Can we find a holomorphic function $g$ on an open disk such that $\sum_{i \in \mathbb{N}} |(f-g)(a_i)|^2 < +\infty$?

Let $ f : \mathbb{C} \to \mathbb{C}$ be a continuous function with $ f(0)=0$ .

Let $ \{a_i\}_{i\in \mathbb{N}}$ be a set of scalars in $ \mathbb{C}$ such that $ $ \exists C > 0 : \forall i\in \mathbb{N} : |a_i| \leq C $ $ Can we always find a holomorphic function $ g$ on $ B(0,C+1)$ (the open disk of radius $ C+1$ ) such that $ g(0)=0$ and $ \sum_{i \in \mathbb{N}} |(f-g)(a_i)|^2 < +\infty$ ?

How to fix “fastboot FAILED (command write failed (No such file or directory))” error?

I was trying to flash TWRP recovery image in my MI 8 SE on my windows 10 laptop. I can detect my device in fastboot mode, but when i try to flash the image i get this error

target didn't report max-download-size sending 'recovery' (25668 KB)... FAILED (command write failed (No such file or directory)) finished. total time: 0.008s 

My phone then exits from the fastboot mode and i get a black screen with a tiny text at the top left corner of the screen “Press any button to exit” or something like that. When i try to flash the same recovery imaget on my windows 7 pc it flashes without any problem. Not sure what the problem is i’ve tried installing drivers for the phone and trying different usb ports but nothing help. I can detect the phone when it’s turned with no problem.

Is there a prime $q$ smaller than a given prime $p>5$ such that the inverse of $q$ modulo $p$ is an integer square?

Let $ p$ be a prime. For each $ k=1,\ldots,p-1$ there is a unique $ \bar k\in\{1,\ldots,p-1\}$ with $ k\bar k\equiv1\pmod p$ , and we call $ \bar k$ the inverse of $ k$ modulo $ p$ . In 2014 I investigated the set $ $ \{\bar q:\ q\ \text{is a prime smaller than}\ p\}$ $ and found that it contains an integer square if $ 5<p<2\times 10^8$ . (See and This led me to formulate the following conjecture.

Conjecture. For any prime $ p>5$ , there is a prime $ q<p$ such that the inverse $ \bar q$ of $ q$ modulo $ p$ is an integer square.

For example, the inverse of $ 13$ modulo $ 23$ is $ 4^2<23$ , the inverse of $ 5$ modulo $ 61$ is $ 7^2<61$ , and the inverse of $ 11$ modulo the prime $ 509$ is $ 18^2<509$ .

QUESTION. What tools in number theory are helpful to prove the above conjecture?

“no such file or directory” when mounting, built using the golang:alpine Docker image

I created a fork of the’s docker-transmission image, adding support for Google Cloud Storage.

I used the Ernest (chiaen)’s docker-gcsfuse project to build gcsfuse, namely, extracting parts of his Dockerfile and added to my own one. gcsfuse is built using the golang:alpine image.

The image builts successfully (including gcsfuse; the Dockerfile instructs to just copy gcsfuse to /usr/local/bin). However, gcsfuse refuses to mount, and the logs outputs Mount: stat /donwloads: no such file or directory, enev if the directory /download actually exists, and the right permissions were set already (set at /etc/cont-init.d/20-config). I even tried to run from the shell, but still fails.

Is there a missing package or parameter in order to get gcsfuse working in my (Alpine) Docker image?

If you want to reproduce, you may bould your own local copy of the image following the instructions at in my repo (you need to upload your json key to the VM) (amitie10g/transmission is also available at Docker Hub).

Logs are available here.

Thanks in advance.

Proper variety such that no complement of closed irreducible subset is affine

Let $ X$ be an integral scheme. Assume there exists a proper morphism $ X\rightarrow \mathrm{Spec}\:\mathbb{C}$ of relative dimension $ d>0$ . Can we find a non-empty affine open subscheme with an irreducible complement?

If $ d=1$ , then the complement of a closed point is necessarily affine so we win.

Is there a point $H$ such that $\frac{AH \cdot DM}{HD^2} = \frac{BH \cdot EN}{HE^2} = \frac{CH \cdot FP}{HF^2}$?

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$ H$ is a point in non-isoceles triangle $ \triangle ABC$ . The intersections of $ AH$ and $ BC$ , $ BH$ and $ CA$ , $ CH$ and $ AB$ are respectively $ D$ , $ E$ , $ F$ . $ AD$ , $ BE$ and $ CF$ cuts $ (A, B, C)$ respectively at $ M$ , $ N$ and $ P$ . Is there a point $ H$ such that the following equality is correct? $ $ \large \frac{AH \cdot DM}{HD^2} = \frac{BH \cdot EN}{HE^2} = \frac{CH \cdot FP}{HF^2}$ $

  • If there is not, prove why.

  • If there is, illustrate how to put down point $ H$ .

Of course, point $ H$ should be one of the triangle centres identified in the Encyclopedia of Triangle Centers. But I don’t which one it is.