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

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1answer
47 views

length of the curve $y=x^n$ in the unit square

Let $l_n$ be the length of the curve $y=x^n$ in $[0,1]\times[0,1]$. Then obviously $\lim_{n\to\infty}l_n = 2$. What about $\lim_{n\to\infty}(n(2-l_n))$ ? The formula $l_n = ...
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1answer
24 views

can use diagonal matrix in a formula to figure out how many characters would occur in all substrings of a string 's'?

Math experts - I'm working through a simple "big O" analysis of algorithms problem comparing two approaches to the longest substring problem. One approach is brute force: checking all possible ...
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1answer
22 views

Little o notation within another little o

To prove $e^{x + o (x)} = 1 + x$ as $x \rightarrow 0$, I can do it directly: $\lim_{x \rightarrow 0} \frac{\log (1 + x) - x}{x} \overset{\text{l'hopital}}{=}\lim_{x \rightarrow 0} \frac{(1 + x)^{- ...
3
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1answer
37 views

Density of linear combination

Let $r_1, \ldots, r_n$ be a set of positive reals. Define \begin{equation*} S = \{a_1r_1+\cdots+a_nr_n : a_i\in \mathbb{N}\}. \end{equation*} Define $\pi(x)= |\{a\in S:a<x\}|$. Is there an ...
2
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0answers
19 views

WKB leading order

I'm learning about the WKB method, and I'm applying it to an assignment. The assignment question asks to find the "leading order" WKB expansion for the particular equation. For WKB you make the ...
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0answers
27 views

how to solve T(n)=T(Logn)+O(1)

Given That $T(1)=1$ Solve following recurrence function $T(n)=T(\log n)+O(1)$ I know the answer is $\log^* n$ but don't know how to prove it. What I tried: $\log(n)+\log(n-1)+\log(n-2)+...+1 = ...
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2answers
55 views

Does $O(\log^2(x))$ imply $O(x)$

Does $O(\log^2(x))$ imply $O(x)$ I have to prove the following: $$\sum\limits_{\substack{n\in\mathbb N\\n\le x}}\Lambda(n)\log(n)=\psi(x)\log(x)+O(x)$$ Applying partial sum I get; ...
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0answers
7 views

Deriving information about asymptotics from finitness of a limit

Let $f_1,f_2:\mathbb{R}\setminus\{0\}\to \mathbb{R^+}$ be two $C^1$ functions and $\alpha:\mathbb{R}\setminus \{0\}\to \mathbb{R}$ be a function from a Zygmund class (in particular it is Holder for ...
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0answers
16 views

How to solve asymptotic recurrence without using Master Theorem

I am working on the following problem. Consider the function $B:\mathbb{N}\to\mathbb{R}$ defined by: $$B(n) = \begin{cases} 1 & \text{if $n\leq 2$,}\\ 3\cdot B(\lceil n/\log_2 n\rceil) + n & ...
4
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1answer
52 views

Product of two sets with density zero has density zero?

Let $A$ and $B$ be two subsets of $\mathbb N$ which have asymptotic density zero. Define $A\times B$ as the set of integers of the form $ab$ with $a\in A$ and $b\in B$. Must $A \times B$ also have ...
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1answer
38 views

Why is $1 - \cos(x)$ of $\mathcal{O}(x^2)$?

I know the definition of order estimates, For the solution to apply we need to show: $ \displaystyle\lim_{x \to 0} \frac{1 - \cos(x)}{x^2} = A \neq 0\space or \space \infty$ But how can one show ...
4
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1answer
38 views

Order of Growth of a Sum

Let $n>k$ and $r$ be arbitrary positive integers. Define $q=k/n$. I want to show that $$ \sum_{i=0}^{rk} \binom{rn}{i}q^i(1-q)^{rn-i}(rk-i)=\Theta(\sqrt{r}) $$ as $r\rightarrow \infty$. I've ...
1
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2answers
12 views

Horizontal and Vertical Asymptotes of functions

So I'm completing a chart analyzing the different properties of three different functions: $f(x)=\log(x^2+6x+9), g(x)=\sqrt{x^2 -1}$ (sorry not sure how to do square roots on here), $h(x)=f(x)(g(x))$ ...
2
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1answer
35 views

Asymptotic of a complex integral

Consider the following integral $$f(x):=\int_x^{+\infty}re^{-(r+ir^2)}dr$$ I want to understand the asymptotic behavior of $f(x)$ as $x\rightarrow +\infty$ Thank you for any suggestion.
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1answer
90 views

Inverse of $x^x$ [duplicate]

Since $x^x$ grows very fast, its inverse should accordingly grow very slow, possibly slower than $\ln(\ln(x))$. I am troubled with finding such an inverse: I only get to the point: $\ln(x)x=\ln(y)$ ...
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1answer
54 views

Evaluate $\int_{x=0}^{\infty}\left(\frac{1}{\sqrt{x}}(1-e^{-x})\right)^{M-1}e^{-x}(1+sx)^{-N}dx,$

I am trying to evaluate $$\int_{x=0}^{\infty}\left(\frac{1}{\sqrt{x}}(1-e^{-x})\right)^{M-1}e^{-x}(1+sx)^{-N}dx,$$ where $s>0$, $M$ and $N$ are positive integers. But seem that the above integral ...
1
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3answers
61 views

Upperbound confusion

Why is the following true? $3n^2-100n+6$ is big $O$ of $n^2$ This has been demonstrated to be true when $c$ is $4$ and $n$ is $10$. $3*100-1000+6 = -694 = 694$ is the absolute value is a big $O$ of ...
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2answers
47 views

Large $a$ asymptotics of $\int_0^{\pi/4} \exp(-a(x^2-\frac{x^4}{3}))$

I'm looking for a way to prove that $\displaystyle \int_0^{\pi/4} \exp(-a(x^2-\frac{x^4}{3}))dx=\int_0^{\pi/4} \exp(-ax^2)dx+o\left(\int_0^{\pi/4} \exp(-ax^2)dx\right)$ as $a$ goes to $\infty$ ...
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0answers
26 views

Proof $\log(cn)$ is in $\Theta(\log(n))$

How can I prove that $\log(cn)$ is in $\Theta(\log(n))$, where $c$ is a constant? I tried to prove $c_1\log(n) \le \log(cn) \le c_2\log(n)$, where $c_1$ and $c_2$ are also constants, but I'm having ...
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3answers
36 views

Show that $\lim_{n\to\infty}\frac{n}{(\log n)^2}=\infty$

I am working on an asymptotic analysis question from a data structures past paper, and I need to show that $$\lim\limits_{n\to\infty}\frac{n}{(\log n)^2}=\infty$$ Could I have a hint for working out ...
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0answers
14 views

Expected size of largest connected component in a binary matrix

Let $C_4(\mathbf M)$ and $C_8(\mathbf M)$ denote the size of binary matrix $\mathbf M$'s largest 4-connected component and 8-connected component of the same value, respectively. For example, the ...
3
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1answer
67 views

A proof involving an infinite sum

I am trying to prove that there exist constants $C_1 > 0$, $C_2>0$ such that$$C_1 \log N \geq\sum_{k=1}^\infty(1 - (1- 1/2^k)^N) \geq C_2\log N$$ where $N\in Z^+$. Could you please give me ...
0
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2answers
36 views

Show that $\sin(\mathcal{o}(x)) = \mathcal{o}(x)$ as $ x\to 0$

So I want to show that $\sin(\mathcal{o}(x)) = \mathcal{o}(x)$ as $ x\to 0$. So far I have thought that my result will come from showing $ \displaystyle \Big|{\frac{\sin(f(x))}{x}}\Big| \to 0$ as ...
1
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3answers
44 views

$N^{1/2}$ and randomness

I apologize if this question is overly vague, but part of the reason I am asking is because I don't know a more precise way of discussing these ideas. To state a general question: What, if any, ...
2
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2answers
36 views

$\int_2^x\frac{dt}{\log^kt}=O\left(\frac{x}{\log^kx}\right)$

I seek to prove the identity $$\int_2^x\frac{dt}{\log^kt}=O\left(\frac{x}{\log^kx}\right)$$ I was given the following hint: Split the integral into $\int_2^{f(x)}+\int_{f(x)}^x$ for a ...
2
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2answers
24 views

Does one of these conditions on a sequence imply the other one?

Let ${(r_n)}_{n \geq 0}$ be a sequence of integers $\geq 2$. Set $q_n=\prod_{i=0}^{n-1} r_i$ (agreeing with $q_0=1$). I want to know whether one of these two conditions implies the other one (I think ...
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1answer
40 views

Compute $(\ln(n!))^2$

In a discrete mathematics past paper, I must solve the following problem: We know (from the Stirling approximation) that ...
1
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1answer
23 views

Asymptotics of $(\cosh(x+c)-\cosh(c))^{-\frac{1}{2}}$

let $c>0$ be a constant and consider the function $$\frac{1}{\sqrt{\cosh(x+c)-\cosh(c)}}, x>0.$$ I'm wondering how the asymptotic expansion for $x\downarrow 0$ look like!? In case of $c=0$ the ...
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0answers
11 views

Graphs Approaching Asymptotes

I've been wondering this for a while. For graphs that approach asymptotes, are there certain formulas that can determine the distance between the graph and the asymptote as $x$ gets infinitely small ...
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0answers
20 views

Approximation of Hermite functions

I'm looking for an "easy" proof of the asymptotic expansion of Hermite functions ($f_n(x)=H_n(x)e^{-x^2/2}$ where $H_n$ is the Hermite polynomials). The asymptotic expansion is $$ f_n(x) \sim_{n ...
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0answers
31 views

How to asses the order of combinations

Let $\{a_i\}_{i=1}^m$ be some increasing sequence, bounded away from zero. How to see that as $n\to\infty$, we obtain $$\begin{pmatrix} n\\ m \end{pmatrix}^{-1}\sum_{i=1}^m\begin{pmatrix} ...
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0answers
32 views

Two-term asymptotic approximation for roots of a polynomial (dominant balance)

I'm trying to find the roots to the following equation: $t^5 - \epsilon t^3 + \epsilon^3 = 0$ as $\epsilon \rightarrow 0$. From expansion $t= \epsilon^{\alpha}t_1 + \epsilon^{2\alpha}t_2 + ...
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1answer
13 views

$ n - \sqrt{n}$ $\Theta$ Complexity

$ n - \sqrt{n} \leq n - \sqrt{n} + \sqrt{n}=n=O(n)$ But I don't know what I should do about $\Omega(.) , \Theta(.)$ Should I try to solve it with lim?
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0answers
13 views

Additive error in Stirling's Approximation

I know that Stirling's Approximation is asymptotic to $n!$ as in the ratio approaches $1$ (which is the definition). But as far as I have noticed, the additive error diverges. Is this a common ...
3
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1answer
25 views

Trapezoidal Rule in Stirling's Approximation

https://en.wikipedia.org/wiki/Stirling's_approximation Here $$\sum \limits_{i=1}^n log(i) $$ is approximated as $$\int^n_1log(x) dx\ +\ 1/2\ log(n)$$ but I would approximate it as ...
5
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1answer
53 views

Decay of Fourier Transform

I encountered the following statement, and I cannot see why it is true(if it is). Suppose $f$ is a nonnegative, bounded, compactly supported and measurable function with the following properties: ...
7
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0answers
45 views

Decay of amplitude integral

Consider the function $$ f(\vec{x}) = \int_{\Bbb R^3} {\frac{ e^{-i\,\vec{x}\cdot\vec{k}}}{\sqrt{\vec{k}^2 + m^2}}} d^3 k $$ from Zee's Quantum Field Theory in a Nutshell. He argues like this: ...
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23 views

Node potentials of minimum cost flow successive shortest path algorithm

I have a simple directed graph $G(V,E)$ that has a source $s$ and sink $t$. Each edge $e$ of $G$ has positive integer capacity $c(e)$ and positive integer cost $a(e)$. I am trying to find the minimum ...
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23 views

Prove or provide a counter example: $f(n) \notin o(g(n))$ and $f(n) \notin \omega(g(n)) \implies f(n) \in \theta (g(n))$

$f(n) \notin o(g(n))$ and $f(n) \notin \omega(g(n)) \implies f(n) \in \theta (g(n))$ Without giving me the answer, please tell me how would you manipulate this? I know that $f(n) \in \theta (g(n)) ...
3
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2answers
41 views

Limit with polylog

How do you show the following limit? $$\lim_{x\to\infty} x\log(-e^x + 1)+\operatorname{Li}_2(e^x)-\frac12x^2=\frac{\pi^2}3$$ Where $\operatorname{Li}_n(x)$ is the polylogarithm. This question is ...
0
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0answers
32 views

Asymptotic analysis of Integrals of powers of sine and their application to intersections of hyperspheres

I am trying to estimate the probability of an event in an algorithm. For simplicity, assume there are two hyperspheres of radius $r$, at a distance $r$ from each other. I am looking to see how the ...
0
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1answer
23 views

comparing expressions confusion

This formula is actually from a big $O$ notation example, but I am confuse about the mathematical formula. I read that: if $n$ and $c$ are $1$, $3n^2 - 100n + 6$ is not a big o of $n^3$ or $cn^3 ...
2
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0answers
39 views

asymptotic expansion of hermite functions

Does anybody know how to proof the first asymptotic expansion of this page: http://en.wikipedia.org/wiki/Hermite_polynomials#Asymptotic_expansion ? (and how the physicist use this asymptotic ...
3
votes
1answer
89 views

$\sum \limits_{n \geq 0}a_n \frac{x^n}{n!}=e^{x+x^2/2}$ implies $a_n \sim \frac1{\sqrt2} n^{\frac n2}e^{ -\frac n2+\sqrt n -\frac14 }$

Prove the following asymptotic formula for the exponential generating function coefficients of $e^{x+x^2/2}$: $\; \; a_n \sim \frac1{\sqrt2} n^{\frac n2}e^{ -\frac n2+\sqrt n -\frac14 }$ Stanley ...
2
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2answers
34 views

is there a concept of asymptotically independent random variables variables?

To prove some results using a standard theorem I need my random variables to be i.i.d. However, my random variables are discrete uniforms emerging from a rank statistics, i.e. not independent: for ...
3
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3answers
64 views

Complete elliptic integral of the first kind $K(m)$ asymptotc expansion at $m = -\infty$

What is the asymptotic behavior of $K\left(-\frac{1}{\delta^2}\right), \delta > 0$ when $\delta$ tends to zero? Here $$ K(m) = \int\limits_0^{\pi/2} \frac{d\theta}{\sqrt{1 - m\sin^2 \theta}}, $$ ...
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vote
2answers
31 views

Is this inductive big O proof possible / Does this question make sense?

Prove that $\sum_{i=j}^k \frac 1i$ is $O(\ln(k)-\ln(j-1))$ using induction for all $i$. The way I understand this question, it's nonsense - $i$ is the iteration variable, not something that can be ...
0
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2answers
22 views

Analysis of Algorithms - Big O Notation Equivalences - Limits

Please see below block question from review for test. True Or False? Justify Your answers A) is 2^(n+1) = O($2^n$) B) is 2^2n = O($2^n$) C) is log($n^2$) = O(logn) D) is ...
0
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0answers
8 views

WKB for a sixth order eigenvalue problem

I have the following 6th order eigenvalue problem: $$ (D^2 - \alpha^2)^3 y(x) = -\alpha^2 \lambda Q(x) \, y(x), \quad 0 < x < 1, \quad \text{+ BCs}, $$ where $D = \mathrm{d}/\mathrm{d} x $, ...