Bounded-rank tensors are defined in bounded degree

Matrices of rank at most k are defined by the vanishing of polynomials of degree k + 1 in their entries (namely, their (k + 1)-times-(k + 1)-subdeterminants), regardless of the size of the matrix. We prove a qualitative analogue of this statement for tensors of arbitrary dimension, where matrices correspond to two-dimensional tensors. More specifically, we prove that for each k there exists an upper bound d = d(k) such that tensors of border rank at most k are defined by the vanishing of polynomials of degree at most d, regardless of the dimension of the tensor and regardless of its sizes in each dimension. Our proof involves passing to an infinite-dimensional limit of tensor powers of a vector space, whose elements we dub infinite-dimensional tensors, and exploiting the symmetries of this limit in crucial way.Comment: Corrected the formulation of Corollary 1.4---thanks to Sonja Petrovic

On the ideals of equivariant tree models

We introduce equivariant tree models in algebraic statistics, which unify and generalise existing tree models such as the general Markov model, the strand symmetric model, and group based models. We focus on the ideals of such models. We show how the ideals for general trees can be determined from the ideals for stars. The main novelty is our proof that this procedure yields the entire ideal, not just an ideal defining the model set-theoretically. A corollary of theoretical importance is that the ideal for a general tree is generated by the ideals of its flattenings at vertices.Comment: 23 pages. Greatly improved exposition, in part following suggestions by a referee--thanks! Also added exampl

ROCK-ALS: Protocol for a Randomized, Placebo-Controlled, Double-Blind Phase IIa Trial of Safety, Tolerability and Efficacy of the Rho Kinase (ROCK) Inhibitor Fasudil in Amyotrophic Lateral Sclerosis

Objectives: Disease-modifying therapies for amyotrophic lateral sclerosis (ALS) are still not satisfactory. The Rho kinase (ROCK) inhibitor fasudil has demonstrated beneficial effects in cell culture and animal models of ALS. For many years, fasudil has been approved in Japan for the treatment of vasospasm in patients with subarachnoid hemorrhage with a favorable safety profile. Here we describe a clinical trial protocol to repurpose fasudil as a disease-modifying therapy for ALS patients.Methods: ROCK-ALS is a multicenter, double-blind, randomized, placebo-controlled phase IIa trial of fasudil in ALS patients (EudraCT: 2017-003676-31, NCT: 03792490). Safety and tolerability are the primary endpoints. Efficacy is a secondary endpoint and will be assessed by the change in ALSFRS-R, ALSAQ-5, slow vital capacity (SVC), ECAS, and the motor unit number index (MUNIX), as well as survival. Efficacy measures will be assessed before (baseline) and immediately after the infusion therapy as well as on days 90 and 180. Patients will receive a daily dose of either 30 or 60 mg fasudil, or placebo in two intravenous applications for a total of 20 days. Regular assessments of safety will be performed throughout the treatment period, and in the follow-up period until day 180. Additionally, we will collect biological fluids to assess target engagement and evaluate potential biomarkers for disease progression. A total of 120 patients with probable or definite ALS (revised El Escorial criteria) and within 6–18 months of the onset of weakness shall be included in 16 centers in Germany, Switzerland and France.Results and conclusions: The ROCK-ALS trial is a phase IIa trial to evaluate the ROCK-inhibitor fasudil in early-stage ALS-patients that started patient recruitment in 2019

Nilpotent subspaces of maximal dimension in semisimple Lie algebras

We show that a linear subspace of a reductive Lie algebra g that consists of nilpotent elements has dimension at most equal to the number of positive roots, and that any nilpotent subspace attaining this upper bound is equal to the nilradical of a Borel subalgebra of g. This generalizes a classical theorem of Gerstenhaber which states this fact for the algebra of n x n matrices

Differentials for Lie algebras

We develop a theory of relative Kähler differentials for Lie algebras. The main result is that the functor of relative differentials is representable, and that the universal object which represents it behaves properly with respect to étale base change. We illustrate how our construction yields a detailed analysis of the structure of derivations of multiloop algebras which is needed for the construction of Extended Affine Lie Algebras.Fil: Kuttler, Jochen. University of Alberta; CanadáFil: Pianzola, Arturo. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. University of Alberta; Canadá. Universidad Centro de Altos Estudios en Ciencia Exactas. Departamento de Matemáticas; Argentin

Lie algebroids arising from simple group schemes

A classical construction of Atiyah assigns to any (real or complex) Lie group G, manifold M and principal homogeneous G-space P over M, a Lie algebroid over M ([1]). The spirit behind our work is to put this work within an algebraic context, replace M by a scheme X and G by a “simple” reductive group scheme G over X (in the sense of Demazure–Grothendieck) that arise naturally with an attached torsor (which plays the role of P) in the study of Extended Affine Lie Algebras (see [9] for an overview). Lie algebroids in an algebraic sense were also considered by Beilinson and Bernstein. We will explain how the present work relates to theirs.Fil: Kuttler, Jochen. University of Alberta; CanadáFil: Pianzola, Arturo. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. University of Alberta; Canadá. Centro de Altos Estudios en Ciencias Exactas; ArgentinaFil: Quallbrunn, Federico. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Centro de Altos Estudios en Ciencias Exactas; Argentina. Universidad de Buenos Aires. Facultad de Ingeniería. Departamento de Matemáticas; Argentin