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On large bipartite graphs of diameter 3
We consider the bipartite version of the {\it degree/diameter problem},
namely, given natural numbers and , find the maximum number
of vertices in a bipartite graph of maximum degree and diameter
. In this context, the bipartite Moore bound \M^b(d,D) represents a
general upper bound for . Bipartite graphs of order \M^b(d,D) are
very rare, and determining still remains an open problem for most
pairs.
This paper is a follow-up to our earlier paper \cite{FPV12}, where a study on
bipartite -graphs (that is, bipartite graphs of order \M^b(d,D)-4)
was carried out. Here we first present some structural properties of bipartite
-graphs, and later prove there are no bipartite -graphs.
This result implies that the known bipartite -graph is optimal, and
therefore . Our approach also bears a proof of the uniqueness of
the known bipartite -graph, and the non-existence of bipartite
-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
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 dual of Banach
and we extend some propositions from Lau and into
dual of Banach . Let be a Banach . By using
some new conditions, we show that and
. 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
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