Bounded geodesics in rank-1 locally symmetric spaces

Let M be a rank 1 locally symmetric space of finite Riemannian volume. It is proved that the set of unit vectors on a non-constant C1 curve in the unit tangent sphere at a point p M for which the corresponding geodesic is bounded (relatively compact) in M, is a set of Hausdorff dimension

Water relations of climbing ivy in a temperate forest

Ivy (Hedera helix) is the most important liana in temperate European forests. We studied water relations of adult ivy in a natural, 35m tall mixed deciduous forest in Switzerland using a construction crane to access the canopy. Predawn leaf water potential at the top of climbing ivy ranged from −0.4 to −0.6MPa, daily minima ranged from −1.3 to −1.7MPa. Leaf water potentials as well as relative sap flow were held surprisingly constant throughout different weather conditions, suggesting a tendency to isohydric behaviour. Maximum stomatal conductance was 200mmolm−2s−1. The use of a potometer experiment allowed us to measure absolute transpiration rates integrated over a whole plant of 0.23mmolm−2s−1. Nightly sap flow of ivy during warm, dry nights accounted for up to 20% of the seasonal maximum. Maximum sap flow rates were reached at ca. 0.5kPa vpd. On the other hand, the host trees showed a less conservative stomatal regulation, maximum sap flow rates were reached at vpd values of ca. 1kPa. Sap flow rates of ivy decreased by ca. 20% in spring after bud break of trees, suggesting that ivy profits strongly from warm sunny days in early spring before budbreak of the host trees and from mild winter days. This species may benefit from rising winter temperatures in Europe and thus become a stronger competitor against its host tree

Bounded geodesics in rank-1 locally symmetric spaces

Let M be a rank 1 locally symmetric space of finite Riemannian volume. It is proved that the set of unit vectors on a non-constant C1 curve in the unit tangent sphere at a point p M for which the corresponding geodesic is bounded (relatively compact) in M, is a set of Hausdorff dimension

Trigonometry of 'complex Hermitian' type homogeneous symmetric spaces

This paper contains a thorough study of the trigonometry of the homogeneous symmetric spaces in the Cayley-Klein-Dickson family of spaces of 'complex Hermitian' type and rank-one. The complex Hermitian elliptic CP^N and hyperbolic CH^N spaces, their analogues with indefinite Hermitian metric and some non-compact symmetric spaces associated to SL(N+1,R) are the generic members in this family. The method encapsulates trigonometry for this whole family of spaces into a single "basic trigonometric group equation", and has 'universality' and '(self)-duality' as its distinctive traits. All previously known results on the trigonometry of CP^N and CH^N follow as particular cases of our general equations. The physical Quantum Space of States of any quantum system belongs, as the complex Hermitian space member, to this parametrised family; hence its trigonometry appears as a rather particular case of the equations we obtain.Comment: 46 pages, LaTe

Trigonometry of spacetimes: a new self-dual approach to a curvature/signature (in)dependent trigonometry

A new method to obtain trigonometry for the real spaces of constant curvature and metric of any (even degenerate) signature is presented. The method encapsulates trigonometry for all these spaces into a single basic trigonometric group equation. This brings to its logical end the idea of an absolute trigonometry, and provides equations which hold true for the nine two-dimensional spaces of constant curvature and any signature. This family of spaces includes both relativistic and non-relativistic homogeneous spacetimes; therefore a complete discussion of trigonometry in the six de Sitter, minkowskian, Newton--Hooke and galilean spacetimes follow as particular instances of the general approach. Any equation previously known for the three classical riemannian spaces also has a version for the remaining six spacetimes; in most cases these equations are new. Distinctive traits of the method are universality and self-duality: every equation is meaningful for the nine spaces at once, and displays explicitly invariance under a duality transformation relating the nine spaces. The derivation of the single basic trigonometric equation at group level, its translation to a set of equations (cosine, sine and dual cosine laws) and the natural apparition of angular and lateral excesses, area and coarea are explicitly discussed in detail. The exposition also aims to introduce the main ideas of this direct group theoretical way to trigonometry, and may well provide a path to systematically study trigonometry for any homogeneous symmetric space.Comment: 51 pages, LaTe

Severe Neuro-COVID is associated with peripheral immune signatures, autoimmunity and neurodegeneration: a prospective cross-sectional study

Growing evidence links COVID-19 with acute and long-term neurological dysfunction. However, the pathophysiological mechanisms resulting in central nervous system involvement remain unclear, posing both diagnostic and therapeutic challenges. Here we show outcomes of a cross-sectional clinical study (NCT04472013) including clinical and imaging data and corresponding multidimensional characterization of immune mediators in the cerebrospinal fluid (CSF) and plasma of patients belonging to different Neuro-COVID severity classes. The most prominent signs of severe Neuro-COVID are blood-brain barrier (BBB) impairment, elevated microglia activation markers and a polyclonal B cell response targeting self-antigens and non-self-antigens. COVID-19 patients show decreased regional brain volumes associating with specific CSF parameters, however, COVID-19 patients characterized by plasma cytokine storm are presenting with a non-inflammatory CSF profile. Post-acute COVID-19 syndrome strongly associates with a distinctive set of CSF and plasma mediators. Collectively, we identify several potentially actionable targets to prevent or intervene with the neurological consequences of SARS-CoV-2 infection