## How can to find asymptotic form of power series?

The Power series is $$R_n(x)=\Sigma_{p=2}^{n+1}\frac{(p-1)(2n-p)!}{n!(n+1-p)!}x^p$$

It says the following:

For large $$n$$, the asymptotic form of $$R_n(1/\beta)$$ can be obtained by looking at the values of $$p$$ which dominate the sum

• $$p$$ of order $$1$$ for $$\beta > 1/2$$,
• $$p$$ of order $$\sqrt n$$ for $$\beta = 1/2$$, and
• $$p \propto (1 – 2\beta)/(1-\beta)$$ for $$\beta < i$$

See This Picture

## Asymptotic Bound on Minimum Epsilon Cover of Arbitrary Manifolds

Let $$M \subset \mathbb{R}^d$$ be a compact smooth $$k$$-dimensional manifold embedded in $$\mathbb{R}^d$$. Let $$\mathcal{N}(\epsilon)$$ denote the size of the minimum $$\epsilon$$ cover $$P$$ of $$M$$; that is for every point $$x \in M$$ there exists a $$p \in P$$ such that $$\| x – p\|_{2}$$.

Is it the case that $$\mathcal{N}(\epsilon) \in \Theta\left(\frac{1}{\epsilon^k}\right)$$? If so, is there a reference?

## Always exists an asymptotic component with large size in a minimal subshift?

Let $$(X, T)$$ be a minimal subshift, i.e. $$X$$ is a closed $$T$$-invariant subset of $$A^\mathbb{Z}$$, where $$T$$ is the shift. A pair $$x,y\in X$$ is asymptotic if $$d(T^nx, T^ny)$$ goes to zero as $$n\to\infty$$. Always exists such a pair when $$X$$ is infinite: for every $$n\geq1$$ there exists $$x^{(n)}, y^{(n)} \in X$$ such that $$x^{(n)}_0\not= y^{(n)}_0$$ and $$x^{(n)}_{[1,N]}= y^{(n)}_{[1,N]}$$ (if not, for some $$n$$, $$x_{[1,n]} = y_{[1,n]}$$ implies $$x_0=y_0$$, i.e., $$x_{[1,n]}$$ determines $$x_0$$, and this forces $$X$$ to be periodic), and any pair of convergent subsequences of $$x^{(n)}$$ and $$y^{(n)}$$ will do the trick.

My question is: do exist $$k$$-tuples of asymptotic points, for every $$k\geq1$$? More precisely, is it true that for every $$k\geq1$$ there exists $$x_1,\dots,x_k\in X$$ such that $$\lim_{n\to\infty}d(T^nx_i, T^nx_j) = 0\ \forall i,j$$

## How can I assign the following functions f2,f3,f4 to the best possible (mostly restricted) asymptotic class?

Try a couple of ways but still having a problem to find the specific asymptotic class for each function. ¿Any reference to find the solution o aproach?

## How can I assign the following functions f1 to the best possible (mostly restricted) asymptotic class?

I’m working in this and so far I been trying to use different methods to assign the specific asymptotic class but always get to a dead end, any insights in the best way to get to the solution?

## Asymptotic analysis with factorial and exponential

I’m solving a complexity question where I have:

$$n!/2^n$$

The goal is to find an upper bound for this.

My idea is using the fact that: $$n! = O(n^n)$$ $$n!/2^n = O((n/2)^n) = O(n^n)$$

But is a correct upper bound, and if so is it the tightest upper bound that can be found? I know that n! is asymptotically larger than 2^n, but I’m struggling to do a tighter analysis.

## Minimization with asymptotic assumption

Given the function

$$g(n,m)=\min\Big\{f(a,b)+f(n-a,c)+f(n,m-bc)\Big|\a,b,c\ \ \text{with} \left\{\begin{matrix} a,\ b,\ n-a,\ c,\ m-bc \geq 0 \ b\leq a! \ c\leq (n-a)! \ \end{matrix}\right. \Big\}$$

Assuming that $$n,m\geq 0,\ ((\lceil n/2\rceil)!)^2\leq m\leq n!,\ f(n,m)=\Omega (n)$$,

is it true that $$g(n,m) \geq 2f(\lfloor n/2\rfloor ,(\lfloor n/2\rfloor)!)+f(n,m-((\lceil n/2\rceil)!)^2)$$ ?

I tried KKT conditions, but can’t derive this (as it contains factorial).

Also, it seems that the condition $$f(n,m)=\Omega (n)$$ implies that $$f$$ is convex on our domain (and thus, satisfies the regularity condition for using KKT), but I managed to prove it only if $$f$$ is polynomial.

So I am fully stuck in this…

Any help would be highly appreciated!

## U statistic asymptotic distribution

Can someone provide me hints as to where to attack part (c) from? I tried using results from U-statistics, but having trouble for connecting distribution of $$L_n(\hat{\theta}_n) – E[1\{Y_i > Y_j\}] \log P_\theta (Y_i > Y_j|X_i,X_j)$$ to distribution of $$\hat{\theta}_n.$$ Others were straightforward.

## conditions for asymptotic comparison to hold

I have the following simple dynamical system: \begin{align} x_1′ &= a – f(x_2)x_1\ x_2′ &= bx_1 – cx_2, \end{align} where all parameters and initial conditions are positive. $$f(x_2)$$ is a positive and increasing function with respect to $$x_2$$. Suppose I want to study the asymptotic behavior of this system and decide to do the following.

First, I note that the more $$x_1$$ I have, the more higher the production rate for $$x_2$$ will be. Secondly, I note that the larger $$x_2$$, the higher $$f(x_2)$$ would be. Thus I choose to replace $$f(x_2)$$ by a function $$g(x_1)$$ such that:

1. $$g(x_1) > 0$$ and $$\frac{dg}{x_1} > 0$$.

2. $$g(x_1)$$ gives the same fixed points for $$x_1$$ (perhaps through something like a quasi-steady-state-approximation for $$x_2$$).

Together, I obtain: $$$$x_1′ = a – g(x_1)x_1.$$$$

Due to the construction, the asymptotic behavior of $$x_1$$ in this equation should be the same as that of $$x_1$$ in the original equation. I tried this out with $$f$$ and $$g$$ being simple hill equation and it works.

This is just a toy example. My question is: for higher dimension and more complicated functional responses, if I only care about asymptotic behavior, when can something similar can be carried out? I would appreciate any references on this topic.

## Comparing different asymptotic notations

Suppose we have 3 algorithms complexity times at the worst case:

• A = $$O(nlogn)$$
• B = $$O(n\sqrt{n})$$
• C = $$\Theta(n)$$

In my opinion, it is not possible to define the best solution, since we don’t know how Cgrows. I’d like to confirm if that’s correct.