We study Sym(∞)-orbit closures of non-necessarily closed points in the Zariski spectrum of the infinite polynomial ring C[xij : i ∈ N, j ∈ [n]]. Among others, we characterize invariant prime ideals in this ring. Furthermore, we study projections of basic equivariant semi-algebraic sets defined by Sym(∞) orbits of polynomials in R[xij : i ∈ N, j ∈ [n]]. For n = 1 we prove a quantifier elimination ...

Measurements of ion cyclotron emission (ICE) are obtained from most large magnetically confined fusion plasma experiments, and may be used in future to quantify properties of the fusion-born alpha-particle population in deuterium-tritium (DT) plasmas in ITER. ICE is driven by spatially localised, strongly non-Maxwellian, minority energetic ion populations which relax collectively under the ...

We would like to start by saying that this is a very interesting paper that outlines a powerful tool that can be applied in a wide range of areas. In our discussion, we focus on how the novel approach potentially can improve classification results in hard tasks within e.g., medicine and geoscience. In such situations, where it is hard to make precise predictions, it is natural to acquire information ...

Associated to a symmetric space there is a canonical connection with zero torsion and parallel curvature. This connection acts as a binary operator on the vector space of smooth sections of the tangent bundle, and it is linear with respect to the real numbers. Thus the smooth section of the tangent bundle together with the connection form an algebra we call the connection algebra. The constraints ...

The exotic aromatic Butcher series were originally introduced for the calculation of order conditions for the high order numerical integration of ergodic stochastic differential equations in Rd and on manifolds. We prove in this paper that exotic aromatic B-series satisfy a universal geometric property, namely that they are characterised by locality and equivariance with respect to orthogonal ...

This work derives upper bounds on the convergence rate of the moment-sum-of-squares hierarchy with correlative sparsity for global minimization of polynomials on compact basic semialgebraic sets. The main conclusion is that both sparse hierarchies based on the Schmüdgen and Putinar Positivstellensätze enjoy a polynomial rate of convergence that depends on the size of the largest clique in the sparsity ...

The Gelfand-Tsetlin and the Feigin–Fourier–Littelmann–Vinberg polytopes for the Grassmannians are defined, from the perspective of representation theory, to parametrize certain bases for highest weight irreducible modules. These polytopes are Newton-Okounkov bodies for the Grassmannian and, in particular, the GT polytope is an example of a string polytope. The polytopes admit a combinatorial ...

A singular <i>k</i>-tuple of a tensor T of format (<i>n₁</i>, ..., <i>n</i>_<i><k></i>) is essentially a complex critical point of the distance function from <i>T</i> constrained to the cone of tensors of format (<i>n</i>₁, ..., <i>n_k</i>) of rank at most one. A generic tensor has finitely many complex singular <i>k</i>-tuples, and their number depends only ...

We state the relation between the variety of binary forms of given rank and the dual of the multiple root loci. This is a new result for the suprageneric rank that appears as a continuation of the cited work by Buczyński, Han, Mella and Teitler. We describe the strata of these varieties and explore their singular loci.

Objectives To identify long-term disease activity trajectories from childhood to adulthood by using the clinical Juvenile Arthritis Disease Activity Score (cJADAS10) in juvenile idiopathic arthritis (JIA). Second, to evaluate the contribution of the cJADAS10 components and explore characteristics associated with active disease at the 18- year follow-up.<p> <p>Methods Patients with onset of ...