## set theory – A monotone countably unbounded function from $omega^omega$ to $omega^{omega_1}$

For a set $$X$$ we endow the set $$omega^X$$ of all functions from $$X$$ to $$omega$$ with the natural partial order $$le$$ defined by $$fle g$$ iff $$f(x)le g(x)$$ for all $$xin X$$. A function $$mu:omega^omegato omega^X$$ is called monotone if for any $$fle g$$ in $$omega^omega$$ we have $$mu(f)lemu(g)$$.

Question. Is there a monotone function $$mu:omega^omegatoomega^{omega_1}$$ which is countably unbounded in the sense that for every countable infinite set $$Asubsetomega_1$$ and function $$finomega^A$$ there exist a function $$ginomega^omega$$ and an infinite set $$Bsubseteq A$$ such that $$f{restriction}_Blemu(g){restriction}_B$$?

Remark. By the answer of Johannes Schurz to this question, for every monotone function $$mu:omega^omegatoomega^{omega_1}$$, there exists a countable set $$Asubsetomega_1$$ and a function $$finomega^A$$ such that $$fnotle mu(g){restriction}A$$ for every $$ginomega^omega$$.

## numerical integration – Can I configure a guaranteed precision for NIntegrate on a monotone function?

I know there are some great posts already about why PrecisionGoal->n doesn’t guarantee the result of NIntegrate will actually have $$n$$-digit precision.

However, for a monotone increasing or decreasing function, it is conceivable to integrate numerically and guarantee a certain precision. For each partition of the region of integration, it is easy to compute an absolute maximum and minimum value for the integral (via Riemann sums), and then the error $$E$$ is bounded by $$max-min$$. Perhaps there are even more sophisticated techniques to deduce a guaranteed error bound.

Is there a way to tell Mathematica that a function is monotone increasing/decreasing and then insist the the result of NIntegrate is within a certain guaranteed precision goal?

## If a polygon is monotonous with respect to a line, how can I determine the two monotone polygonal chains?

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## Limit using the monotone convergence theorem

Show using the monotonic convergence theorem that the limit below convergence, in other words, analytically demonstrate that the limit below convergence:

$$lim_ {x to infty} frac {5 ^ x} {2 ^ {n ^ 2}}$$

I tried to do: $$lim_ {x to infty} frac {x cdot ln {5}} {x ^ 2 cdot ln {2}} = 0$$ but I do not know if this is allowed by the theorem.

## analysis: if $f (x)$ is monotone and $int_a ^ {+ infty} f (x) mathrm {d} x$ is convergent, then $lim_ {x rightarrow + infty} xf (x ) = 0$

Yes $$f (x)$$ it is monotonous and $$int_a ^ {+ infty} f (x) mathrm {d} x$$ is convergent then test
$$lim_ limits {x rightarrow + infty} xf (x) = 0$$

I think of the second theorem of the mean value for integrals
$$int_X ^ {X & # 39;} f ​​(x) mathrm {d} x = f (X) int_X ^ xi f (x) + f (X & # 39;) int_ xi ^ { X & # 39;}$$
because of the monotony of $$f (x)$$. Y
$$forall epsilon> 0, | int_X ^ {X & # 39;} f ​​(x) mathrm {d} x | = | f (X) int_X ^ { xi} mathrm {d} x + f (X & # 39;) int_ xi ^ {X & # 39;} mathrm {d} x | = | (X & # 39; – xi) f (X & # 39;) + ( xi – X) f (X) |$$
Y
$$| X & # 39; f (X & # 39;) – Xf (X) | = | (X & # 39; – xi) f (X & # 39;) + ( xi – X) f (X) + xi f (X & # 39;) – xi f (X) |$$

But I have no idea to continue.

## Algorithms – Find the direction of the monotone polygon (if it exists) in linear time

Implement a program that verifies if there is an address where a simple polygon is monotonous and, in that case, notify that address.
Upper limits: O (n) time, where n is the size of
the polygon

The vertices are in counter-clockwise order.
I have no idea how to solve this problem, if you do not know the solution, even the idea could help you have something to work on.

I have tried to classify the vertices in (start, end, division, combination, regular) as in triangulation by some random direction and do something with it. Because if there are no divisions or mergers, the cop is monotonous in that direction. But I think I came to an impasse.