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 §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
  • 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 $mu(X)<infty$. 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 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.