## Is $f(cn)$ always $O(f(n))$ for constant $c$ and any function $f$?

This seems to be true for any function I can think of, but I’m not quite sure how to prove it. Is there a proof of this proposition for any such function or a counter-example?

## How can I rig a game where dice will always roll poorly?

I’m planning an encounter with the owner of a hotel casino who loads his dice, and for this encounter only I want my players to use special dice to roll poorly. If there is no website to purchase these from or a method to load my own dice, does anyone have an alternative suggestion to make this encounter have the desired effect?

## Is there a way to always call stored procedure on transaction commit?

I want a particular store procedure / function to always be called before a transaction commit happens. Is there a way to enforce this somehow?

## Can one always find a bigger global resolution

Let $$X$$ be a scheme. Let $$E$$ be a perfect complex of coherent sheaves on $$X$$ and suppose it admits two global resolutions $$F$$ and $$F’$$. By global resolution I mean that both $$F$$ and $$F’$$ are quasi-isomorphic to $$E$$ and both are complexes of vector bundles.

$$\bf{Question:}$$ Is it possible to find a global resolution $$H$$ of $$E$$ such that $$F, F’$$ map mono-morphically into $$H$$?

I know how to do this when $$E$$ has perfect amplitude $$[0, 1]$$ but I don’t see how to generalize to say $$[0, n]$$ for arbitrary $$n$$.

## Give a Context Free Language, is the complement of this language always recursive(REC)?

I have seen some people make an argument that given the fact that Context Free Languages are proper subset of REC which is closed under complementation thus complement of a CFL must be in REC. I cannot wrap my head around this argument.

What I know about a CFL(say L) is that the complement of this CFL(say L’) need not necessarily be a CFL. This creates fair bit of skepticism about nature of such a complement in my head and makes me feel that there is a possibility this complement L’ could fly off anywhere in Chomsky Hierarchy as long as it is not a CFL. Why would it be restricted to REC?

In my head, I find myself coming up with arguments like following: “Every CFL is in RE and complement of an RE need not necessarily be RE and thus need not necessarily be REC! Thus there could be CFLs whose complements are not in REC. ” or “Complement of a CFL may fall under RE and there are RE languages which are NOT REC. Thus there is a possibility that a certain complement may not be in REC”

Thread like : What is complement of Context-free languages? did not answer my question.

## Is the transformation matrix of an upper triangular matrix to its Jordan normal form always triangular?

Assume that $$A$$ is an upper triangular matrix. In the case where $$A$$ is 2×2, I’ve checked that a transformation matrix $$P$$ such that $$J = P^{-1}AP$$, with $$J$$ Jordan normal form of $$A$$, is always upper triangular. I think the same is true for every matrix of dimension $$n$$ (probably a proof by induction), but I’ve not seen that anywhere. Any source ?

## Firefox window always starting maximized

Intermittently Firefox reverts to opening its windows maximized. I expect it to remember last size and revert to that.

Ubuntu 18.04, latest patches, Firefox as supplied with Ubuntu, default GNOME desktop. When I was 17.04/Unity desktop (if that matters) it did re-open at same (non-maximized) size as last time I closed it. Now it sometimes does that, and that continues to work for a while, but sometimes reverts to always opening maximized, which I don’t want.

As of today, for example, no matter what I do in the way of closing it, it re-opens maximized. I tried created a new temporary profile, but that didn’t seem to change behaviour. I have done nothing “special” that I am aware of since it last worked, and I cannot figure how to get desired behaviour back.

• Does anyone else suffer this intermittent behaviour?
• Does anyone know what I can do to sort this out?
• If it is saving this setting somewhere in the profile (~/.mozilla/firefox/xyz.default/), which exact file is it in?