A note on the minimum skew rank of a graph

The minimum skew rank $mr^{-}(\mathbb{F},G)$ of a graph $G$ over a field $\mathbb{F}$ is the smallest possible rank among all skew symmetric matrices over $\mathbb{F}$, whose ($i$,$j$)-entry (for $i\neq j$) is nonzero whenever $ij$ is an edge in $G$ and is zero otherwise. We give some new properties of the minimum skew rank of a graph, including a characterization of the graphs $G$ with cut vertices over the infinite field $\mathbb{F}$ such that $mr^{-}(\mathbb{F},G)=4$, determination of the minimum skew rank of $k$-paths over a field $\mathbb{F}$, and an extending of an existing result to show that $mr^{-}(\mathbb{F},G)=2match(G)=MR^{-}(\mathbb{F},G)$ for a connected graph $G$ with no even cycles and a field $\mathbb{F}$, where $match(G)$ is the matching number of $G$, and $MR^{-}(\mathbb{F},G)$ is the largest possible rank among all skew symmetric matrices over $\mathbb{F}$

The Galaxy Velocity Function from MIGHTEE-HI Early Science Data

The velocity function of MIGHTEE-H I Early Science data is presented. This is the first velocity function that is based on a blind radio interferometric survey. As a precursor, understanding the systematics that affect the Early Science velocity function will optimise the full survey's analysis. PYMULTINEST and the Busy Function are employed to estimate the linewidths of the low spectral resolution data. The performance of PYMULTINEST in estimating known linewidths of simulated H I profiles with varying spectral resolution is assessed. The simulation study shows that the estimated linewidths of the Early Science data, using this novel method, are robust and are recovered within the uncertainty. The effects of cosmic variance, instrumental linewidth broadening and Doppler linewidth broadening on the velocity function are quantified within the context of the limitations of the Early Science data. The MIGHTEE-H I Early Science velocity function is compared with the velocity functions from previous large-scale H I surveys, namely the Arecibo Legacy Fast ALFA (ALFALFA) survey and the H I Parkes All-Sky Survey (HIPASS). There is general agreement with the ALFALFA and HIPASS results, when taking linewidth broadening into account, given that the MIGHTEE-H I Early Science data is strongly affected by cosmic variance. In particular, cosmic variance introduces an average uncertainty of ∼ 24% in the measured Early Science volume densities. The larger effective area of the full survey will reduce the impact of cosmic variance. The full survey velocity function can be further optimised by estimating the rotational velocities using kinematic modelling, and correcting the measured linewidths for instrumental broadening, Doppler broadening, turbulent motion and inclination effects

Simple Toroidal Vertex Algebras and Their Irreducible Modules

In this paper, we continue the study on toroidal vertex algebras initiated in \cite{LTW}, to study concrete toroidal vertex algebras associated to toroidal Lie algebra $L_{r}(\hat{\frak{g}})=\hat{\frak{g}}\otimes L_r$, where $\hat{\frak{g}}$ is an untwisted affine Lie algebra and $L_r=$\mathbb{C}[t_{1}^{\pm 1},\ldots,t_{r}^{\pm 1}]$. We first construct an$(r+1)$-toroidal vertex algebra$V(T,0)$and show that the category of restricted$L_{r}(\hat{\frak{g}})$-modules is canonically isomorphic to that of$V(T,0)$-modules.Let$c$denote the standard central element of$\hat{\frak{g}}$and set$S_c=U(L_r(\mathbb{C}c))$. We furthermore study a distinguished subalgebra of$V(T,0)$, denoted by$V(S_c,0)$. We show that (graded) simple quotient toroidal vertex algebras of$V(S_c,0)$are parametrized by a$\mathbb{Z}^r$-graded ring homomorphism$\psi:S_c\rightarrow L_r$such that Im$\psi$is a$\mathbb{Z}^r$-graded simple$S_c$-module. Denote by$L(\psi,0}$the simple$(r+1)$-toroidal vertex algebra of$V(S_c,0)$associated to$\psi$. We determine for which$\psi$,$L(\psi,0)$is an integrable$L_{r}(\hat{\frak{g}})$-module and we then classify irreducible$L(\psi,0)$-modules for such a$\psi$. For our need, we also obtain various general results.Comment: 50 page

Social protection and care: Does the Child Support Grant translate to social justice outcomes for female beneficiaries who receive it on behalf of their children?

No abstract

Effects of the standards of training, certification and watchkeeping for seafarers (STCW) 78 convention, as amended, on Kenya\u27s maritime education and training (MET)

The dissertation is a study of the effects of the STCW 78, as amended, on Kenya’s MET. In this regard, it takes into consideration the recent requirements of the 2010 Manila amendments to the said Convention, and the set International Maritime Organization (IMO) deadline for Governments to comply with the mentioned provisions. Sequentially, the work examines the effects of the Convention, as amended, on Kenya’s Maritime Education and Training (MET) by initially exploring the historical and contemporary status of the country’s MET. Furthermore, it delivers a comparative analysis of Kenya’s strategies in the execution of the STCW 78, as amended, with those of the Philippines, Japan, Egypt and the Netherlands, in order to provide a platform for Kenya to appraise itself and benefit from the experiences of other IMO States, as well as draw lessons for its MET. The study also seeks to envisage the future of Kenya’s MET. Thus, the findings from the study’s data analyses demonstrate that the Convention, as amended, has influenced Kenya’s MET. However, it also identifies certain challenges in the country’s endeavours to comply with the STCW 78, as amended. Additionally, the work also recognizes the significance of strategy use in the implementation of the amended Convention. In conclusion, the study suggests how Kenya can benefit from other nations’ experiences of implementing the STCW 78, as amended, and draws lessons for its MET. If considered, these responses could reinforce Kenya’s STCW compliance efforts and play a vital role in the progress of its future MET