Questions involving asymptotic analysis, including growth of functions, Big-O notation, Big-Omega and Big-Theta notations.

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0
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1answer
29 views

Is square root of n the same as log n for order notation of an algorithm

Given the context of a basic prime number testing algorithm that has the simple optimization of limiting the potential factors to the range from 2 to the square root of the subject number (instead of ...
1
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3answers
57 views

How to evaluate this exponential fraction limit?

I am trying to determine if 3$^n$ grows faster than 2$^{2n}$. One way I found online to do this was, from Growth was to evaluate $\lim_{n\to \infty} \frac{3^n}{2^{2n}}$ and if that limit evaluates ...
0
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1answer
19 views

Prove that $\frac{f(n)+a}{g(n)+b} = O(\frac{f(n)}{g(n)})$

I was reading about algorithm analysis and I saw a similar simplification done in order to find the complexity. I became interested in proving that this simplification is formally correct but I am ...
1
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1answer
17 views

A question regarding the order of an asymptotic estimate

Suppose that $m, n \in \mathbb{N}$ such that \begin{equation} m \cdot \log m = n, \end{equation} where the logarithm is in the natural base. How can we estimate the solution $m = m(n)$ ...
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2answers
48 views

Asymptotic expansion of exp of exp

I am having difficulties trying to find the asymptotic expansion of $I(\lambda)=\int^{\infty}_{1}\frac{1}{x^{2}}\exp(-\lambda\exp(-x))\mathrm{d}x$ as $\lambda\rightarrow\infty$ up to terms of order ...
5
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1answer
208 views

$f ' (x) = f(x - (x+1)^t + 1)$

Let $x > 0 $ and $c $ a given real $> 0.$ Let $t $ be between $0 $ and $1.$ How to find $f(x)$ or good asymptotics for $ f(x)$ such that $$ f ' (x) = f(x - (x+1)^t + 1) $$ And $ f(1) = 1 + ...
2
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1answer
87 views

What does the sign “$=$” exact meanings?

How can I understand the sign "$=$" from the following expression: $$\mathcal{o}f((x))=\mathcal{o}f((x))+\mathcal{o}f((x));$$ $$\mathcal{o}(kf((x)))=\mathcal{o}(f(x));$$ ...
1
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0answers
72 views

Asymptotic distribution and stability?

I am working with asymptotic theory and I have some things I am unsure about. For example if one uses the Central Limit Theorem as an example: $\sqrt{n}\bigg(\bigg(\frac{1}{n}\sum_{i=1}^n ...
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0answers
9 views

How would I compare these differential statements using Big O notation?

I am doing an econ problem. The question asks me to basically discuss in economic terms the effect of increasing or decreasing $\alpha$ on the function $$1= x^\alpha y^{1-\alpha}$$ Anyways, I've ...
1
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1answer
20 views

If $x$ is a $\chi^2_{N-n}$ RV. what is $x/N$ as N goes to infinity

We know that if we have $N-n$ gaussian iid RVs $\{e_i\}$ with mean $0$ and variance $1$, the RV $x = \sum e_i^2$ is $\chi^2$ distributed with $N-n$ degrees of freedom. We have $N$ larger than $n$. I ...
1
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1answer
144 views

Example of pairwise independent random process with expected max load $\sqrt{n}$. [closed]

Throw $n$ balls into $n$ bins. Each bin is selected uniformly at random but the process is only pairwise independent. Call the maximum number of balls in any bin the max load. Lemma 2 in these ...
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2answers
34 views

Asymptotic Algorithm General Approach to Finding $\Theta$ Bound

I'm working on the following asymptotic algorithm bounds problem Find a $\Theta$ bound for $f(n) = \frac{n^2}{2} - \frac{n}{2}$ So I could find the big-$O$ bound fairly easily $$ 0 \leq ...
0
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1answer
27 views

Floor function and little oh notation

Can we replace $o([x]^a)$ where $[x]$ is floor of $x$ and $a$ is a positive number with $o(x^{a})$? And can we replace $o(x^{a})$ with $o([x]^a)$?
0
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1answer
43 views

Confusion on Big $O$

I am so confused on the intuitive idea behind Big $O$ notation. $f(x)=O(g(x))$ iff there is a constant $C>0$ such that for large $x, |f(x)|\leq C|g(x)|$ and I have seen that in many places that ...
1
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1answer
30 views

How to show $n \sum_{k>n} (k^2 \log k)^{-1} \sim (\log n)^{-1}$?

How does one show that $$n \sum_{k>n} \frac{1}{k^2 \log k} \sim \frac{1}{\log n} \quad ?$$ Many thanks for your help.
0
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0answers
17 views

Approximate distribution of product of N normal i.i.d.?

Given $N>30$ i.i.d. $X\approx\mathcal{N}(\mu_X,\sigma_X^2)$, looking for: accurate closed form distribution approximation of $Y=\prod_{n=1}^{N}{X}$ asymptotic normal approximation of same ...
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2answers
82 views

Number of distinct prime divisors of an integer $n$ is $O(\log n/\log\log n)$

I strongly believe that the claim is true; but I'm neither a mathematician nor speaking French and hope that somebody can confirm it, since related questions (here, here and here) either don't have an ...
1
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1answer
60 views

Which way is best to solve: $T(n)=5T(n/5) + n\;?$

I'm not sure which way is best to solve $$T(n)=5T(n/5) + n$$ (recursion tree/master method/recurrence?) I would like some assistance, which way is easier and how can I be sure I got the right answer ...
2
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0answers
23 views

Biggest rate of growth of a sequence in $ℓ^2$

$ℓ^2$ is the space of complex sequences $u_n$ such that $\sum |u_n|^2$ converges. I'm wondering if there are asymptotic results known about such sequences. We have trivially $u_n=o(1)$. Are better ...
1
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1answer
21 views

Is this statement correct $f(n) = \theta(n) \land g(n) = \Omega(n) \Longrightarrow f(n)g(n) = \Omega(n^2)$?

I am having some difficulties understanding what does it mean to "and" $\theta(n)$ and a function $g(n)$, what does it mean in mathematical terms? Specifically, in the following example, I have to ...
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0answers
25 views

Show T(n)=T(n/5)+T(4n/5)+n/2 is $\Omega (n log n)$

I'm tasked with showing T(n)=T(n/5)+T(4n/5)+n/2 is Big-Omega n log n by drawing a recursion tree. The tree shows a lower bound with the following terms: n/2 ... n/10 ... n/50 ... etc. When I solve ...
1
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1answer
35 views

Asymptotic behavior of the confluent hypergeometric function

Consider the following function $$U(a,z)= \frac{1}{\Gamma(a)} \int^{\infty}_0 t^{a-1} \cdot (1+t)^{-a} e^{-zt} dt$$ My Try : Let $\tau= zt$, then : $$ U(a,z)= \frac{z^{-a}}{\Gamma(a)} \int^{\infty}_0 ...
0
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1answer
52 views

How rapidly does $\Gamma(x)$ diverge as $x$ approaches $0$?

Notoriously $$\lim\limits_{x\to0^{\pm}}\Gamma(x)=\pm\infty,$$ but can we be more precise (tightly) bounding from above $\left\lvert \Gamma(x) \right\rvert$ when $x$ is close to $0$? I could not find ...
0
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1answer
38 views
0
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1answer
20 views

Asymptotic-Proof

I am looking at this questions and the proof for it and wondering how this works.Can anyone explain the answer to me or do you have any other way to answer this question.I am new to asymptotic ...
3
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0answers
35 views

Merten's function

I am tasked with applying the Wiener-Ikehara Theorem to achieve a bound of little o(x) on Merten's function $\sum_{n=1}^x \mu (n)$. My problem is the Wiener-Ikehara Theorem applies to Dirichlet series ...
1
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0answers
38 views

Asymptotic Proof

Can someone explain this asymptotic proof to me.I am stuck at the inductive step and get lost around this step $2 × n! < (n + 1) × n!$ $$2n = o(n!)$$ True Proof: In order to $2n = o(n!)$ be true, ...
0
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2answers
35 views

Calculating Running Time of Recurrence Relations

I had to calculate the Running Time of the following Algorithm. ...
2
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1answer
58 views

How to find $s(\exp(d(x)))$ ~ $ x + 2 $?

Let $x$ be a positive real. I want to find a pair of analytic functions $s(x),d(x)$ such that $s(d(x)) = x$ and $ s(\exp(d(x)))$ ~ $ x + 2 $ More presicely I Also want : $$ \lim_{x \to \infty} ...
2
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2answers
21 views

Find the minimum value of $n$ such that $\sin^n(c)<\varepsilon$ for some small constant $\varepsilon>0$

Let $c$ be a constant such that $0 <c \le \pi/2$ and $\sin(c) \ne 0$. Question: What is the minimum value of $n$ such that $\sin^n(c)< \varepsilon$ for some small constant $\varepsilon >0$ ? ...
0
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1answer
22 views

asymptotic complexity of functions

I'm curious if my asymptotic analysis of these functions are correct. I know the process is to strip the constants and then get to where its just comparing functions and taking limit to infinite and ...
5
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1answer
62 views

Why does Titchmarsh say that we can move the derivative under $\frac{2}{\pi}\int_0^\infty \frac{\Xi(t)}{t^2 + \frac{1}{4}} \cosh(\alpha t) \, dt$

If we define the Riemann-Xi function as $$ \Xi(t) = \xi(\frac{1}{2} + it)$$ where $$\xi(s) = \frac{1}{2}s(s-1)\pi^{-\frac{s}{2}}\Gamma(\frac{s}{2})\zeta(s),$$ then according to Titchmarsh in his ...
0
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1answer
31 views

Design an algorithm - Median, computer science

I was wondering if this question belongs here or on StackOverflow, but it is a question of mathematical nature so this seems more appropriate. We have an array $S$ of $n$ different numbers ...
6
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2answers
36 views

If $f(n)$ is $O(g(n))$ and $g(n)$ is $O(f(n))$, is $f(n) = g(n)$?

Question: If $f(n)$ is $O(g(n))$ and $g(n)$ is $O(f(n))$, is $f(n) = g(n)$? I'm studying for a discrete mathematics test, and I'm not sure if this is true or false. Since Big-OH ignores constant ...
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0answers
40 views

Summation involving digamma and floor functions

I am trying to find an asymptotic expansion for the following sum: $$\sum_{n=1}^K \frac{\phi_0( 1/2+n+\lfloor(2n-1)/\sqrt{2}\rfloor)}{(4n-2)}$$ where $\phi_0$ is the digamma function and $\lfloor ...
1
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1answer
45 views

Asymptotic behavior of the zeros of the digamma function

The gamma function has just one extremum on each interval $(k,k+1)$, where $k$ is a negative integer. These extrema occur at the zeros of the derivative of the gamma function. Let $z_n$ denote the ...
2
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0answers
26 views

Asymptotic solution to $m \leqslant e^{\lambda t} (c t^q - \varepsilon)$

What is the smallest $t$ statisfying the inequality: $m \leqslant e^{\lambda t} (c t^q - \varepsilon)$, where $\varepsilon$ is arbitrary small positive number? I believe $t$ must be of the from: $$t = ...
0
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0answers
25 views

Asymptotic analysis of $\int_{0}^{\infty} \frac{\sqrt k J^2_{\ell}(k) \sin{(\tau\sqrt k)}}{(k+1/2)^{n+2}} dk$

Question as the title showed, in which $n$ and $\ell$ are positive integers, $\tau$ is real number and $J$ means Bessel functions. How to do the asymptotic analysis when $\tau$ approaches zero? Any ...
2
votes
1answer
35 views

Simple vs compound interest rates and Taylor expansion

I am having trouble deciphering a portion from my finance text. Let $i = \text{interest rate}$, $n = \text{Some arbitrary time period}$ and $C = \text{Cash invested}$ And also $C(1+i)^n$ ...
3
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3answers
43 views

Rule for calculating big-O plus example I can't figure out

I have the following rule: If $f$ is $O(g)$ for ${x\to\infty}$ and $\lim_{x\to\infty}g(x) = 0$ than also $\lim_{x\to\infty}f(x) = 0$ Then my text proceeds to give an example: ...
6
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3answers
157 views

Aproximation of $a_n$ where $a_{n+1}=a_n+\sqrt {a_n}$

Let $a_1=2$ and we define $a_{n+1}=a_n+\sqrt {a_n},n\geq 1$. Is it possible to get a good aproximation of the $n$th term $a_n$? The first terms are $2,2+\sqrt{2}$, $2+\sqrt{2}+\sqrt{2+\sqrt{2}}$ ... ...
3
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1answer
41 views

Prove that $3\log n$ is $O(\exp(0.001n))$

First time posting here. Hi math stack-exchange community! I have a bonus question on my assignment and I am having trouble proving it. The main reason is that I am only limited to using the rules ...
3
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2answers
96 views

Asymptotic Behaviour Of $\frac{1}{x-1}+\frac{1}{x^2-1}+\frac{1}{x^3-1} + \cdots$ as $x \to 1 $

I define $$ f(x) = \sum_{n=1}^{\infty} \frac{1}{x^n-1} = \frac{1}{x-1} + \frac{1}{x^2-1} + \frac{1}{x^3-1} +\frac{1}{x^4-1} + \frac{1}{x^5-1} + \cdots$$ and I then wish to study the asymptotic ...
2
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3answers
42 views

Proving that $2n^2 + n + 1 = O(n^2)$ and big O proofs in general

Alright so here's the thing, I'm in a class in Computer Science called Algorithm Analysis and it is required for me to learn Big O, Big Omega, etc. While I sort of understand what this is for, I still ...
2
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1answer
17 views

Estimate of tails of sums of reciprocals of a bit more than powers

The tails of sums of reciprocal powers have nice estimates: For $\alpha>1$ the integral test gives $$ \sum_{n=j}^\infty \frac{1}{n^\alpha} \leq \int_{j-1}^\infty \frac{1}{x^\alpha} dx = ...
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0answers
29 views

How would you prove this Big Omega complexity?

We're given $f(n)=\frac{1}{5}n^2-30n-5$ and $g(n)=n^2$, and are asked to prove $f \in \Omega(g)$. The exercise was posted, but no solution is given (this is not an assignment question). So by ...
4
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7answers
447 views

Is it true that $2^n$ is $O(n!)$?

I had a similar problem to this saying: Is it true that $n!$ is $O(2^n)$? I got that to be false because if we look at the dominant power of $n!$ it results in $n^n$. So because the base numbers are ...
3
votes
0answers
36 views

Applying function to both sides of asymptotic expression

I apologize in advance if this has been asked elsewhere, but I couldn't find it. This seems like it should be a pretty simple question, but I'm drawing a blank. If you know that $f(x) \sim g(x)$, ...
1
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3answers
56 views

Big-O notation examples

How do I get c = 4 and n0 = 21, I understand that I could plug in different numbers till f(n) ≤ c * n for all n ≥ n0, but using f(n) how do I arrive at those numbers? ...
0
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1answer
22 views

Counting Primitive Operations

This is the solution I've been given for counting primitive Operation in an algorithm. I think I have my head around how all the operations are found, for instance the 2(n-1), the 2 is the primitive ...