Dec 29, 2019
In a short exact sequence of vector fields 0 → A → B → C→ 0,B must be the direct sum B=A⊕C of the subgroup A and the quotient group C=B/A.
Allen Hatcher
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In a short exact sequence of vector fields 0 → A → B → C→ 0,B must be the direct sum B=A⊕C of the subgroup A and the quotient group C=B/A.
Allen Hatcher
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conformal compactification of two copies of minkowski space along their conformal infinities into S¹ × Sⁿ⁻¹
by user Hachi1 (Yeet Bundle) on Graduate Texts in Memes discord chat.
What is a disjoint union?
The union of {a,d,e} and {b,d,e} would be {a,b,d,e}.
But the disjoint union of those two would be something like {(0,a), (0,d), (0,e), (1,b), (1,d), (1,e)}.
Direct sum ⊕ and formal sum are similar. These ideas come up in tensors—using linearity to treat simple ideas that are given a complex symbology.
They also show up in Ghrist’s E.A.T., Hartshorne appendix C, and Fast Khovanov Homology Computations. Maybe I'll add more examples later.
Around minute 20, a good exposition of ℤ₂ ⊕ ℤ₄.
And I'll add my usual advice for watching maths videos on Youtube: open the file as a 'network stream' in vlc then use ]]] to speed up and shift+→ to skip ahead a few seconds whenever the presenter is slowly writing.)
Direct Sum
This 6-minute video by Lorenzo Sadun explained direct sums to me, after I'd seen the word used many times and wasn't really getting it. The idea is that if you wanted to combine two incompatible things, you could pad each of them with zeroes.
Example:
(a₁, a₂, a₃) + (b₁, b₂, b₃)
(a₁, a₂, a₃, 0, 0, 0) + (0, 0, 0, b₁, b₂, b₃)
(a₁, a₂, a₃, b₁, b₂, b₃)
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Both direct sum and tensor product are standard ways of putting together little Hilbert spaces to form big ones. They are used for different purposes. Suppose we have two physical systems.... Roughly speaking, if ... a physical system's ... states are either of A OR of B, its Hilbert space will be [a] direct sum.... If we have a system whose states are states of A AND states of B, its Hilbert space will be [a] tensor product.... MEASURE SPACE disjoint union Cartesian product HILBERT SPACE direct sum tensor product
John Baez
@isomorphisms We use direct sums and products of small covariance matrices to generate full-size covariance structures in mixed models.
— Luis A. Apiolaza (@zentree) July 12, 2014