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 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 internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure 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 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).

nLab > Latest Changes: Borel's Theorem

Bottom of Page

1 to 12 of 12

  1.  
    • CommentRowNumber1.
    • CommentAuthorTobyBartels
    • CommentTimeFeb 13th 2017

    I wanted to understand Borel's Theorem better, so I wrote out a fairly explicit proof of the one-dimensional case.

  2. Perhaps to a working analyst that argument looks natural, but to me it is extraordinary that anybody could come up with that!

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 14th 2017
    • (edited Feb 14th 2017)

    Possible typo: should “H nx=0H_n x = 0” be “ϕ (k−i)(H nx)=0\phi^{(k-i)}(H_n x)=0”?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeFeb 14th 2017

    I have changed

      [[falling power]]

    to

      [[falling factorial]]

    which seems to be what was intended.

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 14th 2017

    I myself usually call it a “falling power”, and I believe this is the term used by Graham, Knuth, and Patashnik, so I added that terminology to falling factorial as well as a few redirects.

    • CommentRowNumber6.
    • CommentAuthorTobyBartels
    • CommentTimeFeb 15th 2017

    I'm glad that we do have a page on the falling power after all. (To my mind, it's not so much a falling factorial as a partial factorial. Factorials are always falling, after all.)

    Mike, thanks for catching the typo.

    • CommentRowNumber7.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 15th 2017

    I wasn’t aware of this result before; it surprised me somewhat!

    • CommentRowNumber8.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 16th 2017
    • (edited Feb 16th 2017)

    This is a special case of a theorem that says when the restriction map C ∞(M)→C ∞(K)C^\infty(M) \to C^\infty(K), for MM a manifold and appropriate closed subspaces K⊂MK\subset M, is a submersion. For M=ℝ nM=\mathbb{R}^n one has the improved result that it is a split surjection of Fréchet spaces.

    The theorem at hand considers K={0}⊂ℝ=MK=\{0\}\subset \mathbb{R}=M, and by smooth functions on a point one has to consider Whitney jets, which reduce to a mere sequence of numbers, but in general need to satisfy the kind of error estimates that Taylor’s theorem gives.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeFeb 16th 2017

    David, might you have a spare minute to add this kind of comment to the entry?

    • CommentRowNumber10.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 16th 2017
    • (edited Feb 16th 2017)

    Sure. I might be writing a short note about a small extension I have made (from Euclidean space, which is in the literature, to Riemannian manifolds of bounded geometry) at some point, so I should gather the relevant references etc before then :-)

    • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 16th 2017

    I would have called it a shifted factorial, since the difference between it and k!k! is that it starts at xx rather than at kk.

    • CommentRowNumber12.
    • CommentAuthorTobyBartels
    • CommentTimeFeb 16th 2017

    Mike, yes, that also makes sense, especially when xx is not a natural number.

    Of course, there's also the case when kk is negative; then it's a rising power and a shifted reciprocal factorial. (In particular, 1/k!=0 −k̲1/k! = 0^{\underline{-k}}.)

1 to 12 of 12

  1.