
problem Simplifying branched covering surface-knots.
method Chart moves involving black vertices to simplify surface-knots.
result Properties of simplified branched covering surface-knots with branch points.
Simplifies surface-knots by adding 1-handles with chart loops.
problem Complexity of branched covering surface-knots.
method Adding 1-handles with chart loops to simplify charts.
result Properties of simplified charts are investigated.
Minimal charts of type (3,3) are equivalent to a specific subchart.
problem Characterizing minimal charts of type (3,3).
method Analyzing subgraphs and C-move equivalence.
result Minimal charts of type (3,3) are equivalent to a specific subchart.
We define a transformation on harmonic maps from a Riemann surface into the 2-sphere which depends on a complex parameter, the so-called mu-Darboux transformation. In the case when the harmonic map N is the Gauss map of a constant mean curvature surface f and the parameter is real, the mu-Darboux transformation of -N i…
New method linearizes Darboux transformations of discrete curves.
problem Linearizing Darboux transformations of discrete curves.
method Expressing Darboux transformations as parallel sections of discrete connections in quaternionic formalism.
result Closed-form discrete parametrisations of all Darboux transforms and bicycle correspondences.
This paper studies the structure of a specific neighborhood in minimal 2-crossing charts.
problem Understanding the structure of a minimal 2-crossing chart with specific neighborhoods.
method Analyzes the neighborhoods of Γα∪Γβ and proposes a normal form for 2-crossing minimal n-charts. result Proposes a normal form for 2-crossing minimal n-charts. In this study, we give definitions and characterizations of eikonal slant helices, eikonal Darboux helices and non-normed eikonal Darboux helices in 3-dimensional pseudo- Riemannian manifold M . We show that every eikonal slant helix is also an eikonal Darboux helix for timelike and spacelike curves. Furthermore, we ob…
Study periodic solutions using spectral data and Darboux coordinates.
problem Elliptic sinh-Gordon equation periodic solutions.
method Spectral data and Darboux coordinates.
result Existence of Darboux coordinates.
A method to fix radius distortion in generative models on curved spaces.
problem Distortion in geodesic radius measurements across different charts on Riemannian manifolds.
method Radial Compensation (RC) adjusts the tangent-space base distribution to match the geodesic radius law, improving model stability and interpretability.
result RC ensures that the model's geodesic radius matches the intended distribution, improving numerical stability and curvature interpretation.
DDR game charts generated from raw audio tracks.
problem Creating new step charts for songs without existing charts.
method Combining recurrent and convolutional neural networks for step placement and a conditional LSTM for step selection.
result Improved step charts generated from raw audio tracks.
In this study, we give definitions and characterizations of eikonal slant helix curves, eikonal Darboux helices and non-normed eikonal Darboux helices in three dimensional Riemannian manifold 3 M . We show that every eikonal slant helix is also an eikonal Darboux helix. Furthermore, we obtain that if the curve a is a n…
Darboux Wronskian formulas allow to construct Darboux transformations, but Laplace transformations, which are Darboux transformations of order one cannot be represented this way. It has been a long standing problem on what are other exceptions. In our previous work we proved that among transformations of total order on…
Paper improves channel charting using autoencoders with spatial constraints.
problem Improving logical positioning of UEs using channel-state information.
method Representation-constrained autoencoders to enhance channel charts.
result Improved quality of learned channel charts for UE positioning.
New CMC surfaces with dihedral symmetry constructed from Darboux transforms.
problem Constructing closed CMC surfaces with specific symmetries.
method Using Darboux transforms on multiple covers to create new CMC surfaces.
result Explicit parametrisations of new closed CMC surfaces with dihedral symmetry.