131,658 research outputs found
An explicit universal cycle for the (n-1)-permutations of an n-set
We show how to construct an explicit Hamilton cycle in the directed Cayley
graph Cay({\sigma_n, sigma_{n-1}} : \mathbb{S}_n), where \sigma_k = (1 2 >...
k). The existence of such cycles was shown by Jackson (Discrete Mathematics,
149 (1996) 123-129) but the proof only shows that a certain directed graph is
Eulerian, and Knuth (Volume 4 Fascicle 2, Generating All Tuples and
Permutations (2005)) asks for an explicit construction. We show that a simple
recursion describes our Hamilton cycle and that the cycle can be generated by
an iterative algorithm that uses O(n) space. Moreover, the algorithm produces
each successive edge of the cycle in constant time; such algorithms are said to
be loopless
Extremal Lipschitz functions in the deviation inequalities from the mean
We obtain an optimal deviation from the mean upper bound \begin{equation}
D(x)\=\sup_{f\in \F}\mu\{f-\E_{\mu} f\geq x\},\qquad\ \text{for}\
x\in\R\label{abstr} \end{equation} where \F is the class of the integrable,
Lipschitz functions on probability metric (product) spaces. As corollaries we
get exact solutions of \eqref{abstr} for Euclidean unit sphere with
a geodesic distance and a normalized Haar measure, for equipped with a
Gaussian measure and for the multidimensional cube, rectangle, torus or Diamond
graph equipped with uniform measure and Hamming distance. We also prove that in
general probability metric spaces the in \eqref{abstr} is achieved on
a family of distance functions.Comment: 7 page
Maternity leave
A numerous off-days which a woman is legally approved to be absent from work in the weeks prenatal and postnatal recovery phase after giving birth defines maternity leave. It is stated that at least 60 consecutive days of paid maternity leave were entitled to all female workers in Malaysia if they have worked at least 90 days with their current employers in four months leading up to their confinement period, except for exempted categories (Employment Act 1955) During the maternity leave, female workers are entitled to be provided with all relevant contractual benefits and paid with full salary as if they are in an active employment excluding the benefits that are tied to active work. The right to resume working upon their return from maternity leave is also protected
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