fa.functional analysis – Explicit example of a certain weak-* limit

Problem set up:

Consider $C_b$, the Banach space of continuous bounded functions on $[0, infty]$ equipped with the sup norm. Denote by $M$ the set of probability measures on $C_b$, and for $r > 0$ denote by $M_r$ the set of probability measures supported on $[r, infty]$.

Consider the set $mathcal S$ of linear functionals $L in C_b^*$ such that there exist a sequence $r_n$ of real numbers with $r_n to infty$, and a sequence of probability measures $mu_n$ with $mu_n in M_{r_n}$ for all $n$ such that $mu_n to L$ in the weak* topology.

By the Banach-Alaoglu theorem, $mathcal S$ is nonempty.

Question: Is it possible to produce an explicit example of an element of $mathcal S$, and the corresponding probability measures?

ag.algebraic geometry – Explicit Natural Correspondence between Cusps of X(N) and isomorphism classes of Level N structures on Tate(q^N)

In Katz’ paper Antwerp III, section 1.4 (Ka-14) one reads (we assume $n geq 3$ integer):

”The scheme $overline{M}_n – M_n$” over $mathbb{Z}(1/n)$ is finite and étale, and over $mathbb{Z}(1/n,zeta_n)$, it is a disjoint union of sections, called the cusps of $overline{M}_n$,”

I would be interested to see a detailed proof of the next part of that sentence, namely:

“which in a natural way are the set of isomorphism classes of level $n$ structures on the Tate curve $text{Tate}(q^n)$ viewed over $mathbb{Z}((q)) otimes_{mathbb{Z}} mathbb{Z}(1/n,zeta_n)$.”

What I did: I tried to extract the relevant information in Deligne-Rapoport and Katz-Mazur but in each case, certainly for a lack of understanding on my part, I’m not able to establish this correspondence explicitly. I found the discussion of formal completion at (the divisor of) cusps well explained in both references (something which is also addressed in Katz’ paper Antwerp III on page Ka-14), but I couldn’t connect the dots for the natural correspondence above and thus my question. Feel free to ask if you need more details.

In Deligne-Rapoport?

I first looked in Deligne-Rapoport (DR), which was in Antwerp II (and so the natural place to look for the arguments):
http://smtp.math.uni-bonn.de/ag/alggeom/preprints/Lesschemas.pdf

The motivating example on pages DeRa-7 and beginning of page 8 hint to that fact. But it seems that’s not the point of view (DR) take.

“Dans le texte, nous précisons cette interprétation modulaire de l’ensemble des points à l’infini de $mathcal{H}/Gamma(n)$ en une interprétation modulaire de la courbe projective compactifiée $overline{mathcal{H}/Gamma(n)}$ de $mathcal{H}/Gamma(n)$.”

Here: $mathcal{H} = { z in mathbb{C} mid Im(z) > 0 }$ is the upper half-plane.

On page DeRa-10 they do say that $M_n$ can be defined as the normalization, in the field of functions of $M_n^0(1/n)$, of the projective $j$-line over $mathbb{Z}(zeta_n)$. (That’s what Katz and Mazur do in their book on Chapter 8.) (DR) say among other things that they prove that there exists a finite family of points $mathbb{Z}(zeta_n)$-points $f_i : M_n to Spec(mathbb{Z}(zeta_n))$ such that the sections $f_i$ are disjoint (incongruent modulo any prime ideal of $mathbb{Z}(zeta_n)$) and that $M_n^0$ is the complement in $M_n$ of the union of the ”sections at infinity” $f_i$.

The Tate curve is only constructed in chapter VII of (DR). But I don’t find it immediate to deduce initial assertion by Katz in Antwerp III.

In Chapter VII (sections 1 and mostly 2 seem relevant to my question), DeRa-156, (1.16.4) gives me the description of the level $r$-structure of the Tate curve with $r$ edges over $mathbb{Z}((q^{1/r}))$.

Moreover, $text{Tate}(q)$ over $mathbb{Z}((q))$ induces a morphism $tau: Spec(mathbb{Z}((q))) to mathcal{M}_1$ which identifies $mathbb{Z}((q))$ with the formal completion of $mathcal{M}_1$ along the section at infinity $f_1$ (Theorem 2.1).

The Néron $n$-gon $C$ over $mathbb{Z}(zeta_n)$ equipped with its structure of generalized elliptic curve and the natural isomorphism $C(n) = mu_n times mathbb{Z}/nmathbb{Z}$ defines a section at infinity $f_n : Spec(mathbb{Z}(zeta_n)) to mathcal{M}_n$. We also obtain an isomorphism between the $n$-torsion of the Tate curve with $n$ edges and $mu_n times mathbb{Z}/n mathbb{Z}$ and then we geta morphism $Spec(mathbb{Z}(zeta_n)((q^{1/n}))) to mathcal{M}_n$. This latter morphism identifies $mathbb{Z}(zeta_n)((q^{1/n}))$ with the formal completion of $mathcal{M}_n$ along the section at infinity $f_n$.

Finally, Corollary 2.5 says that the completion of $mathcal{M}_n$ along infinity is sum of copies of $Spec(mathbb{Z}(zeta_n)((q^{1/n}))$ indexed by $SL_2(mathbb{Z}/nmathbb{Z})/pm U$, where $U$ is the group of upper unipotent matrices.

It feels like the desired correspondence is there but I couldn’t extract it explicitly.

In Katz-Mazur?
I turned to the book of Katz and Mazur (see https://web.math.princeton.edu/~nmk/katz-mazur.djvu). Again, I feel I’m getting closed, but I’m not sure how to tie up the loose ends.

The point of view in (KM) doesn’t deal (explicitly?) with stacks (as in (DR)). They consider the moduli problem (contravariant functor)

$$
(Gamma(N)) : textbf{Ell} to textbf{Set}
$$

which classifies elliptic curves (proper smooth curves $pi : E to S$ with geometrically connected fibers all of genus one, given with a section $0$, and here $S$ is any scheme.) equipped with a $Gamma(N)$-structure (KM 3.1, page 98).

This functor is relatively representable and flat over $textbf{Ell}$ of constant rank $geq 1$, and regular of dimension $2$. As a functor with source $textbf{Ell}/mathbb{Z}(1/N)$ it is étale on the source. (First Main Theorem 5.1.1, page 129).

When $N geq 3$, $(Gamma(N))$ is in fact representable by some universal elliptic curve $E_text{univ}/Y(N)$, where $Y(N)$ is a smooth affine curve (We have a rigidity.) (See (KM) Cor 2.7.2, 4.7.0 and 4.7.1)

Following (KM 8.6.3 and 8.6.8) we normalize $Y(N)$ near infinity to obtain $X(N)$ (we obtain a smooth proper curve over $mathbb{Z}(1/N)$ which is the normalization of the projective $j$-line in $Y(N)$).

The Tate curve $text{Tate}(q)$ itself represents an appropriate moduli problem $mathcal{S}$. Applying corollary 8.4.4 (p.235) to this and to the moduli problem $(Gamma(N))$ over an excellent noetherian regular ring $R$, we obtain an isomorphim of $R((q))$-schemes

$$
left( (Gamma(N))_{text{Tate}(q)/R((q))} right)/ pm 1 xrightarrow{simeq} Y(N)_{R((q))}
$$

where $Aut(text{Tate}(q)/R((q))) = pm 1$ (see Proposition 8.11.7).

Moreover, the formal completion of $X(N)$ along the (divisor of) cusps $X(N) – Y(N)$, which is a finite $R((q))$-scheme, is the normalization of $R((q))$ in the finite normal $R((q))$-scheme $left( (Gamma(N))_{text{Tate}(q)/R((q))} right)/ pm 1 $.

Finally, we have

Theorem 10.8.2

There is a canonical isomorphism of $mathbb{Z}(zeta_N)((q))$-schemes

$(Gamma(N))_{text{Tate}(q)/mathbb{Z}(zeta_N)((q))} simeq coprod_{text{Hom Surj }((mathbb{Z}/Nmathbb{Z})^2,mathbb{Z}/Nmathbb{Z})} Spec(mathbb{Z}(zeta_N)((q^{1/N})))$

and

Theorem 10.9.1

(1) $text{Cusps}(X(N))$ is the disjoint union of $mid text{Hom Surj }((mathbb{Z}/Nmathbb{Z})^2,mathbb{Z}/Nmathbb{Z}) mid$ sections of $X(N)$ over $mathbb{Z}(zeta_N)$.

(2) There exists an open neighborhood $V$ of the cusps $text{Cusps}((Gamma(N))) subset V subset X(N)$ which is smooth over $mathbb{Z}(zeta_N)$.

(3) The formal completion of $X(N)$ along its cusps is the $mathbb{Z}(zeta_N)$-formal scheme

$coprod_{text{Hom Surj }((mathbb{Z}/Nmathbb{Z})^2,mathbb{Z}/Nmathbb{Z})/pm 1} Spfleft( mathbb{Z}(zeta_N)((q^{1/N})) right)$.

Facebook Live Explicit Content for Art Exhibition

My partner’s child is going to be running a Facebook Live stream for an end-of-the-year show for their college art department. Some of the content in the show is sexually explicit – there are drawings and paintings of various aspects of sexual activity, including illustrations of penetrative intercourse, photographs of naked forms, etc. There will also be a Q&A after the show which will also contain discussions of “explicit” subject matter.

The Facebook Community Standards mention the types of sexual content they find objectionable and/or will remove, but do not at all make it clear about how to mark specific material as “educational.”

Is there a specific protocol (disclaimers, settings, breaking up the stream into 18+ and SFW section, etc?) to ensure that either the content itself or the department’s Facebook page don’t get flagged/reported?

Thank you!

dg.differential geometry – Explicit calculation algorithm for distance function

I study differential geometry. Although there is a lot of study on the local theory, a global description lacks some explicit explanations. I mean, the study of surfaces describes curves, tangent plane, covariant derivative, the geodesic equation. However, I was not able to find a systematic manner to calculate the distance function rather than solve the geodesic equation with begin and end points and integrate its length with extreme as control points. Can you see any manner to find the distance function on an algebraic connected and complete (geodesic may pass everywhere) surface of type $z = f(x, y)$?

Explicit vs transparent proxy – Information Security Stack Exchange

As far as I have understood it:

  • An explicit proxy challenges the user/application within his session.
  • NGFW (transparent proxy) and SSO/identity-based solutions are just letting everything pass that is using the current IP address of the user.

I agree the latter is flexible with regard to roaming users (VPN, Wifi and whatnot) but IMHO similar to machine/IP-based authentication, i.e. a step back from actually challenging the individual application for access. (Note: if you use a captive portal, non-interactive apps will have a hard time authenticating.)

The transparent proxy would let all traffic from your machine go directly to the URL filter, including the potential malware. Whereas in the explicit scenario the malware would need to obtain the user’s credentials, parse the PAC file or somehow else determine the location of the proxy to use etc. Might be considered security through obscurity, still more hurdles can’t hurt…
Additionally, a transparent proxy would require recursive DNS access to the Internet, meaning DNS security would need to be implemented. Whereas when using an explicit proxy, the client needs no DNS access at all, the proxy itself would perform a DNS request once the URL filtering/categorization or any other mechanism has allowed access.

Somehow I fail to see where transparent approach would provide more security than explicit.
The more modern approach (NGFW/transparent) seem to rely more and more on blacklisting and heuristics, while we learned that actual security only comes from denying everything that we do not know i.e. whitelisting. I agree that this is difficult in today’s Internet though.
So which one is more secure, transparent or explicit, or does it only depend on the individual definition of security/risk?

nt.number theory – Explicit construction of division algebras of degree 3 over Q

In his book Introduction to arithmetic groups, Dave Witte Morris implicitly gives a construction of central division algebras of degree 3 over $mathbb{Q}$ in Proposition 6.7.4. More precisely, let $L/mathbb{Q}$ be a cubic Galois extension and $sigma$ a generator of its Galois group.If $p in mathbb{Z}^+$ and $p neq tsigma(t)sigma^2(t)$ for all $t in L$, then
$$ D=left{ begin{pmatrix}
x & y & z\
psigma(z) & sigma(x) & sigma(y)\
psigma^2(y) & psigma^2(z) & sigma^2(x)
end{pmatrix} :(x,y,z)in L^3 right}
$$

is a division algebra.

On page 145, just before Proposition 6.8.8, Morris claims that it is knows that every division algebra of degree 3 arises in this manner. This should follow from the fact that every central division algebra of degree 3 is cyclic. I could not find this explicit construction in my references (e.g. Pierce – Associative Algebras, though maybe I missed something) and I would like to know if there is a reference or a quick way to see that this exhausts all central division algebras of degree 3 over $mathbb{Q}$.

discrete mathematics – Construct an explicit bijection from [x], the equivalence class of x, to Q.

Define a relation $ sim $ on the set of real numbers as follows: For $x, y in mathbb{R}$ : $x sim y$ if $𝑥 −𝑦∈mathbb{Q}$

I have proven that this is an equivalence class:

Reflexitivity

$x – x = 0$

$0 in mathbb{Q}$

so $xsim x$

Symmetry

$x – y = a$, where $a in mathbb{Q}$

$y – x = -a$, where $-a in mathbb{Q}$

so $x sim y, y sim x$

Transitive

$x – y = a$, $a in mathbb{Q}$,

$y – z = b$, $z in mathbb{R}$, $b in mathbb{Q}$

$x – y + y – z = a + b$

$= x – z = a + b$

$= x – z = c$, $c in mathbb{Q}$

Thus, $x sim y$ is an equivalence relation. However, I am having trouble finding the equivalence classes and most importantly, how to construct an explicit bijection from the equivalence class, $[x]$, to $mathbb{Q}$.

ct.category theory – Explicit definition of a pullback of $(2,1)$-categories

In the 1-category of 2-categories, with objects being categories enriched over Cat, and morphisms being 2-functors, is there an explicit way to describe a pullback of two functors $G:Eto D$ and $F:Cto D$? In particular I am interested in the case of $(2,1)$-categories, i.e. categories enriched over groupoids, but the general case

I have been looking for resources, but I could only find literature on higher limits in higher categories, instead of ‘ordinary’ limits in categories with higher categories as objects.

Ideally, the answer would contain of an explicit description of the 0, 1 and 2 cells of the category. It seems reasonable that the 0-cells are just the objects of the ordinary underlying category, and perhaps the 1 and 2 cells given by the pullback of the relevant categories in $mathcal{Cat}$ or $mathcal{Gpd}$, but I can also imagine that there is a larger choice of 1-cells; those which are connected by a 2-cell in $D$. These two possible definitions of $Ctimes_D E$ are, most probably, not equivalent.

Any advice, ideas, or references are more than welcome.

gr.group theory – Explicit abelianization functor for groups

Assume that I have a short exact sequence of finitely presented groups $$1 longrightarrow K longrightarrow H longrightarrow G longrightarrow 1,$$ where $G$ is finite (but I do not know whether this is relevant for what follows). Applying abelianization, we get an exact sequence $$K_{mathrm{ab}} longrightarrow H_{mathrm{ab}} to G_{mathrm{ab}} longrightarrow 0.$$

I would like to have a “quantitative” measure of the lack of left-exactness for the above sequence. For instance, I would like to know if it is possible to find an explicit sequence ${L_i}$ of groups giving a long exact sequence of the form $$ldots longrightarrow L_3 longrightarrow L_2 longrightarrow L_1 longrightarrow K_{mathrm{ab}} longrightarrow H_{mathrm{ab}} to G_{mathrm{ab}} longrightarrow 0.$$

By “explicit” I mean (for instance) that it is, at least in principle, possible to find presentations for the $L_i$ once one has presentations for $K$, $H$, $G$.

Since the category of groups is not abelian (or even additive) we can not perform the usual construction of the derived functors for the abelianization functor. I am aware that some more refined constructions have been presented (cotangent complex, André-Quillen homology, etc, see for instance the comments to this MSE question) but they look very technical and perhaps overkill in the simple case I have in mind.

I am not an expert, but it seems to me that for the case of groups there should be some more down-to-earth construction, related to the usual group (co)homology, but I looked in some standard textbooks and I did not find any. So, let me ask the following

Question. Is it possible to construct groups $L_i$ as above in some explicit and in principle computable way? If so, what are some
references?

domain name system – Android Volley JsonObjectRequest times out, but explicit IP:port request works

I have a REST API in go on my server VM, with gorilla/mux:

router := mux.NewRouter().StrictSlash(true)
log.Fatal(http.ListenAndServe(":49186", router))

I have an exposed IP and the domain forwarding set up on Godaddy as:
(http://), (198.123.123.123:49186), Forward Type: Permanent (301), Settings: Forward Only

In my Kotlin I have:

val url = "http://example.com/getJSONitem/item%20name"
val q = Volley.newRequestQueue(this)
val request = JsonObjectRequest(Request.Method.Get, url, null, Response.Listener<JSONObject> { response ->
    textView.text = response.getString("Name")
    },
    Response.ErrorListener { error ->
        println(error)
})
q.add(request)

When I run this it hangs for a second and I get

I/System.out: com.android.volley.TimeoutError

BUT if I put the IP and port explicitly…

url = "http://198.123.123.123:49186/getJSONitem/item%20name"

…it works and I get my JSON object… At least that tells me that my server is listening. So it is either an issue with my Volley code or I have my DNS forwarding misconfigured, in which case this belongs on SE and I apologize. Thank you for reading.

P.S. I have also tried putting the port after the domain as follows, and receive the same timeout error.

url = "http://example.com:49186/getJSONitem/item%20name"