Fix reals $0<x_1<\cdots<x_n$ and consider $n\ge 3$ player numbered $1,2,\ldots,n$. If player $i$ fights against $j$ then $i$ wins with probability $\frac{x_i}{x_i+x_j}$, and there are no ties.

A player $i_1$ is extracted at random. Then, a second different player $i_2$ is extracted at random and fight against each other. Hence, we extract another player $i_3\neq i_1,i_2$. The winner of the latter round fights against $i_3$.The fights continues until all players have been extracted (so $n-1$ fights in total).

This game has been introduced here, where it is only shown how to calculate the winning probabilities $\pi(i)$ recursively.

**Question.** Is it true that, given $k\ge 3$, the ratio of winning probabilities $\frac{\pi(1)}{\pi(2)}$ is a decreasing function of $x_k$?

(The answer is easily seen to be affirmative for $n=3$.)

Consider a sequence $(x_n)_{n\in\mathbb{N}}$ of 0s and 1s. The asymptotic frequency of 1s in $x$ is usually defined as:

$$f=\lim_{n\to+\infty} \frac{1}{n}{\sum_{i=0}^{n-1} x_i}$$

when this limit exists. But sometimes the limit does not exist yet it sounds "reasonable" to say that the asymptotic frequency still exists. **Is there a general way to define asymptotic frequency so that it applies to a much broader class of sequences?**.

For simplicity, we can focus only on defining $f=0$: how to define there are "infinitely more" 0s than 1s in the sequence. Maybe this could involve probability theory, maybe not. Of course the definition can't be invariant under any permutation since there is nothing to say in terms of cardinal except there are as many 1s as 0s: countably many.

The tricky example I have in mind is successive groups of length $2^k$. In each group the values are equal: all 0s or all 1s. There are infinitely many groups with value 1 but these become more and more sparse for example with frequency $1/k$. It looks like it:

1-00-1111-0000000- ....... groups of length $2^k$: most are 0, with sometimes a group of 1s.

The limit does not exist as the average is greater than 1/2 infinitely many times. Yet, this example can be thought as the realization of a random sequence $(X_n)_{n\in\mathbb{N}}$ that tends to 0 in probability. Hence it could make sense saying the asymptotic frequency is 0.

If $k$ is an uncountable regular cardinal, we want to take a partition of a stationary subset $W\subseteq E_\omega^k=\{\alpha\in k: cf(\alpha)=\omega\}$ in $k$ many stationary sets.
The proof in *Jech, Set Theory*, Lemma 8.8 pag. 94, concludes whit the existence of $k$ disjoint stationary sets $T_\eta$, coming from the Fodor's lemma. But why their union should be all $W$?

Consider the root system of $E_8$, written in its standard "even" coordinate system: i.e., it is the set of all $240$ vectors in $\mathbb{R}^8$ which whose coordinates are either all integers or all integers-plus-a-half, have an even sum, and whose sum of squares is $2$.

For any choice of a set of positive roots, the half-sum of the positive roots is known as the corresponding "Weyl vector". For example, if we take the positive roots to be those whose last nonzero coordinate is positive, the Weyl vector is $(0,1,2,3,4,5,6,23)$. Since the Weyl vector lies in the interior of the Weyl chamber, the Weyl group acts freely on its orbit, and there are $\#W(E_8) = 696\,729\,600$ Weyl vectors, exactly one for each choice of positive roots.

My question is essentially whether we can describe this set of $696\,729\,600$ Weyl vectors in a simple way through its coordinates.

One obvious reduction is that the Weyl group of $D_8$, which is a subgroup (of order $8!\times 2^7 = 5\,160\,960$) of that of $E_8$ acts by permuting the $8$ coordinates in any way and changing the sign of an even number of them. So all that need to be described are the $135 = 696\,729\,600 / 5\,160\,960$ orbits of Weyl vectors modulo this action. It's not difficult to list them explicitly, e.g., a simple computation gives me:

(0, 1, 2, 3, 4, 5, 6, 23) (0, 1, 2, 3, 4, 13, 14, 15) (0, 1, 2, 3, 8, 9, 10, 19) (0, 1, 2, 5, 6, 7, 8, 21) (0, 1, 2, 5, 6, 11, 12, 17) (0, 1, 2, 7, 8, 9, 14, 15) (0, 1, 2, 9, 10, 11, 12, 13) (0, 1, 3, 4, 5, 6, 7, 22) (0, 1, 3, 4, 5, 12, 13, 16) (0, 1, 3, 4, 7, 8, 9, 20) (0, 1, 3, 4, 7, 10, 11, 18) (0, 1, 3, 6, 7, 10, 13, 16) (0, 1, 3, 8, 9, 10, 13, 14) (0, 1, 4, 5, 6, 9, 10, 19) (0, 1, 4, 5, 6, 11, 14, 15) (0, 1, 4, 5, 8, 9, 12, 17) (0, 1, 4, 7, 8, 11, 12, 15) (0, 1, 5, 6, 7, 8, 11, 18) (0, 1, 5, 6, 7, 12, 13, 14) (0, 1, 5, 6, 9, 10, 11, 16) (0, 1, 6, 7, 8, 9, 10, 17) (0, 2, 3, 4, 7, 9, 10, 19) (0, 2, 3, 5, 6, 11, 13, 16) (0, 2, 3, 5, 7, 10, 12, 17) (0, 2, 3, 7, 8, 10, 13, 15) (0, 2, 4, 5, 7, 9, 11, 18) (0, 2, 4, 6, 7, 11, 13, 15) (0, 2, 4, 6, 8, 10, 12, 16) (0, 2, 5, 6, 8, 9, 11, 17) (0, 3, 4, 5, 7, 11, 12, 16) (0, 3, 4, 6, 7, 10, 11, 17) (1/2, 3/2, 5/2, 7/2, 9/2, 11/2, 13/2, -45/2) (1/2, 3/2, 5/2, 7/2, 9/2, 25/2, 27/2, 31/2) (1/2, 3/2, 5/2, 7/2, 15/2, 17/2, 19/2, -39/2) (1/2, 3/2, 5/2, 7/2, 15/2, 19/2, 21/2, 37/2) (1/2, 3/2, 5/2, 9/2, 11/2, 13/2, 15/2, 43/2) (1/2, 3/2, 5/2, 9/2, 11/2, 23/2, 25/2, -33/2) (1/2, 3/2, 5/2, 9/2, 13/2, 15/2, 17/2, 41/2) (1/2, 3/2, 5/2, 9/2, 13/2, 21/2, 23/2, -35/2) (1/2, 3/2, 5/2, 11/2, 13/2, 21/2, 25/2, 33/2) (1/2, 3/2, 5/2, 13/2, 15/2, 19/2, 27/2, -31/2) (1/2, 3/2, 5/2, 15/2, 17/2, 19/2, 27/2, 29/2) (1/2, 3/2, 5/2, 17/2, 19/2, 21/2, 25/2, -27/2) (1/2, 3/2, 7/2, 9/2, 11/2, 23/2, 27/2, 31/2) (1/2, 3/2, 7/2, 9/2, 13/2, 17/2, 19/2, 39/2) (1/2, 3/2, 7/2, 9/2, 13/2, 19/2, 21/2, -37/2) (1/2, 3/2, 7/2, 9/2, 15/2, 19/2, 23/2, 35/2) (1/2, 3/2, 7/2, 11/2, 13/2, 21/2, 27/2, -31/2) (1/2, 3/2, 7/2, 11/2, 15/2, 19/2, 25/2, -33/2) (1/2, 3/2, 7/2, 13/2, 15/2, 21/2, 25/2, 31/2) (1/2, 3/2, 7/2, 15/2, 17/2, 21/2, 25/2, -29/2) (1/2, 3/2, 9/2, 11/2, 13/2, 17/2, 21/2, 37/2) (1/2, 3/2, 9/2, 11/2, 13/2, 23/2, 27/2, 29/2) (1/2, 3/2, 9/2, 11/2, 15/2, 17/2, 23/2, -35/2) (1/2, 3/2, 9/2, 11/2, 17/2, 19/2, 23/2, 33/2) (1/2, 3/2, 9/2, 13/2, 15/2, 23/2, 25/2, -29/2) (1/2, 3/2, 9/2, 13/2, 17/2, 21/2, 23/2, -31/2) (1/2, 3/2, 11/2, 13/2, 15/2, 17/2, 21/2, 35/2) (1/2, 3/2, 11/2, 13/2, 17/2, 19/2, 21/2, -33/2) (1/2, 5/2, 7/2, 9/2, 13/2, 21/2, 25/2, -33/2) (1/2, 5/2, 7/2, 9/2, 15/2, 17/2, 21/2, 37/2) (1/2, 5/2, 7/2, 11/2, 13/2, 19/2, 23/2, -35/2) (1/2, 5/2, 7/2, 11/2, 13/2, 23/2, 25/2, 31/2) (1/2, 5/2, 7/2, 11/2, 15/2, 21/2, 23/2, 33/2) (1/2, 5/2, 7/2, 13/2, 15/2, 21/2, 27/2, -29/2) (1/2, 5/2, 7/2, 13/2, 17/2, 19/2, 25/2, -31/2) (1/2, 5/2, 9/2, 11/2, 15/2, 19/2, 21/2, 35/2) (1/2, 5/2, 9/2, 11/2, 15/2, 21/2, 25/2, -31/2) (1/2, 5/2, 9/2, 13/2, 15/2, 19/2, 23/2, -33/2) (1/2, 7/2, 9/2, 11/2, 13/2, 21/2, 23/2, -33/2) (1, 2, 3, 4, 5, 12, 14, 15) (1, 2, 3, 4, 6, 7, 8, 21) (1, 2, 3, 4, 6, 11, 12, -17) (1, 2, 3, 4, 8, 9, 11, 18) (1, 2, 3, 5, 6, 8, 9, 20) (1, 2, 3, 5, 6, 10, 11, -18) (1, 2, 3, 6, 7, 10, 14, -15) (1, 2, 3, 6, 7, 11, 12, 16) (1, 2, 3, 6, 8, 9, 13, -16) (1, 2, 3, 8, 9, 11, 12, -14) (1, 2, 4, 5, 6, 12, 13, 15) (1, 2, 4, 5, 7, 8, 10, 19) (1, 2, 4, 5, 7, 10, 13, -16) (1, 2, 4, 5, 8, 10, 11, 17) (1, 2, 4, 6, 7, 9, 12, -17) (1, 2, 4, 7, 8, 11, 13, -14) (1, 2, 4, 7, 9, 10, 12, -15) (1, 2, 5, 6, 7, 9, 10, 18) (1, 2, 5, 6, 8, 11, 12, -15) (1, 2, 5, 7, 8, 10, 11, -16) (1, 3, 4, 5, 6, 10, 12, -17) (1, 3, 4, 5, 8, 9, 10, 18) (1, 3, 4, 6, 8, 10, 13, -15) (1, 3, 4, 7, 8, 9, 12, -16) (1, 3, 5, 6, 7, 10, 12, -16) (3/2, 5/2, 7/2, 9/2, 11/2, 15/2, 17/2, 41/2) (3/2, 5/2, 7/2, 9/2, 11/2, 21/2, 23/2, -35/2) (3/2, 5/2, 7/2, 9/2, 11/2, 25/2, 27/2, 29/2) (3/2, 5/2, 7/2, 9/2, 17/2, 19/2, 21/2, 35/2) (3/2, 5/2, 7/2, 11/2, 13/2, 15/2, 19/2, 39/2) (3/2, 5/2, 7/2, 11/2, 15/2, 19/2, 27/2, -31/2) (3/2, 5/2, 7/2, 13/2, 15/2, 17/2, 25/2, -33/2) (3/2, 5/2, 7/2, 15/2, 17/2, 23/2, 25/2, -27/2) (3/2, 5/2, 7/2, 15/2, 19/2, 21/2, 23/2, -29/2) (3/2, 5/2, 9/2, 11/2, 13/2, 19/2, 25/2, -33/2) (3/2, 5/2, 9/2, 11/2, 15/2, 17/2, 19/2, 37/2) (3/2, 5/2, 9/2, 13/2, 17/2, 21/2, 25/2, -29/2) (3/2, 5/2, 9/2, 15/2, 17/2, 19/2, 23/2, -31/2) (3/2, 5/2, 11/2, 13/2, 15/2, 21/2, 23/2, -31/2) (3/2, 7/2, 9/2, 11/2, 17/2, 19/2, 27/2, -29/2) (3/2, 7/2, 9/2, 13/2, 15/2, 19/2, 25/2, -31/2) (2, 3, 4, 5, 6, 7, 9, 20) (2, 3, 4, 5, 8, 9, 14, -15) (2, 3, 4, 6, 7, 8, 9, 19) (2, 3, 4, 6, 7, 9, 13, -16) (2, 3, 4, 7, 9, 11, 12, -14) (2, 3, 4, 8, 9, 10, 11, -15) (2, 3, 5, 6, 9, 10, 13, -14) (2, 3, 5, 7, 8, 10, 12, -15) (2, 4, 5, 6, 8, 9, 13, -15) (5/2, 7/2, 9/2, 11/2, 13/2, 15/2, 17/2, 39/2) (5/2, 7/2, 9/2, 11/2, 15/2, 17/2, 27/2, -31/2) (5/2, 7/2, 9/2, 13/2, 19/2, 21/2, 25/2, -27/2) (5/2, 7/2, 9/2, 15/2, 17/2, 21/2, 23/2, -29/2) (5/2, 7/2, 11/2, 13/2, 17/2, 19/2, 25/2, -29/2) (5/2, 9/2, 11/2, 13/2, 15/2, 17/2, 27/2, -29/2) (3, 4, 5, 6, 7, 8, 14, -15) (3, 4, 5, 6, 10, 11, 12, -13) (3, 4, 5, 7, 9, 10, 12, -14) (3, 4, 6, 7, 8, 9, 13, -14) (7/2, 9/2, 11/2, 13/2, 19/2, 21/2, 23/2, -27/2) (7/2, 9/2, 11/2, 15/2, 17/2, 19/2, 25/2, -27/2) (4, 5, 6, 7, 9, 10, 12, -13) (9/2, 11/2, 13/2, 15/2, 17/2, 21/2, 23/2, -25/2) (5, 6, 7, 8, 9, 10, 11, -12)(Again, any element of this list is defined only up to permutation of the coordinates and an even number of sign changes: here I've sorted the coordinates in absolute value and written the minus sign, if necessary, on the last coordinate, but the representatives in question might not be the best.)

I can see no clear pattern in this list. Maybe I'm looking at it in all the wrong way.

**Question:** How can we describe this set simply?

(A followup question might be whether we can easily multiply two elements of $W(E_8)$ represented as transformations on such Weyl vectors. But the first step is, of course, to recognize them.)

Let $G$ be a finite group and $M$ a finitely generated $\mathbb{Z}G$-module. Then $M \otimes k$ is a representation of $G$ for any field $k$. I am interested in the number of ways we can turn $M \otimes k$ into a (non associative) $kG$-algebra. Equivalently, I would like to compute $\operatorname{Hom}_{kG}(M\otimes M \otimes k, M \otimes k)$.

Suppose that we know that $M\otimes \mathbb{C}$ is irreducible and that $M \otimes M \otimes \mathbb{C}$ contains a unique composition factor isomorphic to $M \otimes \mathbb{C}$. Then we know, by Schur's lemma, that $\operatorname{Hom}_{\mathbb{C}G}(M\otimes M \otimes \mathbb{C}, M \otimes \mathbb{C}) \cong \mathbb{C}$. What can we say about $\operatorname{Hom}_{kG}(M\otimes M \otimes k, M \otimes k)$?

I have the feeling that $\operatorname{Hom}_{kG}(M\otimes M \otimes k, M \otimes k) \cong k$ if $k$ is algebraically closed and $\operatorname{char}(k) \nmid \left| G \right|$. Is this correct and if so, why? In the other cases, I think that a good knowledge of modular representation theory might give a sufficient answer but I am not sure where I should start looking in the literature.

A minimal ideal of a commutative ring $R$ is a nonzero ideal which contains no other nonzero ideal.

Let $X $ be a completely regular topologica space and $C (X) $ the ring of all real velued continuous functions over $X $. Is there any charactrization for minimal ideals of $C (X) $?

Let $I$ be a subinterval of $\lbrack 0,1\rbrack$. How well can one hope to approximate it in the $L_1$-norm (on $\mathbb{R}/\mathbb{Z}$) by a trigonometric polynomial of degree $\leq R$, that is, a function

$$x\mapsto \sum_{r=-R}^R c_r e(r x),$$

where $e(t) = e^{2 \pi i t}$? Are substantially better approximations possible if we allow a linear combination of any $2 R + 1$ different exponentials $e(r x)$ (for not necessarily consecutive values of $r$)?

One can ask the same questions about the sawtooth function x\mapsto {x} instead of $1_{I}$; indeed the two settings feel equivalent. (There are functions related to Beurling-Selberg majorant that may be useful to consider; I've taken a look again at the first chapter of Montgomery's Ten Lectures. It is still unclear to me whether such functions give something approximately optimal with respect to the $L^1$ norm.)

a grocery store organizes a lottery among its customers where you can win a van with 40 randomly selected goods. One can assume that the average value of the goods in the store is 25$ and the standard deviation is 17.

The store sells 1000 lots of which 25 wins. The rest of 975 gives no profit. What price does the shopkeeper need to charge for each lot so that the revenue from lottery sales will exceed the value of the good vans with 99% probability.

the mean is 25 and the standard deviation is 17. The mean for the entire van is 25*50 = 1250. but how do i solve this question? do i use confidence interval?

In the literature, is there any paper or research investigating the invariant subspace problem via differential operators arising from vector fields, namely the derivation operators?

A vector field $X$ on a manifold $M$ defines a derivation on the space of smooth functions. What is a relevant Hilbert space of functions invariant under the $X$-derivation? (A kind of infinite order Sobolov space.) Let's denote this Hilbert space by $H^{\infty}(M)$. Is there any research on the invariant subspaces of the derivations on the infinite order Sobolev space? Of course there is an obvious $1$-dimensional invariant subspace of constant functions, but I guess that this trivial space can be ignored with some Hilbert space techniques. In particular:

Is there a smooth vector field $X$ on the torus $\mathbb{T}^2$ which is tangent to a Kronecker foliation $dx+\theta dy=0$ for some irrational number $\theta$, with an invariant subspace not degenerated to the trivial $1$-dimensional subspace?

Is there an established name for graphs, that can be decomposed into

- a tree with at least three leaf nodes and
- a connected two-regular graph with the tree's leaf nodes as vertices?

examples of those graphs are the edge-graphs of polyhedra with one facet, that is edge-adjacent to all other facets.

Let W be a N-dimensional Brownian motion and P be a N interacting diffusions evolving according SDE $$dx_i=f(x_i,x_j)dt+dB_i$$ Using Girsanov's formula $$\frac{dP}{dW}=exp[\sum_i \int_0^T f dx_i -\frac{1}{2} \int_0^T f^2 dt]$$ if f is bound, the moment condition $$\lim_{n\to \infty} \sup\frac{1}{n} \log E[exp(n\gamma F(Q))]<\infty$$ is set up. When f is not in this case ,how about the moment condition? what type of f can make this condition satisfied?

Working with non-regular topological semigroups, my collegue Oleg Gutik discovered a special space $H$ which we named *Gutik's hedgehog*. It is homeomorphic to the space
$$H:=\{(0,0)\}\cup\{(\tfrac1n,0):n\in\mathbb N\}\cup\{(\tfrac1n,\tfrac1{nm}):n,m\in\mathbb N\},$$ endowed with the topology $\tau$ consisting of sets $U\subset H$ satisfying the following two conditions:

(1) if $(\frac1n,0)\in U$ for some $n\in\mathbb N$, then there exists $m\in\mathbb N$ such that $(\frac1n,\frac1{nk})\in U$ for all $k\ge m$;

(2) if $(0,0)\in U$, then there exists $m\in\mathbb N$ such that $(\frac1n,\frac1{nk})\in U$ for all $n\ge m$ and all $k\in\mathbb N$.

It turns out that Gutik's hedgehog is a test space for regularity in the class of first-countable Hausdorff spaces.

**Theorem.** *A first-countable Hausdorff space is regular if and only if it contains no topological copies of the Gutik hedgehog.*

Because of this fundamental role in testing regularity, I admit that Gutik's hedgehog is known in topology under some different name. I would be grateful for any information in this respect.

**Remark 1.** The Gutik's hedgehog resembles (but is not equal to) the non-regular space of Smirnov, see Example 64 in "Counterexamples in Topology".

For any polynomials of degree $n$ having all its zeros in $|z|\leq K,K\geq 1,$ is it true $\max_{|z|=1}|nP(z)+(a-z)P'(z)|\geq n\min_{|z|=K}|P(z)| $ where $a$ is any complex number with $|a|\geq K?$

We have the following standard theorem : Let $X$ be some set and $g : X^n \rightarrow \mathbb{R}$ be a measurable function such that it satisfies the ``bounded difference property" i.e $\exists$ $\{c_i \geq 0\}_{i=1,..,n}$ s.t for each $i$ we have , $\sup _{x_1,..,x_n, x'_i \in X} \vert g(x_1,.,x_n) - g(x_1,..,x_i',..x_n)\vert \leq c_i $. Then the following is true, $\mathbb{P}[\vert Z - \mathbb{E}[Z] \vert >t] \leq 2e^{-\frac{t^2}{4\sum_{i=1}^nc_i^2}}$ i.e $\vert Z - \mathbb{E}[Z] \vert$ is sub-Gaussian.

- Now how does the above standard theorem imply how the following inequality

$$\mathbb{E}[e^{\lambda (Z - \mathbb{E}[Z])}] \leq e^{\frac{\lambda^2 \sum_{i=1}^nc_i^2 }{2}}$$ $$?$$

I am curious about instances where we can glean nontrivial information about a certain piece of the universe by viewing it as being 'built over' a smaller part of the universe, or alternatively viewing it as a subspace of a larger piece of the universe.

For 'top down' examples we can view any ordered field as a subfield of the Surreals, or any Archimedean ordered field as a subfield of the Reals, or any linearly ordered group as an ordered additive subgroup of $\mathbb{R}^\Gamma$ for $\Gamma$ the collection of Archimedean equivalence classes of the group.

For 'bottom up' examples we can view all abelian groups as $\mathbb{Z}$-modules, or any ordered field as an extension of $\mathbb{Q}$ -- category theory also probably has some nice examples, but it is not my forte.

What are other cool/useful examples of this occurring?

EDIT: In light of the 'uncear' close vote, I will attempt to be more specific.

Suppose we have a class $X$ with some binary operations $\{\odot_i\}_{i<n}\subseteq {}^{X^2}X$ and a collection of subsets $\{U_i\}_{i<\lambda}\subseteq\mathcal{P}(X)$. This may be, for example, a topological group or a valued module.

A-priori we do not necessarily know anything about the structure $S=(X,\{\odot_i\}_{i<n},\{U_i\}_{i<\lambda})$ until we begin writing down some conjectures $\{\phi_i\}_{i<m}$ and seeing if $S\vDash\phi_i$. For nontrivial $\phi_i$, one way to go about this is to begin proving simple or obvious sentences about $S$ and seeing if $\phi_i$ or $\neg\phi_i$ is a consequence of any combination of these simple sentences. If it isn't, we can gradually proceed to prove more and more complex sentences true until we reach a sufficient depth to begin tackling $\phi_i$. A great example here is number theory.

One alternative is to identify $S$ as a substructure of some structure $S'$ that we're very familiar with, then begin using well-known facts about $S'$ and its substructures to tackle $\phi_i$. This is what is meant by a 'top down' approach.

Another alternative route is to identify $S$ as containing a copy of another structure $S''$ we are very familiar with, then view $S$ as an extension of $S''$ in an appropriate sense to begin approaching $\phi_i$ from well known or deep facts abut $S''$. This is what is meant by a 'bottoms up' approach.

Given a partition of an integer *N*, its P-graph is the graph whose vertices are its parts, two of which are joined by an edge if and only if they have a common divisor greater than one (i.e. they are not relatively prime).

For an integer *N*, let k(*N*) be the least number such that a graph on k(*N*)>1 vertices exists that is the P-graph of exactly one partition of *N* into k(*N*) parts. It has been shown, for example, that k(200) = 6. Determining k(*N*) in general seems hopeless, but perhaps satisfactory estimates can be found. In particular, how big is, say, k(1000)? Less than 12, as estimated by this proposer?

Let $\mu(\cdot)$ be the Mobius function, defined on the natural numbers by

$$\displaystyle \mu(n) = \begin{cases} (-1)^{\omega(n)} & \text{if } n \text{ is square-free} \\ 0 & \text{otherwise}.\end{cases}$$

Here $\omega(n)$ is the number of distinct prime divisors of $n$. It is well-known that the Mertens function $M(x)$ defined by

$$\displaystyle M(x) = \sum_{n \leq x} \mu(n)$$

satisfies the asymptotic

$$\displaystyle M(x) = O\left(x \exp(-c \sqrt{\log x}) \right)$$

for some positive number $c$; this is a consequence of the prime number theorem. The Riemann hypothesis is equivalent to the assertion that for any $\varepsilon > 0$ the Mertens function satisfies $M(x) = O_\varepsilon \left(x^{1/2 + \varepsilon}\right)$.

Let $a, q$ be co-prime positive integers, and let $P_{a,q}$ denote the set of primes $p \equiv a \pmod{q}$. Let $N_{a,q}$ be the set of positive integers such that $p | n \Rightarrow p \in P_{a,q}$. Let

$$M_{a,q}(x) = \sum_{\substack{n \leq x \\ n \in N_{a,b}}} \mu(n).$$

Does $M_{a,q}(x)$ satisfy a similar asymptotic as $M(x)$?

Next, let $f \in \mathbb{Z}[x]$. Put $\rho_f(n) = \# \{m \in \mathbb{Z}/n \mathbb{Z} : f(m) \equiv 0 \pmod{n}\}$. Define

$$\displaystyle M_f(x) = \sum_{n \leq x} \mu(n) \rho_f(n).$$

Does $M_f(x)$ satisfy a similar asymptotic upper bound as $M(x)$?

I am interested in the existence of a surface $X$ over $\mathbb{C}$ with the following properties (or a reason for why one cannot exist):

- $X$ is slc (and not-normal)
- There is rational curve $C \subset X$ contained in the double locus (i.e. non-normal locus)
- The cotangent $\Omega^1_Y$ is ample, where $Y \to X$ is the normalization

Note that this last condition implies that $Y$ has no rational or elliptic curves. In particular, since the pre-image of the double locus under the normalization is a 2-to-1 cover branched over the pinch points, Riemann-Hurwitz gives that $C$ contains at least 6 pinch points.

The only examples I know of non-normal surfaces with a rational curve in the double locus containing $> 2$ pinch points have non-positive Kodaira dimension.

Let $\eta$ be a continuous bounded function on $(0, \infty)^{2}$ so that $\eta(0,0)=1$. Let $A$ be a bounded operator on $\ell^{2}(\mathbb{Z}_{\geq 0})=\ell^{2}$ (by bounded operator I will always mean such an object) and let $A_{j,k}$ be its "matrix entries" with respect to the standard basis of $\ell^{2}$.

Suppose now that for any integer $N\geq 1$ and any bounded operator $A$, the infinite matrices $A^{(N)}$ with entries $$A^{(N)}_{j,k}=\eta(\frac{j}{N}, \frac{k}{N})A_{j,k}$$ form a uniformly bounded sequence, meaning that there exists $C>0$, independent of $N$, so that $$\|A^{(N)}\|_{op}\leq C \|A\|_{op},$$ where $\|\cdot\|_{op}$ denotes the usual operator norm.

It is very easy to show that under these assumptions, one can have uniform convergence of $A^{(N)}\to A$ as $N\to \infty$ (i.e. $\|A^{(N)}-A\|_{op}\to 0$) whenever $A$ is a finite matrix (as we have entrywise convergence). By density of the finite matrices and an $\varepsilon/3$-argument it is not so difficult to show that the same happens whenever we restrict to the ideal of compact operators.

What can we say about the convergence of $A^{(N)}\to A$ for an infinite matrix? Of course, the best one can hope for, dropping the compactness of $A$, is convergence in the Strong Operator Topology, but I cannot find an argument that proves my statement. Any help would be appreciated!!!

**Do you know of any very important theorems that remain unknown?** I mean results that could easily make into textbooks or research monographs, but almost
nobody knows about them. If you provide an answer, please:

State only one theorem per answer. When people will vote on your answer they will vote on a particular theorem.

Provide a careful statement and all necessary definitions so that a well educated graduate student working in a related area would understand it.

Provide references to the original paper.

Provide references to more recent and related work.

Just make your answer useful so other people in the mathematical community can use it right away.

Add comments: how you discovered it, why it is important etc.

Please, make sure that your answer is written at least as carefully as mine. I did invest quite a lot of time writing my answers.

As an example I will provide three answers to this question. I discovered these results while searching for papers related to the questions I was working on. I will post two answers right away, but I need slightly more time to write the third one.