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nLab > Latest Changes: power series

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  1. added some details about the ring of power series modeling infinitesimals in fields.

    Anonymous

    diff, v26, current

    • CommentRowNumber2.
    • CommentAuthorGuest
    • CommentTimeMay 26th 2023

    adding section on formal differentiation in the ring of formal power series

    diff, v27, current

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 26th 2023

    Added to the section on formal differentiation by placing it in a context of terminal coalgebra of the endofunctor. I’ll see what more might be usefully added.

    diff, v28, current

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 26th 2023

    Added some more material to illustrate (chain rule, binomial theorem) an abstract approach, with links to other (perhaps not well known) nLab pages.

    diff, v29, current

    • CommentRowNumber5.
    • CommentAuthorGuest
    • CommentTimeMay 27th 2023

    Added a paragraph about formal partial differentiation in power series with more than one indeterminant X iX_i.

    diff, v30, current

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 27th 2023

    I have rolled back to the previous version, because I do not see the point of cluttering up the page with obvious observations. If there is disagreement, let’s discuss.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMay 28th 2023

    It’s hard to see from the edit logs what the contentious material is, because together with the edit the whole Properties-section is being moved around, which defeats Instiki’s ability to identify the added/deleted material.

    But after inspection it seems to me that what Todd wants to not see in the entry is this paragraph:

    In power series rings with multiple indeterminants R[[X 1,X 2,…X n]]R[[X_1, X_2, \ldots X_n]] where RR be a commutative ring, there is a formal partial differentiation operator for every single indeterminant:

    ∂∂X i:R[[X 1,X 2,…X n]]→R[[X 1,X 2,…X n]]\frac{\partial}{\partial X_i}:R[[X_1, X_2, \ldots X_n]] \to R[[X_1, X_2, \ldots X_n]]

    because for each X iX_i, the power series ring R[[X 1,X 2,…X n]]R[[X_1, X_2, \ldots X_n]] is the same as the power series ring R[[X 1,X 2,…X i−1,X i+1…X n]][[X i]]R[[X_1, X_2, \ldots X_{i - 1}, X_{i + 1} \ldots X_n]][[X_i]] due to the commutative property for multiplication in any power series ring. The coefficients a ka_k in the formula

    ∂∂X i(∑ k=0 ∞a kX i k)≔∑ k=0 ∞a k+1(k+1)X i k \frac{\partial}{\partial X_i}\left(\sum_{k = 0}^\infty a_k X_i^k\right) \coloneqq \sum_{k = 0}^\infty a_{k + 1} (k + 1) X_i^k

    are elements of the power series ring R[[X 1,X 2,…X i−1,X i+1…X n]]R[[X_1, X_2, \ldots X_{i - 1}, X_{i + 1} \ldots X_n]].

    Partial derivatives commute with each other: given indeterminants X iX_i and X jX_j for 1≤i,j≤n1 \leq i, j \leq n,

    ∂∂X i∘∂∂X j=∂∂X j∘∂∂X i\frac{\partial}{\partial X_i} \circ \frac{\partial}{\partial X_j} = \frac{\partial}{\partial X_j} \circ \frac{\partial}{\partial X_i}

    Personally, I find this paragraph reasonable thing to say in a section on differentiation in power series rings, it serves completeness. Sure, it’s trivial for readers who already know about differentiation, but those probably wouldn’t be reading this section anyways.

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 28th 2023

    It’s a matter of taste where to strike the balance between completeness and clutter. The definition on display as reproduced in #7 essentially repeats the definition given at the beginning of the section, which is why I didn’t think it needed saying. But I won’t insist, if anyone thinks it’s better to have it in.

    By the way, I don’t consider everything in that section trivial, even for readers who already know about differentiation. Some of those readers, while they will be aware of the chain rule and its analytic proof, might not be aware of how to prove it purely algebraically for formal power series. Trying to brute-force a proof, using the bare-bones definition at the top of the section, quickly turns into a mess. What I added in v29 is discussion of a nice proof that uses universal property arguments. (I’d like that discussion to stand out better, if anyone has any ideas.)

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeMay 28th 2023

    I’d like that discussion to stand out better, if anyone has any ideas.

    Yes: Don’t write prose style but systematically include definitions in Definition-environments, propositions in Proposition-environments, and so forth. For one it allows you to \label and thus point to specific content without relying on readers to hunt the page history for changes you made in v29.

    • CommentRowNumber10.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 28th 2023

    Thanks, I’ll certainly consider it and its feasibility here, although generally I’m not convinced that every part of every entry needs to put everything into environments. I know we’ve had this discussion before; I remember that Mike and I advocated a blend of prose style and environments as sometimes better for achieving narrative flow.

    (Obviously I never intend for general readers to hunt page history; I brought up v29 here in the context of an nForum edit discussion.)

  11. changed higher algebra - contents to algebra - contents in context sidebar

    Anonymouse

    diff, v34, current

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