Effective constructions in plethysms and Weintraub's conjecture

We give a short proof of Weintraub's conjecture by constructing explicit highest weight vectors in the symmetric power of an even exterior power

Preface

The aim of this volume is to advance the understanding of linear spaces of symmetric matrices. These seemingly simple objects play many different roles across several fields of mathematics. For instance, in algebraic statistics these spaces appear as linear Gaussian covariance or concentration models, while in enumerative algebraic geometry they classically represent spaces of smooth quadrics satisfying certain tangency conditions. In semidefinite programming, linear spaces of symmetric matrices define the spectrahedra on which optimization problems are considered, and in nonlinear algebra they encode partially symmetric tensors. It is often the case that one of the above-mentioned fields inspires or pro- vides tools for the advancement of the others. In the articles that follow, the reader will find several examples where this has happened through the common link of linear spaces of symmetric matrices. This volume is the culmination of a collaboration project with the same name, which began at MPI Leipzig in June 2020. Over the course of several months, about 40 researchers gathered on-line to work on the ideas and projects that eventually became the articles of this special issue. We are grateful to Bernd Sturmfels for initiating the project and for being its driving force, in particular for presenting the list of open problems that served as a starting point for the working groups that formed. Many of his conjectures became theorems in this volume. We thank Biagio Ricceri and the editorial team of Le Matematiche for co- ordinating the publication of this volume. Finally, thanks to all participants for their contributions to the talks, discussions, and articles around the project

Very narrow coronal mass ejections producing solar energetic particles

Aims. Our main aim is to study the relationship between low-energy solar particles (energies below 1 MeV) and very narrow coronal mass ejections (“jets” with angular width ≤20 ◦ ). Methods. For this purpose, we considered 125 very narrow coronal mass ejections (CMEs) from 1999 to 2003 that are potentially associated with low-energy solar particles (LESPs). These events were chosen on the basis of their source location. We studied only very narrow CMEs at the western limb, which are expected to have good magnetic connectivity with Earth. Results. We found 24 very narrow CMEs associated with energetic particles such as ions (protons and 3 He), electrons, or both. We show that arrival times at Earth of energetic particles are consistent with onset times of the respective CMEs, and that in the same time intervals, there are no other potential sources of energetic particles. We also demonstrate statistical di ff erences for the angular width distributions using the Kolmogorov–Smirnov test for angular widths for these 24 events. We consider a coherent sample of jets (mostly originating from boundaries of coronal holes) to identify properties of events that produce solar energetic particles (velocities, widths, and position angles). Our study presents a new approach and result: very narrow CMEs can generate low-energy particles in the vicinity of Earth without other activity on the Sun. The results could be very useful for space weather forecasting

Structural abnormalities of the optic nerve and retina in Huntington’s disease pre-clinical and clinical settings

Huntington’s disease (HD) is a fatal neurodegenerative disorder caused by a polyglutamine expansion in the huntingtin protein. HD-related pathological remodelling has been reported in HD mouse models and HD carriers. In this study, we studied structural abnormalities in the optic nerve by employing Spectral Domain Optical Coherence Tomography (SD-OCT) in pre-symptomatic HD carriers of Caucasian origin. Transmission Electron Microscopy (TEM) was used to investigate ultrastructural changes in the optic nerve of the well-established R6/2 mouse model at the symptomatic stage of the disease. We found that pre-symptomatic HD carriers displayed a significant reduction in the retinal nerve fibre layer (RNFL) thickness, including specific quadrants: superior, inferior and temporal, but not nasal. There were no other significant irregularities in the GCC layer, at the macula level and in the optic disc morphology. The ultrastructural analysis of the optic nerve in R6/2 mice revealed a significant thinning of the myelin sheaths, with a lamellar separation of the myelin, and a presence of myelonoid bodies. We also found a significant reduction in the thickness of myelin sheaths in peripheral nerves within the choroids area. Those ultrastructural abnormalities were also observed in HD photoreceptor cells that contained severely damaged membrane disks, with evident vacuolisation and swelling. Moreover, the outer segment of retinal layers showed a progressive disintegration. Our study explored structural changes of the optic nerve in pre- and clinical settings and opens new avenues for the potential development of biomarkers that would be of great interest in HD gene therapies