SQL Server – Is there a way to query the Size and Space available settings for each Database, including system databases on an instance?

I am trying to query the default database properties as seen in the picture below for all databases on an instance, including system databases.

I’ve used various scripts such as the following, but I’m not sure exactly where those two fields are stored.

      database_name = DB_NAME(database_id)
    , log_size_mb = CAST(SUM(CASE WHEN type_desc = 'LOG' THEN size END) * 8. / 1024 AS DECIMAL(8,2))
    , row_size_mb = CAST(SUM(CASE WHEN type_desc = 'ROWS' THEN size END) * 8. / 1024 AS DECIMAL(8,2))
    , total_size_mb = CAST(SUM(size) * 8. / 1024 AS DECIMAL(8,2))
FROM sys.master_files WITH(NOWAIT)
WHERE database_id = DB_ID() -- for current db 
GROUP BY database_id

enter image description here

Thank you for your help.

rt.representation theory – Under what conditions representations of reductive Lie group in Banach space and in its Garding space have the same length?

Let $G$ be a real reductive Lie group (e.g. $G=GL(n,mathbb{R})$). Let $rho$ be a continuous representation of $G$ in a Banach space $V$. Let $V^inftysubset V$ be the subspace of smooth vectors equipped with the Garding topology. Let $rho^infty$ be the natural representation of $G$ in $V^infty$.

Under what precise technical conditions the representations $rho$ and $rho^infty$ have the same length?

A reference would be very helpful.

general topology – Prove that $forall xin E, exists zin Fsubset E, inf_{yin F}|x-y|=|x-z|$, where $F$ is compact and $E$ is a normed vector space.

Let $(E,|cdot|)$ be a normed vector space and let $Fsubseteq E$ be a non-empty compact subset. We define the distance from $x$ to $F$ as
dcolon &Etimesmathcal{P}(E)setminus emptysettomathbb{R}\
&(x,F)mapstoinf_{yin F}|x-y|.

Prove that
$$forall xin E,exists zin F,; d(x,F)=|x-z|.$$

Definition of compact set: A set $F$ is said to be compact if and only if every sequence of elements of $F$ has a subsequence converging on $F$.

From this, it follow that $d(x,F)=0iff xin F$ (this was a previous problem and it only relies on $F$ being closed).
I don’t really know where to start with the proof, any good hints (or incomplete work) would be great.

files – GDM3 is taking most of my SSD space Ubuntu 20.04

I turned on my pc today and got a message that my memory was almost full. Come to find out it is something with GDM3 in /var/log directory. I am not sure what this is and if I can somehow clean it up or remove files to get some memory back?

uname -a
Linux name 5.8.0-50-generic #56~20.04.1-Ubuntu SMP Mon Apr 12 21:46:35 UTC 2021 x86_64 x86_64 x86_64 GNU/Linux

enter image description here

at.algebraic topology – Proofs that the classifying space of Connes’ cycle category $Lambda$ is $mathbb C mathbb P^infty$

Connes showed in Cohomologie cyclique et foncteurs $Ext^n$ (1983) that the classifying space of his cycle category $Lambda$ is $mathbb C mathbb P^infty = B(S^1) = K(mathbb Z,2)$.

Connes’ proof is not quite as conceptual as one might like. He shows that $|Lambda|$ is simply-connected, and then computes its cohomology via an explicit resolution of the constant functor $Lambda to Ab$, $(n) mapsto mathbb Z$ via a double complex which is cooked-up by gluing together the usual way of resolving cyclic groups and the usual way of resolving simplicial objects, with a few modifications. He’s able to get the ring structure on the cohomology using an endomorphism of this double complex. The result follows because $mathbb Cmathbb P^infty$ is characterized among simply-connected spaces by its cohomology ring.

Surely in the last nearly 40 years new proofs that $|Lambda| simeq mathbb C mathbb P^infty$ have been found.

Question 1: What are some alternate proofs that $|Lambda| simeq mathbb C mathbb P^infty$?

Question 2: (somewhat vague) In particular, is there a proof which somehow uses an explicit $S^1$ action on some category or simplicial set related to $Lambda$?

partitioning – “0 bytes free” after partition resizing, but there is much unused space

Using GParted, I resized two Ext4 partitions on my external HDD, making them smaller.

But now the OS says there is no space available on both of them.

This is the outptut of df -H for those partitions.

Filesystem    Size    Used    Avail    Use%
/dev/sdb5     782G    744G        0    100%
/dev/sdb6     635G    616G        0    100%

The very output above makes no sense at all, how can it say there are 0 bytes available when almost 40 gigabytes are not used e.g. in /dev/sdb5 ?

I tried executing both the following commands automatically via GParted
(I show them just for /dev/sdb5 but I tried them on both partitions).

e2fsck -f -y -v -C 0 /dev/sdb5
resize2fs -p /dev/sdb5

but nothing changed.

I can modify existing files in those two partitions, and in particular I can delete files , but even if I delete a big file the OS still says there are 0 bytes available.

What is going on here?

PS: It still says 0 bytes available even if I am root.

untagged – Samsung S21 where is enable auto space after period setting

I cannot find the setting for auto space after period when I’m typing a message on my Samsung S21.

Anybody knows where it is?

*hope it’s not part of the autocorrect, because I have that turned off, because it’s making a mess in my language

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How many subspaces has the space M2(R)2×2 real matrices?

How many subspaces has the space M2(R)2×2 real matrices?