Submanifolds of coordinate finite-type were introduced in HV1. A submanifold of a Euclidean space is called a coordinate finite-type submanifold if its coordinate functions are eigenfunctions of Δ. In the present study we consider coordinate finite-type surfaces in E^4. We give necessary and sufficient conditions for g…
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48 results for “Euclidean coordinates”
Formula for Laplace-Beltrami on orthogonal group in Euclidean coords.
problem Computing Laplace-Beltrami on constrained submanifolds.
method Embedded gradient vector field method, explicit formula derivation.
result Explicit formula for Laplace-Beltrami on orthogonal group.
These lecture notes are based on [arXiv: math/0702714, 0907.4469, 0907.4470]. We introduce and study basic aspects of non-Euclidean geometries from a coordinate-free viewpoint.
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.
Collar parameters for Teichmuller Space & Measured Foliations on a Surface Research Announcementmath.GT
New coordinates for Teichmüller space compactification.
problem Compactification of Teichmüller space.
method Defined new coordinates.
result Thurston compactification is radial compactification of Euclidean space.
In this paper, we consider tubes in the Euclidean 3-space whose Gauss map n is of coordinate finite I-type, i.e., the position vector n satisfies the relation ΔIn = Λn, where ΔI is the Laplace operator with respect to the first fundamental form I of the surface and Λ is a square matrix of order 3. We show that circular…
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.
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…
The paper sharpens a theorem about surfaces with zero Gaussian curvature.
problem Quantifying the isometric property of surfaces with zero Gaussian curvature.
method Asymptotically sharp quantitative version of a classical theorem using isothermal coordinates.
result An isothermal coordinate map from a Riemannian disc to an Euclidean disc is bi-Lipschitz with a constant of exp(4ε).
Formula derived for Laplace-Beltrami on Stiefel manifold.
problem Finding Laplace-Beltrami operator on Stiefel manifold.
method Using the general framework of Laplace operators on constraint manifolds, derived the explicit formula in terms of ambient Euclidean coordinates.
result Extended previously known formulas for sphere and special orthogonal group.
We quantify conditions that ensure that a signed measure on a Riemannian manifold has a well defined centre of mass. We then use this result to quantify the extent of a neighbourhood on which the Riemannian barycentric coordinates of a set of points on an -manifold provide a true coordinate chart, i.e., the ba…
We consider (locally) energy finite coordinates associated with a strongly local regular Dirichlet form on a metric measure space. We give coordinate formulas for substitutes of tangent spaces, for gradient and divergence operators and for the infinitesimal generator. As examples we discuss Euclidean spaces, Riemannian…
Study counts orbits of mapping class group in shearing coordinates.
problem Counting orbits of mapping class group in shearing coordinates.
method Uses shearing coordinates and asymptotics of Teichmüller space.
result Asymptotic behavior of mapping class group orbits in shearing coordinates.
The paper studies special surfaces in pseudo-Euclidean space.
problem Characterizing and classifying -isothermic surfaces in pseudo-Euclidean 3-space.
method Analyzing the pseudo-Calapso equation and providing explicit coordinates for Dupin surfaces.
result Explicit solutions to the pseudo-Calapso equation are provided.
The problem of the invariant classification of the orthogonal coordinate webs defined in Euclidean space is solved within the framework of Felix Klein's Erlangen Program. The results are applied to the problem of integrability of the Calogero-Moser model.
Isothermic parameterizations} are synonyms of isothermal curvature line parameterizations, for surfaces immersed in Euclidean spaces. We provide a method of constructing isothermic coordinate charts on surfaces which admit them, starting from an arbitrary chart. One of the primary applications of this work consists of …
Coordinate-independent convolutions on manifolds avoid reference frame ambiguity.
problem Applying convolutions on non-Euclidean manifolds without reference frame ambiguity.
method Developed coordinate-independent and gauge-equivariant convolutions on Riemannian manifolds.
result Coordinate-independent convolutions are equivariant under local gauge transformations.
We introduce a local coordinate description for the correspondence between the space of oriented affine lines in Euclidean and the tangent bundle to the 2-sphere. These can be utilised to give canonical coordinates on surfaces in , as we illustrate with a number of explicit examples.
Researchers geometrically define asymptotic coordinates in General Relativity.
problem Understanding the asymptotic behavior of relativistic initial data sets.
method Geometrization of asymptotic flatness and analysis of geometric invariants.
result Geometrically defined asymptotic coordinates for mass, energy, momentum, and angular momentum.
Characterizes metrics on triangulated surfaces using glued Euclidean triangles.
problem Describing metrics on triangulated surfaces constructed from glued Euclidean triangles.
method Carefully constructing polyhedral metrics and proving their uniqueness.
result Polyhedral metrics are the only intrinsic metrics preserving Euclidean triangle lengths.
We define a class of Euclidean distances on weighted graphs, enabling to perform thermodynamic soft graph clustering. The class can be constructed form the "raw coordinates" encountered in spectral clustering, and can be extended by means of higher-dimensional embeddings (Schoenberg transformations). Geographical flow …
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.
The paper proves stability of certain graph types in Euclidean space with specific densities.
problem Stability of vertical and radial graphs in Euclidean space with certain densities.
method Techniques of calibrations used to prove stability and minimization.
result Vertical and radial graphs are strongly stable for specific densities.
The center of mass in General Relativity is hard to define due to coordinate freedom.
problem Defining the center of mass in General Relativity rigorously and consistently.
method Analyzing the challenges in Newtonian Gravity and using Bartnik's asymptotic harmonic coordinates.
result Examples of initial data sets in General Relativity that do not satisfy center of mass definitions.
PGPCA improves PCA for nonlinear data in neuroscience.
problem Nonlinear data distribution in neuroscience.
method Developed PGPCA for nonlinear manifolds, incorporating EM algorithm.
result PGPCA outperforms PPCA in modeling data around nonlinear manifolds.
In this paper, we consider lower order eigenvalues of Laplacian operator with any order in Euclidean domains. By choosing special rectangular coordinates, we obtain two estimates for lower order eigenvalues.
A proof that hyperbolic plane cannot be immersed in Euclidean 3-space.
problem Proving the impossibility of isometrically immersing the hyperbolic plane in Euclidean 3-space.
method Applying ideas from undergraduate mathematics, including moving frames and connection forms, to simplify the proof.
result A key transition from principal directions to asymptotic directions simplifies the proof and yields a coordinate system.
Expressing the curvature tensor and connection of a given metric in terms of those of another metricmath.DG
Let be a Riemannian manifold, and be a second metric on . We give expressions of 's associated connection, and Riemann curvature tensor , in terms of and certain combinations of covariant derivatives of (with respect to the Levi-Civita connection associated with ). The formulas turn …
We study a natural intrinsic definition of geometric simplices in Riemannian manifolds of arbitrary dimension , and exploit these simplices to obtain criteria for triangulating compact Riemannian manifolds. These geometric simplices are defined using Karcher means. Given a finite set of vertices in a convex set on t…
A fundamental question in data analysis, machine learning and signal processing is how to compare between data points. The choice of the distance metric is specifically challenging for high-dimensional data sets, where the problem of meaningfulness is more prominent (e.g. the Euclidean distance between images). In this…
New GL-GP models learn covariance respecting domain geometry.
problem Suboptimal results from nonparametric regression on restricted domains.
method Graph Laplacian based Gaussian Processes (GL-GPs) with Nyström extension.
result Performance gains in various applications.
We consider ruled and quadric surfaces in the 3-dimensional Euclidean space which are of coordinate finite type with respect to the third fundamental form , i.e., their position vector satisfies the relation where is a square matrix o…
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.
Log-Ratio Propagation on the Simplex: A Theory of Cellwise Contamination for Compositional Datastat.ML
A theory of cellwise contamination for compositional data using log-ratios.
problem Contamination in compositional data analysis.
method Develops a theory combining contamination model and propagation theorem.
result Reduction in cellwise breakdown value by for certain estimators.
New method speeds up optimization over probability measures.
problem High computational overhead in optimizing probability measures.
method Randomized coordinate descent on Wasserstein space.
result Significant speedups over full-gradient methods.
New algorithm for nonconvex optimization on constrained Riemannian manifolds converges quickly.
problem Optimization on constrained Riemannian manifolds.
method Block majorization-minimization (BMM) for smooth nonconvex objectives with Riemannian constraints.
result Converges to stationary points within iterations.
We consider surfaces of revolution in the three-dimensional Euclidean space which are of coordinate finite type with respect to the third fundamental form. We show that a surface of revolution satisfying the preceding relation is a catenoid or part of a sphere.
We define and study a family of distributions with domain complete Riemannian manifold. They are obtained by projection onto a fixed tangent space via the inverse exponential map. This construction is a popular choice in the literature for it makes it easy to generalize well known multivariate Euclidean distributions. …
In the field of statistics, many kind of divergence functions have been studied as an amount which measures the discrepancy between two probability distributions. In the differential geometrical approach in statistics (information geometry), dually flat spaces play a key role. In a dually flat space, there exist dual a…
A method models nonlinear dynamics from data using barycentric coordinates and memory.
problem Modeling complex dynamical systems from data.
method SPA for data projection, barycentric coordinates, delay-embedding theorem for memory.
result Stable models of chaotic dynamics and attractors are reproduced.
We define barycentric coordinates on a Riemannian manifold using Karcher's center of mass technique applied to point masses for n+1 sufficiently close points, determining an n-dimensional Riemannian simplex defined as a "Karcher simplex." Specifically, a set of weights is mapped to the Riemannian center of mass for the…
Log-Euclidean Lie Groupsmath.DG
Develops log-Euclidean Lie groups for SPD and correlation matrices.
problem Unifies various log-Euclidean constructions for SPD and correlation matrices.
method Theory and explicit isometries linking different log-Euclidean metrics.
result Explicit log-Euclidean metrics on SPD and correlation matrices.
The paper proves new inequalities in hyperbolic space using Euclidean methods.
problem Proving weighted isoperimetric inequalities in hyperbolic space.
method Using isoperimetric inequality with log-convex density in Euclidean space.
result Removed horo-convex assumption and proved new inequalities for star-shaped domains.
On Transitive Consistency for Linear Invertible Transformations between Euclidean Coordinate Systemsmath.OC
Transitive consistency is an intrinsic property for collections of linear invertible transformations between Euclidean coordinate frames. In practice, when the transformations are estimated from data, this property is lacking. This work addresses the problem of synchronizing transformations that are not transitively co…
CAVI converges for log-concave measures via optimal transport.
problem Finding the closest product measure to a log-concave measure via CAVI.
method Adapting coordinate descent techniques from Euclidean space to optimal transport for log-concave densities.
result Proves convergence of CAVI for log-concave densities and provides rates of convergence under additional conditions.
Random scan CAVI converges linearly under log-concave assumptions.
problem Analyzing the convergence rate of random scan Coordinate Ascent Variational Inference (CAVI) under log-concave conditions.
method Building on previous work, we analyze the random scan version of CAVI using optimal transport geometry.
result We obtain tight linear convergence rates for the random scan version of CAVI.
We study the problem asking if one can embed manifolds into finite dimensional Euclidean spaces by taking finite number of eigenvector fields of the connection Laplacian. This problem is essential for the dimension reduction problem in massive data analysis. Singer-Wu proposed the vector diffusion map which embeds mani…
Knots in Euclidean space which may be parameterized by a single cosine function in each coordinate are called Lissajous knots. We show that twist knots are Lissajous knots if and only if their Arf invariants are zero. We further prove that all 2-bridge knots and all (3,q)-torus knots have Lissajous projections.