## banach spaces – Finite-dimensional subspaces of \$l_{p}\$ and \$c_{0}\$

Let $$M$$ be a finite-dimensional subspace of $$X=l_{p}$$ or $$c_{0}$$. Let $$epsilon>0$$. Does there exist a projection from $$X$$, of norm $$leq 1+epsilon$$, onto a subspace $$N$$ of $$X$$ with $$Msubseteq N$$ and the Banach-Mazur distance $$textrm{d}(N,l_{p}^{n})$$(resp.$$textrm{d}(N,l_{infty}^{n}))leq 1+epsilon$$, where $$n=textrm{dim}N$$ ?

Thank you!

## banach spaces – Uniform smoothness and twice-differentiability of norms

To get to the simplest case, consider a norm $$|cdot|$$ over $$R^n$$ that is uniformly convex of power-type 2, that is, there is a constant $$C$$ such that $$frac{|x+y| + |x – y|}{2} le 1 + C |y|^2$$ for all $$x$$ with $$|x| = 1$$ and for all $$y$$.

Question: Does this guarantee that $$|cdot|$$ has a second-order Taylor expansion on $$R^n setminus {0}$$, that is, there is a vector $$g$$ and a symmetric matrix $$A$$ such that $$|x + y| = |x| + langle g, y rangle + frac{1}{2} langle Ay, y rangle + o(|y|^2)$$ for all $$x neq 0$$. (Apparently this is a weaker requirement than twice-differentiability of $$|cdot|$$ on $$R^n setminus {0}$$)

It is easy to see that $$|cdot|$$ is differentiable on $$R^n setminus {0}$$, and a classic result of Alexandrov guarantees that the above second-order Taylor expansion holds for any convex function on almost every point $$x$$. It is also known that the norm of any separable Banach space can be approximated arbitrarily well by a power-type 2 norm that is twice differentiable on $$R^n setminus {0}$$ (see Lemma 2.6 here). But I wonder if the original norm itself has a second-order Taylor expansion.

## infinity categories – Using the universal property of spaces

The $$infty$$-category of spaces is known to be the $$infty$$-category obtained from the (ordinary) category of finite sets by freely adding sifted colimits. (See e.g. Cesnavicius-Scholze https://arxiv.org/abs/1912.10932 §5.1 for a review of this notion and for pointers to Lurie’s HTT where this is proven.)

Can this characterization be used (ideally without referring to the model of quasi-categories) to show other properties, such as:

• that colimits in spaces are universal (proven by Lurie in HTT Lemma 6.1.3.14)?
• possibly even that $$Cat_infty$$ is compactly generated by $$*$$ and $$Delta^1$$?

## Split text in Google Sheets with a delimiter which contains spaces

There are many ways to achieve the result you are looking. Here is one of the oldies, using RIGHT, LEN and FIND

``````=ArrayFormula(RIGHT(A:A,LEN(A:A) - FIND(" on ",A:A) - 3)
``````

Another way, in this case using SPLIT, SUBSTITUTE and INDEX

``````=ArrayFormula(INDEX(SPLIT(SUBSTITUTE(A:A," on ","|"),"|"),1,2))
``````

One way using regular expressions

``````=ArrayFormula(REGEXEXTRACT(A:A," on (.+)"))
``````

P.S. The simplest way to avoid having errors for blanks cells below the las row having a value on A, delete all the bottom blank rows.

## nt.number theory – Classification of vector spaces with a quadratic form and an order n automorphism

Introductory general nonsense (for motivation: feel free to skip): Let $$G$$ be a finite group and $$k$$ be a field of characteristic $$0$$. Consider the set $$mathcal{S}$$ of isomorphism classes of finite dimensional vector spaces $$V$$ over $$k$$ endowed with (a) a nondegenerate quadratic form $$qcolon Vto k$$, and (b) a linear action $$G to mathit{GL}(V)$$, which are compatible in the sense that $$G$$ preserves $$q$$ (viꝫ. $$q(gcdot v)) = q(v)$$ for $$gin G$$). Note that we can take direct sums and tensor products of such data, giving $$mathcal{S}$$ a semiring (“ring without subtraction”) structure; we can also form the Grothendieck group $$mathcal{R}$$ of $$mathcal{S}$$, which is a ring.

Classifying (a) alone and (b) alone is well studied: (a) gives the Grothendieck-Witt ring of $$k$$, and (b) gives the representation ring of $$G$$ over $$k$$. I’m curious about what can be said about both data simultaneously (and compatibly). We have obvious ring homomorphisms from $$mathcal{R}$$ to the Grothendieck-Witt ring of $$k$$ and to the representation ring of $$G$$ over $$k$$, but I think $$mathcal{R}$$ (generally) isn’t a fiber product of them, and I suppose there isn’t much we can say at this level of generality (though I’d be happy to be wrong!).

I might still point out that if $$V = V_1 oplus V_2$$ is a decomposition of $$V$$ as representations of $$G$$ and there is no irreducible factor common in $$V_1$$ and the dual $$V_2^vee$$ of $$V_2$$, then necessarily $$V_1$$ and $$V_2$$ are orthogonal for $$q$$ (proof: apply Schur’s lemma to the $$G$$-invariant linear map $$V_1 to V_2^vee$$ obtained from $$q$$). So we are reduced to classifying elements of $$mathcal{R}$$ (or $$mathcal{S}$$) whose underlying representation is of the form $$U^r$$ for $$U$$ an irreducible self-dual representation of $$G$$, or of the form $$(Uoplus U^vee)^r$$ for an irreducible non-self-dual representation $$U$$.

Anyway, let me concentrate on the important special case where $$k=mathbb{Q}$$ and $$G=mathbb{Z}/nmathbb{Z}$$. The irreducible representations of $$mathbb{Z}/nmathbb{Z}$$ over $$mathbb{Q}$$ are of the form $$U_d$$ (self-dual) for $$d$$ dividing $$n$$ where $$U_d$$ splits over $$mathbb{C}$$ as sum of one-dimensional representations on which a chosen generator acts through each of the primitive $$d$$-th roots of unity. So I ask:

Actual question: Given $$d,n,rgeq 1$$ be integers such that $$d$$ divides $$n$$, let $$U_d$$ be the irreductible representation of $$mathbb{Z}/nmathbb{Z}$$ over $$mathbb{Q}$$ such that the generators act with characteristic polynomial given by the $$d$$-th cyclotomic polynomial. Can we classify quadratic forms on $$(U_d)^r$$ which are invariant under the action of $$mathbb{Z}/nmathbb{Z}$$ (i.e., describe the corresponding elements of the (known) Grothendieck-Witt ring of $$mathbb{Q}$$)? Or equivalently, in the other direction, given a quadratic form $$(V,q)$$ over $$mathbb{Q}$$ (through its image in the G-W ring), can classify $$mathbb{Z}/nmathbb{Z}$$-actions (over $$V$$, linear, preserving $$q$$) which make $$V$$ isomorphic to $$(U_d)^r$$?

I don’t even know the answer when $$q$$ is the standard Euclidean form (viꝫ. $$mathbb{Q}^m$$ with the quadratic form $$x_1^2 + cdots + x_m^2$$): for which $$d,n,r$$ is there a $$mathbb{Z}/nmathbb{Z}$$-invariant quadratic form on $$(U_d)^r$$ that is isomorphic to this?

Note: there is a $$mathbb{Z}/nmathbb{Z}$$-invariant standard Euclidean structure on $$mathbb{Q}^n = bigoplus_{d|n} U_d$$ with cyclic permutation of the coordinates, which induces a quadratic form on each of the $$U_d$$ so that this direct sum is orthogonal (the class of this form in the G-W ring can be computed by the Möbius inversion formula; because there is a scaling involved, it depends on $$n$$, not just $$d$$). It might be tempting to think that all $$mathbb{Z}/nmathbb{Z}$$-invariant quadratic forms on $$U_d$$ are obtained in this way: if my Witt ring calculations are correct, this is not the case: $$U_{30}$$ does not get a standard Euclidean structure that way; but there is a standard Euclidean structure on $$U_{30}$$, namely, take the Coxeter element of the Weyl group of $$E_8$$ as acting on $$mathbb{Q}^8$$ with its standard Euclidean structure, which is then $$U_{30}$$ as a representation of $$mathbb{Z}/30mathbb{Z}$$.

## vector spaces – understand of span, linear independence and basis by using dimension

Before start explaining what makes me confused, I’m sorry about my poor English. I’m not good at English. lol

If V is Finite-dimensional vector space, let {v1,v2, ⋯,vn} is abritary basis of V.
(a)The set of vectors that have more than n is linearly dependent set.
(b)The set of vectors that have less than n can’t span V

I wanna know that (b) means if we have more than n vectors when the basis of the V is n, we can just span V.

I’m confused because I’m not sure about the concept of span and linear combination exactly.

If S={v1,v2,⋯vr}, S’={w1,w2,⋯wk} is a vector set that is included in vector space V, if and only if span{v1,v2,⋯vr}=span{w1,w2,⋯wk} is that each vector of S is linear combination of w1,w2,⋯wk and also each vector of S’ is linear combination of v1,v2,⋯vr.

I wanna know this sentence is right.
`If there are ***n**-times basis* *(I mean the number of the basis vector is n)* and n≦r, n≦k then S,S' can span vector space V and also S,S' is linearly dependent set.`

## measure theory – Inclusions between \$L^p\$ spaces.

Theorem. Let $$mu(X). Then $$1le p le qleinftyimplies L^q(X,mathcal{A},mu)subseteq L^p(X,mathcal{A},mu)$$

Def. Let $$Omegasubseteqmathbb{R}^n$$ be an open subset, $$fcolonOmegato (-infty,+infty)$$ q.o defined. We said that the function $$f$$ is locally integrable in $$Omega$$ if $$fin L^1(G,mathcal{L}(mathbb{R}^n)cap G,lambda)$$ for all $$Ginmathcal{L}(mathbb{R^n})$$ such that $$overline{G}subseteqOmega.$$

In the definition $$lambda$$ is the Lebesgue measure on $$mathbb{R}^n$$.

We denote the set of locally integrable function with $$L^1_{text{loc}}$$

I must prove that $$L^p(Omega)subseteq L^1_{text{loc}}(Omega)quadtext{for all};pin(1,+infty)$$

using the previous theorem.

Naturally $$L^1(Omega)subseteq L^1_{text{loc}}(Omega)$$

Now I don’t know how to proceed. Could anyone give me a suggestion? Thanks!

## 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. My purpose for this conversion is to get the colors look reasonably correct in a typical non-professional display (PC, tablet etc) when I save as png or jpeg for example.

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 with D65 illuminant is the most common working color space of consumer displays. Which of the inverseM matrices shall I be using to get this right?

After the linear conversion above, I know that I must also apply the gamma-companding.

Thank you!

## ag.algebraic geometry – Moduli spaces of rational curves in rational surfaces

Let $$S$$ be a smooth rational surface. Consider the scheme $$X_A(S)^o$$ parametrizing irreducible rational curves in a fixed linear system $$|D|$$ on $$S$$ passing through a bunch of general points of $$S$$, and having multiplicity two at another bunch of general points of $$S$$.

Let $$X_d(S)$$ be the closure of $$X_d(S)^o$$ in the scheme of all rational curves in $$S$$.

In general does $$X_d(S)$$ have more than one irreducible component whose general point represents an irreducible curve?

Thank you.

## geometry – Hyperbolic spaces – Mathematics Stack Exchange

Suppose X is a δ-hyperbolic metric space and α is a geodesic in X. Let
P : X → α denote any nearest point projection map, i.e. for all x ∈ X,
d(x, α) = d(x, P(x)). Show that the P coarsely L-Lipschitz where L is a
constant depending only on δ.

My attempt is: Considering the geodesic line joining P(z) and P(z’), I am subdividing it so that it’s distance is less than z and z’. But I don’t how to proceed further.