The examples I have are: $S$ is equal to the spectrum of a global field; or a proper non-empty open subscheme of the spectrum of the ring of integers $\mathcal O_{K}$ of a number field $K$ (proper means $S$ is not all of ${\rm Spec}\,\mathcal O_{K}$); or $S$ is a non-empty open subscheme of a smooth, complete and irreducible curve over a finite field. Can anyone supply other examples, please?

Let $(M_n,\tau_n)_{n\geq 1}$ be a non-commutative probability space. Suppose each $M_n$ is hyperfinite. Is it true that $\overline{\otimes}_{n=1}^\infty M_n$ is again hyperfinite? How to prove or disprove this?

my questions relates to the following talk of Tsuji:

https://www.youtube.com/watch?v=2brDj26phP0

At around 10:30 of the video, Tsuji is interrupted by a man stating that his construction does not work if $k$ is not algebraically closed. I do not understand this remark at all.

For those of you that do not want to have a look at the video here is a quick summary of the part in question:

The assumptions are as follows: $K$ a complete valuation field of mixed characteristic (0,p) with ring of integers $\mathcal{O}_K$ uniformizer $\pi$ and residue field $k$ (*algebraically closed*). $A$ an semistable algebra i.e. $$ A \underset{étale}{\leftarrow} \mathcal{O}_K[T_1,\cdots, T_c, \cdots, T_d]/(T_1 T_2 \cdots T_c-\pi)$$ let $\mathcal{K}=Frac(A)$ and $\overline{\mathcal{K}}$ an algebraical closure with $\overline{K} \subset \overline{\mathcal{K}}$. For every finite extension $\mathcal{K} \subset \mathcal{L} \subset \overline{\mathcal{K}}$ denote by $A_{\mathcal{L}}$ the integral closure of $A$ in $\mathcal{L}$. Define the set $$ S:=\big\{\mathcal{L} \, \vert \, A_{\mathcal{L}}[\frac{1}{T_1\cdots T_d}] /A[\frac{1}{T_1\cdots T_d}] \, \text{étale}\big\}$$ such that $Spec(A)$ is connected and $Spec(A/\pi A)\neq 0$ and $Spec(A/\sum_{i \in I} T_iA)$ is irreducible or empty for all $I \subset\{1,\cdots,d\}$. Set $\mathcal{K}^{ur}=\cup_{\mathcal{L}\in S} \mathcal{L}$.
He then states that $$ Gal(\mathcal{K}^{ur}/\mathcal{K}\overline{K})= \pi_1(A[1/T_1\cdots T_d] \otimes _{\mathcal{O}_K} \overline{K},Spec(\overline{K})).$$ Why is this not true if $k$ is not algebraically closed? Any help would be very much appreciated.

In 1974, W. B. Johnson and E. Odell observed that there are subspaces $X$ of $L_{1}$ with the Schur property. In 1980, J. Bourgain and H. P. Rosenthal constructed a subspace $X$ of $L_{1}$ such that $X$ has the Schur property, but $X$ is not isomorphic to a subspace of $l_{1}$. Hence, I have the first question as follows:

Question 1. Is every Banach space with the Schur property isomorphic to a subspace of $L_{1}(\mu)$ for some measure $\mu$?

Moreover, W. B. Johnson and E. Odell gave natural non-trivial conditions that a subspace of $L_{p}$ embeds into $l_{p}$ for $1<p<\infty, p\neq 2$. But

Question 2. Are there conditions that a subspace of $L_{1}$ embeds into $l_{1}$ ?

Thank you!

Let $R$ be a regular, local $\mathbb{Q}$-algebra with a regular system of parameters $x_1, \dotsc, x_n$, and let $$f \colon \mathbb{Q}[X_1, \dotsc, X_n]_{(X_1, \dotsc, X_n)} \rightarrow R$$ be the map given by $X_i \mapsto x_i$. Then $f$ is flat (for instance, by Bourbaki, cf. EGA III, 0.10.2.2).

Is $f$ a regular morphism, that is, are the (geometric) fibers of $f$ regular? Certainly, the closed fiber is regular, but how about the others?

The answer should be yes, and I would appreciate an argument for this. In the case when $R$ is excellent, the positive answer seems to be a special case of EGA IV, 7.9.8, but a less contrived argument (and one that would not use an additional excellence assumption) would be greatly appreciated.

Why Is The Category Of Sets A Grothendieck Topos? How can just object and relation lead to a Grothendieck Topos which {!} 'suggests the possibility of synthesis of algebraic geometry, topology, and arithmetic' [ReS, The vision].

I am reading a paper by Diening and Kreuzer where they consider the convergence of finite element approximations for $p$-Laplace equation when using a certain algorithm.

In the paper, they assume that the domain $\Omega$ is polyhedral. It is not clear to me why exactly this is assumed. **What is the significance of the polyhedral domain assumption for results of this type?**

My first guess is that this is related to discretization of the boundary $\partial \Omega$. If it is not piecewise linear, then different meshes will cover different domains, which hopefully approximate $\Omega$ in some way.

- Is there a standard technique or result that allows extending the types of results Diening and Kreuzer get to non-polyhedral domains? I might want to use different meshes on the same domain.
- In particular, can the error caused by discretizing a domain with non-flat boundary with triangular elements be controlled in some way?
- Or is there an actual problem with convergence of FEM in non-polyhedral (say, Lipschitz) domains?

I would like to know if there exists a Liouville theorem for solutions $u : \mathbb{R}^n \to \mathbb{R}$ of uniformly elliptic equations of the kind $$ D_i \left( a_{ij} D_j u \right) + b_i D_i u = 0. $$ I assume the coefficients $a_{ij},b_i \in C^{\infty}(\mathbb R^n) \cap L^{\infty}(\mathbb{R}^n)$.

Any hint/reference would be highly appreciated!

Does anybody has a copy of the following paper: Alfred van der Poorten, Some facts that should be better known, especially about rational functions; Number Theory and Applications”, Richard A. Mollin (ed.), Kluwer Academic Publishers, Dordrecht, 1989?

I have following expression

$\frac{P(x, t+dt)}{(dt)^2} $

And i need to get rid of division by (dt)², any ideas how to approximate function?

What is meant by a classical solution of a fractional laplacian in $ (-\Delta)^su= f(u)\text{ in } \mathbb{R}^N$ with no condition at infinity. If one can show that u is a weak solution of the above solution, how do one show it is classical.

I noticed and found only first three cases:

We can write $1$ as difference of two composites that have one prime factor $$3^2-2^3=1$$

and as difference of two composites that have two prime factors $$3\cdot 5 - 7\cdot 2 = 1$$

and as difference of two composites that have three prime factors $$2^2 \cdot 3^2 \cdot 43-7 \cdot 13 \cdot 17=1$$

I believe that this holds for every $k \in \mathbb N$, that is, that for every $k \in \mathbb N$ there exist composites $a_k$ and $b_k$ that have exactly $k$ prime factors and are such that we have $a_k-b_k=1$.

Is my belief true? Is this known? What is known about all of this and similar problems? Can someone find solutions for some larger $k$´s?

Smallest examples found so far:

for $k=1$ $$3^2-2^3=1$$ for $k=2$ $$3 \cdot 5 - 7 \cdot 2=1$$ for $k=3$ $$3\cdot7\cdot11-2\cdot5\cdot23=1$$ for $k=4$ $$5 \cdot 7 \cdot 11 \cdot 19 - 2 \cdot 3 \cdot 23 \cdot 53=1$$

Let $G=(V,E)$ be a finite, simple, undirected graph. For $v\in V$ we set $N_0(v) = N(v) = $ $\{w\in V: \{v,w\} \in E\}$ and for $k\in \omega$ let $$N_{k+1}(v) = N_k(v) \cup \bigcup\big\{N(z): z\in N_k(v)\big\}.$$
We define the *neighborhood fingerprint* of $G$ as $F_G: V\times \omega \to \omega$ defined by $(v,k) \mapsto |N_k(v)|.$

It is clear that if two graphs $G,H$ are isomorphic, then there is a bijection $\varphi:V(G)\to V(H)$ such that for all $v\in V(H)$ and all $k\in \omega$ we have $F_G(v,k) = F_H(\varphi(v), k)$.

Does the converse hold? More precisely, if there is a bijection $\varphi$ as described above, are the graphs $G,H$ isomorphic?

I have been thinking for a while how to determine all the orthogonal coordinate systems in linear spaces of an arbitrary dimension $n$.

The motivation for such a task comes from physics: I am studying a separability of Hamilton-Jacobi equations in "some orthogonal systems". The merit is to examine how does an orthogonality relate to the separation in general.

Let $n$ be a natural and suppose $\mathbf{q}=(q^1, q^2, ..., q^n)$ are coordinates describing $\mathbb{R}^n$. The coordinates are orthogonal if and only if the coordinate surfaces all meet at right angles. This means that while setting for each $i \in \{1,2, ..., n\}$ $q^i = const$ the hyperplanes obtained all meet at right angles.

For example in $\mathbb{R}^n$ the cartesian coordinates are orthogonal. Surely, we can describe $\mathbb{R}^n$ using generalized spherical coordinates etc.

I tried to figure out how to find "all" the orthogonal coordinates in $\mathbb{R}^n$. We can obtain some of them trivially from those we already know using rotations, shrinking and in general any orthogonal linear transformations. But how to find the coordinates which couldn't be obtained like that?

I had a look at the transformations from a cartesian set of coordinates in $\mathbb{R}^n$ $\mathbf{x}=(x^1, x^2, ..., x^n)$ which preserves the orthogonality. Let $f$ be a transformation of the coordinate system $\mathbf{x}$ to a new one, say $\mathbf{y}=(y^1, y^2, ..., y^n)$. We would like f to be regular and defined at domain of f $\mathrm{dom}(f)=\mathbb{R}^n \setminus M$, where the set $M$ has Lebesgue measure zero. Locally, the orthogonality condition requires that the linear approximation of vectors $y^i$ where $i \in \{1,2, ..., n\}$ created an orthogonal set. The linear approximation of $y^i$ at an arbitrary fixed point $\mathbf{x_0}$ is it's derivative at $\mathbf{x_0}$ $(y^i)'(\mathbf{x_0})$ . (Here the derivative of a mapping r from a Banach space E to a Banach space F at a point t is understood as a linear mapping r'(t) which is tangent to r.) This means: for each par of $i, j \in \{1,2, ..., n\}$ and each point $\mathbf{x} \in \mathrm{dom}(f)$ the dot product of $(y^i)'(\mathbf{x}) \cdot (y^j)'(\mathbf{x})$ equals zero for $i \neq j$ and equals a positive number otherwise.

In the notation of Jacobian matrices: $J_f$ of $f$: $(J_f)^TJ_f=A$, where $A$ is a diagonal matrix. This should be true in a more general case when f is considered to be a mapping of an arbitrary orthonormal coordinate system to a new orthogonal system.

This method gives a system of nonlinear PDE's which is quite unsolvable. (Or at least it seems to me.)

Do you have any other idea how to find orthogonal systems describing linear spaces? Do you know about an interesting book covering the topic of orthogonal systems deeper then giving a list of the known ones in $\mathbb{R}^2$ and $\mathbb{R}^3$. I would like to get some insight in separation of differential equations also, so I appreciate tips for articles covering this subject too.

Thanks a lot, Filip.

PS: This is my first question here. I am not sure whether I posted it right.

There are some articles about rational points of varieties over number field (with many open problems). Finite field is simpler than number field, and I am interested in arithmetic of algebraic varieties over finite field beyond Weil conjectures and Abel varieties which are well studied, but I am unable to find comprehensive materials for self-learning (This article includes some geometrical propositions while not much for arithmetic problems). As there are good theories for abelian varieties (such as structure theorem for rational points or Deligne's description for category of ordinary abelian varieties over a finite field), I will avoid them.

In particular, I am interested in those kinds of varieties over finite fields and information about the existence and number of their rational points:

- Hyperelliptic curves or more general high-genus curves
- Del Pezzo surface
- K3 surface
- Linear algebraic groups
- Good moduli spaces (Grassmannians, Modular curves, moduli of supersingular objects)

………

(If there are other examples with interesting results please tell me about them)

Besides counting rational point, I am also interested in counting rational curves or special varieties (e.g number of supersingular elliptic curves)...

So my question is: is there any good material for systematic study? Are there some interesting results in this area?

Some results I have seen before:

- Finite fields are $C_1$, moreover every smooth projective rationally chain connected variety, for example every Fano variety, over a finite field $k$ has a $k$-rational point.

(Finite field is not PAC as there exists geometrically connected varieties without rational points over $k$, an example may be $x^q-x+y^q-y=1$ over $k=\Bbb F_q$ or some $K3$ surfaces)

Results improving Weil bounds for curves and construction of examples with many rational points

For smooth cubic surface $S$ over $\Bbb F_q$, $\#S(\Bbb F_q)=1+aq+q^2$ for some $a$ in $\{-2,-1,0,1,2,3,4,5,7\}$

Results about particular $K3$ surface (like counting points for Fermat quartic or existence of rational points of those can be lift to char zero, and the unirationality for supersingular K3 surfaces with char$>3$)

Counting formula for some linear algebraic groups, such as $\mathrm {O} (2n+1,q)|=2q^{n}\prod _{i=0}^{n-1}(q^{2n}-q^{2i})$ (Here is a relevant course, I don't know the result for general linear algebraic groups)

Counting formula for number of isomorphism classes of some objects (like subspaces, supersingular elliptic curves……)

(If there are other interesting results please tell me about them.)

Apologies if this question is a bit simplistic/vague for MO:

I'm looking for an all-purpose definition in the literature of when a sufficiently generic filter "canonically codes" a generic real. Examples of what I mean come from everyone's favorite notions of forcing to add certain reals "on purpose":

In Cohen forcing, the generic real is obtained by taking the union of the generic real.

In random forcing, the generic real is obtained by taking the intersection of the generic filter.

In Mathias or Laver forcing, the generic real is obtained by taking the union of stems in the generic filter.

In Sacks forcing, the generic real is obtained by taking the intersection of (branch spaces of trees in) the generic filter... etc, etc.

What I am **not** looking for is anything having to do with the "other" reals that are added "by accident" through such forcing.

I can, of course, come up with a bespoke definition that covers the above cases, but I was hoping for something general and known.

**Edit:** To be more precise, I am looking for something that is an *injective function* from (sufficiently generic) filters to reals. A preliminary definition might be something like: "there is a definable (from the poset $\mathbb{P}$) injection from generic filters for $\mathbb{P}$ to reals". In the case of Cohen forcing, the function would be the map which just takes the union of the generic filter.

Assume you have an array of length $n$ filled with the numbers $1,2,...,n$. (Actually, it only matters that all numbers are different.) This corresponds to a Dirac delta distribution for the number prevalency. Now, synchroneously, all slots of the array choose a uniform random other slot (this includes choosing itself or duplicates) and take this as new value. You get a Poisson distribution (with $\mu=1$) for the prevalency. Rinse and repeat (this is the Voter protocol) until you converged to consensus (another Dirac delta).

Since there is a mass of literature on this protocol, I drowned in references (mainly dealing with runtime and such) but was not able to find one giving an analytic formula for the distributions that come after the Poisson. Can you help me out? (Or solve the one after the Poisson in your head? :-)

EDIT: To make clearer how the process runs, here is an example for $n=4$. $ABCD[abcd]$ means that slot $1$, having opinion $A$, "asks" slot $a$ for his opinion, and so on. Votes drawn from the hat :-)

$1234 [2133]\rightarrow 2133[4322] \rightarrow 3311[1114] \rightarrow 3331[2112] \rightarrow 3333$, stop.

Let $T:D(T)\rightarrow H$ be a (densely-defined) self adjoint operator. If $0\not \in \sigma(T)$, then $T$ is invertible, and we know that if $T^{-1}$ is compact, then $T$ has discrete spectrum. I came across a couple of papers which seem to be using the following statement:

Suppose that the Rayleigh quotient, $\frac{\left<v,v\right>}{\left<v,Tv\right>}$, generates a compact operator. Then, this is equivalent to the fact that $T^{-1}$ is compact, which means that $T$ has discrete spectrum.

There are a few things I don't understand in that statement. Firstly, in what sense does the quotient $\frac{\left<v,v\right>}{\left<v,Tv\right>}$ (which is the inverse of the Rayleigh quotient, according to Wikipedia) generate an operator? Secondly, how is the compactness of the operator it generates equivalent to the compactness of $T^{-1}$?

The question is regarding this paper, Theorem 5.3.

(I know this is perhaps only tangentially related to mathematics research, but I'm hoping it is worthy of consideration as a community wiki question.)

Today, I was reminded of the existence of this paper: *Terminating Decimals in the Cantor Set*.

https://www.fq.math.ca/Scanned/28-2/wall.pdf

It is a concise paper (3.5 pages) that employs nothing too sophisticated, just some modular arithmetic and careful casing. I thought it would be a wonderful to spend a few class meetings with undergraduate math majors reading this paper for understanding. We could practice reading the dense writing and filling in the induction proofs that are "left for the reader". It would also be an occasion to remind the students of concepts learned in important courses in their major (e.g. modular arithmetic in an algebra course, or the topology of the real number line in an analysis course). And by the end of it, I hope the students would gain some satisfaction from realizing that they read and understood the entirety of a scholarly article in mathematics.

I am looking for other papers that could fill this role. Ideally, I think it would be great to construct a semester-long course based on reading, say, 5-10 papers like this and spending a week or two on each one. **Each paper should...**

**be relatively short, say less than 6 pages**(but obviously the density of writing plays a big factor). Ideally, we should be able to digest it the whole thing within a couple of weeks of careful reading.**be interesting and not too esoteric or specific.**For instance, the result in the Cantor Set example above is surprising and interesting, and although I may have to remind students of what the Cantor Set is, I won't need to presume deep background knowledge in a specific topic. Or, Niven's proof that pi is irrational would be a good example. However, many of the "Proofs without words" in MAA journals, while perhaps fun, would not really give students practice with reading scholarly writing and the results may be too specific to inspire their interest.**tie together, or remind students of, some knowledge from core courses in the undergraduate curriculum,**like algebra, analysis, calculus, combinatorics, or probability.**be published in a book or journal**. (The Cantor Set example above was a footnote in the book*Chaos and Fractals: New Frontiers of Science*, according to this reddit post.)

*Meta comment:* The only similar question I could find on this site is *MO.88946* ("Readings for an honors liberal art math course"), but it focuses on books for a popular audience and not scholarly mathematics per se. Also, MathEducators tends to avoid "community wiki"-style questions, so I decided to post here.)

Set $M=\{(\cos(\theta),\sin(\theta),z):\theta\in[0,2\pi],z\in[0,1]\}$. A bending of $M$ is a smooth map $\Gamma:M\times [0,1]\rightarrow \mathbb{R}^3$ such that

1) $\Gamma[M\times\{t\}]$ is a submanifold with boundary of $\mathbb{R}^3$ (I call it $M_t$)

2) For every $m\in M$, we have $\Gamma(m,0)=m$

3) For every $t\in[0,1]$, the map $m\rightarrow \Gamma(m,t)$ is an isometry between $M$ and $M_t$

Question: Must any bending of $M$ leave the bases planar?

The only smooth bendings I can think of so far are those which change the shaper of the circular bases to elipses

Thank you