The paper studies f-polyharmonic maps and their properties.
problem Understanding and characterizing f-polyharmonic maps. method Deriving the Euler-Lagrange equation and analyzing specific cases.
result Every f-polyharmonic function on a closed Riemannian manifold is constant. We derive the stress-energy tensor for polyharmonic maps between Riemannian manifolds. Moreover, we employ the stress-energy tensor to characterize polyharmonic maps where we pay special attention to triharmonic maps.
The article discusses conservation laws for polyharmonic maps and their applications.
problem Understanding conservation laws for polyharmonic maps.
method Recalling the stress-energy tensor and showing conservation laws with Killing vector fields.
result Conservation laws for polyharmonic maps and their applications.
The study classifies conformal biharmonic and k-polyharmonic maps between space forms.
problem Classifying conformal biharmonic and k-polyharmonic maps between space forms.
method Proving conditions for proper biharmonic and k-polyharmonic maps between space forms.
result Proper k-polyharmonic conformal maps exist if and only if the dimension is 2k.
We prove the energy identity and the no neck property for a sequence of smooth extrinsic polyharmonic maps with bounded total energy.
New conservation laws found for polyharmonic maps in critical dimension.
problem Existence of conservation laws for polyharmonic maps in critical dimension.
method Small perturbation of Uhlenbeck's gauge fixing matrix.
result Existence of conservation laws for elliptic systems of even order in critical dimension.
We prove that polyharmonic maps of arbitrary order from complete nonparabolic Riemannian manifolds to arbitrary Riemannian manifolds must be harmonic if certain smallness and integrability conditions hold.
We study polyharmonic (k-harmonic) maps between Riemannian manifolds with finite j-energies (j=1, cdots, 2k-2). We show if the domain is complete and the target is the Euclidean space, then such a map is harmonic.
The paper explores unique continuation properties for polyharmonic maps between Riemannian manifolds.
problem Investigating unique continuation principles for polyharmonic maps.
method Analyzing critical points of higher order functionals to prove extensions of known results in harmonic and biharmonic cases.
result Proving extensions of unique continuation principles for k-harmonic maps.
We prove that for any two closed Riemannian manifolds M2m (m≥1) and N, there exists a minimizing (extrinsic) m-polyharmonic map for every free homotopy class in [M2m,N], provided that the homotopy group π2m(N) is trivial. This generalizes the celebrated existence results for harmonic maps and …
We consider polyharmonic maps φ:(M,g)→\mathbb{E}^noforderkfromacompleteRiemannianmanifoldintotheEuclideanspaceandletpbearealconstantsatisfying1<p<\infty.(i)If,\int_M|W^{k-1}|^p dv_g<\infty,and\int_M|\bar \nabla W^{k-2}|^2dv_g<\infty.Thenφ$ is a polyharmonic map of orde…
Survey on conservation laws for geometric PDEs.
problem Modeling polyharmonic maps.
method Conservation law approach.
result Overview of conservation laws in geometric PDEs.
We establish both local and global well-posedness for the heat flow of polyharmonic maps from Rn to a compact Riemannian manifold without boundary for initial data with small BMO norms.
Study classifies polyharmonic helices in various space forms.
problem Classifying polyharmonic helices in different space forms.
method Derived classification results for polyharmonic helices in space forms.
result Polyharmonic helices of arbitrary order in space forms of negative curvature are geodesics.
Study on polyharmonic curves on spheres and space forms.
problem Classifying polyharmonic curves of constant curvature.
method Analyzing curves on spheres and space forms, deriving explicit families.
result New insights into higher order variational problems.
The paper proves a hierarchy of Liouville theorems for polyharmonic functions on manifolds with nonnegative Ricci curvature.
problem Establishing Liouville theorems for polyharmonic functions on manifolds with nonnegative Ricci curvature.
method A new L2 estimate for the Laplacian of a polyharmonic function, obtained by induction through a cutoff construction combined with a hole-filling argument. result All polyharmonic functions of sublinear growth on manifolds of nonnegative Ricci curvature are constant.
We study eigenvalues of polyharmonic operators on compact Riemannian manifolds with boundary (possibly empty). In particular, we prove a universal inequality for the eigenvalues of the polyharmonic operators on compact domains in a Euclidean space. This inequality controls the kth eigenvalue by the lower eigenvalues,…
The paper explores polyharmonic curves in semi-Riemannian manifolds.
problem Investigating polyharmonic curves in semi-Riemannian manifolds.
method Analyzing Frenet curves in semi-Riemannian manifolds of various types.
result Existence, non-existence, and classification results for polyharmonic curves.
The paper studies Q-curvature and volume entropy on manifolds, proving polynomial growth polyharmonic function finiteness and rigidity.
problem Investigating Q-curvature and volume entropy on conformally flat manifolds.
method Introducing a new volume entropy, establishing identities, and proving rigidity results.
result Each polynomial growth polyharmonic function on such manifolds is of finite dimension, and the Cohn-Vossen inequality achieves equality under specific conditions.
The paper investigates polyharmonic helices in 3D solvable Lie group Sol_3 and Euclidean spheres.
problem Existence and classification of polyharmonic helices of order r.
method Analytical and geometric approaches, including Lie group theory and Euclidean sphere analysis.
result Complete classification of proper r-harmonic helices in Sol_3 and new examples in Bianchi-Cartan-Vranceanu spaces.
Optimized GAN discriminator using polyharmonic interpolation.
problem Optimizing the discriminator in GANs with higher-order gradient regularization.
method Polyharmonic interpolation and variational calculus.
result The optimal discriminator is a polyharmonic radial basis function.
Develops a framework for generating harmonic maps from a unit ball to a sphere.
problem Creating families of harmonic maps from a unit ball to a sphere.
method Based on Toth's machinery for generating eigenmaps, combined with a generalized radial projection.
result Provides theoretical background for constructing solutions to variational problems.
In this paper we consider the existence and regularity of weakly polyharmonic almost complex structures on a compact almost Hermitian manifold M2m. Such objects satisfy the elliptic system weakly [J,ΔmJ]=0. We prove a very general regularity theorem for semilinear systems in critical dimensions (with \emph{cr…
Polyharmonic, or r-harmonic, maps are a natural generalization of harmonic maps whose study was proposed by Eells-Lemaire in 1983. The main aim of this paper is to construct new examples of proper r-harmonic immersions into spheres. In particular, we shall prove that the canonical inclusion i:Sn−1(R)→Sn i…
In this paper we consider the polyharmonic heat flow of a closed curve in the plane. Our main result is that closed initial data with initially small normalised oscillation of curvature and isoperimetric defect flows exponentially fast in the C^infty-topology to a simple circle. Our results yield a characterisation of …
Sharp bounds derived for eigenvalues on specific geometric spaces.
problem Eigenvalue problems on asymptotically hyperbolic manifolds and submanifolds.
method Sharp bounds derived for three types of eigenvalue problems: p-Dirichlet, polyharmonic, and weakly Poincaré-Einstein. result Sharp bounds and their implications for asymptotic sectional curvatures and mean curvature.
The paper explores polyharmonic hypersurfaces in pseudo-Riemannian space forms.
problem Characterizing polyharmonic hypersurfaces in pseudo-Riemannian space forms.
method Analyzing hypersurfaces with specific properties under given conditions.
result Existence of new families of proper r-harmonic hypersurfaces.
Biharmonic or polyharmonic curves and surfaces in 3-dimensional contact manifolds are investigated.
Characterizes metrics with finite total Q-curvature and introduces new volume entropy.
problem Understanding metrics with finite total Q-curvature and their geometric properties.
method Characterization of metrics through total Q-curvature and introduction of new volume entropy.
result Controlled volume growth for complete metrics with finite total Q-curvature and bounded scalar curvature.
In this paper, we study the benefits of using polyharmonic splines and node layouts with smoothly varying density for developing robust and efficient radial basis function generated finite difference (RBF-FD) methods for pricing of financial derivatives. We present a significantly improved RBF-FD scheme and successfull…
In this paper we shall assume that the ambient manifold is a space form Nm+1(c) and we shall consider polyharmonic hypersurfaces of order r (briefly, r-harmonic), where r≥3 is an integer. For this class of hypersurfaces we shall prove that, if c≤0, then any r-harmonic hypersurface must be minima…
New method uses minimal assumptions for machine learning, improving performance and speed.
problem Current machine learning methods require specific model assumptions that are not derived from prior knowledge.
method Assumes scale invariance principles and differentiability of the true function to derive a novel stochastic process.
result The method achieves equal performance to Gaussian process regression but is less arbitrary, faster, and has better extrapolation.
The study characterizes and studies stability of biharmonic hypersurfaces in complex space forms.
problem Characterizing and studying biharmonic hypersurfaces in complex space forms.
method Characterizing hypersurfaces as critical points of a higher order energy functional.
result Existence and non-existence results for CPn and CHn. Study classifies triharmonic surfaces in 3D homogeneous spaces.
problem Classifying triharmonic surfaces in 3D homogeneous spaces.
method Classification through isoparametric and CMC surfaces.
result Complete classification of CMC r-harmonic Hopf cylinders in BCV-spaces.
In this paper, we obtain a new abstract formula relating eigenvalues of a self-adjoint operator to two families of symmetric and skew-symmetric operators and their commutators. This formula generalizes earlier ones obtained by Harrell, Stubbe, Hook, Ashbaugh, Hermi, Levitin and Parnovski. We also show how one can use t…
We study the curve diffusion flow for closed curves immersed in the Minkowski plane M, which is equivalent to the Euclidean plane endowed with a closed, symmetric, convex curve called an indicatrix that scales the length of a vector in M depending on its length. The indiactrix $\partial\mathcal{…
In this paper, we prove that nonnegative polyharmonic functions on the upper half space satisfying a conformally invariant nonlinear boundary condition have to be the "\emph{polynomials} plus \emph{bubbles}" form. The nonlinear problem is motivated by the recent studies of boundary GJMS operators and the Q-curvature …
New neural architectures with multivariate nonlinearities are optimal in function space.
problem Optimality of neural architectures with multivariate nonlinearities.
method Construction of Banach spaces via k-plane transform and sparsity-promoting norm, proving representer theorem. result Neural architectures with multivariate nonlinearities are optimal in function space.
Complex-valued neural networks can approximate any continuous function.
problem Generalizing the universal approximation theorem to complex-valued networks.
method Characterizing activation functions for complex networks to approximate any continuous function.
result Different activation functions are required for deep vs shallow complex networks to achieve universal approximation.
Complex-valued neural networks can approximate any continuous function with bounded widths and depths.
problem Approximating continuous functions with complex-valued neural networks of bounded widths and depths.
method Analyzing activation functions and proving universality for complex-valued networks.
result Deep narrow complex-valued networks are universal if and only if their activation function is neither holomorphic, nor antiholomorphic, nor R-affine. Gradient descent training of neural networks leads to solutions close to natural cubic splines.
problem Understanding the implicit bias of gradient descent in neural networks.
method Analysis of gradient descent training for wide neural networks, focusing on the curvature penalty and initialization schemes.
result The solutions of gradient descent training are polyharmonic splines for certain initialization schemes.
CVNNs improve performance in tasks with complex-valued inputs.
problem Improving performance in tasks with complex-valued inputs.
method Analyze the approximation properties of complex-valued neural networks (CVNNs).
result Quantitative approximation bounds for CVNNs, showing error scales as m−k/(2n). Spider GAN accelerates GAN training with a new approach.
problem Stable training of Generative adversarial networks (GANs).
method Spider GAN leverages a novel approach to identify closely related datasets (friendly neighborhoods) and uses a new measure (signed inception distance) to accelerate GAN training.
result Spider GAN achieves faster convergence and state-of-the-art FID values with one-fifth of the training iterations.
The article explores the mapping class group using unicellular maps and provides filtrations.
problem Understanding the structure of the mapping class group.
method Using unicellular maps and surgeries, the article describes the mapping class group.
result Provides filtrations of the mapping class group.
Constructs a moment map flow for isotropic maps on surfaces.
problem Understanding isotropic maps on surfaces and their properties.
method Develops a Kähler moment map geometry and a modified moment map flow.
result Polyhedral modified moment map flow induces a strong deformation retraction.
Maps are an important medium that enable people to comprehensively understand the configuration of cultural activities and natural elements over different times and places. Although massive maps are available in the digital era, how to effectively and accurately access the required map remains a challenge today. Previo…
The paper constructs biharmonic maps between spheres using polynomial maps.
problem Creating biharmonic maps between spheres.
method Using harmonic homogeneous polynomial maps of different degrees to generate proper biharmonic maps.
result Established a method for constructing proper biharmonic product maps.
Both bi-harmonic map and f-harmonic map have nice physical motivation and applications. In this paper, by combination of these two harmonic maps, we introduce and study f-bi-harmonic maps as the critical points of the f-bi-energy functional 21∫Mf∣τ(φ)∣2dvg. This class of maps generalizes both …