Of triangles there are two kinds; one having the opposite sides equal (isosceles), the other with unequal sides (scalene).
"Timaeus" by Plato
He is a scalene triangle.
"The Caxtons, Complete" by Edward Bulwer-Lytton
A scalene or scaly member has no one side which is equal to his own interest.
"Punch, or the London Charivari, Vol. 1, October 9, 1841" by Various
A scalene or scaly member has no one side which is equal to his own interest.
"Punch, or the London Charivari, Volume 1, Complete" by Various
On the other hand, triangles can be divided into equilateral, isosceles and scalene, since no other kind of triangle can exist.
"Encyclopaedia Britannica, 11th Edition, Volume 8, Slice 5" by Various
This is bounded by eight scalene triangles and has the indices {hkl}.
"Encyclopaedia Britannica, 11th Edition, Volume 7, Slice 7" by Various
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These students can then be asked to develop test cases for a really trivial program that reads three integers as the sides of a triangle and print a message that states whether the triangle is scalene, isosceles or equilateral (posed by Myers in 1978) .
Learning by Test-infecting Symmetric Ciphers
The proposed statistics are a ﬁrst step toward the automated detection of subtle characteristics of white-matter microstructure, that is, scalene diffusions (Figure 1) or asymmetry in decay in a ﬁxed axis.
Nonparametric tests of structure for high angular resolution diffusion imaging in Q-space
Both properties, scalene diffusion and asymmetry, have been found in a forking ﬁber structure (Figure 1), and may be important summaries to feed into ﬁber-tracking algorithms [Mori and van Zijl (2002)].
Nonparametric tests of structure for high angular resolution diffusion imaging in Q-space
Scalene diffusion is when two ﬁber bund les of similar mass cross in perpendicular directions (λ1 ≈ λ2 ≫ λ3 ).
Nonparametric tests of structure for high angular resolution diffusion imaging in Q-space
Ellipsoidal diffusions are an important class of diffusions and the scalene structure of the diffusion PDF is particularly important when combining voxel-wise information [Seunarine et al. (2007)].
Nonparametric tests of structure for high angular resolution diffusion imaging in Q-space
The aforementioned work showed that the scalene structure of the peak is related to the peak anisotropy in space and important for treating bending and fanning ﬁbers (Figure 1).
Nonparametric tests of structure for high angular resolution diffusion imaging in Q-space
For diffusions with ellipsoidal decay, their minor axes are well deﬁned by this (scalene) decay structure, while for nonellipsoidal diffusions the minor axes correspond to a set of axes in the plane of the dominant great circle, parameterizing locations on the dominant great circle.
Nonparametric tests of structure for high angular resolution diffusion imaging in Q-space
We examine the scalene structure of the diffusion PDF, which is quantiﬁed by the difference in decay in the two spatial minor axes, deﬁned as such also for nonellipsoid diffusions.
Nonparametric tests of structure for high angular resolution diffusion imaging in Q-space
Common processes, such as prolate and scalene diffusion, are given as well as more exotic examples, such as a mixture of prolate diffusion processes and a process that cannot be represented using a Gaussian diffusion model.
Nonparametric tests of structure for high angular resolution diffusion imaging in Q-space
The scalene structure of the diffusion PDF corresponds to variation on the belt (Figure 2h), while the asymmetry of Figure 2d and e are mapped onto the local structure of the delineation of the belt in Figure 2j and k.
Nonparametric tests of structure for high angular resolution diffusion imaging in Q-space
This is apparent by the asymmetric decay in great circles perpendicular to the dominant great circle in the (α, β ) plane. (d, e, f ) A scalene diffusion process with eigenvalues (λ1 ≈ λ2 ≫ λ3 ).
Nonparametric tests of structure for high angular resolution diffusion imaging in Q-space
Having established methodology to discriminate the number of peaks in the diffusion PDF at a single voxel, we now provide additional methodology to characterize the diffusion PDF as scalene versus other forms of asymmetry, for example, to observe the indication of forking or fanning white-matter structure.
Nonparametric tests of structure for high angular resolution diffusion imaging in Q-space
The q -space version of these two populations shows a scalene distribution developing in voxels (ii,b)–(ii,e).
Nonparametric tests of structure for high angular resolution diffusion imaging in Q-space
As the forking progresses, from left-to-right, it appears both highly warped and scalene until the distribution clearly displays multiple ﬁbers in voxel (iii,g).
Nonparametric tests of structure for high angular resolution diffusion imaging in Q-space
Detecting the scalene diffusion depends on the SNR, while the isotropic diffusion is clearly distinguishable from its alternatives under the full range of SNR using the U -statistics.
Nonparametric tests of structure for high angular resolution diffusion imaging in Q-space
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