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In this paper, we study the convergence of the spectral embeddings obtained from the leading eigenvectors of certain similarity matrices to their population counterparts. We opt to study this convergence in a uniform (instead of average)…
The interpolation-regression approximation is a powerful tool in numerical analysis for reconstructing functions defined on square or triangular domains from their evaluations at a regular set of nodes. The importance of this technique lies…
Recent experimental studies focusing on the morphological properties of surfaces eroded by ion-bombardment report the observation of self-affine fractal surfaces, while others provide evidence about the development of a periodic ripple…
Multiphysics simulations frequently require transferring solution fields between subproblems with non-matching spatial discretizations, typically using interpolation techniques. Standard methods are usually based on measuring the closeness…
We present {\tt radpol} - a numerical scheme for integrating multifrequency polarized radiative transfer equations along rays propagating in a curved spacetime. The scheme includes radiative processes such as synchrotron emission,…
For $h>0$ and positive integers $m$, $d$, such that $m>d/2$, we study non-stationary interpolation at the points of the scaled grid $h\mathbb{Z}^d$ via the Mat\'{e}rn kernel $\Phi_{m,d}$---the fundamental solution of $(1-\Delta)^m$ in…
Two methods are developed for solving the steady-state spherically symmetric radiative transfer equation for resonance line radiation emitted by a point source in the Intergalactic Medium. One method is based on solving the ray and moment…
This paper constructs polynomial bases that capture the structure of the de Rham complex with boundary conditions in disks and cylinders (both periodic and finite) in a way that respects rotational symmetry. The starting point is explicit…
We generalize the transfinite triangular interpolant of (Nielson, 1987) in order to generate visually smooth (not necessarily polynomial) local interpolating quasi-optimal triangular spline surfaces. Given as input a triangular mesh stored…
For every finite collection of curves on a surface, we define an associated (semi-)norm on the first homology group of the surface. The unit ball of the dual norm is the convex hull of its integer points. We give an interpretation of these…
We present a topological recursion formula for calculating the intersection numbers defined on the moduli space of open Riemann surfaces. The spectral curve is $x = \frac{1}{2}y^2$, the same as spectral curve used to calculate intersection…
We study the parameter space of cnoidal waves -- the periodic solitons of the Korteweg-de Vries equation -- from the perspective of Virasoro coadjoint orbits. The monodromy method familiar from inverse scattering implies that many, but not…
We study points of moderately low degree on a curve $C$ over a number field, which is embedded on a nice toric surface $S$. Recently, Smith and Vogt related the linear equivalence classes of such points to intersections of $C$ with curves…
We extend the univariate Newton interpolation algorithm to arbitrary spatial dimensions and for any choice of downward-closed polynomial space, while preserving its quadratic runtime and linear storage cost. The generalisation supports any…
Any two equivalent discrete curves must have the same invariants at the corresponding points under an affine transformation. In this paper, we construct the moving frame and invariants for the discrete centroaffine curves, which could be…
Many applications of geometry modeling and computer graphics necessite accurate curvature estimations of curves on the plane or on manifolds. In this paper, we define the notion of the discrete geodesic curvature of a geodesic polygon on a…
We describe a numerical scheme for studying time-dependent, multifluid, magnetohydrodynamic shock waves in weakly ionized interstellar clouds and cores. Shocks are modeled as propagating perpendicular to the magnetic field and consist of a…
The marriage of recurrent neural networks and neural ordinary differential networks (ODE-RNN) is effective in modeling irregularly-observed sequences. While ODE produces the smooth hidden states between observation intervals, the RNN will…
Research on smooth vector graphics is separated into two independent research threads: one on interpolation-based gradient meshes and the other on diffusion-based curve formulations. With this paper, we propose a mathematical formulation…
Classical approximation bases such as Chebyshev polynomials provide principled and interpretable representations, but their multivariate tensor-product constructions scale exponentially with dimension and impose axis-aligned structure that…