Related papers: Cubatures on Grassmannians: moments, dimension red…

The decomposition of polynomial spaces on unions of Grassmannians $\mathcal G_{{k_1},d}\cup\ldots\cup \mathcal G_{{k_r},d}$ into irreducible orthogonally invariant subspaces and their reproducing kernels are investigated. We also generalize…

Utilizing recently developed abstract notions of sectional curvature, we introduce a method for constructing a curvature-based geometric profile of discrete metric spaces. The curvature concept that we use here captures the metric relations…

Metrics on Grassmannians have a wide array of applications: machine learning, wireless communication, computer vision, etc. But the available distances between subspaces of distinct dimensions present problems, and the dimensional asymmetry…

The goal of the paper is to describe essentially optimal cubature formulas on compact Riemannian manifolds which are exact on spaces of band- limited functions.

The Grassmann angle improves upon similar angles between subspaces that measure volume contraction in orthogonal projections. It works in real or complex spaces, with important differences, and is asymmetric, what makes it more efficient…

We describe a new method to compute general cubature formulae. The problem is initially transformed into the computation of truncated Hankel operators with flat extensions. We then analyse the algebraic properties associated to flat…

We construct cubature methods on scattered data via resampling on the support of known algebraic cubature formulas, by different kinds of adaptive interpolation (polynomial, RBF, PUM). This approach gives a promising alternative to other…

Recently it has been shown that cumulants significantly simplify the analysis of multipartite weak measurements. Here we consider the mathematical structure that underlies this, and find that it can be formulated in terms of what we call…

In this tutorial, we provide an overview of many of the established combinatorial and algebraic tools of Schubert calculus, the modern area of enumerative geometry that encapsulates a wide variety of topics involving intersections of linear…

Cubature formulas and geometrical designs are described in terms of reproducing kernels for Hilbert spaces of functions on the one hand, and Markov operators associated to orthogonal group representations on the other hand. In this way,…

The goal of the paper is to establish cubature formulas on combinatorial graphs. Two types of cubature formulas are developed. Cubature formulas of the first type are exact on spaces of variational splines on graphs. Since badlimited…

A number of topics involving metrics and measures are discussed, including some of the special structure associated with ultrametrics.

We obtain an explicit determinantal formula for the multiplicity of any point on a classical Schubert variety.

In numerical integration, cubature methods are effective, especially when the integrands can be well-approximated by known test functions, such as polynomials. However, the construction of cubature formulas has not generally been known, and…

This paper contains the material discussed in the series of three lectures that I gave during the workshop of the ICRA 2018 in Prague. I will introduce the reader to some of the techniques used in the study of the geometry of quiver…

Many real life problems can be reduced to the solution of a complex exponentials approximation problem which is usually ill posed. Recently a new transform for solving this problem, formulated as a specific moments problem in the plane, has…

This work presents the cubature scheme for the fitting of spatio-temporal Poisson point processes. The methodology is implemented in the R Core Team (2024) package stopp (D'Angelo and Adelfio, 2023), published on the Comprehensive R Archive…

Numerical integration formulas in $n$-dimensional Euclidean space of degree three are discussed. For the integrals with permutation symmetry we present a method to construct its third-degree integration formulas with $2n$ real points. We…

The cubature on Wiener space method, a high-order weak approximation scheme, is established for SPDEs in the case of unbounded characteristics and unbounded payoffs. We first introduce a recently described flexible functional analytic…

We introduce a stochastic analysis of Grassmann random variables suitable for the stochastic quantization of Euclidean fermionic quantum field theories. Analysis on Grassmann algebras is developed here from the point of view of quantum…