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

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Growth of binomial recurrence with different initial conditions

The binomial coefficients $\binom{n}{r}$ satisfies $\binom{n}{r}=\binom{n-1}{r}+\binom{n-1}{r-1}$. This means it is a solution of the equation $f(n,r)=f(n-1,r)+f(n-1,r-1)$, with initial conditions ...
3
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41 views

Asymptotic expansion of a Fourier Transform as $\omega\rightarrow 0$

First of all, I do apologise if the question is not formulated in precise mathematical terms, but as a physics student I lack a formal background on rigorous functional analysis. Suppose we have a ...
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0answers
16 views

Multivariate Delta Method

If I have a $\sqrt{N}$ asymptotic normal estimator (call it $\boldsymbol{\theta}$, possibly a vector). Say I want to find the asymptotic distribution of $g(\boldsymbol{\theta})$ and suppose ...
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0answers
26 views

A necessary condition for boundedness in probability

I understand that it is straightforward to show (via Markov's inequality and standard arguments) that \begin{equation} E(X_n)=O(a_n) \end{equation} implies \begin{equation} X_n=O_P(a_n) \end{equation} ...
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2answers
88 views

Solve the recurrence $T(n) = 2T(n-1)+n^2$

Solve the recurrence $$T(1) = 1, T(2) = 1, T(3) = 1,T(n) = 2T(n-1)+n^2, n > 3$$ I have now, $$T(n) = 2T(n-1)+^2 $$ $$= 2(2T(n-2)+(n-1)^2+n^2$$ $$=4T(n-2)+2(n-1)^2+n^2$$ $$....$$ ...
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2answers
18 views

On estimating a prime related Diophantine equation related to a partition .

A friend gave me the following algebraic combinatorics question which I couldn't solve Let $p$ be a prime number and $f(p)$ the smallest integer for which there exists a partition of the set $\{2,3, ...
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1answer
26 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 ...
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1answer
28 views

Divisor number asymptotic? [duplicate]

I have got an interesting task, but I can't solve it: We use $d(n)$ as the number of divisors for the positive $n$ integer. We have: $$a(n)=\sum_{i=1}^n d(i)$$ How much is $a(n)$ asymptotic? $a(1) ...
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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 ...
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2answers
137 views

Asymptotic integral expansion of $\int_{0}^{\pi/4}{d\theta \over \epsilon^2+\sin^2\theta}$ for $\epsilon \to 0$

I am studying how to evaluate the integral $$\int_{0}^{\pi/4}{d\theta \over \epsilon^2+\sin^2\theta}$$ as $\epsilon \rightarrow 0$ with asymptotic methods. The book perturbation methods by Hinch ...
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1answer
174 views

Integral asymptotic expansion of $\int_{0}^{\infty} \frac{e^{-x \cosh t}}{\sqrt{\sinh t}}dt$ for $x \to \infty$

$$\int_{0}^{\infty} \frac{e^{-x \cosh t}}{\sqrt{\sinh t}}dt$$ I'm trying to use Laplace's method to find the leading asymptotic behavior as $x$ goes to positive infinity, but I'm having some trouble. ...
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0answers
86 views

Find the Laplace approximation of $\frac{1}{2\pi} \int_{-\pi}^{\pi }e^{x\cos(\theta )}d \theta$ for $ x \to \infty$

Let's have integral $$ I(x) = \frac{1}{2\pi} \int \limits_{-\pi}^{\pi}e^{x\cos(\theta )}d \theta, \quad x \to +\infty . $$ How to use Laplace approximation for this integral and find first two ...
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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 ...
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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 ...
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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 ...
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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 $m = m(n)$ ...
2
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1answer
86 views

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

How can I understand the sign "$=$" from the following expression: $$\mathcal{o}f((x))=\mathcal{o}f((x))+\mathcal{o}f((x));$$ $$\mathcal{o}(kf((x)))=\mathcal{o}(f(x));$$ ...
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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 ...
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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 ...
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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 $o([x]^a)$?
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1answer
42 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 ...
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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.
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2answers
77 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 ...
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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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1answer
59 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 ...
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34 views

If I colour $n$ vertices independently, randomly with $n^{(1-x)}$ colours, why is the size of the colour classes $(1+o(1))n^x$?

By $o(1)$, I mean 'little-o' of $1$. A paper I'm reading uses this result, but I can't see where it comes from. Thanks.
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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 ...
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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 ...
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2answers
96 views

Laplace's method with nontrivial parameter dependency

I need to approximate the following integral using Laplace's method: $$ \int_0^{\infty} \frac{x^{\lambda} \lambda^{-x}}{(1+x^2)^\lambda} dx \\ = \int_0^{\infty} \exp\left(\lambda \log(x) - ...
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1answer
19 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 ...
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 ...
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1answer
22 views

limiting variances of iid sample mean

In the book Statistical Inference (George Casella 2nd ed.), page 470, there is an example: $\bar{X}_n$ is the mean of $n$ iid observations, and E$X=\mu$, $\operatorname{Var}X=\sigma^2$. "If we take ...
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1answer
71 views

Is it possible to find the least common divisor of a two numbers that are not relatively prime in polynomial time?

As the question states: Is it possible to find the least common divisor of two number that are not relatively prime in polynomial time? If so, how? Thanks!
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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 ...
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34 views
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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, ...
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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} ...
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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 $\varepsilon >0$ ? ...
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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 ...
3
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2answers
93 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 ...
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1answer
30 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 ...
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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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1answer
44 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 ...
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35 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 ...
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 = ...
2
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0answers
39 views

Asymptotic behavior of $1/(a^2+\epsilon^2)$ as $\epsilon\to0$

A limit that often arises in physics is $$ \lim_{\epsilon \to 0} \frac{ \epsilon }{ a^2 + \epsilon^2 } = \pi \delta(a) ............ (1) $$ Is there a similar sort of limit for $$ \lim_{\epsilon \to 0} ...