Sentence examples for density of a set from inspiring English sources

Szemerédi's theorem relies on the concept of density of a set of natural numbers.

(d) The uniform density of a set A ⊂ N is defined as follows.

Monte Carlo localisation generally requires a metrical map of the environment to calculate a robots position from the posterior probability density of a set of weighted samples.

A lower density of a set (Asubset N) is the number begin{aligned} d_(A)=lim_{ktoinfty}inffrac{1}{k+1} operatorname{card}{0 leq jleq k:jin A}. end{aligned}.

We shall use cardA to denote the cardinality of A. An upper density of a set (Asubset N) is the number begin{aligned} d^(A)=lim_{ktoinfty}supfrac{1}{k+1} operatorname{card}{0 leq jleq k:jin A}. end{aligned}.

The natural density (or asymptotic density) of a set A⊂ (mathbb{N}) is defined by (delta ( A ) =lim_{nrightarrow infty } frac{1}{n}vert { kleq n kin A } vert ) if the limit exists, where (vert A(n)vert ) is cardinality of the set (A(n)) (see [5]).

The f-density of a set (A subsetmathbb{N}) is defined by d^{f} (A) = lim_{n toinfty} frac{f (|{k leq n: k in A }| )}{f(n)} in case this limit exists.

The α-density of a set (A subsetmathbb{N}) is defined by d_{alpha} (A) = lim_{n toinfty} frac{1}{n^{alpha}}bigl|{k leq n: k in A }bigr| in case this limit exists.

For any unbounded modulus f, the f-density of a set (E subset mathbb{N}) is denoted by (d^{f} (E)) and is defined by d^{f} (E) = lim_{n toinfty} frac{f (vert {k leq n: k in E }vert )}{f(n)} in the case this limit exists.

It was also observed that some particles particularly in the ultrafine (PM0.18) fraction were larger than the stated cutpoint, although this may be reconciled by understanding that the diameter relates to the aerodynamic behavior equivalent to a sphere of unit-density of a set diameter.

We found that a higher frequency of IR in human is associated with individual introns that have weaker splice sites, genes with shorter intron lengths, higher expression levels and lower density of both a set of exon splicing silencers (ESSs) and the intronic splicing enhancer GGG.