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In each function, r represents the distance between two random points, and D is the specimen edge length.
For this reason, the "upper bound approach" has been proposed which relies on the fact that the Euclidean distance between two random points on a plane is necessarily less than (or equal to) the corresponding geodesics, which is known as inextensibility constraint.
In a simple continuous space, you'd need to write out infinitely many decimal places just to specify the exact distance between two random points, but physics titan John Wheeler showed that quantum effects probably make any digits after the 35th decimal place meaningless, because our whole classical notion of space breaks down on smaller scales, perhaps being replaced by a strange foamy structure.
After 25 minutes, the distance between the centre of two successive pictures is smaller than the mean distance between two random points of a square corresponding to the nest area (after 25 minutes, one sample t test, t75 = 68, p<0.0001, n = 77).
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Ohbuchi et al. [56] investigate shape histograms that are discretely parametrized along the principal axes of inertia of the model and also extended D2 shape function by considering the angle between the surfaces on which two random points are located [57].
Next, we measured the amplitude between two random time points during the downstate (background uEPSP) with the time difference between these two points being the same as for the uEPSPs.
Interestingly the D2 shape function (the mean distance between two random surface points), which was successfully incorporated into several classifiers, was adapted from a general purpose comparison approach designed to discriminate between 3D objects that have nothing to do with teeth (such as CAD models for cars).
The mean fraction of the nearest neighbours between two random sets of points is (26) F NN = (f A f B 4 + f B f A 4 ) ∑ k = 0 ∞ (k + 1 ) (f A f B ) k + 2 f A 2 f B 2 ∑ k = 1 ∞ k (f A f B ) k = (f A f B 4 + f B f A 4 ) (1 − f A f B ) 2 + 2 f A 3 f B 3 (1 − f A f B ) 2 In particular, if f A = f B = 0.5, the mean fraction is F NN = 1/6.
Five random points from within each zone were measured for a total of at least 10 seconds per point, and a measurement of aerated seawater was taken between each point.
But then, there are also measures of co-movement between two random variables.
In this crossover approach, two random crossover points are generated.
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com