Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics planar pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
  1. added some details about the ring of power series modeling infinitesimals in fields.


    diff, v26, current

    • CommentRowNumber2.
    • CommentAuthorGuest
    • CommentTime5 days ago

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

    diff, v27, current

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTime5 days ago

    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
    • CommentTime5 days ago

    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
    • CommentTime5 days ago

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

    diff, v30, current

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTime4 days ago

    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
    • CommentTime4 days ago

    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 i1,X i+1X 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 i1,X i+1X 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 1i,jn1 \leq i, j \leq n,

    X iX j=X jX 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
    • CommentTime3 days ago

    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
    • CommentTime3 days ago

    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
    • CommentTime3 days ago

    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.)