reference request – Closed manifolds of nonnegative curvature operator are symmetric spaces

In an online webinar, I heard (not directly) the statement that (closed) manifolds of nonnegative curvature operator $mathcal{R}geq 0$ are symmetric spaces. Is this a valid theorem? Any reference that contains its proof? I am not sure that in the above statement whether “positive curvature” is a part of assumptions or not.

So by the above claim, it seems that $mathcal{R}geq 0iff$ $M$ is symmetric space!

Can the Upgrade: header have spaces between the comma?

This is probably best answered by the RFC defining HTTP headers: RFC 2616.

In section 4.2, defining the general form of headers, is written:

Any LWS (linear white space)
that occurs between field-content MAY be replaced with a single SP
before interpreting the field value or forwarding the message
downstream.

And the header format:

   message-header = field-name ":" ( field-value )
   field-name     = token
   field-value    = *( field-content | LWS )
   field-content  = <the OCTETs making up the field-value
                    and consisting of either *TEXT or combinations
                    of token, separators, and quoted-string>

Note the section about field-content.
The section describing the upgrade header in special, 14.42, does not mention any further restrictions. This is why my answer to your question is:
Yes, space in between the values should be legal.

(EDIT)
Sorry about citing the wrong RFC, the correct one is RFC7231 – but my answer is still valid, see Appendix B. This section lists the changes from the obsolete RFC 2616, and a change of the header format is not listed.

The new definition, which is quite similiar, is found in RFC 7230, Secion 3.2:

 header-field   = field-name ":" OWS field-value OWS

 field-name     = token
 field-value    = *( field-content / obs-fold )
 field-content  = field-vchar ( 1*( SP / HTAB ) field-vchar )
 field-vchar    = VCHAR / obs-text

 obs-fold       = CRLF 1*( SP / HTAB )
                ; obsolete line folding
                ; see Section 3.2.4

templates – strange spaces in the Footer HTML/CSS

templates – strange spaces in the Footer HTML/CSS – Stack Overflow

reference request – Definition of pointed Gromov-Hausdorff convergence for metric spaces

Whereas the definition of Gromov-Hausdorff convergence for compact metric spaces seems to be standard, difference sources seem to give slightly different definitions of pointed Gromov-Hausdorff convergence for (noncompact) pointed metric spaces.

For example, A course in metric geometry (D. Burago, Yu. Burago, and S. Ivanov) at page 272 gives this definition

Definition A A sequence ${(X_n,d_n,p_n)}$ of pointed metric spaces converges to the pointed metric space ${(X,d,p)}$ if for all $varepsilon,r>0$ there exists a natural number $n_0$ such that for every $n>n_0$ there is a map $f:B(p_n,r)rightarrow X$ such that

  1. $f(p_n)=p$
  2. $text{dis} f<varepsilon$ (i.e. $sup_{B(p_n,r)}|d_n(x_1,x_2)-d(f(x_1),f(x_2))|<varepsilon$)
  3. the $varepsilon$-neighbourhood of $f(B(p_n,r))$ contains the ball $B(p,r-varepsilon)$

On the other hand, Petersen’s Riemannian Geometry (3rd edition) in Chapter 11.1.2 (page 401) restricts to proper metric spaces, introduces a pointed Gromov-Hausdorff distance as follows:
$$d_{text{GH}}left((X,p),(Y,q)right)=inf{d_H(X,Y)+d(p,q)}$$
where the inf is over all metrics $d$ on the disjoint union $Xsqcup Y$ which extend the metrics on $X$ and $Y$ and $d_H$ denotes the Hausdorff distance of $X$ and $Y$ as subsets of $Xsqcup Y$ and gives the following definition

Definition B A sequence ${(X_n,d_n,p_n)}$ of pointed metric spaces converges to the pointed metric space ${(X,d,p)}$ if for all $r>0$ there exists a sequence $r_nrightarrow r$ such that
$$d_{text{GH}}left((overline{B}(p_n,r_n),p_n),(overline{B}(p,r),p)right)rightarrow 0$$

Finally, Gromov’s own book Metric Structures for Riemannian and Non-Riemannian Spaces (Def. 3.1.4, at page 85) uses essentially the same definition as Petersen’s, but restricts to (complete) locally compact length metric spaces.

And these are not all the definitions I have found in the literature.
For example, this paper by Dorothea Jansen states (Definition 2.1)

Definition C Let $(X, d_X , p)$ and $(X_n, d_{X_n} , p_n)$, $ninmathbb{N}$, be pointed proper
metric spaces. If $$d_{text{GH}}left((overline{B}(p_n,r),p_n),(overline{B}(p,r),p)right)rightarrow 0$$ for all $r>0$ where the balls are equipped with the restricted metric, then $(X_n, p_n)$ converges to $(X, p)$ in the pointed Gromov-Hausdorff sense.

Not all of them are equivalent. For example, say $X_n$ is the space consisting of the two points ${0,1+frac{1}{n}}$ and $X={0,1}$ (both with the metric inherited from $mathbb{R}$).
Here all $X_n$ (and $X$) are proper, but they are not length spaces. It is easy to see that $(X_n,0)rightarrow (X,0)$ according to definitions A and B. However for all $n$, $overline{B}_{X_n}(0,1)={0}$ and $overline{B}_X(0,1)={0,1}$, so $(overline{B}_{X_n}(0,1),0)notto (overline{B}_X(0,1),0)$ and $(X_n,0)notto (X,0)$ according to definition C.

Also, these notes (which adopt Petersen’s definition B) claim that if ${(X_n,d_n,p_n)}$ ($ninmathbb{N}$) and ${(X,d,p)}$ are compact pointed spaces such that $d_{text{GH}}left((X_n,p_n),(X,p)right)to 0$, then $(X_n,p_n)to (X,p)$ in the sense of definition B, i.e. for each $r>0$ there exist $r_nto r$ such that $$d_{text{GH}}left((overline{B}(p_n,r_n),p_n),(overline{B}(p,r),p)right)rightarrow 0$$
without ever assuming the spaces are length spaces.

My questions are:

  • When are these definitions equivalent?
  • What are the advantages of restricting to proper spaces? What are the advantages of restricting to length spaces?
  • Is there a reference which deals with these issues?

Linear Algebra – Spanning Subspaces and Spaces Plz help!

Exercise 1. Let Mn(R) denote the vector space (actually an R-algebra) of n × n matrices
with entries in R. Given A ∈ Mn(R), let
HA = {B ∈ Mn(R)| AB = BA}.
Show that HA is a subspace of Mn(R).

Exercise 2. Given A ∈ Mn(R), let
JA = {f(A)| f(X) ∈ R(X)} ⊆ Mn(R)
Show that JA is a subspace of Mn(R) by showing that
JA = Span{I, A, A2, A3, . . .}.

Exercise 3. In the notation of the preceding two exercises, show that JA ≤ HA.

Exercise 4. Let H = {f(X) ∈ R(X) | f(0) = 0}.
a. Show that
H = X · R(X) = {X · f(X)| f(X) ∈ R(X)} = Span{X, X2
, X3
, . . .}.
b. Use the preceding part to show that H is a subspace of R(X). Find a basis for H.
c. What can you say about Ha = {f(X) ∈ R(X) | f(a) = 0}, where a ∈ R is arbitrary?

Exercise 5. Let V denote the vector space of all sequences {an}

n=1 of real numbers. We
will say that {an} is eventually zero if there is an N ∈ N so that an = 0 for all n ≥ N (N
can vary with the sequence). That is,
{an} = {a1, a2, . . . , aN−1, 0, 0, 0, . . .}.
Let H denote the set of all eventually zero sequences. Show that H is a subspace of V . Find
a basis of H.

Exercise 6. Let a, b ∈ R and set
H = {e^(ax)(p(x) cos bx + q(x) sin bx)| p(x), q(x) ∈ R(x)} .
Show that H is a subspace of C
1
(R). Show that this conclusion remains valid if we replace
R(x) in the definition of H with Pn. Find a basis for H in both cases. (Remark. You’ll need
to know that the function tan x is not a rational function of x. Why is this true? Think in
terms of vertical asymptotes.)

Exercise 7. Show that the set (field) of complex numbers C is an R-vector space. Find an
R-basis for C.

linear algebra – Prove that $V^n$ and $mathcal{L}(mathbf{F}^n,V)$ are isomorphic vector spaces

For $n$ positive integer, define $V^n$ by $V^n=underbrace{Vtimes…times V}_{n times}$. Prove that $V^n$ and $mathcal{L}(mathbf{F}^n,V)$ are isomorphic vector spaces. I would like to know if my proof holds and to have a feedback, please.

Let $(v_1,…,v_n)$ be a basis of $V$. So, each element in $V$ can be expressed as $lambda_1 v_1+…+lambda_n v_n$ for $lambda_1,…,lambda_n in mathbf{F}$.

Let $xi:mathbf{F}^nto V$, $xi(lambda_1,…,lambda_n)=lambda_1 v_1+…+lambda_n v_n$ and define $psi: V^nto mathcal{L}(mathbf{F}^n,V)$ as $psi (lambda_1 v_1+…+lambda_n v_n,…,lambda_1 v_1+…+lambda_n v_n)=xi(lambda_1,…,lambda_n)$.

Clearly $psi$ is a linear application (it is easy to check). We show now that $psi$ is injective.

$psi(lambda_1 v_1+…+lambda_n v_n)=xi(lambda_1+…+lambda_n)=lambda_1v_1+..+lambda_nv_n=0 iff lambda_1=…=lambda_n=0$ because $(v_1,…,v_n)$ is linearly independent in $V$. So, $lambda_1 v_1+…+lambda_n v_n=0$ (so null vector in $V$) and we conclude that $psi$ is injective.

Moreover, the dimension of $V^n$ is equal to a dimension of $mathcal{L}(mathbf{F}^n,V)$. Thus, by fundamental theorem we conclude that $psi$ is surjective. Therefore, $psi$ is an isomorphism

ag.algebraic geometry – Open immersion of affinoid adic spaces

If $R$ and $S$ are complete Huber rings with $varphi: R to S$ a continuous map, then is it true in general that if $mathrm{Spa}(S, S^circ) to mathrm{Spa}(R, R^circ)$ is an open immersion of adic spaces (here $S^circ$ and $R^circ$ are the power-bounded subrings) then $mathrm{Spec}(S) to mathrm{Spec}(R)$ is injective?

For example, this is true if $R$ and $S$ both have the discrete topology, because if $frak p$ and $frak q$ are two prime ideals in $S$ which are equal after restricting to $R$ then $(frak p, |cdot|_{rm triv})$ and $(frak q, |cdot|_{rm triv})$ (trivial valuations), which are both points in $mathrm{Spa}(S,S)$, restrict to the trivial valuation on $R/varphi^{-1}(frak p)$.

But I’m not sure how generally to expect that this is true.

reference request – Kernels with finite dimensional feature spaces

Suppose $x,y in mathbb{R}^n$ for some given fixed n.

Consider a kernel $K(x,y) = f(langle x, y rangle)$, I’d like to know which functions $f$ admit a finite dimensional feature map. In other words, for $x,y in mathbb{R}^n$, what functions $f$ does there exist an $m$ and $phi: mathbb{R^n} rightarrow mathbb{R}^m$ with

$f(langle x, y rangle ) = langle phi(x), phi(y)rangle?$

I can show that $f$ must be polynomial if $m < 2^n$, but I’m sure there must exist a more comprehensive result.

real analysis – Supremum norm for convolution in sequence spaces

Question:

Suppose that $1 leq p leq infty$, and the convolution $x ast y$ exists. For sequences $x in ell^p(mathbb{Z})$ and $y in ell^q(mathbb{Z})$, we have

$$||x ast y ||_{infty} leq ||x||_p||y||_q,$$

and $x ast y in ell^{infty}$.


We know $p=1$ and $p=infty$ follow by Young’s Inequality ($||x ast y||_p leq ||x||_p||y||_1$).

For $1 < p < infty$, I’m trying take the supremum over what Hölder’s inequality gives. We can define the $n$-th “convolution” as

$$(x ast y)_n = sum_{i=-infty}^{infty} x_iy_{n-i},$$

and so Hölder’s gives

$$||(x ast y)_n||_1 leq ||x||_pleft(sum_{i=-infty}^{infty} |y_{n-i}|^qright)^{frac{1}{q}}.$$

Taking the sup over $n$,

$$||(x ast y)_n||_{infty} leq ||x||_p sup_{n in mathbb{Z}} left(sum_{i = -infty}^{infty} |y_{n-i}|^qright)^{frac{1}{q}} stackrel{?}{=} ||x||_p||y||_q.$$

Can I just reindex this in some manner, $i mapsto n – i$? Is this equivalent to the translation invariant that one would do for the convolution in function spaces?

Confused about D50/D65 conversion going from Lab to sRGB color spaces

I am writing code for some color conversion work and have a confusion. Here are my steps:

  • I utilize a color calibration target (ISA ColorChecker) with reference values provided by the manufacturer in Lab space with D50 white point.
  • I capture a raw image of the target and demosaic the Bayern pattern arriving at RawRGB values for each of the Calibration Target’s patches (average value is taken). To calculate a color correction matrix, I want to find the (not-gamma-corrected) sRGB values starting from the Lab reference values of the Target.
  • I use the formulas in http://www.brucelindbloom.com. First step is going from Lab to XYZ I use the D50 white reference point XYZReference = 0.9504,1.0000,1.0888
  • Second step is going from XYZ to sRGB and this is where the confusion is: I arrived at the XYZ values using a D50 white point, but sRGB standard seems to use D65. Which of the inverseM matrices shall I be using to get this right?

As additional info: My purpose for this conversion is to get the colors look reasonably correct in a typical commercial display (PC, tablet etc) when I save as png or jpeg for example. After the linear conversion above, I know that I must also apply the gamma-companding.

Thank you!

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