On large bipartite graphs of diameter 3

We consider the bipartite version of the {\it degree/diameter problem}, namely, given natural numbers $d\ge2$ and $D\ge2$, find the maximum number $\N^b(d,D)$ of vertices in a bipartite graph of maximum degree $d$ and diameter $D$. In this context, the bipartite Moore bound \M^b(d,D) represents a general upper bound for $\N^b(d,D)$. Bipartite graphs of order \M^b(d,D) are very rare, and determining $\N^b(d,D)$ still remains an open problem for most $(d,D)$ pairs. This paper is a follow-up to our earlier paper \cite{FPV12}, where a study on bipartite $(d,D,-4)$-graphs (that is, bipartite graphs of order \M^b(d,D)-4) was carried out. Here we first present some structural properties of bipartite $(d,3,-4)$-graphs, and later prove there are no bipartite $(7,3,-4)$-graphs. This result implies that the known bipartite $(7,3,-6)$-graph is optimal, and therefore $\N^b(7,3)=80$. Our approach also bears a proof of the uniqueness of the known bipartite $(5,3,-4)$-graph, and the non-existence of bipartite $(6,3,-4)$-graphs. In addition, we discover three new largest known bipartite (and also vertex-transitive) graphs of degree 11, diameter 3 and order 190, result which improves by 4 vertices the previous lower bound for $\N^b(11,3)$

The evaluation of a tabular application of the NICE guidelines for universal interpretation of non-stress test (NST) and cardiotocograph (CTG)

Thesis (M.Med.(Obstetrics and Gynaecology))--University of the Witwatersrand, Faculty of Health Sciences, 2015Objective: To assess consensus in the interpretation of cardiotocographs (CTGs) and non-stress tests (NSTs) between different grades of obstetric clinical staff by comparing assessment of traces by non-systematic eyeballing with assessment of traces using a tabular approach suggested by the National Institute of Health and Clinical Excellence (NICE) guidelines for interpretation of CTGs and NSTs, and to identify components ofNSTs and CTGs where medical personnel experience difficulty with interpretation. Design: Prospective observational study. Setting: Maternity units of the tertiary care hospitals for the teaching and training of the Witwatersrand University postgraduates, interns and midwives. Participants: Midwives, advanced midwives, interns, medical officers, registrars and specialists working in the above-mentioned maternity units. Method: Participants were recruited at the time of formal gatherings and departmental meetings in the various institutions. Each participant was given five traces that were a combination of NSTs and CTGs to interpret and assess in a non-systematic way using three categories: baby well; baby requires further surveillance; and baby needs immediate delivery. The same participants were then given the same set of traces in a different sequence for interpreting in a systematic way using the tabular approach from the NICE guidelines on electronic fetal monitoring with a scoring modification. Main outcome measure: Differences in interpretation of CTGs by different grades of staff, and degree of certainty between study participants in the different assessment systems. Results: Twenty seven specialists, 25 registrars, 21 medical officers, 10 interns and 15 midwives participated. There were varying interpretations by individuals in both the non-systematic assessment and the systematic assessment using the NICE tabular application, with best agreement in Trace 3 (77% and 84% respectively). In the non-systematic assessment, there was a statistically significant difference in the assessment of traces 1, 2 and 4 between the different grades of staff(P-values<0.01, 0.03 and <0.01 respectively). There was no statistically significant difference when the traces were assessed using the NICE guidelines tabular application (P-values; Trace1 >0.99, Trace2=0.27, Trace 3 = 0.76, Trace 4 = 0.15 and Trace5 = 0.35).Certainty of the evaluation by the participants was determined if75% or more of the participants agreed on a classification. Using the NICE guidelines, there was uncertainty (failure to agree on classification by 75% or more of the participants) with baseline variability, accelerations, decelerations and overall assessment of the CTG in most of the traces. Conclusion: There is no uniformity in the assessment of traces by midwives, interns, medical officers, registrars and specialists. Some uniformity in the interpretation of traces and reduction in inter-observer variation is attained by the use of the NICE guidelines tabular application. However, baseline variability, accelerations, and decelerations remain a problem in the interpretation of NSTs and CTGs using the NICE guidelines

The Topological Centers Of Module Actions

In this paper, we will study the topological centers of $n-th$ dual of Banach $A-module$ and we extend some propositions from Lau and $\ddot{U}lger$ into $n-th$ dual of Banach $A-modules$. Let $B$ be a Banach $A-bimodule$. By using some new conditions, we show that ${{Z}^\ell}_{A^{(n)}}(B^{(n)})=B^{(n)}$ and ${{Z}^\ell}_{B^{(n)}}(A^{(n)})=A^{(n)}$. We also have some conclusions in dual groups

I/O-Efficient Planar Range Skyline and Attrition Priority Queues

In the planar range skyline reporting problem, we store a set P of n 2D points in a structure such that, given a query rectangle Q = [a_1, a_2] x [b_1, b_2], the maxima (a.k.a. skyline) of P \cap Q can be reported efficiently. The query is 3-sided if an edge of Q is grounded, giving rise to two variants: top-open (b_2 = \infty) and left-open (a_1 = -\infty) queries. All our results are in external memory under the O(n/B) space budget, for both the static and dynamic settings: * For static P, we give structures that answer top-open queries in O(log_B n + k/B), O(loglog_B U + k/B), and O(1 + k/B) I/Os when the universe is R^2, a U x U grid, and a rank space grid [O(n)]^2, respectively (where k is the number of reported points). The query complexity is optimal in all cases. * We show that the left-open case is harder, such that any linear-size structure must incur \Omega((n/B)^e + k/B) I/Os for a query. We show that this case is as difficult as the general 4-sided queries, for which we give a static structure with the optimal query cost O((n/B)^e + k/B). * We give a dynamic structure that supports top-open queries in O(log_2B^e (n/B) + k/B^1-e) I/Os, and updates in O(log_2B^e (n/B)) I/Os, for any e satisfying 0 \le e \le 1. This leads to a dynamic structure for 4-sided queries with optimal query cost O((n/B)^e + k/B), and amortized update cost O(log (n/B)). As a contribution of independent interest, we propose an I/O-efficient version of the fundamental structure priority queue with attrition (PQA). Our PQA supports FindMin, DeleteMin, and InsertAndAttrite all in O(1) worst case I/Os, and O(1/B) amortized I/Os per operation. We also add the new CatenateAndAttrite operation that catenates two PQAs in O(1) worst case and O(1/B) amortized I/Os. This operation is a non-trivial extension to the classic PQA of Sundar, even in internal memory.Comment: Appeared at PODS 2013, New York, 19 pages, 10 figures. arXiv admin note: text overlap with arXiv:1208.4511, arXiv:1207.234