35,793 research outputs found
Counting Humps in Motzkin paths
In this paper we study the number of humps (peaks) in Dyck, Motzkin and
Schr\"{o}der paths. Recently A. Regev noticed that the number of peaks in all
Dyck paths of order is one half of the number of super Dyck paths of order
. He also computed the number of humps in Motzkin paths and found a similar
relation, and asked for bijective proofs. We give a bijection and prove these
results. Using this bijection we also give a new proof that the number of Dyck
paths of order with peaks is the Narayana number. By double counting
super Schr\"{o}der paths, we also get an identity involving products of
binomial coefficients.Comment: 8 pages, 2 Figure
Metal-free activation of H<sub>2</sub>O<sub>2</sub> by g-C<sub>3</sub>N<sub>4</sub> under visible light irradiation for the degradation of organic pollutants
A Criterion for Comparing Measurement Results and Determining Conformity with Specifications
In this paper a new criterion for comparing measurement results and determining conformity with specifications is proposed, which essentially is a strategy of estimating the empirical relationships of objects. Comparing with traditional methods given in GUM: 2008 and ISO 14253-1, this criterion improves the resolution of comparison by reducing the sizes of the coverage intervals to be compared. Interval order (a binary relation) is used for comparing the coverage intervals of the measurand and represents the empirical relations. The systematic effects of measurement are classified into two types: monotonic and non-monotonic effects, so that, without correcting the monotonic effects, a biased measurand can be specified to represent the empirical relations. Thereby the uncertainty components arising from the monotonic effects can be removed from the combined uncertainty. A strategy is given for determining the relationships among measurement results and specification limits. An example is given to demonstrate the application of the criterion
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