Richardson v Oracle more than half a decade on: Did the ‘ground break’ for victim compensation?

In 2014, the decision in Richardson v Oracle significantly increased compensation awards for victims of discrimination and sexual harassment. Commentators referred to the decision as ‘ground-breaking’ and hypothesised that the floodgates would open for sexual harassment and discrimination cases. This article reviews these claims seven years on and asks: did the floodgates open? Case law review and analysis concludes that, while the ground did not break open for victims of discrimination and sexual harassment, it has certainly cracked – although unevenly. As such, more is required to create truly just outcomes for victims in discrimination law.

Energy storage and coherence in closed and open quantum batteries

We study the role of coherence in closed and open quantum batteries. We obtain upper bounds to the work performed or energy exchanged by both closed and open quantum batteries in terms of coherence. Specifically, we show that the energy storage can be bounded by the Hilbert-Schmidt coherence of the density matrix in the spectral basis of the unitary operator that encodes the evolution of the battery. We also show that an analogous bound can be obtained in terms of the battery's Hamiltonian coherence in the basis of the unitary operator by evaluating their commutator. We apply these bounds to a 4-state quantum system and the anisotropic XY Ising model in the closed system case, and the Spin-Boson model in the open case.

Pulmonary Koch in 4 Months Old: Congenital or Acquired?

4 months old female child presented to us with complaints of fever, cough, cold, and poor weight gain. On examination the child had failure to thrive (weight 2.75kg, length 52cms, head circumference 35cms, weight for length <-3SD), cachexic look, severe respiratory distress (bilateral chest retractions and nasal flaring), hepatosplenomegaly, delayed milestones with low birth weight (2.07kg), continuous low grade fever. Mother was an open case of pulmonary Koch, taking antitubercular treatment since 6 months. Investigations revealed anemia, neutrophilia with reactive CRP, raised transaminases, pulmonary fibrosis and cavitary lesions in chest x-ray with normal CSF examination. Mantoux was reactive with gastric aspirate and cerebrospinal fluid negative for TB bacilli in GeneXpert. So the question arises, is it congenital or acquired? Key words: congenital TB, hepatosplenomegaly, respiratory distress, failure to thrive.

Toughness, Forbidden Subgraphs and Pancyclicity

Motivated by several conjectures due to Nikoghosyan, in a recent article due to Li et al., the aim was to characterize all possible graphs H such that every 1-tough H-free graph is hamiltonian. The almost complete answer was given there by the conclusion that every proper induced subgraph H of $$K_1\cup P_4$$ K 1 ∪ P 4 can act as a forbidden subgraph to ensure that every 1-tough H-free graph is hamiltonian, and that there is no other forbidden subgraph with this property, except possibly for the graph $$K_1\cup P_4$$ K 1 ∪ P 4 itself. The hamiltonicity of 1-tough $$K_1\cup P_4$$ K 1 ∪ P 4 -free graphs, as conjectured by Nikoghosyan, was left there as an open case. In this paper, we consider the stronger property of pancyclicity under the same condition. We find that the results are completely analogous to the hamiltonian case: every graph H such that any 1-tough H-free graph is hamiltonian also ensures that every 1-tough H-free graph is pancyclic, except for a few specific classes of graphs. Moreover, there is no other forbidden subgraph having this property. With respect to the open case for hamiltonicity of 1-tough $$K_1\cup P_4$$ K 1 ∪ P 4 -free graphs we give infinite families of graphs that are not pancyclic.

Chiral smoothings of knots

Can smoothing a single crossing in a diagram for a knot convert it into a diagram of the knot's mirror image? Zeković found such a smoothing for the torus knot T(2, 5), and Moore–Vazquez proved that such smoothings do not exist for other torus knots T(2, m) with m odd and square free. The existence of such a smoothing implies that K # K bounds a Mobius band in B4. We use Casson–Gordon theory to provide new obstructions to the existence of such chiral smoothings. In particular, we remove the constraint that m be square free in the Moore–Vazquez theorem, with the exception of m = 9, which remains an open case. Heegaard Floer theory provides further obstructions; these do not give new information in the case of torus knots of the form T(2, m), but they do provide strong constraints for other families of torus knots. A more general question asks, for each pair of knots K and J, what is the minimum number of smoothings that are required to convert a diagram of K into one for J. The methods presented here can be applied to provide lower bounds on this number.

Serre’s Property (FA) for automorphism groups of free products

We provide some necessary and some sufficient conditions for the automorphism group of a free product of (freely indecomposable, not infinite cyclic) groups to have Property (FA). The additional sufficient conditions are all met by finite groups, and so this case is fully characterised. Therefore, this paper generalises the work of N. Leder [Serre’s Property FA for automorphism groups of free products, preprint (2018), https://arxiv.org/abs/1810.06287v1]. for finite cyclic groups, as well as resolving the open case of that paper.

A Sufficient Condition for Planar Graphs of Maximum Degree 6 to be Totally 7-Colorable

A total k-coloring of a graph is an assignment of k colors to its vertices and edges such that no two adjacent or incident elements receive the same color. The total coloring conjecture (TCC) states that every simple graph G has a total ΔG+2-coloring, where ΔG is the maximum degree of G. This conjecture has been confirmed for planar graphs with maximum degree at least 7 or at most 5, i.e., the only open case of TCC is that of maximum degree 6. It is known that every planar graph G of ΔG≥9 or ΔG∈7,8 with some restrictions has a total ΔG+1-coloring. In particular, in (Shen and Wang, 2009), the authors proved that every planar graph with maximum degree 6 and without 4-cycles has a total 7-coloring. In this paper, we improve this result by showing that every diamond-free and house-free planar graph of maximum degree 6 is totally 7-colorable if every 6-vertex is not incident with two adjacent four cycles or three cycles of size p,q,ℓ for some p,q,ℓ∈3,4,4,3,3,4.