## Dimension of a tangent space of the normalization of a domain

Let $$A$$ be a finitely generated (commutative) domain over an algebraically closed field $$k.$$ Let $$B$$ be the normalization of $$A$$. Let $$m$$ be a maximal ideal in $$A$$, and $$m’$$ be a maximal ideal in $$B$$ lying over $$m.$$ Then is it true that $$dim_{k}(m/m^2)\geq dim_{k}(m’/m’^2)$$?

## Is close quarters combat with firearms in a confined space considered a hazardous task?

I know firing indoors has the potential to hit walls, hulls and equipment, but from the rules this is potentially an additional armor value if a player or npc has cover. I also see where some combat tasks are hazardous if untrained. What I can’t find is if firing indoors should be considered hazardous and failures (or fumbles) roll on the failure and potentially mishap tables. (Like a hull breach if it penetrates or a ricochet otherwise.)

Is this covered anywhere? I have both the original Traveller 2300 rule books (Referee’s and Players’ guides) and the 2300AD CDROM canon (which includes the Adventurers’ and Director’s guides).

Traveller 2300 was renamed to 2300AD, some changes were made and quite a few rules were expanded or added. I’m only curious about Traveller 2300 independently if rules that cover this are only in one edition or are different between the editions.

## Regarding orthogonality in Banach space

Let $$(X,\|.\|)$$ be a Banach space over the real line. Let $$x\neq 0$$ and $$y\neq 0$$ be in $$X$$, then $$x$$ is said to be orthogonal to $$y$$ if $$\|x+\lambda y\|\geq\|x\|$$ for every real number $$\lambda$$. It is said that given any $$x$$ and $$y$$ there exists a real number $$\alpha$$ such that $$x$$ is orthogonal to $$\alpha x + y$$. Can anyone tell how?

I can see it diagrammatically, that is we need to show that $$x$$ is orthogonal to a point in the line passing through $$y$$ and in the direction of $$x$$. But I am not able to find this $$\alpha$$ rigourously

## Does a Google Drive shared folder occupy storage space on all accounts?

Related to another question of mine about Dropbox, I’d like to know if a shared folder on Google Drive occupy space even in the accounts of the “receivers” and not only in the “sharer” one.

Example:

“A” creates a folder whose total size is, let’s say, 200 MB; the total storage space of “A” is of course reduced by 200 MB; if “A” shares this folder with “B”, “C” and “D”, even their total storage space is reduced accordingly, or the folder is considered to be “owned” only by “A”, and only his account is affected?

## /tmp/ is somehow out of space and contains an irremovable file

When tab-autocompleting in terminal, I’m getting the error:

-bash: cannot create temp file for here-document: No space left on device 

This would appear to mean that /tmp/ is full, but it’s mounted on my hard disk, which itself has lots of space left.

/tmp only contains one thing: a folder called /tmp/.mount_VCeNjK/
I can’t find out anything about it, because even sudo and su can’t chmod it, read it, umountit, rm it, or stat it. They complain about permissions and say that it’s busy.

What can I do? I’m nothing without my autocomplete…

## Is the metric completion of a Riemannian manifold always a geodesic space?

A length space is a metric space $$X$$, where the distance between two points is the infimum of the lengths of curves joining them. The length of a curve $$c: [0,1] \rightarrow X$$ is the sup of $$d(c(0), c(t_1)) + d(c(t_1), d(t_2)) + \cdots + d(c(t_{N-1}), c(1))$$ over all $$0 < t_1 < t_2\cdots < t_{N-1} < 1$$ and $$N > 0$$.

A geodesic space is a length space, where for each $$x,y \in X$$, there is a curve $$c$$ connecting $$x$$ to $$y$$ whose length is equal to $$d(x,y)$$.

A Riemannian manifold $$M$$ and its metric completion $$\overline{M}$$ are length spaces. If the Riemannian manifold is complete, then it is a geodesic space.

But is $$\overline{M}$$ necessarily a geodesic space? If not, what is a counterexample?

This was motivated by my flawed answer to Minimizing geodesics in incomplete Riemannian manifolds

Also, note that if $$\overline{M}$$ is locally compact, then it is a geodesic space by the usual proof. One example of $$M$$, where $$\overline{M}$$ is not locally compact is the universal cover of the punctured flat plane. However, this is still a geodesic space.

## Norm in a Vector Space

A vector space with norm $$\parallel\cdot\parallel$$ Satisfy for two vectors the following

$$\parallel x+y\parallel=\parallel x\parallel +\parallel y\parallel$$

i need to proof the fallowing statement

$$\parallel \alpha x+\beta y\parallel=\alpha\parallel x\parallel +\beta\parallel y\parallel$$

for all $$\alpha\geq0$$ and $$\beta\geq0$$

The fact that

$$\alpha\parallel x\parallel +\beta\parallel y\parallel\geq\parallel \alpha x + \beta y\parallel$$

it’s clear because the property of the norm, what i can’t proof it’s the fact that

$$\alpha\parallel x\parallel +\beta\parallel y\parallel\leq\parallel \alpha x + \beta y\parallel$$

i will appreciate any help.

## Reference for compact embedding between (weighted) Holder space on $\mathbb{R}^n$

Suppose $$0<\alpha<\beta<1$$, and $$\Omega$$ is a bounded subset of $$\mathbb{R}^n$$. Then the Holder space $$C^{\beta}(\Omega)$$ is compactly embedded into $$C^{\alpha}(\Omega)$$. But if $$\Omega=\mathbb{R}^n$$, then the compact embedding is not true.

However, if we consider the weaker weighted Holder space $$C^{\alpha, -\delta}(\mathbb{R}^n)$$ (for any $$\delta>0$$) instead of $$C^{\alpha}(\mathbb{R}^n)$$. Then is $$C^{\beta}(\mathbb{R}^n)$$ compactly embedded to $$C^{\alpha, -\delta}(\mathbb{R}^n)$$?

Here $$\|f\|_{C^{\alpha, -\delta}}=\|(1+|\cdot|^2)^{-\frac{\delta}{2}}f\|_{C^{\alpha}}.$$

I could not find a precise reference from some books on functional analysis. Any comment is welcome.