## Using O-notation for asymptotic estimation of the number of additions in recursive function

the number of additions that are executed during the calculation is a(n). How can i find an asymptotic estimation for the function mystery(n) with the help of the O-notation and master theorem. Note: here the question is not asked for the value of mystery(n), but rather for the number of additions!

def mystery(n):     if n==0:         return n * n     return 2 * mystery(n/3) + 4 * n 

## What is the asymptotic bound for $1n + 2(n-1) + 3(n-2) + … + (n-1)2 + n$?

My best guess is that the series $$\sum_{i=1}^n i(n-(i-1))$$ becomes $$2 \Bigg[ n + 2(n-1) + … + \frac{n}{2} \bigg(n-\bigg(\frac{n}{2}-1\bigg)\Bigg)\Bigg]$$ So the highest term is $$n^2$$ and there are $$n$$ terms. Does that mean its $$O(n^3)$$? That seems high. Intuitively, it seems like it should be closer to $$O(n^2)$$ but I can’t find a way to bring it down mathematically.

## Asymptotic value of the Shannon entropy

I would like to evaluate the asymptotic value of following sum:

$$\frac{1}{2^N}\sum_{n=0}^{N} \begin{pmatrix}N\ n \end{pmatrix} \text{log}_{2} \begin{pmatrix}N\ n \end{pmatrix}$$. This is related to the computation of the Shannon entropy. Any help would be greatly appreciated!

## Asymptotic analysis of a complex expression.

$$A = -\frac{((a + k_{3}(\alpha + t_{1}v)^n c^n(\eta + 1) – 2)^2 – (a-1)^2 + 4l_{6}(\eta + 1)b – 1)c^n}{4l_{6}c^{2n}}$$

$$a = k_{3}(\eta + 1)b$$

$$b = ((\alpha + t_{1}v)*c)^n$$

$$c = u_{1}v + \frac{1}{\alpha}$$

I am trying to simplify this expression $$A$$ taking $$n$$ to be large and $$\eta$$,$$\mu$$,$$v$$ to be very small. The calculation is getting tedious. Any other way of seeing this or source where I can put this and obtaina simplified expression?

## Interpretation of an asymptotic notation

Assume that we measure the complexity of an algorithm (for some problem) by two parameters $$n$$ and $$m$$ (where $$m \le n$$). What is the formal interpretation of the following claim: there is no algorithm that solves the given problem in $$o(m + \log{n})$$?

In particular, does it mean that an $$O(\log{n})$$ algorithm is possible?

## Asymptotic of integral $\int_{1}^{e^n}(1-\frac{ln(x)}{n})^n$

How could we find the large-$$n$$ asymptotic of $$\int_{1}^{e^n}\left(1-\frac{\ln x}{n}\right)^n\,dx$$. I have a suspicion that this is $$\sqrt{n}$$.

## Recurrence relation for the asymptotic expansion of an ODE

I want to solve for the asymptotic solution of the following differential equation

$$\left(y^2+1\right) R”(y)+y\left(2-p \left(b_{0} \sqrt{y^2+1}\right)^{-p}\right) R'(y)-l (l+1) R(y)=0$$

as $$y\rightarrow \infty$$, where $$p>0$$. I did the standard way by obtaining a series solution by the Frobenius method prescription in the form

$$R(y)=\sum_{n=0}^\infty \frac{a_{n}}{y^{n+k}}$$ where $$k=l+1$$ is the indicial exponent. I had difficulty finding for a recurrence relation for the coefficients $$a_n$$ for arbitrary value of the parameter $$p$$. Right now, I am just doing the brute force method of solving individual $$a_n$$ for every value of $$p$$. But I am just wondering whether the recurrence relation is possible to solve. Any help is appreciated.

## Asymptotic rate for the expected value of the square root of sample average

I have iid random variables $$X_1, \dots, X_n$$ with $$X_i \geq 0$$, $$E[X_i]=1$$ and $$V[X_i] = \sigma^2$$. Let $$S_n = \frac{\sum_{i=1}^n X_i}{n}$$.

I’d like to say that $$E[\sqrt{S_n}] = 1-O(1/n)$$.

My first approach was to write $$E[\sqrt{S_n}] = \sqrt{E[S_n] – V[\sqrt{S_n}]} = \sqrt{1-V[\sqrt{S_n}]}$$.

I’m then left with showing that $$V[\sqrt{S_n}] = O(1/n)$$.

I’m unsure how to go about this. First, can I hope to prove such an asymptotic bound in general? If not, are there extra assumptions that can be made on the $$X_i$$ so that this holds true?

## Asymptotic complexity of function with two Input variables

Suppose I have a function with two input below.

$$f(m,n) = \log {n^m} + 100n \log \log {m^5} + 150m + 4n^2 + 1000$$.

Is it safe to say that $$f(m,n)$$ is $$\mathcal{O}(m \log n)$$, or is it $$\mathcal{O}(n^2)$$ instead? I think the first one is more representative as it includes the variable $$m$$. But that may not be the case if $$n$$ is relatively very larger than $$m$$.

## Adiabatic limit for metric asymptotic to the Taub-NUT metric

Let $$(M^4,g)$$ be a Ricci-flat Riemannian manifold asymptotic to the Taub-NUT metric. That is, there exists a circle fibration $$\pi:M\backslash K \to \mathbb R^3\backslash B(0,R)$$ which is the Hopf fibration if restricted on $$S^2$$. Moreover, the metric $$g$$ satisfies $$g=h+O(r^{-\tau}) ,\quad \nabla ^{h,k} g=O(r^{-\tau-k})$$ for some $$\tau>0$$ and any integer $$k \ge 1$$, where $$h$$ is the Taub-NUT metric and $$r=\sqrt{x_1^2+x_2^2+x_3^3} \circ \pi$$.

Now we define $$D_{s}=\{x\in M \mid r(x) \le s\}$$ and consider the Hirzebruch signature formula: $$\tau(M)= \frac{1}{12 \pi^2} \int_{D_{s}} |W_+|^2-|W_-|^2\, dV+I(s)+\eta(\partial D_s)$$ where $$I(s)$$ involves the boundary integral of the second fundamental form and $$I(s) \to 0$$ if $$s \to \infty$$.

Can we prove that $$\lim_{s \to \infty}\eta(\partial D_s)=\frac{2}{3}$$?