28 Matching Annotations
  1. Jul 2018
  2. arxiv.org arxiv.org
    1. Forsimplicity, let us assume that the boundary of Ω has only one component.Letι: Σ :=∂Ω→Rnbe its isometric embedding. Letν:ι(Σ)→Sn−1be the outer unit normal. Sinceι(Σ) is assumed to be a strictly convexhypersurface inRnthere is a smooth family of embeddingsF: Σ×[0,∞]→RnwhereFt(σ) =F(σ, t) =ι(σ) +tν(ι(σ)).Note thatFt(Σ) are the ‘outer’ distance surfaces ofι(Σ). IfˆΩ denotes thebounded domain enclosed byι(Σ), then{Ft(Σ)}t≥0foliatesRn\ˆΩ and theEuclidean metric on this set can be written asG=dt2+gt,wheregtis the first fundamental form of the embeddingFt: Σ→Rn.
  3. May 2018
    1. we use the dilationinvariance of weighted H ̈older norms together with suitable curvature conditionsto obtain uniform bounds of solutions to the initial value problem (1) with initialconditionu−1(1 +ǫ,·) on [1 +ǫ,∞). By Arzela-Ascoli Theorem, there exists aweak solution to (1) withu−1(1,·) = 0 (Theorem 2). S
    2. We introduce the scaling transformation ̃u(t) =√tt+ 1u(t+ 1) wheret∈(0,∞).
  4. Apr 2018
    1. Theorem 2.1.The initial value problem (2.1) has a unique solutionuonΣ0×[0,∞)such that(a)u(z) = 1 +m0ρn−2+vwherem0is a constant andvsatisfies|v|=Oρ1−nand|∇0v|=O(ρ−n);(b)The metricds2=u2dr2+gris asymptotically flat in the sense of (2.23) with scalarcurvatureR≡0outsideΣ0;(c)The ADM massmADMofds2is given byc(n)mADM= (n−1)ωn−1m0= limr→∞ZΣrH0(1−u−1)dσr= limr→∞ZΣr(H0−H)dσr,for some positive constantc(n), whereH0andHare the mean curvatures ofσrwith respect to the Euclidean metric andds2respectively.

      A menos de uma normalização, o valor constante no item (c) pode ser escolhido como sendo $$c(n)= 2(n-1) \omega_{n-1}$$

    2. ∇0and∇20are the gradient and Hessianoperator of the Euclidean metric respectively. If we writeu2dr2+gr=∑i,jgijdzidzj.Then direct computations show (see the computations in (2.24), (2.27) below, for example):(2.23)|gij−δij|+ρ|∇0gij|+ρ2|∇20gij|≤Cρ2−n.By the result in [B1], the ADM mass of the metricds2=u2dr2+gris well defined, becausethe scalar curvature ofds2is zero outside a compact set.
    3. he second fun-damental formhijof Σρwith respect tods2is given by(1.6)hij=u−1h0ij.
    4. he second fundamental formh0ij,1≤i,j≤n−1 of Σρwith respect to the normalen=∂∂ρis given by(1.3)ωni=n−1∑j=1h0ijωj.
  5. Mar 2018
    1. It would be fair to characterize Beaker as “a novel application of Bittorrent’s concepts to the Web platform.” If Beaker had been started in 2006, it would be using Bittorrent as its primary protocol. However, as of 2016, new variants have appeared with better properties.
  6. Jan 2018
  7. Sep 2017
    1. Theorem 1.1.LetMn1andMn2be hypersurfaces ofNn+1that are tan-gent atpand let0be a unitary vector that is normal toMn1atp. SupposethatMn1remains aboveMn2in a neighborhood ofpwith respect to0. De-note byH1r(x)andH2r(x)ther-mean curvature atx2WofMn1andMn2,respectively. Assume that, for somer,1rn, we haveH2r(x)H1r(x)in a neighborhood of zero; ifr2, assume also that2(0), the principal cur-vature vector ofM2at zero, belongs tor. ThenMn1andMn2coincide in aneighborhood ofp

      Princípio da tangência no interior, para as curvaturas médias de ordem superior.

    2. LetMn1andMn2be hypersurfaces ofNn+1that are tangentatp, i.e., which satisfyTpM1=TpM2. Fix a unitary vector0that is normaltoMn1atp. We say thatMn1remains aboveMn2in a neighborhood ofpwith respect to0if, when we parametrizeMn1andMn2by'1and'2asin (1.1), the corresponding functions1and2satisfy1(x)2(x) in aneighborhood of zero.

      O conceito de uma hipersuperfície está (localmente) acima ou abaixo de uma outra.

    1. Lemma 4.2.The functionm(r) =ZΣrH0(1−u−1)dσris nonincreasing inr, whereH0is the mean curvature ofΣrinRn.

      Essa fórmula de monoticidade de fato vale em um cenário mais amplo, vide essa anotação, por exemplo.

    2. we can solve (2.1)with initial valueu−10= 0. In fact, by Lemma 2.2,u0satisfies:1−exp−Zr0ψ(s)ds−12≤u0(x,r)≤1−exp−Zr0φ(s)ds−12.This means that Σ0is a minimal surface with respect to the asymptotically flat metricu2dr2+gr.

      Esse é um ingrediente fundamental na nossa abordagem para a desigualdade de Alexandrov-Frenchel via desigualdade de Penrose.

    3. solve (2.1) and show that the metricds2=u2dr2+gris asymptotically flatoutside Σ0. We will also compute the mass ofds2.

      Vide teorema 2.1, no final da sessão.

    4. Let Σ0be a compact strictly convex hypersurface inRn,Xbe the position vector ofa point on Σ0, and letNbe the unit outward normal of Σ0atX. Let Σrbe the convexhypersurface described byY=X+rN, withr≥0. The Euclidean space outside Σ0canbe represented by(Σ0×(0,∞),dr2+gr)wheregris the induced metric on Σr. Consider the following initial value problem(2.1)2H0∂u∂r= 2u2∆ru+ (u−u3)Rron Σ0×[0,∞)u(x,0) =u0(x)whereu0(x)>0 is a smooth function on Σ0,H0andRrare the mean curvature and scalarcurvature of Σrrespectively, and ∆ris the Laplacian operator on Σr.

      Note que de agora em diante o autor se detém a estudar esse caso particular, onde estão inteiramente determinadas as geometrias intrínseca e extrínseca das folhas do semi cilindro, obtido folheando-se pelas paralelas o exterior da hipersuperfície estritamente convexa dada a priori.

    5. u2dρ2+gρhas the scalar curvatureR, if and onlyifusatisfies(1.10)H0∂u∂ρ=u2∆ρu+12(u−u3)Rρ−12uR0+u32R.

      Observe que essa equação fica inteiramente determinada pela especificação da geometria intrínseca e extrínseca das folhas.

      Para uma ideia do que é essencial se saber sobre a geometria das folhas do semi cilindro, vide essa anotação.

    6. Given a functionRonN, we want to find the equation forusuch that(1.2)ds2=u2dρ2+gρhas scalar curvatureR.

      O papel da aplicação \( u: N \longrightarrow \mathbb{R} \) é distorcer as fibras do semi cilindro \( N \), por dilatações e torções, deixando a geometria intrínseca das folhas invariante, de tal forma que o resultado seja um semi cilindro com a curvatura escalar prescrita \( \mathcal{R} \).

    7. Let Σ be a smooth compact manifold without boundary with dimensionn−1 and letN= [a,∞)×Σ equipped with a Riemannian metric of the form(1.1)ds20=dρ2+gρfor a point (ρ,x)∈N. Heregρis the induced metric on Σρwhich is the level surfaceρ=constant

      Isso significa que a construção a seguir é feita a partir de um semi cilindro em que a geometria das folhas é dada a priori.

      Esse artigo não trata da construção desse semi cilindro inicial.

  8. arxiv.org arxiv.org
    1. By (4), we haveddtZΣ×{t}(Hη−Hu)dσt!=ZΣ×{t}(η−1−u−1)H21+K(η−u)−12(η−1−u−1)(H21+|h1|2)dσt.(9)By the Gauss equation and the assumption thatRic(gη) = 0, we have(10)2K=H2η−|hη|2=η−2(H21−|h1|2).Therefore, it follows from (9) and (10) that(11)ddtZΣ×{t}(Hη−Hu)dσt!=−ZΣ×{t}K(η−u)2u−1dσt≤0,where we also used the assumption thatK >0.
    2. Assumption:The scalar curvatureR(gt) =: 2Kofgtand the meancurvatureH1of the leaves Σ×{t}with respect tog1are everywhere positive.Proposition 2(cf. [2], [23], [22]).Under the above assumption, given anypositive functionu0onΣ×{0}, there is a smooth positive functionuonΣ×[0, t0]such that the scalar curvatureR(gu)ofguis identically zero andu|t=0=u0.

      A prova dessa proposição deixa mais claro o que é essencial saber sobre a geometria das folhas do semi cilindro reto, para que seja possível deformar suas fibras prescrevendo a curvatura escalar, conforme foi descrito (com mais generalidade) por Shi-Tam.

  9. Apr 2016
    1. One thing I held on to during fedwiki was that it wasn’t intended to be wikipedia, and to me that meant it wasn’t intended to produce articles so much as to sustain and connect ideas in formation that might find their way into article-like things on other platforms.
  10. Feb 2016
  11. Jan 2016
    1. Here’s what the Finns, who don’t begin formal reading instruction until around age 7, have to say about preparing preschoolers to read: “The basis for the beginnings of literacy is that children have heard and listened … They have spoken and been spoken to, people have discussed [things] with them … They have asked questions and received answers.”
  12. Nov 2015
    1. Trying to override the getter on the prototype of an element is somewhat pointless as WebKit does not use getters for DOM properties. Technically we could make it do so, but that basically means throwing away performance for no good reason. Comment 2 T. Brains 2010-03-21 14:23:58 PDT That means there's absolutely no way to override or extend the default get or set behavior of any of the built-in DOM properties, which is very limiting, and like I mentioned contradicts the behavior of other browsers.

      When this was written, overriding getters for native DOM properties was not possible.

  13. Sep 2015
  14. Jun 2015
  15. May 2015
    1. https://via.hypothes.is/http://www.autostraddle.com/the-new-yorkers-skewed-history-of-trans-exclusionary-radical-feminism-ignores-actual-trans-women-247642/

      Error 1000 Ray ID: 1e70d2a751550d7f • 2015-05-15 18:14:30 UTC

      DNS points to prohibited IP

      What happened?

      You've requested a page on a website (www.autostraddle.com) that is on the CloudFlare network. Unfortunately, it is resolving to an IP address that is creating a conflict within CloudFlare's system.

  16. Feb 2015
    1. A "non-transparent proxy" is a proxy that modifies the request or response in order to provide some added service to the user agent, such as group annotation services, media type transformation, protocol reduction, or anonymity filtering.

      Hey look!!1! "group annotation services"!

      Here's one: http://via.hypothes.is/