I have a doubt, Roth's discrepancy theorem [1] says that there is a subset of arithmetic progressions $A \in [n]$ where any function $f:N\rightarrow \left \{ -1,1 \right \} $ implies

$ \left | \sum _{a\in A}f(a) \right |\geq cn^{1/4} $

(Does not specify if the arithmetic progression is homogeneous or not homogeneous)

In 2015 Terence Tao [2] proved for any homogeneous arithmetic progression we have

$Sup_{n,d\in N}\left | \sum^{n} _{j=1}f(jd) \right |=\infty$

It occurred to me to ask if the discrepancy is infinite for non-homogeneous arithmetic progressions or is it an open question

[1]Roth, K. F. Remark concerning integer sequences. Acta Arith. 9 1964 257–260.

[2] Tao, Terence (2016). "The Erdős discrepancy problem". Discrete Analysis: 1–29. arXiv:1509.05363

Let $X$ be a proper scheme over $k$ where the characteristic of $k$ is $p>0$.

Consider the etale sheaves $\mathbb{G}_m$ over $X$ and consider the $p$-th power map from $\mathbb{G}_m \to \mathbb{G}_m$ ( $x \in \mathbb{G}_m(U)$ goes to $x^p \in \mathbb{G}_m(U)$. When is this map of map of etale sheaves? It is true $X$ is smooth over $k$, is it if $X$ is just proper?

I am trying to identify a solution for a multiplicative problem that involves taking two large numbers and multiplying them together.

Has the 3-tag system investigated by Emil Post $(0->00, 1->1101)$ been solved? Is there a decision algorithm to determine which starting strings terminate, which end up in a cycle, and which (if any) grow without bound?

Also, what cycle structures are there? Setting a='00' and b='1101', the only cycles I know of begin with $ab, b^2 a^2$, combinations of these, and $a^2 b^3 (a^3 b^3)^n$. Are there any more?

Define: $$H^*_{\alpha}=\{x | \ \forall y \in TC(\{x\}) (|y| \leq |\alpha|)\}$$

Where: $TC(x)= \{y| \ \forall t (transitive(t) \wedge x \subseteq t \to y \in t)\},$

$transitive(t) \iff \forall r,s (r \in s \in t \to r \in t)$, $``| x|"$ signify the cardinality of $x$ which is the smallest Von Neumann ordinal bijective to $x$.

In English: $H^*_{\alpha}$ is the set of all sets that are hereditarily subnumuerous $(\leq)$ to $\alpha$.

Define recursively: $$\daleth_0=\omega_0$$

$$\daleth _{i+1}= |H^*_{\daleth_i}|$$

$$\ \ \daleth_j = \bigcup_{i<j} (\daleth_i) , \text { if }\not \exists k(k+1=j)$$ .

Questions:

1) is $\daleth_1 = \aleph_1$ satisfied in $L$?

2) is $\daleth_i = \aleph_i$ satisfied in $L$?

where $L$ is Gödel's constructible universe.

In paper: **Three manifolds with positive ricci curvature**

In theorme 9.4, p.281, What does ``Suppose the null eigenvalue of $M_{ij}$ occurs in the top position" mean? Does it mean the null eignvector of $M_{ij}$ is $e_1$ (first vector in the standard ordered basis)? Also diagonalizing with respect to some basis gives us $R_{ij}=\begin{bmatrix} \lambda&&\\ &\mu&\\ &&\nu\\ \end{bmatrix}$, why does diagonalizing with respect to the same basis gives us $g_{ij}=I_3$?

This question is related to rough path theory. Consider we have obtained signature obtained from a set discrete data points postulating linear from one data point to another. Such signature are used in some machine learning applications. A link to such application is : https://sutdbrain.files.wordpress.com/2017/07/nengli_lim_sutd_technical_talk-first-half.pdf

I am looking for possibility of extracting information from the signature to construct the signal. This would similar to filtering as in digital signal processing https://en.wikipedia.org/wiki/Digital_signal_processing.

Are anyone aware of such application of signature? More importantly is there any model (in the literature) of the original discrete data points based on the linear (or other) combination of signature elements.

I was wondering... Is every symplectic connection $\nabla$ on some symplectic manifold $(M,ω)$ the Levi-Civita connection of some metric $g$ on $M$? What about the local statement?

**Definition.** A finite group $G$ is called *multifactorizable* if for any positive integer numbers $a_1,\dots,a_n$ with $a_1\cdots a_n=|G|$ there are subsets $A_1,\dots,A_n\subset G$ such that $A_1\cdots A_n=G$ and $|A_i|=a_i$ for all $i\le n$.

In this case we shall write that the group $G$ is *$a_1{\times}\cdots{\times}a_n$-factorizable.*

It can be shown that each finite Abelian group is multifactorizable.

**Problem 1.** Is each finite (simple) group multifactorizable?

As was observed by Geoff Robinson in his answer to this question, each finite nilpotent group is multifactorizable.

**Problem 2.** Is each finite solvable group multifactorizable?

**Added in Edit.** It turns out that the alternating (solvable) group $A_4$ is not multifactorizable, more precisely, $A_4$ is not $2{\times}3{\times}2$-factorizable.

Now it remains to find an example of a finite simple group which is not multifactorizable.

**Problem 3.** Is the alternating group $A_5$ multifactorizable? In particular, is $A_5$ $2{\times}15{\times}2$-factorizable?

Suppose $B$ is the ball of center $a$ and radius $R>0$ in $ \mathbb{R}^{n} $ $n>1$. Suppose also that $u$ is subharmonic and real analytic on a neighborhood of the closure of $B$. We know that according to Riesz decomposition theorem there is a harmonic function $h$ on $B$ and a potential $p$ such that \begin{equation} u(x)=h(x)-p(x) \end{equation} Here, the potential is given by $p(x)=\int_{B}G(x,\zeta) \Delta u(\zeta)d\zeta$, where $G(x,\zeta)=\log|x-\zeta|$ for $n=2$, and $G(x,\zeta)=|x-\zeta|^{2-n}$ for $n>2$.

My problem is the following: since $u$ is real analytic, by the Riesz decomposition theorem, so is $p$. But if I try to expand $p(x)$, about, say $x=a$, it is impossible that the expansion of $G(.,\zeta)$ about $a$ be convergent (because $\zeta$ varies in the whole $B$ and among other things, it can even be $\infty$ at $a$) and the laplacian $\Delta u(\zeta)$ under the integral sign has no effect on the expansion of $p(x)$ (which is in terms of $x$). So: where does the power series expansion of $p$ come form? From that of $h$ only??? Is there someone who can explain? Thanks.

One more motivated by recent questions of Zhi-Wei Sun.

Let $S_n$ be the group of permutations of $\{1,2,\ldots, n\}$.

Is it true that, for every $n \ge 8$, there is at least one even permutation $\pi \in S_n$ and at least one odd permutation $\tau \in S_n$ with $$\sum_{k=1}^n \frac{1}{k \, \pi(k)} = \sum_{k=1}^n \frac{1}{k \, \tau(k)} = 1?$$

One case for each $n$ is not hard; I made it a math.stackexchange question that was successfully answered in 12 minutes. Hopefully the other case is more interesting.

Clarification: As per the MSE question referenced above and Zhi-Wei's comment, the $n$-cycle $(1,2, \dots, n) \in S_n$ satisfies the sum condition. An $n$-cycle is an odd permutation for even $n$ and an even permutation for odd $n$.

Here are the remaining parts of the conjecture.

a. For $n$ even and $n \ge 8$, there is an even $\pi \in S_n$ satisfying $\sum_{k=1}^n \frac{1}{k \, \pi(k)} = 1$.

b. For $n$ odd and $n \ge 9$, there is an odd $\tau \in S_n$ satisfying $\sum_{k=1}^n \frac{1}{k \, \tau(k)} = 1$.

Here are the numbers of even and odd permutations satisfying the sum condition for small $n$.

\begin{array}{c|rr} n\backslash \text{sgn} & +1 & -1 \\ \hline 1 & 1 & 0 \\ 2 & 0 & 1 \\ 3 & 2 & 0 \\ 4 & 0 & 2 \\ 5 & 4 & 0 \\ 6 & 0 & 2 \\ 7 & 4 & 0\\ 8 & 6 & 4\\ 9 & 12 & 24\\ 10 & 90 & 88 \end{array}

One of the first ``interesting'' permutations is the even permutation (in cycle notation) $(1,2,5,8,7,6)(3,4) \in S_8$ which gives \begin{align*} \frac{1}{1\cdot2} + \frac{1}{2\cdot5}+ \frac{1}{3\cdot4}+ \frac{1}{4\cdot3}+ \frac{1}{5\cdot8}+ \frac{1}{6\cdot1}+ \frac{1}{7\cdot6}+ \frac{1}{8\cdot7}\\ = \frac{1}{2} + \frac{1}{10}+ \frac{1}{12}+ \frac{1}{12}+ \frac{1}{40}+ \frac{1}{6}+ \frac{1}{42}+ \frac{1}{56}=1. \end{align*} Not coincidentally, $n=8$ is the smallest value for which there are non-$n$-cycle permutations that satisfy the sum condition.