## complexity theory – What data structure or algorithms to solve this problem

Given an unsorted array of ex-file sizes [40,10,20,30,100,60] and a maximum of ex-150 free disk space. Adjust in all files, what will be the maximum size that you can integrate.

The simplest thing to do was to take the average of the maximum available size = 150/6 = 25 .. but since there are files whose size is less than 25, we must use a maximum of space, which means is not the right answer. Sort and scroll through the table, continue to subtract and calculate the following average until we see an above-average size. There will be special cases if all sizes are below average and we should simply return the average in this case.

The answer in this particular case is 30. I am not able to link it to a data structure or algorithm. I looked on Hackerrank and leetcode but I did not find anything similar.

## nt.number theory – See the character of Dirichlet as a character in the Galois group

In Jaclyn Lang's article "On the image of the Galois representation associated with the Hida non CM family", section 2, the Dirichlet character $$chi$$ module $$N$$ is also considered a character $$chi colon textrm {Gal} ( overline { mathbb {Q}} / mathbb {Q}) rightarrow overline { mathbb {Q}}$$ without explicit explanation.

Is there a natural way to see this?

## Theory of Mining – If I have 51% of the capacity of the entire network, can I exploit all the blocks absolutely faster than the network?

Can you exploit me absolutely? If you have a 51% share of the network hash rate, the probability (no certainty) that you will exploit the next block is 51%. It is theoretically possible that you never find a solution to the blockage. However, if you continue to exploit the blocks that only you generate, then you can practically build a chain made up entirely of blocks that you exploit after the point of junction.

## gr.group theory – on standard 2-cocycle

Let $$G$$ to be a group acts trivially on an abelian group $$A$$. Let
$$varepsilon$$ to be a normalized 2-cocycle in $$Z ^ {2} (G, A)$$. assume
this $$G = H_ {1} times H_ {2}$$ to be the direct product of $$H_ {1}$$ and
$$H_ {2}$$.

1. Is it true that $$varepsilon$$ can be expressed as
$$varepsilon ((h, g), (h, g)) = varepsilon_ {1} (h, h) times varepsilon_ {2} (g, g) times alpha (g, h)$$ such as $$varepsilon_ {1} = res_ {H_ {1} times H_ {1}} ( varepsilon) in Z ^ {2} (H_ {1}, A)$$, $$varepsilon_ {2} = res_ {H_ {2} times H_ {2}} ( varepsilon) in Z ^ {2} (H_ {2}, A)$$ and $$alpha in Hom (H_ {2} times H_ {1}, A)$$.
2. If the answer is no, what are the possible expressions of $$varepsilon$$ it depends on $$varepsilon_ {1}$$ and $$varepsilon_ {2}$$.

The reader can see (Tahara, Proposition 1)

Any help would be so appreciated. Thank you all.

## Number Theory – Primitive Roots for \$ x ^ k = a \$ for mod n

(1) Existence of a root for the equation $$x ^ k = a$$ (mod$$n$$)

(Here there are primitive roots for mod $$n$$.)

Then, between the two statements below, which is the instruction iff for (1)?

Let $$d = gcd (k, phi (n))$$

1) $$a ^ { phi (n) over d}$$ = 1 (mod $$phi (n)$$ )

2) $$a ^ { phi (n) over d}$$ = 1 (mod $$n$$ )

Personally, I vote for the 2nd statement is right. But I'm not sure that my decision is correct. What do you think?

## ct.category Theory – What are the actual epimorphisms of the \$ infty \$ -presentables categories?

Let $$mathcal C$$ to be kind enough $$infty$$-category, and leave $$f: U to X$$ to be a morphism in $$mathcal C$$. Remind that $$f$$ is said to be a effective epimorphism if the induced card $$| U ^ { times_X ( bullet + 1)} | to X$$ is an equivalence, where we took the geometric realization of the Cech nerve from $$f$$, whose $$n$$the level is $$U ^ { times_X (n + 1)} = U times_X dots times_X U$$.

Question 1: Let $$mathcal C = Pr ^ R$$ Be the $$infty$$-category of presentable $$infty$$-categories and right assistant functors. Yes $$f_ ast: mathcal U to mathcal X$$ is a morphism in $$Pr ^ R$$is there a more basic characterization of the moment when $$f_ ast$$ is an effective epimorphism in $$Pr ^ R$$?

Since the limits in both $$Pr ^ R$$ and the opposite category $$Pr ^ L$$ are calculated at the level of the underlying categories, we can at least say that the $$n$$The level of the Cech nerve is $$mathcal U ^ { times _ { mathcal X} (n + 1)} = mathcal U times _ { mathcal X} points # times _ { mathcal X} mathcal U$$ where the withdrawal is taken $$Cat$$ on the functor $$f_ ast$$; its geometric realization in $$Pr ^ R$$ is the totalization in $$Cat$$ of the cosimplicial category at these levels, whose cosimplicial coface cards are given by the left joins of the various projections. But it seems difficult to explicitly describe these left joins, which hinders the analysis for me.

The only case I think I understand is when $$f_ ast$$ is a co-location, ie his / her left assistant $$f ^ ast$$ is totally faithful. In this case, I think the left fits the projections $$mathcal U times _ { mathcal X} mathcal U ^ to_ to mathcal U$$ are given by $$U mapsto (U, f ^ ast u)$$ and $$U mapsto (U, U)$$, and I think it looks like $$f_ ast$$ indeed it turns out to be an effective epimorphism.

Question 2: Are colocalisations examples of effective epimorphisms in $$Pr ^ R$$?

Even if it were, I would be hesitant to guess that these are all examples.

Question 3: A natural assumption is that $$f_ ast$$ is an effective epimorphism $$Pr ^ R$$ if and only if his left assistant $$f ^ ast$$ is conservative. Is it correct?

## gr.group theory – Generate the monoid of free group injections per se

Let $$F$$ to be the group of free rank $$2$$ (or any rank over if it does not matter). The set of injections $$F to F$$ form a monoid $$M$$ by compositions. is there simple search set of generators $$M$$?

It's natural to ask that. For example, if we would like to show a property $$P$$ valid for all injections $$M$$ and is preserved under compositions, it is natural to prove $$P$$ for a set of generators. So by simple search, I mean it would really save us some work by proving that $$P$$ for generators, for example, the generators are all alike (such as the generation of mapping class groups by Dehn twists) or divide into a large number of (infinite) classes.

obviously, $$M$$ contains $$mathrm {Aut} (F)$$, which has a nice, finite generator set: Nielsen's elementary transformations.

Also note that $$M$$ is not finished, since every homomorphism $$F to F$$ induces a homomorphism on the abelianization $$mathbb {Z} ^ 2 to mathbb {Z} ^ 2$$whose determinant could be any integer. Thus, the determinants of any generator of $$M$$ should be a generator of the monoid $$mathbb {Z}$$, where the operation is a multiplication.

I do not have a good idea to think about it. Is there a reference or related theory? Thank you!

## simplify the expressions – disappear with the terms derived from the perturbation theory

After the question here, I would now like to remove the terms related to derivatives. To explain briefly, given a decomposed variable as follows:

$$a (t, r) = a (r) + delta a (t, r)$$

in my long calculations several derivatives of the term $$delta a (t, r)$$ Appears like $$partial_t delta a (t, r)$$ or $$partial_r delta a (t, r)$$. But I need that multiplied $$delta$$-terms with another to disappear, as

$$partial_r delta a (t, r) partial_r delta b (t, r) = 0$$
or
$$( partial_r delta a (t, r)) ^ 2 = 0$$
or
$$partial_r delta a (t, r) partial_t delta a (t, r)$$.

In my previous question, the method shown was able to answer only when there were no derived terms and I would like to generalize it. If I am the method shown, Mathematica takes the derivative of the variable $$delta a (t, r)$$ as:

``````partial_r delta(a(t,r)) = delta'(a(t,r)) partial_r a(t,r).
``````

following the chain rule. And, of course, this does not give zero when two terms of this type are multiplied.

## Elementary Number Theory – Show that \$ frac {(m + n-1)!} {M! n!} in mathbb {Z} \$, if \$ m, n \$ coprime (homework problem)

My attempt: to assume $$m> n$$. Since $$binom {x} {y} in mathbb {Z}$$ for each $$x geq y$$we have that

$$K: = binom {m + n-1} {n} = frac {(m + n-1)!} {(M-1)! N!} In mathbb {Z}$$

It remains to show that $$m | K$$. I know this is kind of a result of $$gcd (m, n) = 1$$, but I do not know how. All suggestions / suggestions are appreciated.

## Theory of complexity – How to prove NP hardness from scratch?

I am working on a problem whose complexity is unknown.
Due to the nature of the problem, I can not use the long edges as I see fit, so 3SAT and its variants are almost impossible to use.

Finally, I decided to choose the most primitive method, Turing Machines.

Curiously, I did not find any examples of a reduction in the hardness of NPs directly by modeling the problem in language form and showing that a deterministic Turing machine can not decide if a given instance belongs to this language (j might have ruined the terminology here).

So, assuming there is no problem in reducing NP hardness, how do you prove that a problem is NP-hard? Are there publications that do that?