## 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!

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.

``````   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>
``````

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. $$text{dis} f (i.e. $$sup_{B(p_n,r)}|d_n(x_1,x_2)-d(f(x_1),f(x_2))|)
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!