arXiv research

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…

New method uses minimal assumptions for machine learning, improving performance and speed.

Gradient descent training of neural networks leads to solutions close to natural cubic splines.

Study classifies polyharmonic helices in various space forms.

Study on polyharmonic curves on spheres and space forms.

The paper proves a hierarchy of Liouville theorems for polyharmonic functions on manifolds with nonnegative Ricci curvature.

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 paper studies $f$-polyharmonic maps and their properties.

The article discusses conservation laws for polyharmonic maps and their applications.

The study classifies conformal biharmonic and k-polyharmonic maps between space forms.

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 $k$th eigenvalue by the lower eigenvalues,…

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.

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.

The paper explores polyharmonic curves in semi-Riemannian manifolds.

We consider polyharmonic maps $φ:(M,g)\rightarrow$\mathbb{E}^n$of order k from a complete Riemannian manifold into the Euclidean space and let$p$be a real constant satisfying$1<p<\infty$. (i) If,$\int_M|W^{k-1}|^p dv_g<\infty,$and$\int_M|\bar \nabla W^{k-2}|^2dv_g<\infty.$Then$φ$ is a polyharmonic map of orde…

The paper studies Q-curvature and volume entropy on manifolds, proving polynomial growth polyharmonic function finiteness and rigidity.

The paper investigates polyharmonic helices in 3D solvable Lie group Sol_3 and Euclidean spheres.

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.

Optimized GAN discriminator using polyharmonic interpolation.

Survey on conservation laws for geometric PDEs.

We establish both local and global well-posedness for the heat flow of polyharmonic maps from $R^n$ to a compact Riemannian manifold without boundary for initial data with small BMO norms.

We prove that for any two closed Riemannian manifolds $M^{2m}$ ($m\geq 1$) and $N$, there exists a minimizing (extrinsic) $m$-polyharmonic map for every free homotopy class in $[M^{2m}, N]$, provided that the homotopy group $π_{2m}(N)$ is trivial. This generalizes the celebrated existence results for harmonic maps and …

The paper explores unique continuation properties for polyharmonic maps between Riemannian manifolds.

In this paper we consider the existence and regularity of weakly polyharmonic almost complex structures on a compact almost Hermitian manifold $M^{2m}$. Such objects satisfy the elliptic system weakly $[J, Δ^m J]=0$. We prove a very general regularity theorem for semilinear systems in critical dimensions (with \emph{cr…

The paper studies polyharmonic hypersurfaces in space forms, proving their minimal properties and characterizing specific cases.

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.

The paper explores polyharmonic hypersurfaces in pseudo-Riemannian space forms.

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.

New neural architectures with multivariate nonlinearities are optimal in function space.

Paper finds maximum curvature of Bézier-spline curves.

Revisits stochastic collocation with exponential splines for option pricing.

Paper proves regularity and existence of Riemannian splines.

Improves spline quality and accuracy in computational microscopy.

This paper develops a new method for constructing splines on Lie groups using Poisson equation solutions.

Sig-Splines model uses signatures and splines for time series data, achieving universality and convexity.

We extend the adaptive regression spline model by incorporating saturation, the natural requirement that a function extend as a constant outside a certain range. We fit saturating splines to data using a convex optimization problem over a space of measures, which we solve using an efficient algorithm based on the condi…

With the renewed and growing interest in geometric continuity in mind, this article gives a general definition of geometrically continuous polygonal surfaces and geometrically continuous spline functions on them. Polynomial splines defined by G1 gluing data in terms of rational functions are analyzed further. A general…

Sinh-acceleration speeds up B-spline option pricing.

Combines spline interpolation and ARIMA for stock market forecasting.

A new method evolves point clouds using B-splines for smooth surfaces.

A new nonparametric approach for system identification has been recently proposed where the impulse response is seen as the realization of a zero--mean Gaussian process whose covariance, the so--called stable spline kernel, guarantees that the impulse response is almost surely stable. Maximum entropy properties of the …

A new spline method for manifold learning using Hessian-based curvature penalties.

Locally-verifiable conditions ensure exactness of spline discrete de Rham complex.

We use splines and the Sasaki metric to analyze and compare manifold-valued trajectories.

A new knot selection method for GAMs reduces model complexity.