A new method splits surface flow discretizations into streamfunctions and harmonic fields.
problem Discretizing incompressible flows on surfaces with pressure and saddle-point structure.
method Discrete Helmholtz-Hodge decomposition for BDM elements on surfaces.
result Eliminates pressure and saddle-point structure, ensuring exact tangentiality and divergence-freeness.
We classify all harmonic maps with finite uniton number from a Riemann surface into an arbitrary compact simple Lie group G, whether G has trivial centre or not, in terms of certain pieces of the Bruhat decomposition of the group ΩalgG of algebraic loops in G and corresponding canonical elements. Th…
Making use of Murakami's classification of outer involutions in a Lie algebra and following the Morse-theoretic approach to harmonic two-spheres in Lie groups introduced by Burstall and Guest, we obtain a new classification of harmonic two-spheres in outer symmetric spaces and a Weierstrass-type representation for such…
We describe algorithms for finding harmonic cochains, an essential ingredient for solving elliptic partial differential equations in exterior calculus. Harmonic cochains are also useful in computational topology and computer graphics. We focus on finding harmonic cochains cohomologous to a given cocycle. Amongst other …
Harmonic maps link Teichmüller spaces to framed representations.
problem Connecting Teichmüller spaces with framed representations.
method Uses harmonic map heat flow to find unique maps.
result Unique harmonic maps exist under specified conditions.
The study examines eigenfunctions on lens spaces using harmonic-counting measures.
problem Understanding the asymptotic behavior of eigenfunctions on lens spaces.
method Introduced harmonic-counting measures and applied them to lens spaces.
result Determined the asymptotic behavior of eigenfunctions associated with lattice points.
Defines Φ-harmonic maps and studies their unstable manifolds.
problem Characterizing unstable manifolds of Φ-harmonic maps. method Extrinsic average variational method in calculus of variations.
result Every compact Φ-superstrongly unstable manifold is Φ-strongly unstable. Constructs harmonic maps near retractions in hyperbolic spaces.
problem Finding harmonic maps near retractions in hyperbolic spaces.
method Constructs harmonic maps to the hyperbolic plane from quasidisks, and to convex hulls from sets in the boundary at infinity of pinched Hadamard manifolds.
result Harmonic maps are bounded from nearest-point retractions in hyperbolic spaces.
Hodge theory proven for elliptic complexes over compact operator C∗-algebras.
problem Proving Hodge theory for elliptic complexes over compact operator C∗-algebras. method Analyzing pseudodifferential operators and finitely generated projective Hilbert bundles.
result Topological isomorphism between cohomology groups and harmonic elements.
We study here the action of subgroups of PSL(2,R) on the space of harmonic functions on the unit disc bounded by a common constant, as well as the relationship this action has with the foliated Liouville problem: Given a foliation of a compact manifold by Riemannian leaves and a leafwise harmonic continuous function on…
The paper constructs metrics with non-negative curvature and harmonic maps.
problem Constructing metrics with non-negative curvature and harmonic maps.
method Using Clifford systems and characteristic maps, the paper constructs metrics with non-negative curvature and harmonic representatives of certain elements in homotopy groups of spheres.
result The construction of a metric of non-negative curvature on S(η) which is diffeomorphic to the inhomogeneous focal submanifold M+ of OT-FKM type isoparametric hypersurfaces. Paper presents a consistent discretization for Hodge decomposition on volumetric meshes.
problem Discretization of Hodge decomposition for vector fields on volumetric meshes.
method Edge-based Nedelec elements and face-based Crouzeix-Raviart elements interplay.
result Stable and efficient method for large-sized models with good performance.
Adapting \cite{strz3}, we define generalized p-harmonic maps into Riemannian homogeneous targets, a notion of solutions not belonging to the energy space. Restricting our attention to the subcritical range p greater than the domain dimension n, we show a uniform C1,α-regularity result for a sequence of such …
Non-trivial elements in homotopy groups lead to metrics of positive curvature.
problem Constructing non-trivial elements in homotopy groups to understand metrics of positive curvature.
method Using Gromoll filtration, Toda brackets, and Morlet's homotopy equivalence.
result Non-trivial elements in homotopy groups act on spaces of metrics of positive curvature.
Projective loops generate rational loop groups without needing nilpotent loops.
problem Generating rational loop groups with noncompact reality conditions.
method Used projective loops to prove rational loop groups can be generated.
result Projective loops alone are sufficient to generate rational loop groups.
Given a measure on the Thurston boundary of Teichmueller space, one can pick a geodesic ray joining some basepoint to a randomly chosen point on the boundary. Different choices of measures may yield typical geodesics with different geometric properties. In particular, we consider two families of measures: the ones whic…
We introduce new finite-dimensional cohomologies on symplectic manifolds. Each exhibits Lefschetz decomposition and contains a unique harmonic representative within each class. Associated with each cohomology is a primitive cohomology defined purely on the space of primitive forms. We identify the dual currents of lagr…
Study on 3-manifolds finds regular conformal metrics for rough metrics.
problem Characterize conformal metrics for rough Riemannian metrics on 3-manifolds.
method Analogous to the Yamabe problem, study conformal classes and regularity.
result Characterize when a more regular representative exists in the conformal class.
The study uses music chords to predict Brazilian music genres.
problem Classifying popular Brazilian music genres based on harmonic structures.
method Extracted and engineered harmonically related features from chords data, used random forest model for classification.
result Features from harmonic elements can predict Brazilian music genres.
Wavelet boosts gradient boosting, improving accuracy, especially in imbalanced data.
problem Improving gradient boosting performance, especially in imbalanced data.
method Wavelet decomposition of trees in gradient boosting.
result Wavelet-based gradient boosting outperforms existing methods, especially in imbalanced data.
We introduce a flow of maps from a compact surface of arbitrary genus to an arbitrary Riemannian manifold which has elements in common with both the harmonic map flow and the mean curvature flow, but is more effective at finding minimal surfaces. In the genus 0 case, our flow is just the harmonic map flow, and it tries…
We define a discrete Laplace-Beltrami operator for simplicial surfaces. It depends only on the intrinsic geometry of the surface and its edge weights are positive. Our Laplace operator is similar to the well known finite-elements Laplacian (the so called ``cotan formula'') except that it is based on the intrinsic Delau…
Develop a variational framework for statistical inference on cyclic interactions.
problem Estimating and comparing large-scale recurrent organization in directed interactions.
method Represent directed interactions as edge flows on a simplicial complex and evolve under an energy-minimizing dynamical system.
result Separate transient interaction components from persistent harmonic flows, yielding a low-dimensional cycle space.
The energy of harmonic sections of flat bundles of nonpositively curved (NPC) length spaces over a Riemann surface S is a function Eρ on Teichmüller space $\Teich$ which is a qualitative invariant of the holonomy representation ρ of π1(S). Adapting ideas of Sacks-Uhlenbeck, Schoen-Yau and Tromba, we show that…
Attention tokens are group elements, negated algebra norms.
problem Attention mechanism for matrix Lie groups
method Lie-Algebra Attention
result Score matches learned kernel on invariant poses, outperforms vector-token baseline
This paper contains two main results. The first is the existence of an equivariant Weil-Petersson geodesic in Teichmueller space for any choice of pseudo-Anosov mapping class. As a consequence one obtains a classification of the elements of the mapping class group as Weil-Petersson isometries which is parallel to the T…
Positive curvature manifolds from smaller ones using division algebras and geodesic flow.
problem Understanding positive curvature manifolds and their properties.
method Using Conner-Kobayashi reductions and division algebras to build up positive curvature manifolds.
result Existence of two linearly independent harmonic k-forms on positive curvature manifolds.
Study on minimal energy problems for Riesz kernels on manifolds.
problem Minimal energy problems for strongly singular Riesz kernels on manifolds.
method Formulation of natural regularization using Hadamard's partie finie integral operator and analysis of measures with finite energy in Sobolev space.
result The minimal energy problem admits a unique solution and is related to discrete minimal energy problems.
Abstract theory of flows of geometric structures on manifolds.
problem Deformations and flows of geometric structures on manifolds.
method Develops an abstract theory using infinitesimal action of GL(n,R) and bundle decompositions.
result Almost monotonicity formula for harmonic flow of H-structures, leading to regularity and energy gap results.
Study cohomology classes related to harmonic maps on submersions.
problem Understanding harmonic maps on submersions and their cohomology.
method Extending previous results on Riemannian submersions and p-harmonic morphisms to F-harmonic and f-harmonic maps.
result Extend results on Riemannian submersions to F-harmonic and f-harmonic maps.
Study on harmonic maps on weighted Riemannian foliations.
problem Characterize harmonic maps on weighted foliations.
method Analyze transversally f-harmonic and (F,F′)f-harmonic maps. result Equivalence of transversally f-harmonic and (F,F′)f-harmonic maps in minimal foliations. The paper examines triviality of Ricci-Bourguignon harmonic solitons.
problem Investigating triviality of Ricci-Bourguignon harmonic solitons.
method Utilizing results from V-harmonic map to study Ricci harmonic soliton properties.
result Triviality of Ricci-Bourguignon harmonic solitons examined.
Introduces generalized equations based on subequations and Dirichlet duality.
problem Solving generalized partial differential equations (PDEs).
method Viscosity solutions and subequations, with a focus on the Dirichlet problem.
result Uniqueness and existence of solutions depend on the absence of interior in the generalized equation.
Study cohomology classes related to n-harmonic morphisms and F-harmonic maps.
problem Understanding cohomology classes associated with n-harmonic morphisms and F-harmonic maps. method Utilizing the n-conservation law (2.6) to obtain sharp results. result Sharp results on cohomology classes related to n-harmonic morphisms and F-harmonic maps. The paper proves nonexistence of harmonic and bi-harmonic maps under specific conditions.
problem Existence of harmonic and bi-harmonic maps into certain Riemannian manifolds.
method Analysis of manifolds with conformal vector fields or Ricci solitons.
result Nonexistence of harmonic and bi-harmonic maps in specified conditions.
f-Harmonic maps were first introduced and studied by Lichnerowicz in \cite{Li} (see also Section 10.20 in Eells-Lemaire's report \cite{EL}). In this paper, we study a subclass of f-harmonic maps called f-harmonic morphisms which pull back local harmonic functions to local f-harmonic functions. We prove that a map betwe…
Finite energy solutions of 4-harmonic and ES-4-harmonic maps are trivial.
problem Characterizing the qualitative behavior of 4-harmonic and ES-4-harmonic maps.
method Proving triviality of finite energy solutions for both maps.
result Finite energy solutions of both 4-harmonic and ES-4-harmonic maps are trivial.
We propose a new notion called \emph{infinity-harmonic maps}between Riemannain manifolds. These are natural generalizations of the well known notion of infinity harmonic functions and are also the limiting case of p% -harmonic maps as p→∞. Infinity harmoncity appears in many familiar contexts. For example,…
The paper examines properties of generalized τ-quasi Ricci-harmonic metrics and proves rigidity results.
problem Characterizing and proving rigidity of generalized τ-quasi Ricci-harmonic metrics.
method Exploring conditions for harmonic-Einstein metrics, obtaining rigidity results, and proving gap theorems.
result Rigidity results for compact generalized τ-quasi Ricci-harmonic metrics.
Constructs explicit harmonic functions and morphisms on complex and quaternionic Grassmannians.
problem Creating explicit solutions for p-harmonic functions and harmonic morphisms. method Using joint eigenfunctions of the Laplace-Beltrami and conformality operators.
result Induces solutions on dual non-compact Riemannian symmetric spaces.
New p-harmonic and harmonic morphisms found on Lie groups.
problem Constructing explicit p-harmonic and harmonic morphisms on Lie groups.
method Using the method of eigenfamilies to construct explicit complex-valued p-harmonic functions and harmonic morphisms.
result Explicit complex-valued p-harmonic functions and harmonic morphisms constructed on non-compact classical Lie groups.
Study examines maximal domains of radial harmonic functions across different curvature types.
problem Understanding maximal domains of radial harmonic functions in various curvature settings.
method Analysis of harmonic spaces with positive, zero, and negative curvature.
result Characterization of maximal domains for radial harmonic functions in different curvature contexts.
Study of generalized harmonic morphisms and biharmonic maps.
problem Characterizing and constructing generalized harmonic morphisms.
method Characterizations and two methods of constructions.
result Complete classification of generalized harmonic morphisms among projections of a warped product space.
J. Eells and L. Lemaire introduced k-harmonic maps, and Wang Shaobo showed the first variational formula. When, k=2, it is called biharmonic maps (2-harmonic maps). There have been extensive studies in the area. In this paper, We study k-harmonic immersion into a sphere, and get the rerationship between radious and "k"…
Extends p-harmonic map theory for new properties.
problem No specific problem stated; extends existing theory.
method Extended p-harmonic and biharmonic map definitions.
result New properties of generalized stable p-harmonic maps.
A new clustering method versatile linkage improves on existing strategies.
problem Improving agglomerative hierarchical clustering methods.
method Introducing versatile linkage, a family of clustering strategies based on generalized means.
result Versatile linkage is space-conserving compared to existing methods.
∞-Harmonic maps are a generalization of ∞-harmonic functions. They can be viewed as the limiting cases of p-harmonic maps as p goes to infinity. In this paper, we give complete classifications of linear and quadratic ∞-harmonic maps from and into a sphere, quadratic ∞-harmonic maps between E…
Study on bi-f-harmonic curves and hypersurfaces extending harmonic and biharmonic concepts.
problem Extending harmonic and biharmonic concepts to new geometric structures.
method Derive bi-f-harmonic equations for various geometric settings. result New equations for bi-f-harmonic maps in different geometric contexts.