Banach-like analysis on metric spaces

Some time ago, I was thinking about whether it would be possible to generalize some results from functional analysis on Banach spaces to some metric spaces. Specifically, I wondered whether if one were to assume that the metric space possesses some parametrized segment-like structure (which, for example, in Banach spaces would consist of functions of the form $(0,1) ni t mapsto (1-t)x + ty$, where $x, y$ are elements of some Banach space), we could define something analogous to continuous linear (or, more appropriately, affine) operators by requiring that such operator would “preserve” this structure.

I’d like to ask if there are any papers or books which might be related to this topic and, if so, where should I look for them.

ag.algebraic geometry – Unirational subvarieties of moduli spaces of curves

Let $overline{M}_{g,n}$ be the Deligne-Mumford compactification of the moduli space $M_{g,n}$ of $n$-pointed genus $g$ smooth curves, and $Xsubsetoverline{M}_{g,n}$ a unirational variety intersecting $M_{g,n}$.

Is there an upper bound on the dimension of $X$ depending on $g,n$ (at least for $g$ big enough when $overline{M}_{g,n}$ is of general type)?

Thank you.

export – Exporting multiple lists (x1,y1…n) (x2,y2..n)of different lengths into same excel files separated by spaces of two columns between each (x,y) list?

I have multiple lists like this this with different lengths and different x, y values,

list1 = {{3., 1.49463}, {3.1, 1.49238}, {3.2, 1.49027}, {3.3, 1.48814}, {3.4, 
  1.48592}, {3.5, 1.48366}, {3.6, 1.48158}, {3.7, 1.47997}, {3.8, 
  1.47894}, {3.9, 1.47848}, {4., 1.47824}, {4.1, 1.47892}, {4.2, 
  1.47973}, {4.3, 1.4809}, {4.4, 1.48231}, {4.5, 1.48374}, {4.6, 
  1.48498}, {4.7, 1.48589}, {4.8, 1.4865}, {4.9, 1.4869}, {5., 

list2 = {{0.3, 1.70796}, {0.4, 1.69032}, {0.5, 1.66887}, {0.6, 1.65187}, {0.7,
   1.64455}, {0.8, 1.64575}, {0.9, 1.65089}, {1., 1.65581}}

list 3 = {4.1, 1.47892}, {4.2, 1.47973}, {4.3, 1.4809}, {4.4, 1.48231}, {4.5, 
  1.48374}, {4.6, 1.48498}, {4.7, 1.48589}, {4.8, 1.4865}, {4.9, 
  1.4869}, {5., 1.48717}}

and so on until 10 lists and all of them have different lengths with some having (x,y) sets as much as 30 (x1,y1),(x2,y2) … (x30,y30) like that,

i want to list them list this in single excel sheet with two different options whichever you can guide,

list 1 x,y values (next list after two columns separation) –list 2 x,y values– (next list after two columns separation) -**list 3 x,y values****

list 1 x,y values (no separation)- list 2 x,y values– (no separation)- list 3 x,y values

reference request – Number of homeomorphisms between spaces

I need help with the following.

We know that those spaces that have a trivial group of homeomorphisms are those called rigid, there are even some references in the group about them here.

My first question was about the existence of spaces that are not homeomorphic to more than one space and I think that with the above we have an answer. My question is, will you have any reference where it talks about spaces that admit only 2 homeomorphic spaces to it, or in general, spaces that only admit $n$ homeomorphic spaces to it?

sobolev spaces – Perimeter continuity of $BV$ sets on any sequence from $W^{1,1}$

In the book of S.Osher & R.Fedkiw – Level set methods and dynamic implicit surfaces, at page 15, is stated without proof, a formula like that:

$$text{Per}_{Omega}(omega)=limlimits_{varepsilonto 0} int_{Omega}delta_{varepsilon}(phi(x)) |nablaphi(x)| dx,$$

where $omega={xinOmega | phi(x)>0}$ and $phi:Omegatomathbb{R}$ is a smooth level set function. By $text{Per}_{Omega}(omega)$ I denote the perimeter of $omega$ that lies inside $Omega$. Here $Omegasubsetmathbb{R}^2$ is an open and bounded set.

Also $delta_{varepsilon}:mathbb{R}tomathbb{R}, delta_{varepsilon}(x)=dfrac{varepsilon}{pi (varepsilon^2+x^2)}$ is a smooth approximation of the $delta$-Dirac (generalized) function.

My question is: Is it true that for any $chi_{omega}in BV(Omega)$ ($chi_omega$ being the characteristic function of $omega$ – so by this condition we require $omega$ to be a set with finite perimeter), whenever $chi_nto chi_{omega}$ in the $L^1(Omega)$ norm and $chi_nin W^{1,1}(Omega)$ for each $n$, we have that:
$$bigvee_{Omega} chi_{omega}=limlimits_{ntoinfty} bigvee_{Omega}chi_n(x)$$

I denote by $bigvee_{Omega} f$ the total variation of $f$ in $Omega$.


$bullet$ It is known that:

$$bigvee_{Omega} f = supleft{int_{Omega} f(x)text{div}(varphi(x)) dx | varphiin C^{infty}_{c}(Omega;mathbb{R}^2) text{and} VertvarphiVert_{L^{infty}(Omega;mathbb{R}^2)}leq 1right }$$

$bullet$ It is also known that for $chiin W^{1,1}(Omega)subset BV(Omega)$ we have that:

$$bigvee_{Omega} chi=int_{Omega} |nablachi(x)| dx.$$

$bullet$ We have that $text{Per}_{Omega}(omega)=bigvee_{Omega}chi_{omega}$ for any $omegasubseteqOmega$.

$bullet$ One part of the inequality is well-know (lower semicontinuity of the total variation), so:

$$bigvee_{Omega} chi_{omega}leqliminflimits_{ntoinfty} bigvee_{Omega}chi_n(x)$$

P.S. In this course, at page 10, Theoreme 1.3 gives an approximation result of the $BV$ functions with $C^{infty}_c$ functions. Maybe it is useful.

I tried to prove it. It is for many examples that I take, but I can’t figure out why. Intuitively I understand it but technically I’m blocked.

I’m interested in this type of formulas because if it is indeed true we will have that:

$$text{Per}_{Omega}(omega)=limlimits_{varepsilonto 0}int_{Omega} H’_{varepsilon}(phi(x))|nablaphi(x)| dx,$$

for any function $H_{varepsilon}:mathbb{R}tomathbb{R}, H_{varepsilon}in W^{1,1}(mathbb{R})$ that approximates in $L^1(Omega)$ the Heaviside function $H(x)=begin{cases} 1, x>0 \ 0, x<0end{cases}$. In the above example $H_{varepsilon}(x)=dfrac{1}{2}+dfrac{1}{pi}cdottan^{-1}(x/varepsilon)$, for each $varepsilon>0$ and $delta_{varepsilon}(x)=H’_{varepsilon}(x)$.

complete spaces – Definition of ‘completeness’ of a valued field

I have on a number of occasions seen reference to ‘complete valued fields’ without any explanation of precisely what completeness means in this context. It is definitely not referring to maximal completeness or spherical completeness, but rather something along the lines of Cauchy completeness. Although most of the literature refers to complete discrete valued fields, some of it appears to indicate that completeness also works in a non-discrete setting.

Most importantly I need to understand the notion of completeness referred to here, where it is claimed that it follows from maximal completeness. I add the definition of a valued field that I am working with below to avoid confusion:

A valued field is a field $K$ equipped with a valuation, which is a surjective map $v:Kto Gcuplbraceinftyrbrace$ where $v(x)=inftyLeftrightarrow x=0$ and $$v(xy)=v(x)+v(y),$$ $$v(x+y)geqminlbrace v(x),v(y)rbrace$$ for all $x,yin K$ and where $G$ is a totally ordered abelian group, called its value group.

Is there a common understanding of what ‘completeness’ means in the context of valued fields?

general topology – Examples of homeomorphic spaces

Are the following pairs of spaces homeomorphic?

(i) $mathbb Q cup (0, 1)$ and $mathbb R setminus mathbb Q$ (as subspaces of $(mathbb R, tau_{usual}))$;

(ii) $(mathbb C, tau_{disc})$ and $(mathbb C, tau_{cocountable})$;

(iii) a circle with one point removed (in $mathbb R^2$ with $tau_{usual}$) and $(mathbb R, tau_{usual})$;

(iv) $mathbb N$ with topology ${1, 2}, {3, 4}, {5, 6}, {7, 8}, ldots$ and all unions of these, and $mathbb N$ with topology ${1, 3}, {2, 4}, {5, 7}, {6, 8}, {9, 11}, {10, 12}, ldots$ and all unions of these.

The following have answers

(v) $(mathbb R, tau_{cocountable}), (mathbb R, tau_{cofinite})$

(vi) $mathbb R^2$ and the surface of a sphere with one point removed (natural metric
topologies here)

(vii) $(mathbb Q,l_x), (mathbb Q, l_y)$, where $x, y$ are two distinct rational numbers and $l_p$ is included-point topology

Book’s answers to (v) and (vii):

(v) No. The statement ‘there is a countably infinite closed subset’ is
(obviously) a homeomorphic invariant, is true in $(mathbb R, tau_{cocountable})$ and is false in $(mathbb R, tau_{cofinite})$

(vii) Yes. The map $h : mathbb Q → mathbb Q$ given by $h(x) = y, h(y) = x, h(z) = z$
when $z ne x$ or $y$ is routinely checkable to be a homeomorphism.

My answers/questions regarding question above:

(i) $mathbb Q, (0, 1)$ are not compact and so their union is not compact meaning there’s no continuous inverse function from $mathbb Q cup (0, 1)$

(ii) Any function from a discrete space to an indiscrete space is continuous. Since idenity $i$ is continuous and bijective, $i$ should work here to show homeomorphism

(iii) I think this is a special case of Stereographic projection

(iv) Any inverse image of an open set in $mathbb N$ is a union of $2$-sets. Since the given $2$-sets are open, their union is open as well. This works in both directions, so a continuous function with its continuous inverse likely exists. I am not sure how to find a concrete homeomorphism, but would $f(x, y) = (x, y – 1)$ if $y$ is odd and $f(x, y) = (x + 1, y)$ if $x$ is even work?

Are my answers to (i) through (iv) correct? If not (or if incomplete), how do I improve them?

(v) This question might be trivial, but I am still new to the very basics of topology. Consider $f: (mathbb R, tau_{cofinite}) to (mathbb R, tau_{cocountable})$. Every $Y in (mathbb R, tau_{cocountable})$ must have a pre-image $X in (mathbb R, tau_{cofinite})$. By definition, every $X$ is open, but for $f$ to be continuous $X$ must be cocountable. Correct?

(vi) It’s yet another special case of Stereographic projection. Correct?

(vii) Is $h(y)$ a typo? Did they mean $h^{-1}(y)?$


How to remove extra spaces from powershell generated file

I am trying to use PowerShell to extract a series of lines from a source file. The code works, even looks fine, but when viewed from FTP software, and on the actual machine, it has too many spaces between characters.

BPS,1,150,3, /billinfo 11234567 108,13457890
BPS,1,150,3, /billinfo 12234567 108,32457890
BPS,1,150,3, /billinfo 12334567 108,34357890
BPS,1,150,3, /billinfo 12344567 108,34547890
BPS,1,150,3, /billinfo 12345567 108,34575890


BPS,1,150,3, /billinfo 12234567 108,3457890
BPS,1,150,3, /billinfo 12345567 108,3457890
BPS,1,150,3, /billinfo 12344567 108,3457890

Output received:
ÿþS R C _ N A M E , N O T F _ U C _ T Y P E , N O T F _ S C N _ I D , D L V _ A D D R _ T Y P E , D L V _ A D D R _ I D , A c c o u n t N o
B P S , 1 , 1 5 0 , 3 , 0 . 0 . 0 . 1 / b i l l i n f o 1 2 2 3 4 5 6 7 1 0 8 , 3 4 5 7 8 9 0
B P S , 1 ,1 5 0 , 3 , 0 . 0 . 0 . 1 / b i l l i n f o 1 2 3 4 5 5 6 7 1 0 8 , 3 4 5 7 8 9 0
B P S , 1 ,1 5 0 , 3 , 0 . 0 . 0 . 1 / b i l l i n f o 1 2 3 4 4 5 6 7 1 0 8 , 3 4 5 7 8 9 0

Script used:
Get-Content .source.csv -First 1 > ExpectedOutput.csv -Encoding ASCII #copy the header
get-content .source.csv | select-string -Pattern (Get-Content .search.txt) | Add-Content ExpectedOutput.csv -Encoding ASCII #search the contents of search.txt in source.csv and append to the ExpectedOutput.csv file

at.algebraic topology – (Homotopy) inverse limits of towers of spaces or simplicial sets – reference request

Suppose we have a tower of Kan fibrations between Kan complexes:
$$ X_0 xleftarrow{f_0} X_1 xleftarrow{f_1} X_2 xleftarrow{f_2} dotsb $$
From this we get a commutative diagram of topological spaces:
left|text{lim}_nX_nright| @>{alpha}>> left|text{holim}_nX_nright| \
@V{beta}VV @VV{gamma}V \
text{lim}_n|X_n| @>>{delta}> text{holim}_n|X_n|

It is known that $alpha,beta,gamma$ and $delta$ are weak equivalences. What is a convenient reference for this?

Some ingredients:

  1. The geometric realization of a Kan fibration is a Serre fibration, by a short paper of Quillen with precisely that title. Thus, the spaces $|X_n|$ form a tower of Serre fibrations.
  2. The map $alpha$ is the geometric realization of a map $alpha_0$ of simplicial sets, and $alpha$ is a weak equivalence iff $alpha_0$ is a weak equivalence.
  3. There are various relevant things in the Bousfield-Kan book. In particular, Example XI.4.1(v) essentially asserts that $alpha_0$ is a weak equivalence, but they do not really spell out a proof.
  4. If $Z$ is any one of the four spaces in the diagram, then after saying the right things about basepoints we get a short exact sequence relating $pi_*(Z)$ to $text{lim}$ and $text{lim}^1$ of ${pi_*(X_n)}_{ngeq 0}$. For $|text{lim}_nX_n|$ this is Theorem IX.3.1 in Bousfield-Kan, and for $|text{holim}_nX_n|$ it is essentially XII.7.4, although few details are given. The spaces on the bottom row can be dealt with in a similar way. Phil Hirschhorn wrote a careful and elementary argument for $text{lim}_n|X_n|$.

This is enough to piece together a proof, but it would be nice to have a single reference where the claim is stated and proved explicitly.

dg.differential geometry – Infinitely many nonempty Seiberg-Witten moduli spaces

The classic “finiteness” statement in Seiberg-Witten (SW) theory is that, for a fixed closed (smooth connected) 4-manifold, there are only finitely many spin-c structures with nontrivial SW invariants. The literature typically goes further to claim that there are only finitely many spin-c structures with nonempty SW moduli spaces, but we need to be careful with the statement:

Is there a 4-manifold $X$ with $b^2_+(X)>1$, and a non-generic pair of Riemannian metric $g$ and closed self-dual 2-form perturbation $mu$ (to the SW curvature equations) on $X$, for which there are infinitely many nonempty SW moduli spaces (indexed over the set of spin-c structures)?

Fix a pair $(g,mu)$. For each spin-c structure $mathfrak s$ the Weitzenbock formula and SW equations yield a uniform upper bound on the SW spinors (independent of $mathfrak s$), hence on the self-dual part of the SW curvatures (using the SW curvature equation), hence on $c_1(mathfrak s)^2=frac{1}{4pi^2}int_X(|F_A^+|^2- |F_A^-|^2)text{dvol}_g$ via Chern-Weil theory. When the pair is generic, there are no SW solutions when its virtual dimension is negative, yielding a uniform lower bound on $c_1(mathfrak s)^2$ in terms of the topology of the 4-manifold, $c_1(mathfrak s)^2ge2chi(X)+3sigma(X)$. This is enough to ‘bound’ $c_1(mathfrak s)$ in $H^2(X;mathbb Z)$ and get the desired finiteness statement.

(Aside: When $b^2_+(X)=1$, we learn that each perturbation can only lie in finitely many chambers with nontrivial SW invariants.)