Minimum Weight Resolving Sets of Grid Graphs

For a simple graph $G=(V,E)$ and for a pair of vertices $u,v \in V$, we say that a vertex $w \in V$ resolves $u$ and $v$ if the shortest path from $w$ to $u$ is of a different length than the shortest path from $w$ to $v$. A set of vertices ${R \subseteq V}$ is a resolving set if for every pair of vertices $u$ and $v$ in $G$, there exists a vertex $w \in R$ that resolves $u$ and $v$. The minimum weight resolving set problem is to find a resolving set $M$ for a weighted graph $G$ such that$\sum_{v \in M} w(v)$ is minimum, where $w(v)$ is the weight of vertex $v$. In this paper, we explore the possible solutions of this problem for grid graphs $P_n \square P_m$ where $3\leq n \leq m$. We give a complete characterisation of solutions whose cardinalities are 2 or 3, and show that the maximum cardinality of a solution is $2n-2$. We also provide a characterisation of a class of minimals whose cardinalities range from $4$ to $2n-2$.Comment: 21 pages, 10 figure

The two-child limit & 'choices' over family size: when policy presentation collides with lived experiences

The two-child limit restricts the child element in Universal Credit and Tax Credits to two children in a household (for children born after April 2017). One objective of the two-child limit is to influence the fertility decisions of parents in (or at risk of) poverty; therefore it is especially important to explore and understand its fertility effects. Previous analysis of administrative birth records suggests that the two-child limit had only a very small impact on the fertility of third and subsequent births in England and Wales. In this paper, we contrast the policy assumptions underpinning the two-child limit with everyday realities of fertility decision making. To do this, we draw on qualitative interviews conducted with those directly affected by the policy. This reveals a series of mismatches between policy presentation and lived realities, which help explain the absence of sizeable fertility effects. This also points to the importance of better and more sustained engagement with qualitative evidence in the design and review of policies. It is especially vital to continue to monitor the impact of the two-child limit, given the extent of the harms it can cause, and its status as an internationally unusual and significant policy

Chiral density wave versus pion condensation at finite density

The quark-meson model is often used as an effective low-energy model for QCD to study the chiral transition at finite temperature $T$, baryon chemical potential $\mu_B$, and isospin chemical potential $\mu_I$. The parameters of the model are determined by matching the meson and quark masses, as well as the pion decay constant to their physical values using the on-shell and modified minimal subtraction schemes. In this paper, we study the possibility of different phases at zero temperature. In particular, we investigate the competition between an inhomogeneous chiral condensate and a pion condensate. For the inhomogeneity, we use a chiral-density wave ansatz. For a sigma mass of $600$ MeV, we find that an inhomogeneous chiral condensate exist only for pion masses below approximately 37 MeV. We also show that due to our parameter fixing, the onset of pion condensation takes place exactly at $\mu_I={1\over2}m_{\pi}$ in accordance with exact results.Comment: 14 pages and 11 figs. v2: Minor changes, matches published version in PR

Inhomogeneous chiral condensate in the quark-meson model

acceptedVersionpublishedVersio

On-shell parameter fixing in the quark-meson model

The quark-meson model is often used as an effective low-energy model for QCD to study the chiral transition at finite temperature T and baryon chemical potential μB. The parameters in the quark-meson model can be found by expressing them in terms of the sigma mass mσ, the pion mass mπ, the constituent quark mass mq and the pion decay constant fπ. In practice, this matching is done at tree level, which is inconsistent once loop effects of the effective potential are taken into account. We show how to properly perform the matching in the quark-meson model by using the on-shell and the minimal subtraction renormalization schemes relating the physical masses and the pion decay constant to the running mass parameter and couplings. We map out the phase diagram in the μB–T plane and compare our results with other approximations.acceptedVersio

Pion condensation and phase diagram in the Polyakov-loop quark-meson model

We use the Polyakov-loop extended two-flavor quark-meson model as a low-energy effective model for QCD to study the phase diagram in the μI–T planewhere μI is the isospin chemical potential. In particular, we focus on the Bose condensation of charged pions. At T ¼ 0, the onset of pion condensation is at μI ¼ 1 2mπ in accordance with exact results. The phase transition to a Bose-condensed phase is of second order for all values of μI and in the Oð2Þuniversality class. The chiral critical line joins the critical line for pion condensation at a point whose position depends on the Polyakov-loop potential and the sigma mass. For larger values of μI these curves are on top of each other. The deconfinement line enters smoothly the phase with the broken Oð2Þ symmetry. We compare our results with recent lattice simulations and find overall good agreement.publishedVersio

THE EFFECT OF TALO-CRURAL JOINT MANIPULATION ON RANGE OF MOTION AT THE ANKLE JOINT IN SUBJECTS WITH A HISTORY OF ANKLE INJURY

Introduction: There is little research available on the effects of peripheral joint manipulation. Only a few studies have examined the effect of manipulation on ankle range of motion, with conflicting results. This study aimed to determine whether a single high-velocity, low-amplitude (HVLA) thrust manipulation to the talo-crural joint altered ankle range of motion in subjects with a history of lateral ligament sprain