nt.number theory – A question on a real sequence

Let ${a_n}_{nge1}$ be a real sequence that decays faster than any algebraic speed, that is, $lim_{nto infty} n^pa_n = 0$ for every positive integer $p$. Assume that $$sum_{nge 1}(n+1)^kn^ka_n = 0$$ for every integer $k ge 0$.

Question: Can we conclude that $a_n equiv 0$?

graph theory – Generating triangulations with given topology

I am looking for information about the problem of identifying the heaviest minimal subset $Fsubset E$ of the edgeset $E$ of a complete symmetric graph $G(V,E)$ with randomly weighted edges such that $Gsetminus F$ has a given topology, e.g. is planar, and a triangulation, i.e. every remaining edge is an edge of a 3-cycle.

An exemplary problem would be to find a triangulation of a complete graph, whose vertices resemble points on a torus and edge weigths equal to euclidean distance, that allows for a planar embedding.

Being able to calculate the lightest planar triangulation of graphs with arbitrarily weighted edges would e.g. yield a meaningful definition of the planar convex hull of such graphs and thus yield an initial tour that can be expanded to a lightest Hamilton cycle by successive integration of vertices.

gr.group theory – What does it matter if a group has a non-elementary hyperbolic quotient?

My adviser recently shared a problem with me that seeks to establish non-elementary* hyperbolic quotients for mapping class groups. They told me that this could be useful for establishing results on separability or omnipotence, and that these could be relevant for examining profinite rigidity of hyperbolic 3-manifolds. Unfortunately, I’m not fully read up on these topics.

In this recent paper, Behrstock et. al. also seek to make headway on the question of hyperbolic quotients for mapping class groups. They have a discussion in their introduction of the relevance of this question, mentioning again separability and omnipotence, profinite rigidity, and placing things in the context of the virtual Haken conjecture. Again, I’m a little ignorant of these topics and their history of this point. So my question:

Q: Can anyone explain with some detail (or point me to some nice references as to) why it’s relevant that mapping class groups have hyperbolic quotients? Or why it’s helpful that any group has such a quotient?

*A Gromov-hyperbolic group is non-elementary if it is not virtually cyclic, i.e. is infinite and not virtually $mathbb{Z}$.

complexity theory – Why would $P = NP$?

The overwhelming consensus of the CS community seems to be that $text{P} neq text{NP}$. You can easily find many resources giving arguments to believe why that would be the case. This includes blog posts like this one by Scott Aaronson or even questions on Stack Exchange.

Personally, over time, the more results in complexity theory I’ve seen, the more hardness proofs I’ve seen, the more I’ve learned about this topic, the more I myself also became convinced that $text{P} neq text{NP}$.

Now, I’d like to turn this question around. What reasons are there to believe $text{P} = text{NP}$? What, if any, compelling arguments could be made in favor of $text{P} = text{NP}$? Naturally, the consensus of of the community points to there not being many truly compelling reasons, but I’d be curious about what one can say as the devil’s advocate.

nt.number theory – Natural functions giving zero on integer partitions

I’m working with integer partitions $ eta$ of $w$, with parts of size at most $n$, so that $eta$ satisfies $sum_{i = 1}^{n} i, eta_i = w$.

I have a set of functions $c_{n,w}(eta)$ on integer partitions in computer algebra in Mathematica for a given $n$ and $w$, and I want to try to reconstruct the form of the functions for general $n$ and $w$. The functions are valued on rational numbers, ie. $c_{n,w}(eta) in mathbb{Q}$.

For a given $n$ and $w$ some of the $c_{n,w}(eta)$ are zero, and I’m trying to work out the zeros first.
For example, for some $n$ and $w$ I have a $mathbb{Z}_2$ symmetry which forces the $c_{n,w}$ to be zero, depending on whether $sum_{i = 1}^{n} eta_i$ is even or odd. So I know that in those cases, I have that

c_{n,w}(eta) sim 1+(-1)^{sum_{i = 1}^{n} eta_i}.$$

or $$
c_{n,w}(eta) sim 1-(-1)^{sum_{i = 1}^{n} eta_i}.$$

I also have some other cases with $c_{n,w}(eta) = 0$, but in those cases it’s not so clear to me to see what the pattern is. I’m looking for suggestions about natural functions $c_{n,w}(eta)$ which have zeros, additional to the ones I gave above.

I’ve also thought of $$c_{n,w}(eta) = sum_{i = 1}^{n} (-1)^i eta_i,$$ and $$c_{n,w}(eta) = 1-(-1)^{f(eta)}$$ for some different functions $f$, maybe such as $f(eta) = sum_{i = 1}^{n} i^a (eta_i)^b$ for $a,b in mathbb{N}.$ Neither of those seem to work for my current problem. I also considered taking the sums up to some $m < n$ in the functional forms I gave above so that I consider only some of the parts of the partition, but that didn’t seem to help either, and I’m not really sure how natural it would be to do that either.

I’m looking for functions constructed of only addition, subtraction, multiplication and integer powers of the componenets of $eta$. (So combinatorial functions like factorial or binomial coefficients would also be fine). In the end I sum the $c_{n,w}(eta)$ over $eta$ and multiply by $prod_{i=1}^{n}a_i^{eta_i}$ to construct a polynomial with a well-defined scaling weight $w$, if that helps at all in suggesting relevant functions.

As an example, I was looking at a case with $w = 9$, $n=6$. In that case, I have all of the $c_{n,w}(eta) = 0$ when $sum_{i = 1}^{n} eta_i$ is even, so I know I need the relevant function above as a factor (I also have the $mathbb{Z}_2$ symmetry in this case). I was surprised to find that there are also two additional zeros, when $eta = (3,0,2,0,0,0)$ and when $eta = (1,1,0,0,0,1)$, so I think I need to multiply in an additional factor which is zero on these two $eta$, non-zero on $eta$ where $sum_{i = 1}^{n} eta_i$ is odd, and which I don’t have any information about when $sum_{i = 1}^{n} eta_i$ is even. Does this give enough information for someone to suggest a relevant function in this particular situation?

I appreciate that this is unlikely to fully specify the function that I’m interested in. My philosophy is to try some of the more simple possible such functions, and see if they match against the form I need in Mathematica. Hopefully this way I’ll be able to reconstruct the full functional form that I’m looking for, and this should make it easier to go back and prove that form later on. (Also I enjoy doing it this way!)

Any references which may be relevant for me to read about this would also be very welcome. I have some basic understanding of integer partitions, but I’ve not worked with them before.

homotopy theory – Elementary proof of the exactness of Čech complex associated to a hypercovering (“Illusie’s Conjecture”)

Let $mathcal{E}$ be a sheaf of abelian groups on a topological space (or a site). For an open covering $mathfrak{U} = (U_i)_i$, it is well known that the augmented Čech complex $0 to mathcal{E} to mathcal{check C}^bullet(mathfrak{U}, mathcal{E})$ is a resolution of $mathcal{E}$. (Depending on the sheaf cohomology of $mathcal{E}$ on the intersections of the members of $mathfrak{U}$, this resolution might compute $mathbb{R}Gamma(mathcal{E})$, but this need not concern us here.)

A simple proof runs as follows: Let $s in mathcal{check C}^{n+1}(mathfrak{U}, mathcal{E})(V)$ such that $ds = 0$. Then $t = (s_{i_text{fix},i_0,ldots,i_n})_{i_0,ldots,i_n} in mathcal{check C}^{n}(mathfrak{U}, mathcal{E})(V cap U_{i_text{fix}})$ is a local preimage of $s$ under $d$, as $(dt)_{i_0, ldots, i_{n+1}} = s_{i_0, ldots, i_{n+1}} – (ds)_{i_text{fix}, i_0, ldots, i_{n+1}} = s_{i_0, ldots, i_{n+1}}$. As $V = bigcup_{i_text{fix}} (V cap U_text{fix})$, that’s all what’s needed.

I’m looking for a similarly elementary proof in the case that $mathfrak{U}$ is a hypercovering. Apparently this fact is sometimes referred to as Illusie’s Conjecture. A proof involving trivial Kan fibrations and an induction reducing to the case of ordinary covers is in Section 01GA of the Stacks Project, but I’d prefer either a more direct proof or some insight as to why a much simpler argument isn’t likely to exist.

complexity theory – Asymptotic notation between two set of variables

I have problems interpreting the definition of asymptotic notation where the functions involve two different set of variables. I am quite confident with the definition of $f(n) = O(g(n))$ and its extension to the multivariable case ($f(n, m) = O(g(n, m))$). However I don’t acutally understand the meaning of $f(n) = O(g(m))$ since $f$ and $g$ work with two different variables.

If for exame I have that $n = O(m)$, the intuition behind this is that $m$ upper bounds $n$ but I’m not quite sure how to apply the formal definition of big O. The first idea was to use a dummy function approach by defining $f(n, m) = n$ and $g(n, m) = m$ but this does not work out well.

Another idea was to assume that both $n$ and $m$ are function of some unkown variables that depends on the problem at hand. In this case $n = n(x_1, ldots, x_k)$, $m = m(x_1, ldots, x_k)$ and $n(x_1, ldots, x_k) = O(m(x_1, ldots, x_k))$.

Is this right? Am I missing something?

set theory – Show that a set is countable by specifying an alphabet for it

Consider the set $S$ = { ${sqrt(n) | n in N_0}$}
This is a countable set certainly as we can produce a simple one-to-one function
for this set to natural numbers. But I want to do it another way.
There is a theorem that given an alphabet set the language generated
over that alphabet is countable. So for using this theorem, I have to
come up with an alphabet for $S$

soft question – Book on logic similar in tone/difficulty to halmos’ naive set theory

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What was the first result that belongs to the field of analytic number theory?

Euclid proved that there are infinitely many primes via a clever algebraic argument that most of us are familiar with. Euler proved that $lim_{s to 1^+} sum_p frac{1}{p^s} = +infty$ which gives us analytic proof of Euclid’s result. Is this is the first known result in analytic number theory?