11 Matching Annotations
  1. Last 7 days
    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
  2. Dec 2018
  3. Nov 2018
    1. Grassmannian Learning: Embedding Geometry Awareness in Shallow and Deep Learning


      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

  4. Mar 2018
  5. Mar 2016
  6. 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.

  7. 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.
  8. 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?


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

  9. 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