Search

In recent years there has been an explosion of interest in digital twinning in many disciplines, including the manufacturing, geospatial, and medical domains. A core topic of importance in modelling digital twins, is reconstruction of geometric models from raw data. Despite the diversity of requirements in the vast space of digital twin applications, methods for geometric reconstruction can often be transferred between disciplines with only minor modifications. In this paper we present some recent results related to how advances in machine learning over the last decade can be used for data-driven geometric reconstruction in the medical, geospatial and manufacturing domains.

For the space of $C^3$ quintics on the Powell-Sabin 12-split of a triangle, we determine explicitly the six symmetric simplex spline bases that reduce to a B-spline basis on each edge, have a positive partition of unity, a Marsden identity that splits into real linear factors, and an intuitive domain mesh. The bases are stable in the $L_\infty$ norm with a condition number independent of the geometry, have a well-conditioned Lagrange interpolant at the domain points, and a quasi-interpolant with local approximation order 6. We show an $h^2$ bound for the distance between the control points and the values of a spline at the corresponding domain points. For one of these bases we derive $C^0$, $C^1$, $C^2$ and $C^3$ conditions on the control points of two splines on adjacent macrotriangles., Comment: 26 pages

We review the construction and a few properties of the S-basis, a simplex spline basis for the $C^1$ quadratic splines on the Powell-Sabin 12-split., Comment: Oberwolfach report for the conference Multivariate Splines and Algebraic Geometry

A novel and deterministic algorithm is presented to detect whether two given planar rational curves are related by means of a similarity, which is a central question in Pattern Recognition. As a by-product it finds all such similarities, and the particular case of equal curves yields all symmetries. A complete theoretical description of the method is provided, and the method has been implemented and tested in the SAGE system., Ministerio de Ciencia e Innovación

Under general conditions, the equation $g(x,y) = 0$ implicitly defines $y$ locally as a function of $x$. In this short note we study the combinatorial structure underlying a recently discovered formula for the divided differences of $y$ expressed in terms of bivariate divided differences of $g$, by analyzing the number of terms $a_n$ in this formula. The main result describes six equivalent characterizations of the sequence $\{a_n\}$., 11 pages, 2 figures

This paper addresses the pattern of damage, and investigates its properties, of a theoretical hail storm which gathers in intensity before subsiding, and which travels linearly across the landscape at constant velocity. We start by assuming a simpler model, that of a storm which does not move, restricted to having an uncorrelated binormal distribution of damage. This model, expressed in the natural polar coordinates, leads to a 1-dimensional pattern of damage as a function of the marginal radial distance conforming to the $\chi$-distribution with two degrees of freedom. The model is then extended to the traveling form, allowing further for a correlation of the variables, extending, as well, to the multidimensional case. In its full florescence the model produces hyperellipsoidal hypersurfaces of equal intensity for the correlated multinormal assumption. We provide closed-form solutions for the totality of damages upon these hypersurfaces as proxies for the insurance claims to follow. Finally the model is applied to extensive datasets of hail events, as detected by the NEXRAD network of weather radars., Comment: 15 pages, 3 figures, 2 tables

Differential equations with the Painlev�� property have been studied extensively due to their appearance in many branches of mathematics and their applicability in physics. Although a modern, differential algebraic treatment of the order one equations appeared before, the connection with the classical theory did not. Using techniques from algebraic geometry we provide the link between the classical and the modern treatment, and with the help of differential Galois theory a new classification is derived, both for characteristic 0 and p., 1 figure