Archimedes proves that the volume of the solid so cut off is one sixth part of the volume of the parallelepiped.
"Archimedes" by Thomas Little Heath
A rectangular parallelepiped has, as a rule, the three edges unequal, which meet at a point.
"Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6" by Various
The same may be said for the proposition about the diagonal plane of a parallelepiped.
"The Teaching of Geometry" by David Eugene Smith
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This formula remains correct if applied to a rectangular parallelepiped and a derivative ∂a paral lel to the edges of the domain.
Understanding the Random Displacement Model: From Ground-State Properties to Localization
While the curvature matrix becomes singular along the corners and edges of the parallelepiped one may argue that these singularities do not contribute to the right hand side of (45) because the derivatives of u0 vanish in the directions in which K is singular. A direct argument for this case is provided in .
Understanding the Random Displacement Model: From Ground-State Properties to Localization
In this case no curvature term appears as the faces of the parallelepiped are ﬂat.
Understanding the Random Displacement Model: From Ground-State Properties to Localization
Nevertheless, by a reﬁned analysis we can show that these statements remain true for the case where the domain is a rectangular parallelepiped.
Understanding the Random Displacement Model: From Ground-State Properties to Localization
The second term in (45) does not vanish unless the domain is a rectangular parallelepiped! Of course, in this case one would assume that the potential shares the symmetries of the domain.
Understanding the Random Displacement Model: From Ground-State Properties to Localization
However, ﬁnding a pro jection onto a convex set (except for simple bodies such as hyper-parallelepipeds) can be as diﬃcult as the original optimization problem.
Optimization of Convex Functions with Random Pursuit
When applying Lemma 1 for ﬁxed Λ and T we shall take as P1 and P2 a parallelepiped B(s) scaled by the factors λ1 (B(s)) and λp (B(s)), respectively.
A simple proof of Schmidt-Summerer's inequality
We also denote this parallelepiped by Λ = ΛL and by VL = L3 eαL its volume.
Random point field approach to analysis of anisotropic Bose-Einstein condensations
The starting point of the method of perpendiculars is the elementary fact that the magnitude of the determinant of n real vectors in n dimensions is equal to the volume of the parallelepiped spanned by those vectors.
The logarithmic law of random determinant
For the case of a block deformed into a parallelepiped, the tractions on the inclined faces necessary to maintain the derived deformation are calculated.
Simple shear is not so simple
In the above, we have considered a ﬁnite system whose shape is a rectangular parallelepiped of linear dimensions L⊥ ×L⊥ ×Lz , Lz being the length of the system along the ordering (z ) direction.
Smectic Liquid Crystals in Random Environments
Here we use the well-known fact that the determinant of a positive-deﬁnite matrix is at most the product of its diagonal entries (equivalently, the volume of a parallelepiped is at most the product of its side lengths).
Random Matrices and Random Permutations
We further take the 3-surface ∂Ω as the standard-oriented hyper-parallelepiped made up of two spacelike surfaces {Σx0 , Σx0+∆x0 } plus timelike surfaces {Σxi , Σxi+∆xi } that join the two temporal slices together.
Numerical hydrodynamics in general relativity
For this purpose, it will suﬃce to evaluate both measures on the inﬁnitesimal parallelepiped B in Slk (R) centered at the identity matrix and spanned by the tangent vectors εeij for i 6= j and ε(eii − ekk ) for i < k .
Volumes of symmetric spaces via lattice points
Rectangular parallelepipeds. 7 More Work is Needed.
Astrophysics in 2006
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