lore – What locations do these Official Forgotten Realms inspired Magic the Gathering cards reference?

There is a Magic: the Gathering set, which has cards inspired by The Forgotten Realms, but which locations (if any) are on these land cards?

a glowing sun-like sphere in an underground grotto. Blue round crystals rise out of the surrounding sand.

Venturing beneath the desert sands, you’ve discovered an alien power pulsing with inconceivable energy.

a tornado of water with a battered ship near the top

Alarmed by the news you brought, the storm giant king of the Maelstrom has called his kin to council

a black must covered ruin underground

You expected to meet hostile drow in this ancient ruin… But they fled long ago. What darkness could have driven them out?

ruins carved out of a mountain, with clouds obscuring the ground

As you make camp near the ancient ruin, you hear the sound of drums echoing in halls no dwarf has lived in for centuries.

houses and spiralling walk ways built amongst the tops of trees

Before the elf queen will aide you, you must cure the strange rot afflicting the roots of the ancient trees


Which locations are on these land cards?

reference request – Embedding theorems for fractional Sobolev spaces $W^{s,p}(Gamma)$ where $Gamma$ is closed piecewise $C^1$ curve in $Bbb R^2$

I am interested in embedding theorems for the fractional Sobolev space $W^{s,p}(Gamma)$ where $Gamma$ is closed piecewise $C^1$ curve in $Bbb R^2$ such as the boundary of a triangle or rectangle. What are basic results for this? Also, is there some $p$ for which $W_{s,p}(Gamma)subset C(Gamma)?$

Let’s use $Gamma=partial Omega$ where $Omega$ is a square as our basic example.
This is supposedly a smooth manifold as it is homemorphic to a circle, is that right? So we can define $W^{s,p}(Gamma)$ as Brezis mentions in the comments of Chapter 9 in his functional analysis books.

elliptic pde – Reference request on Pucci extremal operators

While reading (1), I encountered with the concept “Pucci extremal operator” which is defined by:
$$M_Lambda^-(N):=left(sumtext{positive eigenvalues of }Nright)+Lambdaleft(sumtext{negative eigenvalues of }Nright),text{ and}$$
$$M_Lambda^+(N):=-M_Lambda^-(-N),$$
where $Nintext{Sym}_{ntimes n}$ and $Lambdageq 1$.

Then the author claims that the problem
$$M_Lambda^-(D^2u)leq 0leq M_Lambda^+(D^2u)$$
in viscosity sense includes all $C^2$ solutions to uniformly elliptic equations of the form $tr(A(x)D^2u)=0$ where $Ileq A(x)leqLambda I$. Since there is no citation on this, it seems this is well-known result in the field, but I am new to this and I want to know about its motivation, history, etc. I did some article search but I could not find a good reference book or paper that introduces the concept of Pucci extremal operator (of course this may be due to my lacking of searching skill). I will appreciate if anyone would explain the concept or give me some good reference on it. Thank you in advance.

(1) Mooney, Connor, A proof of the Krylov-Safonov theorem without localization, Commun. Partial Differ. Equations 44, No. 8, 681-690 (2019). ZBL1426.35124.

reference request – Cohomology of commutative monoid acting on module

I have a some naive questions about how to define the cohomology of a commutative monoid.

One way to express the cohomology of a group $G$ with coefficients in a module $A$ is as $text{Ext}^i_{mathbb{Z}(G)}(mathbb{Z},A)$. If we have a commutative monoid $M$ (you can also assume it’s cancellative if you want), we can follow the exact same recipe over the monoid algebra $mathbb{Z}(M)$; I think this gives derived functors in the category of $M$-modules of “taking $M$-invariants,” which is what I’d expect and want. I was wondering if this theory was developed anywhere, in terms of what analogues of standard group cohomology constructions/theorems exist, vanishing theorems, etc.

Incidentally I’ve found by googling there are various other monoid cohomologies, but constructed in ways that seem arcane to me, e.g. Leech or Gillet symmetric cohomology. I guess you could also take the cohomology of the classifying space of the monoid as a category. Do any of them restrict to/agree with the construction above when restricted to some nice class of commutative monoids? What are the relations between them?

Confused in choosing constant reference or constant value to avoid post modification on variables in C++

I am relatively new to C++. When learning OpenCV I need to name a window. The title should not be changed afterwards, so I should make it constant. There are two options:

const std::string title = "testing...";

or

const std::string& title = "testing...";

Both do not allow us to modify the declared or defined variables.

Could you tell me which one should I choose?

#include <iostream>

#include <opencv2/opencv.hpp>

int main(int, char **)
{
    cv::Mat image = cv::imread("../test.png");
    const std::string title = "testing...";
    // const std::string& title = "testing...";
    cv::imshow(title, image);
    cv::waitKey();
    return 0;
}

When do reference cycles occur other than doubly linked lists?

I’m trying to find examples of reference cycles and look for common patterns. The only one I’ve managed to find so far are doubly linked lists again and again.. any other common patterns or really any times you’ve encountered them in real life?

reference request – A bijective proof for the odd companion of Shapiro’s Catalan convolution

Shapiro’s Catalan convolution is the following formula (where $C_n$ is the $n$th Catalan number):
$$
sum_{k=0}^{n}{C_{2k}C_{2(n-k)}}=4^nC_n.
$$

In other words, letting $C(z)=sum_{n=0}^{infty}{C_nz^n}$, and $C_mathrm{even}(z^2)=dfrac{C(z)+C(-z)}{2}$, we have $(C_mathrm{even}(z))^2=C(4z)$. See, for example, Curious Catalan convolutions and Proofs of some combinatorial identities for further discussion of this and links to bijective proofs of this identity.

I am wondering a similar bijective proof exists in the literature for the following odd-index companion to the Shapiro’s Catalan convolution. Namely, let $C_mathrm{odd}(z^2)=dfrac{C(z)-C(-z)}{2z}$. Then
$$
(zC(z))circ(zC(4z))=zC_mathrm{odd}(z),
$$

where $circ$ denotes composition of functions. Comparing the coefficients of both sides and cancelling a few factors yields
$$
sum_{k=0}^{n}{binom{2n-2k}{n-k}binom{n+k}{k}4^k}=binom{4n+1}{2n},
$$

or, equivalently,
$$
sum_{k=0}^{n}{binom{2k}{k}binom{2n-k}{n-k}4^{n-k}}=binom{4n+1}{2n}.
$$

reference request – Hopficity of Baumslag-Solitar groups

I am struggling to find the exact source (with proofs) of the following ”well-known” statement:

the Baumslag-Solitar group $BS(m,n)=langle a,t mid ta^m t^{-1}=a^nrangle$ is Hopfian if and only if $pi(m)=pi(n)$, i.e., the non-zero integers $m$ and $n$ have the same prime divisors.

This fact is commonly attributed to Collins and Levin (Collins, Donald J.; Levin, Frank, Automorphisms and Hopficity of certain Baumslag-Solitar groups, Arch. Math. 40, 385-400 (1983). ZBL0498.20021.). However, that paper seems to primarily deal with the case when either $m$ divides $n$ or $n$ divides $m$. So, my question is

Where can I find a published proof of the following theorem?

Theorem: if $m,n in mathbb{Z}$, $pi(m)=pi(n)$ and neither $m$ divides $n$ nor $n$ divides $m$ then $BS(m,n)$ is Hopfian.

In particular,

Who was the first to prove that $BS(12,18)$ is Hopfian?

Collins and Levin seem to attribute this theorem to Meskin (Meskin, Stephen, Nonresidually finite one-relator groups, Trans. Am. Math. Soc. 164, 105-114 (1972). ZBL0245.20028.).
But I do not actually see any proofs of Hopficity in Meskin’s paper, covering the desired non-residually finite case.

Autocomplete is not working for entity reference

Autocomplete is not working for new data in entity reference field but if I am changing the wiget type to list in that case all data is coming.

reference request – Oka-Grauert principle, up to the boundary

Let $Zsubset mathbb{C}^n$ a domain of holomorphy with smooth boundary $partial Z$ and closure $bar Z$. There is a natural notion of holomorphic vector bundle over $bar Z$, given in terms of transition functions $(U_{alphabeta}:h_{alphabeta}rightarrow GL(m,mathbb{C}))$ which are holomorphic in $U_{alphabeta}backslash partial Z$ and smooth up to the boundary.

Suppose $Erightarrow bar Z$ is a holomorphic vector bundle, which is trivial as continuous vector bundle. The Oka-Grauert principle then implies that $Evert_{Z}$ is holomorphically trivial, but does not say anything about whether there exists a global holomorphic frame that extends smoothly up to $partial Z$. As Donaldson remarks in this paper from 1992, ‘the result is almost certainly true’, but like him I have ‘unfortunately not been able to find such a result in the literature’ (he gives an ad hoc proof in $n=2$).

In this paper by Leiterer (1990) there is actually almost the right thing: Theorem 10.1 gives the result for holomorphic vector bundles, which are continuous up to $partial Z$. That means in the situation above we obtain a global frame that extends continuously to $partial Z$, but not smoothly.

Questions.

  • Has somebody since made the effort to write up some sort of Oka-Grauert principle for holomorphic vector bundles that are smooth up to the boundary?
  • Is Leiterer’s paper (or his original article in German, referenced in there) the best reference for the continuous case?
  • Are there nice elementary approaches for special cases of $Z$‘s? (E.g. Donaldson gives an argument for $n=2$ and for $Z$ homeomorphic to a ball.)