In mathematics, concentration of measure is a principle that is applied in measure theory, probability and combinatorics, and has consequences for other fields such as Banach space theory. It is illustrated by the phenomenon of the measure near an equatorial section of the n-sphere, where n is large. If one estimates what proportion of the measure on the n-sphere, for the usual rotationally-invariant measure, is near any one (n − 1)-sphere that is equatorial in it (analogous to a great circle on the Earth's surface, when n = 2), the answer is that almost all of the measure is concentrated near it. In other words, the 'poles' and their neighbourhoods down to very small latitudes account for a tiny proportion of the generalised 'surface area', in a sense that can be made precise as n → ∞.
This principle can be made the basis of many proofs; recognition of it is often attributed to Paul Lévy. For example it can be used to prove Dvoretsky's theorem on subspaces of Banach spaces that are almost Hilbert spaces.
The measuring volume of the profiler is determined by one of two things, the bandwidth of the output electronics, and the pulse-width of the transducer excitation pulse.
The return signal from a low concentration of large sand grains may be the same as the signal from a high concentration of small sand grains.
The profiler was calibrated in a vertical duct for sediment concentrations between 0.1 and 1 percent by mass.
However, mass concentrationmeasurements may be applicable for evaluating occupational exposure to nanometer aerosols where a good correlation between the surface area of the aerosol and mass concentration can be determined or if toxicity data based on mass dose are available for a specific nanometer aerosol associated with a known process (e.g., diesel exhaust particulate).
Although using nanoparticle number concentration as an exposure measurement may not be consistent with exposure metrics being used in animal toxicity studies, such measurements may be a useful indicator for identifying nanoparticle emissions and determining the efficacy of control measures.
Field measurements indicate that estimates are within a factor of three of the active surface-area, particularly at higher concentrations.
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