Revisiting cosmologies in teleparallelism

We discuss the most general field equations for cosmological spacetimes for theories of gravity based on non-linear extensions of the non-metricity scalar and the torsion scalar. Our approach is based on a systematic symmetry-reduction of the metric-affine geometry which underlies these theories. While for the simplest conceivable case the connection disappears from the field equations and one obtains the Friedmann equations of General Relativity, we show that in $f(\mathbb{Q})$ cosmology the connection generically modifies the metric field equations and that some of the connection components become dynamical. We show that $f(\mathbb{Q})$ cosmology contains the exact General Relativity solutions and also exact solutions which go beyond. In $f(\mathbb{T})$~cosmology, however, the connection is completely fixed and not dynamical.

Electroweak monopoles with a non-linearly realized weak hypercharge

We present a finite-energy electroweak-monopole solution obtained by considering non-linear extensions of the hypercharge sector of the Electroweak Theory, based on logarithmic and exponential versions of electrodynamics. We find constraints for a class of non-linear extensions and also work out an estimate for the monopole mass in this scenario. We finally derive a lower bound for the energy of the monopole and discuss the simpler case of a Dirac magnetic charge.

Genotype, Nursery Design, and Depth Influence the Growth of Acropora cervicornis Fragments

Growing fragments of corals in nurseries and outplanting them to supplement declining natural populations have gained significant traction worldwide. In the Caribbean, for example, this approach provides colonies of Acropora cervicornis with minimal impacts to existing wild colonies. Given the impetus to scale up production to augment limited natural recovery, managers and researchers should consider how the design and location of the nurseries affect the growth of different genotypes of corals and the effort required for maintenance. To elucidate such influences, we grew fragments of different genotypes (five varieties) on differing structures (trees and frames) at two depths (6–8 and 16–18 m). The sum of the lengths of all branches or total linear extensions (TLEs) and accumulation of biofouling were measured over 198 days from May to December 2016 to assess the growth of fragments and the effort required to maintain nurseries. TLEs for all fragments increased linearly throughout the incubation period. Mean daily incremental growth rates varied among the genotypes, with one genotype growing significantly faster than all others, two genotypes growing at intermediate rates, and two genotypes growing more slowly. Mean daily incremental growth rates were higher for all genotypes suspended from vertical frames at both sites, and mean daily incremental growth rates were higher for all fragments held on both types of nurseries in deeper water. If linear growth continued, a fragment of the fastest growing genotype held on a frame in deeper water was estimated to increase the sum of the length of all its branches by an average of 88 cm y–1, which was over two times higher than the estimated mean annual growth rate for a fragment of the slowest growing genotype held on a tree in shallow water. Nurseries in deeper water had significantly less biofouling and appeared to be buffered against daily fluctuations in temperature. Overall, the results demonstrated that increased production and reduced maintenance can result from considering the genotype of fragments to be cultured and the design and location of nurseries.

Accidental Gauge Symmetries of Minkowski Spacetime in Teleparallel Theories

In this paper, we provide a general framework for the construction of the Einstein frame within non-linear extensions of the teleparallel equivalents of General Relativity. These include the metric teleparallel and the symmetric teleparallel, but also the general teleparallel theories. We write the actions in a form where we separate the Einstein–Hilbert term, the conformal mode due to the non-linear nature of the theories (which is analogous to the extra degree of freedom in f(R) theories), and the sector that manifestly shows the dynamics arising from the breaking of local symmetries. This frame is then used to study the theories around the Minkowski background, and we show how all the non-linear extensions share the same quadratic action around Minkowski. As a matter of fact, we find that the gauge symmetries that are lost by going to the non-linear generalisations of the teleparallel General Relativity equivalents arise as accidental symmetries in the linear theory around Minkowski. Remarkably, we also find that the conformal mode can be absorbed into a Weyl rescaling of the metric at this order and, consequently, it disappears from the linear spectrum so only the usual massless spin 2 perturbation propagates. These findings unify in a common framework the known fact that no additional modes propagate on Minkowski backgrounds, and we can trace it back to the existence of accidental gauge symmetries of such a background.

De Finetti Lattices and Magog Triangles

The order ideal $B_{n,2}$ of the Boolean lattice $B_n$ consists of all subsets of size at most $2$. Let $F_{n,2}$ denote the poset refinement of $B_{n,2}$ induced by the rules: $i < j$ implies $\{i \} \prec \{ j \}$ and $\{i,k \} \prec \{j,k\}$. We give an elementary bijection from the set $\mathcal{F}_{n,2}$ of linear extensions of $F_{n,2}$ to the set of shifted standard Young tableau of shape $(n, n-1, \ldots, 1)$, which are counted by the strict-sense ballot numbers. We find a more surprising result when considering the set $\mathcal{F}_{n,2}^{1}$ of minimal poset refinements in which each singleton is comparable with all of the doubletons. We show that $\mathcal{F}_{n,2}^{1}$ is in bijection with magog triangles, and therefore is equinumerous with alternating sign matrices. We adopt our proof techniques to show that row reversal of an alternating sign matrix corresponds to a natural involution on gog triangles.

Counting Linear Extensions of Restricted Posets

We prove the 1991 conjecture by Brightwell and Winkler that counting the number of linear extensions for posets of height two is #P-complete. We further extend this result to incidence posets of graphs.