# Recent MathOverflow Questions

### Decreasing ratio of winning probabilities

Math Overflow Recent Questions - Wed, 03/28/2018 - 08:30

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 $n=3$.)

### Definition of asymptotic frequency

Math Overflow Recent Questions - Wed, 03/28/2018 - 08:13

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.

### Solovay's theorem about decomposition of stationary sets

Math Overflow Recent Questions - Wed, 03/28/2018 - 08:04

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 $W$?

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### Coordinates of the Weyl vector of $E_8$ (and the 135 classes of $W(E_8)/W(D_8)$)

Math Overflow Recent Questions - Wed, 03/28/2018 - 07:59

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.)

### $G$-invariant bilinear maps

Math Overflow Recent Questions - Wed, 03/28/2018 - 07:49

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.

### Minimal ideals of the ring of continuous functions

Math Overflow Recent Questions - Wed, 03/28/2018 - 07:25

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 $C (X)$?

### Approximating $1_I$, $I\subset \lbrack 0,1\rbrack$, by trigonometric polynomials

Math Overflow Recent Questions - Wed, 03/28/2018 - 05:48

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.)

### What price should i take for lottery tickets to cover costs with 99% probability [on hold]

Math Overflow Recent Questions - Wed, 03/28/2018 - 05:35

### Do you know important theorems that remain unknown?

Math Overflow Recent Questions - Tue, 03/27/2018 - 08:26

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:

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

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

3. Provide references to the original paper.

4. Provide references to more recent and related work.

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