## reference request – Maximization of a possibly non-negative monotonous function

I have been doing a small review of the literature on submodular optimization.
Something seemed strange to me: although there is a greedy algorithm that provides limited optimization for a monotonous submodular function, I could not find much of what has been said about maximizing a non-monotonous submodular function, or at least one that is not required. To be not negative.

Some definitions: let $$S$$ Be a finite set.
$$f: 2 ^ S rightarrow mathbb {R}$$ is submodular yes for every $$U, V subseteq S$$,
$$f (U cup V) + f (U cap V) leq f (U) + f (V)$$
$$f$$ is monotonous Yes always $$U subseteq V$$, $$f (U) leq f (V)$$.
$$f$$ is not negative Yes $$F (U) geq 0$$ for all $$U subseteq S$$.

What is my motivation? If they give us a graph, $$mathcal {G} = ( mathcal {E}, mathcal {V})$$, with possibly negative edge weights $$w: mathcal {E} rightarrow mathbb {R}$$clearly states that
$$f: 2 ^ mathcal {E} rightarrow mathbb {R}$$
given by
$$U mapsto sum_ {i j en U} w ^ {i j}$$
it's submodular (in fact modular Since inclusion-exclusion remains in the nose!) Regardless of whether the weights are positive or negative.

Is there a greedy algorithm to maximize such functions?
It is so, I would appreciate an explanation or a reference below.

## reference request – Diofanthin equations and consistency of ZFC

There is a diofanthin equation that has no solution if and only if ZFC is consistent.

Question 1: Is that diofanthin equation written somewhere? Is anyone (probably with a computer) trying to find solutions?

Question 2: Suppose we have a Diofanthin equation P = 0 and some intelligent person would find an elementary proof that it has no solution. It can be so elementary that only some transformations of the equation are needed to get it in shape and see that | P |> 0 or something like that. What would be the consequences of that? ZFC cannot prove its own consistency, but the test used only elementary things, so this would be a contradiction. Therefore, something has to be wrong in this case and it seems to me that this would mean that basically "everything" is wrong, even basic finite mathematics.

Question 3: If someone finds a solution (for example, with a computer), this would mean that ZFC is inconsistent. Would you get some information about the problem and maybe information about weaker theories, such as Peano's artihmetic?

## reference request – The \$ S \$ -integers rings are finely generated as rings

Leave $$K$$ be a global field (numeric field or field of algebraic function over a finite field), $$mathcal {V}$$ the set of $$mathbb {Z}$$-valuations on $$K$$, $$S subseteq mathcal {V}$$ A finite set. The ring of $$S$$-integers is the replacement of $$K$$ defined as
$$mathcal {O} _S = lbrace x in K mid forall v in mathcal {V} setminus S: v (x) geq 0 rbrace.$$
Such a ring is always generated finely as a ring (that is, as a ring $$mathbb {Z}$$-algebra). Where should I search to find a reference for this statement?

## networking: command & # 39; sudo nping -xxxx & # 39; of the script. how to disable sudo password request

I'm using the `nping` tool to run tests on several networks.
I also wrote some scripts to automate the parameters of such tests.
I also need to avoid `sudo` to request the password every time a new instance of the terminal is started (the tool will be included in a custom boot USB that some colleagues, not Linux users, will use to test remote networks, static LANs)

I plan to include a password deactivation line in `/etc/sudoers.d` for the nping command

``````%someUsers ALL=(ALL) NOPASSWD: //nping
``````

However, I cannot find where the nping command is placed.

I'm not an experienced Linux user either, so I apologize in advance if the question sounds silly

## javascript: CORS error in the request with axes

Well guys, I have a problem making a request with axes of the Riot games API. The response returns code 200, which is the code when the request worked, but I cannot handle the object returned by json due to an error that is occurring.
NOTE: I am a little beginner in JS

Code:
let axios = require (& # 39; axios & # 39;)

axios.get (& # 39; https: //en1.api.riotgames.com/lol/summoner/v4/summoners/by-name/user? api_key = myApiKey & # 39;)
.catch (error => console.log (error)) ## Reference request: Does taking the module preserve the weak \$ p \$ -summability of sequences in \$ L_q \$?

For this question, all Banach spaces are above the real ones.

Leave $$1 leq p < infty$$. Remember that a sequence $$(x_n)$$ in a Banach space $$E$$ is weakly $$p$$-summable yes
$$Vert (x_n) Vert_ {p, w}: = sup _ { gamma in E ^ * colon Vert gamma Vert leq 1} left ( sum_ {n = 1} ^ infty vert gamma (x_n) | ^ p right) ^ {1 / p} < infty.$$

Another way of thinking about this: for a Banach space $$X$$, bounded linear maps $$X to ell_p$$ corresponds (isometrically) to weakly $$p$$-summable sequences in $$X ^ *$$.

Now suppose $$1 and let $$q$$ be the conjugate index of $$p$$.

Question. Leave $$(x_n)$$ be a weak $$p$$sequence sequential in $$L_q (0,1)$$. Is the sequence $$(| x_n |)$$ also weakly $$p$$-summable?

I suspect the answer is negative, but only because I had no luck finding a "soft" proof of a positive response. On the other hand, the question seems natural enough that it should be in the literature in one way or another.

## reference request – Explicit brief presentation of a universal group generated by 2?

A result of Higman states that there is a finely presented group $$G$$ in which all the other groups presented finitely are integrated, I will call that universal group. Each accounting group is integrated into a group generated by 2, so there are universal groups generated by 2.

I was told that someone somewhere wrote some explicit presentations of such universal groups generated in 2. Where can I find such presentations? Is the minimum number of rapporteurs known for a universal group of 2 generated? Lower limit?

## United Kingdom: I request a tourist visa sponsor from my British friend from Scotland, I am from the Philippines and unemployed

hello have a good day madam / sir

I just want to ask if it is possible for me to obtain a standard tourist visa, even if I am an unemployed person. I live in the Philippines and I have my long-term friend for almost 4 years so far and we already met, he decided to sponsor me to apply for a visa to be his sponsor, if there is the possibility of obtaining a standard visa, what requirements do I need, even if I am unemployed ?, do I have to submit all the documents of the sponsors? Is that enough?