In the game of Number Master, the player controls a number starting with $1$ and hits the other numbers one by one on the road. If the player hits a number smaller or equal to the current controlling number, then the controlling number will become the sum of itself and the number he/she just hit....

Suppose that $(z_n) \subset \mathbb C$ is a sequence (repetitions allowed) such that $$ F(z) = \prod_n \left ( 1-\frac{z}{z_n} \right ) $$ defines an entire function of exponential type, that is, $|F(z)| \leq ae^{b|z|}$ for $a,b > 0$. Further, let $(p_n) \subseteq \mathbb C$ be a bounded sequence...

Let $G$ be a group. A subgroup $H$ of $G$ is said to be fully invariant if for every endomorphism $\phi $ of $G$, we have $\phi(H) \subseteq H$. For a finitely generated residually finite group $G$, let $N$ be the intersection of all finite index fully invariant subgroup of $G$. It seems in all t...

Reading through Jean-Eric Pin's "Mathematical Foundations of Automata Theory". Love this book. However, I am confused by the following section, and am hoping for some clarity and more examples if possible. I need this for my research. Syntactic order definition I am not sure exactly how it is use...

Assume that $n$ is a sufficiently large number. Is there a Hadamaed matrix $H_{4n \times 4n}=(h_{ij})$ with the last row and the last cloumn $J$ (thet is, for every $k$, $h_{k,4n}=1$ and $h_{4n, k}=1$) such that the resulting numbers in other rows ($\overline{h_{k,1} ... h_{k, 4n}})_2$ and colum...

I am looking for any information about the following property for a compact Hausdorff space $K$: For any sequence $\left(U_{n}\right)$ of nonempty open sets (not necessarily distinct) there is a disjoint sequence $\left(V_{n}\right)$ of nonempty open sets such that $V_{n}\subset U_{n}$, for every...

I am trying to understand fundamental characters of the tame inertia group $I_{p,t}$ of $\mathrm{Gal}(\overline{\mathbb Q}_p/\mathbb Q_p)$ and I have some questions. I'm following Gabor Wiese notes on Galois representations, page 65 (https://math.uni.lu/~wiese/notes/GalRep.pdf). I start explainin...