Complex and quaternionic projective spaces lack local orthogonal coordinates.
problem Lack of local orthogonal coordinates in complex and quaternionic projective spaces.
method Analysis of Riemannian manifolds and canonical metrics.
result Complex and quaternionic projective spaces do not have local systems of orthogonal coordinates.
We find necessary and sufficient conditions under which the complex coordinates on a flag manifold of a classical group described in [2] are Bochner coordinates.
Deroin and Tholozan's representations are mapped to complex projective space via action-angle coordinates.
problem Mapping representations of a punctured sphere into PSL(2,R) to a simpler geometric space. method Polygonal model and chains of triangles to extract action-angle coordinates.
result Action-angle coordinates give an explicit isomorphism and almost global Darboux coordinates.
The Newlander-Nirenberg theorem links complex coordinates to the vanishing of the Nijenhuis tensor.
problem Existence of complex coordinates associated with almost complex structures.
method Simple explicit proof and supersymmetric interpretation.
result The vanishing of the Nijenhuis tensor is both necessary and sufficient for complex coordinates to exist.
PURE-CD algorithm proves complexity bounds for convex-concave problems.
problem Solving convex-concave min-max problems with bilinear coupling.
method Primal-dual algorithm with random extrapolation and coordinate descent (PURE-CD).
result Complexity bounds match or improve existing results for dense and sparse problems.
Paper classifies rational 3-tangles using normal forms and minimal coordinates.
problem Classifying rational 3-tangles up to isotopy.
method Defined normal form and normal coordinate, investigated minimal coordinates, constructed contractible simplicial complex.
result Simplicial complex of normal forms is contractible, leading to classification of rational 3-tangles.
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 bound on Rademacher complexity for vector functions.
problem Bounding Rademacher complexity for vector-valued functions.
method Bounding Rademacher complexity by coordinate-wise complexity with a factor of sqrt(K).
result Rademacher complexity is bounded by the maximum coordinate-wise complexity times sqrt(K).
New findings on Kähler manifolds restrict orthogonal coordinates existence.
problem Existence of orthogonal coordinates on Kähler manifolds.
method Algebraic and geometric techniques applied to Kähler manifolds.
result No nontrivial self-dual Kähler 4-manifolds or Ricci-flat Kähler 4-manifolds support orthogonal coordinates.
Rust library solves complex equations on abstract simplicial complexes.
problem Solving partial differential equations on abstract simplicial complexes.
method Finite Element Exterior Calculus, intrinsic Riemannian metric, first-order Whitney basis functions.
result Verification through convergence studies on elliptic Hodge-Laplace eigenvalue and source problems.
A new sampling method reduces computational cost for high-dimensional log-concave distributions.
problem High computational cost of ULMC in high dimensions.
method Random Coordinate ULMC (RC-ULMC) selects a single coordinate per iteration.
result RC-ULMC is cheaper than classical ULMC, especially in highly skewed and high-dimensional problems.
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.
Canonical coordinates defined for minimal time-like surfaces in n-dimensional Minkowski space.
problem Characterizing canonical coordinates on minimal time-like surfaces.
method Introducing canonical coordinates and proving their existence and uniqueness; using analysis over the algebra of double numbers.
result Canonical coordinates on minimal time-like surfaces are characterized by a natural condition for a complex function over the algebra of double numbers.
In this paper we provide two ways of constructing complex coordinates on the moduli space of pairs of a Riemann surface and a stable holomorphic vector bundle centred around any such pair. We compute the transformation between the coordinates to second order at the center of the coordinates. We conclude that they agree…
Study on diagonal and separating coordinates for symmetric spaces of rank 1.
problem Existence and nonexistence of diagonal and separating coordinates for symmetric spaces of rank 1.
method Generalization of results by Gauduchon and Moroianu, 2020, and analysis of constant sectional curvature and orthogonal separation of variables.
result Diagonal coordinates exist if and only if the symmetric space has constant sectional curvature.
Kashaev algebra associated to a surface is a noncommutative deformation of the algebra of rational functions of Kashaev coordinates. For two arbitrary complex numbers, there is a generalized Kashaev algebra. The relationship between the shear coordinates and Kashaev coordinates induces a natural relationship between th…
Paper speeds up IoT device detection and data decoding.
problem Efficiently detect and decode massive IoT devices in grant-free random access.
method Develops multi-armed bandit approaches for more efficient detection via coordinate descent.
result Proposed bandit based algorithms achieve faster convergence rates with lower time complexity.
We propose accelerated randomized coordinate descent algorithms for stochastic optimization and online learning. Our algorithms have significantly less per-iteration complexity than the known accelerated gradient algorithms. The proposed algorithms for online learning have better regret performance than the known rando…
We present an algorithm for calculating the geometric intersection number of two multicurves on the n-punctured disk, taking as input their Dynnikov coordinates. The algorithm has complexity O(m2n4), where m is the sum of the absolute values of the Dynnikov coordinates of the two multicurves. The main ingredien…
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.
The paper defines a new coordinate system for anti-de Sitter structures.
problem Understanding the geometry of anti-de Sitter spaces.
method Combining recent work on anti-de Sitter structures with classical theory.
result Introduced a coordinate system resembling Fenchel-Nielsen coordinates.
Sharp Hölder regularity found for complex Frobenius theorem coordinates.
problem Finding optimal Hölder-Zygmund regularity for complex Frobenius theorem coordinates.
method Analyzing necessary and sufficient conditions for coordinate charts achieving the theorem's structure.
result The optimal Hölder-Zygmund regularity for coordinate charts is shown to be α. We propose and analyze a new parallel coordinate descent method---`NSync---in which at each iteration a random subset of coordinates is updated, in parallel, allowing for the subsets to be chosen non-uniformly. We derive convergence rates under a strong convexity assumption, and comment on how to assign probabilities t…
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…
New algorithm for robust circular coordinates in recurrent time series data.
problem Inefficient and sensitive methods for finding circular coordinates on recurrent data.
method Subsampling, aligning, and averaging to correct uneven sampling density.
result More robust and efficient circular coordinates for neuronal recordings.
We give a necessary and sufficient condition for a non-degenerate symmetric 3-differential with nonzero Blaschke curvature on a complex surface to be locally representable as a product of three closed holomorphic 1-forms. We give two versions of this condition corresponding to different choices of coordinates, one of w…
A new sampling method, RC-LMC, reduces computational cost for high-dimensional log-concave distributions.
problem High computational cost of LMC in high dimensions.
method RC-LMC updates only one coordinate at a time, adding noise.
result RC-LMC is more efficient than LMC in high dimensions, especially for skewed distributions.
Develops fractional de Rham theory for Maxwell equations.
problem Formulating fractional calculus for Maxwell equations.
method Fractional tangent functionals, Riemann-Liouville integral, polynomial algebra, exterior algebra.
result Fractional de Rham complex for Maxwell equations.
In every point of a Kähler manifold there exist special holomorphic coordinates well adapted to the underlying geometry. Comparing these Kähler normal coordinates with the Riemannian normal coordinates defined via the exponential map we prove that their difference is a universal power series in the curvature tensor and…
Constructs coordinates to diagonalize Toda flow on matrices with simple spectrum.
problem Diagonalizing the Toda flow on matrices with simple spectrum.
method Lie theoretic methods applied to complex semisimple Lie algebras and their real forms.
result Decouples the Toda vector field into simpler components.
Let G be a simply connected, simple, complex Lie group of rank 2. We give explicit Fock-Goncharov coordinates for configurations of triples and quadruples of affine flags in G. We show that the action on triples by orientation preserving permutations corresponds to explicit quiver mutations, and that the same holds for…
Let $M\subset{\complex}^n$ be a complex domain of ${\complex}^n$ endowed with a rotation invariant \K form ωΦ=2i∂∂ˉΦ. In this paper we describe sufficient conditions on the \K potential Φ for (M,ωΦ) to admit a symplectic embedding (explicitely described in terms of Φ) into a compl…
The pathwise coordinate optimization is one of the most important computational frameworks for high dimensional convex and nonconvex sparse learning problems. It differs from the classical coordinate optimization algorithms in three salient features: {\it warm start initialization}, {\it active set updating}, and {\it …
CD methods tackle nonconvex optimization with three terms, achieving critical points.
problem Minimizing nonconvex functions with specific structure.
method Developed randomized CD, randomly permuted CD, and accelerated CD methods.
result CD methods converge to critical points with sublinear complexity.
Differentially private random block coordinate descent improves utility in machine learning.
problem Lack of privacy in classical CD methods when handling sensitive information.
method Proposes a differentially private random block coordinate descent method using sketch matrices and importance sampling.
result Demonstrates improved convergence rates and utility guarantees compared to non-private methods.
A general class of Newton algorithms on Graßmann and Lagrange-Graßmann manifolds is introduced, that depends on an arbitrary pair of local coordinates. Local quadratic convergence of the algorithm is shown under a suitable condition on the choice of coordinate systems. Our result extends and unifies previous convergenc…
PolarBM models complex-valued audio signals in polar coordinates, improving over conventional methods.
problem Discarding structural information in complex-valued problems simplifies models but loses important amplitude-phase relationships.
method Proposes PolarBM, a novel Boltzmann machine for complex-valued variables in polar coordinates, and LogPolarBM for logarithmic amplitude.
result PolarBM and LogPolarBM achieve superior modeling accuracy compared to conventional models, including deep neural networks.
Local coordinates for non-singular pairs in complex and quaternionic hyperbolic groups.
problem Classifying conjugation orbits of non-singular pairs in complex and quaternionic hyperbolic groups.
method Extending the notion of non-singular pairs, classifying orbits, proving smallness, constructing twist-bend parameters.
result Local parametrization of non-singular pairs in G(3), extending to generic representations of surface groups. Kahn and Markovic \cite{KahnMark} proved that the fundamental group of each closed hyperbolic three manifold contains a closed surface subgroup. One of the main ingredients in their proof is a theorem which states that an assignment of nearly real, complex Fenchel-Nielsen coordinates to the cuffs of a pants decompositi…
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.
The paper extends Newlander-Nirenberg theorem to domains with C2 boundary.
problem Extending Newlander-Nirenberg theorem to domains with C2 boundary. method Analyzing formally integrable complex structures on domains with C2 boundary. result Existence of global holomorphic coordinate systems on the closure of a bounded strictly pseudoconvex domain.
Analytic plane curves determine unique conformal coordinates.
problem Determining a conformal coordinate system for analytic plane curves.
method Holomorphic continuation of the Frenet curvature form.
result Holomorphic continuation of the curvature form uniquely determines a conformal coordinate net.
Study warped product metrics on hyperbolic and complex hyperbolic manifolds.
problem Understanding curvature of warped product metrics.
method Prove curvature formulas for warped product metrics on hyperbolic and complex hyperbolic manifolds.
result Curvature formulas expressed in spherical coordinates about totally geodesic submanifolds.
Normal and almost normal surfaces are essential tools for algorithmic 3-manifold topology, but to use them requires exponentially slow enumeration algorithms in a high-dimensional vector space. The quadrilateral coordinates of Tollefson alleviate this problem considerably for normal surfaces, by reducing the dimension …
Sequential coordinate ascent is more robust in high-dimensional linear regression.
problem Behavior difference between sequential and parallel coordinate ascent in variational inference.
method Comparison of sequential and parallel coordinate ascent algorithms in high-dimensional linear regression.
result Sequential algorithm converges under more relaxed conditions than parallel algorithm.
Newlander-Nirenberg theorem extended to complex b-manifolds.
problem Characterizing complex b-manifolds.
method Involutive splitting of b-tangent bundle, formal local invariants, singular coordinate change.
result Complex b-manifolds have a single local model.
The paper simplifies complex 2D functions near their critical points.
problem Simplifying smooth functions on 2-manifolds near critical points.
method Explicit construction of coordinate changes to canonical form.
result Estimates the radius of required neighbourhoods for specific singularity types.
A new algorithm selects independent coordinates for complex manifolds.
problem Embedding algorithms fail with large aspect ratio manifolds.
method IES algorithm selects smooth embeddings using carefully chosen eigenfunctions of the Laplace-Beltrami operator.
result The IES algorithm successfully embeds synthetic and real data.