Extended Field Equations for Conformally Curved Spacetime

The recent Planck Legacy release has confirmed the presence of an enhanced lensing amplitude in the cosmic microwave background power spectra, which prefers a positively curved early Universe with a confidence level greater than 99%. In addition, the spacetime curvature of the entire galaxy differs from one galaxy to another due to their diverse energy densities. This study considers both the implied positive curvature of the early Universe and the curvature across the entire galaxy as the curvature of ‘the background or the 4D bulk’ and distinguishes it from the localized curvature that is induced in the bulk by the presence of comparably smaller celestial objects that are regarded as ‘relativistic 4D branes’. Branes in different galaxies experience different bulk curvatures, thus their background or bulk curvature should be taken into consideration along with their energy densities when finding their induced curvatures. To account for the interaction between the bulk and branes, this paper presents extended field equations in terms of brane-world modified gravity consisting of conformal Einstein field equations with a boundary term, which could remove the singularities and satisfy a conformal invariance theory. A visualization of the evolution of the 4D relativistic branes over the conformal space-time of the 4D bulk is presented.

Folding-like techniques for CAT(0) cube complexes

In a seminal paper, Stallings introduced folding of morphisms of graphs. One consequence of folding is the representation of finitely-generated subgroups of a finite-rank free group as immersions of finite graphs. Stallings’s methods allow one to construct this representation algorithmically, giving effective, algorithmic answers and proofs to classical questions about subgroups of free groups. Recently Dani–Levcovitz used Stallings-like methods to study subgroups of right-angled Coxeter groups, which act geometrically on CAT(0) cube complexes. In this paper we extend their techniques to fundamental groups of non-positively curved cube complexes.

Mechanistic insight into bacterial entrapment by septin cage reconstitution

Septins are cytoskeletal proteins that assemble into hetero-oligomeric complexes and sense micron-scale membrane curvature. During infection with Shigella flexneri, an invasive enteropathogen, septins restrict actin tail formation by entrapping bacteria in cage-like structures. Here, we reconstitute septin cages in vitro using purified recombinant septin complexes (SEPT2-SEPT6-SEPT7), and study how these recognize bacterial cells and assemble on their surface. We show that septin complexes recognize the pole of growing Shigella cells. An amphipathic helix domain in human SEPT6 enables septins to sense positively curved membranes and entrap bacterial cells. Shigella strains lacking lipopolysaccharide components are more efficiently entrapped in septin cages. Finally, cryo-electron tomography of in vitro cages reveals how septins assemble as filaments on the bacterial cell surface.

Geodesic orbit Finsler spaces with K ≥ 0 and the (FP) condition

We study the interaction between the g.o. property and certain flag curvature conditions. A Finsler manifold is called g.o. if each constant speed geodesic is the orbit of a one-parameter subgroup. Besides the non-negatively curved condition, we also consider the condition (FP) for the flag curvature, i.e. in any flag we find a flag pole such that the flag curvature is positive. By our main theorem, if a g.o. Finsler space (M, F) has non-negative flag curvature and satisfies (FP), then M is compact. If M = G/H where G has a compact Lie algebra, then the rank inequality rk 𝔤 ≤ rk 𝔥+1 holds. As an application,we prove that any even-dimensional g.o. Finsler space which has non-negative flag curvature and satisfies (FP) is a smooth coset space admitting a positively curved homogeneous Riemannian or Finsler metric.