New examples of Z/2 harmonic 1-forms and their branching sets are explored.
problem Exploring the properties and examples of Z/2 harmonic 1-forms and their branching sets.
method Elementary constructions and families of Z2 harmonic 1-forms. result The branching set Σ of a Z2 harmonic 1-form can exhibit various features including non-trivial links, multiple covers, and immersed structures. This paper constrains Gaussian processes to arbitrary domains using harmonic features.
problem Constraining Gaussian processes to arbitrary domains with boundary conditions.
method Solves a Fourier-like generalised harmonic feature representation of the GP prior, scaling as O(nm^2) in prediction and O(m^3) in hyperparameter learning.
result The method allows for efficient inference and handling of non-Gaussian likelihoods.
Revisits Gaussian process model with spherical harmonics for scalable deep learning.
problem Scaling Gaussian process models to large input dimensions with high frequency learning.
method Introduces new kernels related to deep models, variational learning of spherical harmonic phases, and sparseness in eigenbasis.
result Enables scaling to larger input dimensions and learning of high frequency variations.
Existence of an infinite sequence of harmonic maps between spheres of certain dimensions was proven by Bizon and Chmaj. This sequence shares many features of the Bartnik-McKinnon sequence of solutions to the Einstein-Yang-Mills equations as well as sequences of solutions that have arisen in other physical models. We ap…
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.
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 …
A new GP model uses spherical harmonics for faster inference.
problem Efficiently fitting large datasets with Gaussian processes.
method Sparse Gaussian processes with spherical harmonic features.
result Significant speed-up in inference for large datasets.
We present effective methods to compute equivariant harmonic maps from the universal cover of a surface into a nonpositively curved space. By discretizing the theory appropriately, we show that the energy functional is strongly convex and derive convergence of the discrete heat flow to the energy minimizer, with explic…
Embedded minimal surfaces of finite total curvature in R3 are reasonably well understood: From far away, they look like intersecting catenoids and planes, suitably desingularized. We consider the larger class of harmonic embeddings in R3 of compact Riemann surfaces with finitely many punctures…
Unified approach to Laplace and Steklov eigenvalues via n-harmonic maps.
problem Eigenvalue problems on manifolds of arbitrary dimension.
method Unified description using n-harmonic maps. result Uncovering two new features of Steklov eigenvalues.
Paper introduces MSP Network for harmonizing images from different scanners.
problem Improving predictive performance and learning efficiency in data harmonization.
method Multi-Stage Prediction (MSP) Network integrating neural networks of different architectures.
result MSP Network shows 20% improvement in patch-based mean-squared error over state-of-the-art methods.
BacHMMachine harmonizes Baroque chorales using theory-driven principles and Hidden Markov Models.
problem Algorithmic harmonization of Baroque chorales.
method Theory-driven approach guided by music composition principles, combined with data-driven learning of key and chord transitions.
result BacHMMachine generates musically coherent harmonizations with reduced computational burden and greater interpretability.
Translating or rotating an input image should not affect the results of many computer vision tasks. Convolutional neural networks (CNNs) are already translation equivariant: input image translations produce proportionate feature map translations. This is not the case for rotations. Global rotation equivariance is typic…
Enhances ASC using time- and frequency-liked CNNs and bilinear pooling.
problem Improving acoustic scene classification accuracy.
method Harmonic and percussive source separation, two-stream CNN architecture, bilinear pooling.
result Improved accuracy on DCASE 2019 sub task 1a dataset.
We propose a novel framework for combining datasets via alignment of their intrinsic geometry. This alignment can be used to fuse data originating from disparate modalities, or to correct batch effects while preserving intrinsic data structure. Importantly, we do not assume any pointwise correspondence between datasets…
Evaluating human brain potentials during watching different images can be used for memory evaluation, information retrieving, guilty-innocent identification and examining the brain response. In this study, the effects of watching images, with different levels of familiarity, on subjects' Electroencephalogram (EEG) have…
Approximating non-linear kernels using feature maps has gained a lot of interest in recent years due to applications in reducing training and testing times of SVM classifiers and other kernel based learning algorithms. We extend this line of work and present low distortion embeddings for dot product kernels into linear…
This paper explores vortices and harmonic flows on compact surfaces, using Hodge decomposition.
problem Understanding the interplay between vortices and harmonic flows on compact surfaces.
method Hodge decomposition of Euler's equations, focusing on point vortices on compact Riemann surfaces.
result The harmonic part of the flow is constant on flat tori but not on non-flat tori.
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.
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.
The Lie group Sol(p,q) is the semidirect product induced by the action of the real numbers R on the plane R^2 which is given by (x,y) --> (exp{p z} x, exp{-q z} y), where z is in R. Viewing Sol(p,q) as a 3-dimensional manifold, it carries a natural Riemannian metric and Laplace-Beltrami operator. We add a linear drift …
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.
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.
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.
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.
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.
FSL-Net detects and localizes feature shifts in large, high-dimensional datasets.
problem Feature shifts between data sources lead to erroneous features in various applications.
method FSL-Net is a neural network trained on multiple datasets to localize feature shifts.
result FSL-Net accurately localizes feature shifts from unseen datasets without re-training.
∞-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…
Dirac-harmonic maps are uncoupled under certain conditions.
problem Understanding the uncoupling of Dirac-harmonic maps.
method Critical points of a super-symmetric energy functional, with focus on harmonic maps.
result Dirac-harmonic maps are uncoupled under minimality assumption.
Study existence of harmonic and Dirac-harmonic maps from degenerating surfaces.
problem Existence of harmonic and Dirac-harmonic maps from degenerating surfaces.
method Using the Sacks and Uhlenbeck scheme, analyze a sequence of maps from degenerating surfaces to non-positive curved manifolds.
result Existence of limiting harmonic and Dirac-harmonic maps under certain conditions.
Harmonic morphisms are maps between Riemannian manifolds that pull back harmonic functions to harmonic functions. These maps are characterized as horizontally weakly conformal harmonic maps and they have many interesting links and applications to several areas in mathematics (see the book by Baird and Wood for details)…
The paper investigates Liouville type theorems for various harmonic forms on Riemannian manifolds.
problem Investigating Liouville type properties of harmonic forms on Riemannian manifolds.
method Normalized integral Ricci curvature and BiRic curvature.
result Established Liouville theorems for p-harmonic function, p-harmonic 1 form, and harmonic q form (with q≥2). Paper proves existence of smooth nontrivial Dirac-harmonic maps.
problem Existence of nontrivial Dirac-harmonic maps from closed surfaces.
method Proves existence using ε-regularity and perturbations.
result Existence of smooth nontrivial Dirac-harmonic maps.
Analyzes harmonic and biharmonic maps from gradient Ricci solitons.
problem Characterizing maps from gradient Ricci solitons.
method Derives conditions for maps to be constant or harmonic.
result Biharmonic maps of finite energy from the two-dimensional cigar soliton are harmonic.
Study heat flow for half-harmonic maps and harmonic maps with free boundary.
problem Integrability and regularity of half-harmonic maps and harmonic maps with free boundary.
method Introduced a heat flow associated to half-harmonic maps and constructed weak solutions via Ginzburg-Landau approximation.
result Proved partial regularity of weak solutions in space and time.
The paper explores harmonic and asymptotically harmonic Finsler manifolds and their properties.
problem Characterizing and understanding harmonic Finsler manifolds.
method Investigation of various types of harmonic Finsler manifolds, characterizations via mean curvature and Laplacian, and construction techniques.
result Certain harmonic Finsler manifolds are of Einstein type and examples of non-Riemannian Finsler harmonic manifolds are provided.
Study examines harmonic functions in sub-Riemannian and RCD settings.
problem Characterizing harmonic functions in sub-Riemannian and RCD settings.
method Analyzes weak and strong asymptotically mean value harmonic functions.
result Weakly amv-harmonic functions are equivalent to harmonicity in Carnot groups.
The paper characterizes harmonic maps and studies their properties on Riemannian manifolds.
problem Characterizing and analyzing harmonic maps between Riemannian manifolds.
method Analytic and geometric methods, including L2-orthogonal decomposition and energy density analysis.
result A criterion for harmonic submersions and diffeomorphisms, and new results linking harmonic symmetric bilinear forms and metrics.
Study on harmonic maps between cones, linking degrees to graph Laplacian eigenvalues.
problem Understanding harmonic maps between singular spaces.
method Analyzing homogeneous harmonic maps between simplicial cones and their degrees.
result Degrees of homogeneous harmonic maps are related to eigenvalues of discrete graph Laplacians.
Harmonic metallic structures on compact manifolds are equivalent to vanishing of their metallic structure.
problem Characterizing harmonic metallic structures on compact manifolds.
method Proving equivalence of harmonicity to dJ=0 and conditions for preservation by harmonic maps. result Conditions for harmonic metallic structures on compact manifolds.
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 consider the relationship between biharmonic maps and k-harmonic maps, and show non-existe…