32 Matching Annotations
  1. Dec 2021
  2. Oct 2021
  3. Jun 2021
    1. If you have a vector space, any vector space, you can define linear functions on that space. The set of all those functions is the dual space of the vector space. The important point here is that it doesn't matter what this original vector space is. You have a vector space V

      One of the better "simple" discussions of dual spaces I've seen:

      If you have a vector space, any vector space, you can define linear functions on that space. The set of all those functions is the dual space of the vector space. The important point here is that it doesn't matter what this original vector space is. You have a vector space V, you have a corresponding dual V∗.

      OK, now you have linear functions. Now if you add two linear functions, you get again a linear function. Also if you multiply a linear function with a factor, you get again a linear function. Indeed, you can check that linear functions fulfill all the vector space axioms this way. Or in short, the dual space is a vector space in its own right.

      But if V∗ is a vector space, then it comes with everything a vector space comes with. But as we have seen in the beginning, one thing every vector space comes with is a dual space, the space of all linear functions on it. Therefore also the dual space V∗ has a corresponding dual space, V∗∗, which is called double dual space (because "dual space of the dual space" is a bit long).

      So we have the dual space, but we also want to know what sort of functions are in that double dual space. Well, such a function takes a vector from V∗, that is, a linear function on V, and maps that to a scalar (that is, to a member of the field the vector space is based on). Now, if you have a linear function on V, you already know a way to get a scalar from that: Just apply it to a vector from V. Indeed, it is not hard to show that if you just choose an arbitrary fixed element v∈V, then the function Fv:ϕ↦ϕ(v) indeed is a linear function on V∗, and thus a member of the double dual V∗∗. That way we have not only identified certain members of V∗∗ but in addition a natural mapping from V to V∗∗, namely F:v↦Fv. It is not hard to prove that this mapping is linear and injective, so that the functions in V∗∗ corresponding to vectors in V form a subspace of V∗∗. Indeed, if V is finite dimensional, it's even all of V∗∗. That's easy to see if you know that dim(V∗)=dimV and therefore dim(V∗∗)=dimV∗=dimV. On the other hand, since F is injective, dim(F(V))=dim(V). However for finite dimensional vector spaces, the only subspace of the same dimension as the full space is the full space itself. However if V is infinite dimensional, V∗∗ is larger than V. In other words, there are functions in V∗∗ which are not of the form Fv with v∈V.

      Note that since V∗∗again is a vector space, it also has a dual space, which again has a dual space, and so on. So in principle you have an infinite series of duals (although only for infinite vector spaces they are all different).

    1. There are some very beautiful and easily accessible applications of duality, adjointness, etc. in Rota's modern reformulation of the Umbral Calculus. You'll quickly gain an appreciation for the power of such duality once you see how easily this approach unifies hundreds of diverse special-function identities, and makes their derivation essentially trivial. For a nice introduction see Steven Roman's book "The Umbral Calculus".

      Note to self: Look at [[Steven Roman]]'s book [[The Umbral Calculus]] to follow up on having a more intuitive idea of what a dual space is and how it's useful

    2. Dual spaces also appear in geometry as the natural setting for certain objects. For example, a differentiable function f:M→R

      Dual spaces also appear in geometry as the natural setting for certain objects. For example, a differentiable function f:M→R where M is a smooth manifold is an object that produces, for any point p∈M and tangent vector v∈TpM, a number, the directional derivative, in a linear way. In other words, ==a differentiable function defines an element of the dual to the tangent space (the cotangent space) at each point of the manifold.==

    3. The dual is intuitively the space of "rulers" (or measurement-instruments) of our vector space. Its elements measure vectors. This is what makes the dual space and its relatives so important in Differential Geometry, for instance.

      A more intuitive description of why dual spaces are useful or interesting.

    1. To put it succinctly, differential topology studies structures on manifolds that, in a sense, have no interesting local structure. Differential geometry studies structures on manifolds that do have an interesting local (or sometimes even infinitesimal) structure.

      Differential topology take a more global view and studies structures on manifolds that have no interesting local structure while differential geometry studies structures on manifolds that have interesting local structures.

  4. Apr 2021
  5. Mar 2021
  6. Nov 2020
  7. icla2020b.jonreeve.com icla2020b.jonreeve.com
    1. the word gnomon in the Euclid and the word simony in the Catechism

      Joyce seems to like putting elements from mathematics in his works. He referred the "Ithaca" episode of Ulysses as a "mathematical catechism" in his letter. Maybe he found some connections between geometry and Catechism as he was a very knowledgeable writer.

  8. Oct 2020
    1. Finally, and as fundamentally as there is a numerical memory and a dia-lectical memory, there is a geometry of memory too. Almost every monas-tic mnemotechnical scheme—ladders, roses, buildings, maps—was based ongeometrical figures: squares, rectangles, triangles, circles, and complex refor-mations of these, including three-dimensional structures

      She doesn't mention it, but they're not only placing things in order for potential memory purposes, but they're also placing an order on their world as well.

      Ladders and steps were frequently used to create an order of beings as in the scala naturae or the Great Chain of Being.

      Some of this is also seen in Ramon Lull's Ladder of Ascent and Descent of the Mind, 1305 (Ars Magna)

  9. May 2020
  10. Feb 2020
  11. Oct 2019
  12. Jun 2019
    1. In 1953 I realized that the straight line leads to the downfall of mankind. But the straight line has become an absolute tyranny. The straight line is something cowardly drawn with a rule, without thought or feeling; it is the line which does not exist in nature. And that line is the rotten foundation of our doomed civilization. Even if there are places where it is recognized that this line is rapidly leading to perdition, its course continues to be plotted. ..Any design undertaken with the straight line will be stillborn. Today we are witnessing the triumph of rationalist know-how and yet, at the same time, we find ourselves confronted with emptiness. An aesthetic void, dessert of uniformity, criminal sterility, loss of creative power. Even creativity is prefabricated. We are no longer able to create. That is our real illiteracy.   Friedensreich Hundertwasser
  13. Dec 2018
  14. Nov 2018
    1. Grassmannian Learning: Embedding Geometry Awareness in Shallow and Deep Learning

      就喜欢这些用微分流形讲机器学习的~!

      应该写个 Paper Summary 表示尊敬~~~

    2. Classification and Geometry of General Perceptual Manifolds

      一篇Physical Review X 上的文章~ 读读看~

      Paper Summary

  15. Mar 2018
  16. Mar 2016
  17. arxiv.org arxiv.org
    1. Letβ:V×V→Wbe a symmetric bilinear form whereVand (W,h,i) arereal vector spaces of finite dimensionnandp, respectively, equipped withinner products.Thes-nullityνsofβfor any integer 1≤s≤pis defined byνs= maxUs⊂Wdim{x∈V:βUs(x, y) = 0 for ally∈V}.HereβUs=πUs◦βwhereUsis anys-dimensional subspace ofWandπUs:W→Usdenotes the orthogonal projection.LetR:V×V×V×V→Rbe the multilinear map with the algebraicproperties of the curvature tensor defined byR(x, y, z, w) =hβ(x, w), β(y, z)i − hβ(x, z), β(y, w)i.Lemma 4.Assume that2p < nandνs< n−2sfor all1≤s≤p. LetV=V1⊕V2be an orthogonal splitting such thatR(x, y, z, u) =R(x, y, u, v) =R(x, u, v, w) = 0for anyx, y, z∈V1andu, v, w∈V2. Then,S=span{β(x, y) :x∈V1andy∈V2}= 0.
    1. second fundamental_form h satisfies h(TpL,xTpLj =0 forallp E M

      Para o nosso caso, assumir essa hipótese com respeito a decomposição do espaço tangente ao longo do bordo.

  18. Dec 2015
    1. Let M be an rc-dimensional manifold of class C°° and g any given Riemannian metric on M. We will consider the following classical problem motivated by differential geometry. Does there exist an embedding u = (w1,..., uq) : M -> R9 such that the usual euclidian metric of R9 induces on the submanifold u(M) the given metric gl In other words, w must satisfy E(w) := du-du = g, (1) or in local coordinates 9 du1 du1 _ ,tîâ?â?"Qij' The dot in (1) denotes the usual scalar product of R9. The notion embedding means, that w is locally an immersion and globally a homeomorphism of M onto the subspace u(M) of R*. If an embedding w : M -• R9 satisfies (1) on the whole M, we speak of an isometric embedding. If w is an immersion and a solution of (1) in a (possibly small) neighbourhood of any point of M, we speak of a local isometric embedding.
  19. Mar 2015
    1. θ dμ ≥ p 16 π | Σ |

      Qual a relação dessa desigualdade com a dita desigualdade de Penrose Riemanniana provada por Huisken-Ilmanen e Bray?

    2. GIBBONS-PENROSE INEQUALITY

      Qual a relação dessa desigualdade com a dita desigualdade de Penrose Riemanniana provada por Huisken-Ilmanen e Bray?

  20. Oct 2013
    1. Order, in the first place, is necessary in geometry, and is it not also necessary in eloquence?

      comparison between geometry and rhetoric