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

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28 views

How to study the asymptotic behavior of $f(r)=\int_0^1 dx\, \text{Li}_2\Big(1-\frac{r}{x(1-x)}\Big)$ for small $r$?

How does one study the asymptotic behavior of the integral $$f(r)=\int_0^1 dx\, \text{Li}_2\Big(1-\frac{r}{x(1-x)}\Big)$$ as $r\rightarrow0$ from positive values? Here $\text{Li}_2$ is the ...
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1answer
28 views

Number of rationals with denominator less than $N$

This is probably a duplicate since it seems like elementary number theory, but didn't find it after a cursory search. Let $r(N)$ be the number of rationals in $[0,1]$ with denominator less than or ...
2
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2answers
50 views

Order estimates question and big O notation

How can I show that $y(x) = 1 - \cos(x)$ is $\mathcal{O}(x^2)$ for $|x| <<1$ ? Additionally, with the $|x| << 1$ is there a precise definition? I tried to google it but nothing conclusive ...
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0answers
34 views

$f(n)=3f(\frac{n}{3})+O(logn)$

I was asked to figure out the time complexity analysis for the following recurrence relation: $f(n)=3f(\frac{n}{3})+O(logn)$ I worked it out as O(nlgn), Would like to know if this is right or ...
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0answers
22 views

Asymptotic expansion of integrals of the form $\int_{\mathcal{D}} \exp(\lambda\, \phi(x))\, g(x) \,dx$ for small $\lambda.$

In the limit $\lambda\to\infty$ the asymptotic expansion of integrals of the form $\int_{\mathcal{D}}\exp(\lambda\,\phi(x))\,g(x)\,dx$ (where $\mathcal{D}\subseteq \mathbb{R^n}$ denotes the domain of ...
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2answers
36 views

Big O notation and limits

I'm wanting to take the $\lim_{x\to \infty} \frac {O(1)}{x^s}$, where $O(1)$ is Big O notation and $s>1$. I can see that it will be zero but I'm wanting to do it somewhat rigorously. Can I take the ...
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1answer
27 views

Show a function similar to $(1/x)lnx$ becomes small as x grows

I am tasked with showing that, for $l \gg k$, $$ t = \frac{1}{l-k}\ln(l/k) $$ is small (it is given that $t$ is positive). Intuitively, this seems correct because it is 'similar' to $$ \frac{1}{x} ...
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0answers
46 views

Proving $n^{10\log(n)} = O((\log^n(n))$

I need to decide which of the following is correct: $n^{10\log(n)} = O((\log^n(n))$ $n^{10\log(n)} = \Theta((\log^n(n))$ $n^{10\log(n)} = \Omega((\log^n(n))$ So I'm saying $n^{10\log(n)} = ...
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0answers
13 views

Question about big omega proof

I'm not sure if I should post it here or in StackOverflow, but anyway... Prove that: $n^5-2\log{n}=\Omega{(n^5)}$. Proof: We need to find $c, n_0 \geq0$ such that, for all $n \geq n_0$, ...
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0answers
27 views

Asymptotic expansion of a Laplace-type integral with a “manifold of maxima”

Consider the integral $$ I(\alpha)=\int_0^1 dx_1 \int_0^1 dy_1\int_{x_1}^1dx_2\int_{y_1}^1dy_2\,e^{-\alpha(x_2-x_1)(y_2-y_1)} $$ in the limit $\alpha\rightarrow\infty$. To find the asymptotic ...
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0answers
54 views

Asymptotic large order approximation for Bessel function expression

How does one find the asymptotic large order approximation for $\sup_{0\le x\le\infty} \left(\sqrt{x} J_n(x)\right)$, where $J_n$ is the Bessel function of the first kind and order $n$. This is NOT a ...
4
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0answers
66 views

Growth of $n!!\dots !$

The asymptotic growth of the factorial function $n!$ is famously given by Stirling's formula as $$n! \sim \sqrt{2 \pi n}\left(\frac{n}{e}\right)^n$$ Is there a similar formula for the iterated ...
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1answer
27 views

Suppose that $f (x)$ is $O(g(x))$. Does it follow that $2^{f(x)}$ is $O(2^{g(x)})$?

Suppose that $f(x)$ is $O(g(x))$. Does it follow that ? First, I start from for some $c$ is a real number. Then, I find . But, if i start from , I just find . I confused with that different form.
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20 views

How to do this asymptotic task?

Let a(n) be the amount of natural numbers, which are smaller than n, and their prime divisors are only 2 and 3. For example: 6 is good, because it only has 2 and 3 has prime divisors, but 10 is not ...
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1answer
28 views

Suppose that $f (x) =O(g(x))$. Does it follow that $\log |f (x)| =O(log |g(x)|)$?

Suppose that $f(x)=O(g(x))$. Does it follow that $\log |f (x)|=O(log |g(x)|)$? I start from $f(x)=O(g(x))$, until I get Does it mean $\log |f (x)|=O(log |g(x)|)$?
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0answers
21 views

Non-deterministic multiplication algorithms

Are there any algorithms for non-deterministic Turing machines that can compute the decision problem $mn=x$ (where $m=O(n),x=O(n^2)$) faster than the equivalent deterministic algorithm? Equivalently, ...
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1answer
38 views

How to solve this recurrence, $T(n) = T(\sqrt{n}) + n$ using recursive tree method?

How to solve this recurrence, $ T(n) = T(\sqrt{n}) + n $ using recursive tree method? I draw the tree and got a sum, $ T(n) = T(1) + ( n + n^{\frac 12} +n^{\frac 14}+n^{\frac 18}+\ldots +1) $ I need ...
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2answers
33 views

How to prove that $2^{n+1} = \Theta(2^n)$?

I have a problem were I need to prove big theta. $f(n) = 2^{n+1} = Θ(2^n)$. I proved that this was true for big O but I'm not sure how to go about proving big Theta.
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4answers
51 views

Does the inequality $ n! > A \cdot B^{2n+1}$ hold for sufficiently large $n$?

Suppose $A,B >0$ are given constants. Is it possible to find a large enough $n \in \mathbb{N}$ such that $$ n! > A \cdot B^{2n+1}?$$
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33 views

Proof $\mathcal{O}(f(n)) = \mathcal{O}(g(n)) \iff f(n) \in O(g(n)) \land g(n) \in \mathcal{O}(f(n))$

There is an exercise that ask me to prove this logic formula about the complexity of algorithms: $\mathcal{O}(f(n)) = \mathcal{O}(g(n)) \iff f(n) \in O(g(n)) \land g(n) \in \mathcal{O}(f(n))$ ...
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2answers
20 views

How would I go about proving these.

I need to prove or disprove these two problems, but I'm not sure I did it right. $$(a).\quad f(n) = 2^n+1 = O(2^n)\\ (b).\quad f(n) = 2^n+1 = Θ(2^n) .$$ What I tried for the first one is, ...
1
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1answer
23 views

Simplifying $f(n)$ by substituting, for $n$, an appropriately chosen function $n(x)$ to observe limiting behaviour of $f(n)$. Is this justified?

Say, I'm comparing two functions $f(n) = (ln(n))^2$ and $ g(n) = n^{0.01}$ as $n \rightarrow \infty$, by evaluating $\lim_{n \rightarrow \infty } \frac{f(n)}{g(n)} = \lim_{n \rightarrow ...
12
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2answers
248 views

When is this sum of perfect powers bounded

For any positive integers $n,d$, let $$ A_d(n)=\frac{\sum_{k=1}^n k^{2d}}{n(n+1)(2n+1)} $$ It is easy to see (and well-known) that for fixed $d$, $A_d(.)$ is a polynomial of degree $2d-2$. Writing ...
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0answers
8 views

Comparing Growth Rates

Suppose I want to compare the growth rate of some function $f(x)$ as $x\to a$ and another function $g(x)$ as $x\to b$. How do I go about doing that? For example, I want to decide whether ...
4
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1answer
47 views

Upper bound for $\prod_{ 5 \leq p <n} p^{\frac{n}{p-1}}$

Does anyone know how I could get a good upper bound for the following: $$R := \prod_{\substack{ p \; \text{prime} \\ 5 \leq p < n}}p^{\frac{n}{p-1}}$$ I'm not that skilled at asymptotic analysis ...
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0answers
21 views

finding the n in Aymptotic notations

consider any quadratic function $f(n) = an^2 + bn + c$, where a, b, and c are constants and $a > 0$. Throwing away the lower-order terms and ignoring the constant yields $f(n)= \theta(n2)$. ...
1
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1answer
97 views

Expectation and Variance of random walks

Consider random walks of fixed length (e.g. $5$) starting at node $u$ in an undirected and connected graph with $N$ vertices. If a node $k$ has $N_k$ edges, the probability of the walk reaching any of ...
3
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1answer
70 views

Estimating an unusual infinite sum

I came across the following summation, which I would like to estimate. I only need an answer which is correct up to a constant multiple; one can assume that $a, b, c$ are real numbers in the range ...
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3answers
50 views

Prove line asymptotic to curve

I have a function denoted as: $f(x) = \frac{x}{1+e^\frac{1}{x}}$ I want to prove the line: $g(x)= \frac{x}{2} - \frac{1}{4}$ Is asymptotic (slant asymptote) to the above function when approaching ...
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3answers
40 views

Asymptotic notation (big Theta)

I'm currently in the process of analyzing runtimes for some given code (Karatsuba-ofman algorithm). I'm wondering if I'm correct in saying that $\Theta(\left\lceil n/2\right\rceil) + \Theta(n)$ is ...
1
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1answer
57 views

Big-theta notation

I was wondering about big-theta ($\Theta$) notation. A) Is $\Theta(n/2) \leq \Theta(n)$ for $n$ being an integer? I know that $n/2 = O(n)$, but does it also mean that $\Theta(n/2) \leq \Theta(n)$? ...
2
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3answers
44 views

Lower bound for the falling factorial $(2n)_{n}$

I'm looking for a lower bound for the falling factorial $$(2n)_{n}:= \frac{(2n)!}{n!}$$ I saw on Wikipedia that $n! > \sqrt{2{\pi}n}(\frac{n}{e})^n$ . So I assume that a possible lower bound ...
1
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1answer
33 views

Dealing with floor function in binomial coefficients

I'm trying to estimate $\binom{n}{\left \lfloor{\alpha n}\right \rfloor }$ asymptotically using Stirling's formula. However, I'm a little lost with what to do about the floor function here. In the ...
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1answer
38 views

Stuttering Subsequence Problem - Explain the example

I'm reading an article that deals with solving the stuttering subsequence problem in $\Theta (n)$. The article can be found here: http://www.cse.yorku.ca/~andy/pubs/Stutter.pdf Some background on ...
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3answers
48 views

Why does $\lim_{ t\to 0} \frac{o(t^2)}{t} = 0$?

Why does $\lim_{ t\to 0} \frac{o(t^2)}{t} = 0$? $\sqrt t = o(t^2) \implies \lim_{t\to 0} \frac{\sqrt t}{t} = \infty$ Maybe I don't understand completely the little-o notation.
2
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1answer
46 views

I need to show the following two limits

First, for $a>-1$: $$\lim_{n\to\infty}\frac{a+1}{n^{a+1}}\sum_{j=1}^nj^a = 1$$ Second, for $p>0$: $$\lim_{n\to\infty}\frac{e^a-1}{e^{a(n+1)}}\sum_{j=1}^ne^{aj} = 1$$ In particular, why do we ...
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1answer
16 views

Is Θ(⌈x/4⌉) = Θ(x)?

I'm currently working on aysmptotic notation. I know the basic laws of big theta, O, and omega. But I'm having a little confunsion in understanding simplifying the expressions (if that's even ...
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0answers
17 views

Leading behaviour of DE at infinity

This is taken from the book of Bender and Orszag, problem 3.44. Find the leading behavior as $x\rightarrow+\infty$ of the differential equation: $x^3y'' - (2x^3 -x^2)y' +(x^3-x^2-1)y=0$ Explain ...
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0answers
26 views

How to find asymptotic cost of matrix filling algorithm . Big O Notation

So I have a list X of N strings each of length M that will be called x_i for the ith index in X Example ...
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0answers
31 views

What are some examples of asymptotic expansions of integrals displaying the Stokes phenomenon?

With the term Stokes phenomenon we refer to how the asymptotic behaviour of a function can differ in different regions of the complex plane. What are some examples of asymptotic expansions of ...
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1answer
46 views

Big-O Constants Rule Question for not-monotonically non-decreasing functions

I know that for positive monotonically non-decreasing functions, f(n) and g(n), f(n) = O(g(n) + c) entails f (n) = O(g(n)) Why does this always true only for ...
2
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1answer
20 views

Expansion of cumulant transform

Verify the following expansion for a cumulant generating function of a random variable $X$. \begin{align} \kappa(t) & = \mu t + \frac{1}{2}\sigma^2t^2+\frac{1}{6}\rho_3\sigma^3t^3 + ...
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0answers
41 views

How does the Stokes phenomenon appear in the asymptotic expansion of $\int_0^\infty \frac{e^{-zt}}{1+t^4} dt$ for $z \to \infty$?

Consider the asymptotic $z \to \infty$ behaviour of the function $$ \tag 1 I_1(z) \equiv \int_0^\infty \frac{e^{-zt}}{1+t^4} dt.$$ This converges for $\Re(z) > 0$, and the asymptotic expansion $$ ...
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0answers
53 views

How to determine a Big-O estimate for an algorithm

This question has been mentioned in the forum but with a different approach. I need to determine a Big-O estimate for the number of operations of the algorithm below taking into account only additions ...
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3answers
92 views

Propose an algorithm to find a “celebrity”

A celebrity is a person that everyone knows, but he doesn't know anyone. If we think of a group of people as a graph, where if there is an arrow from $A$ to $B$ that means "$A$ knows $B$", then a ...
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2answers
51 views

Probability that colored balls are separated

Say we throw $b$ blue balls and $r$ red balls uniformly into $n$ boxes. The probability that no box contains a red as well as a blue ball is then, by the inclusion exclusion principle: $$p = ...
1
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1answer
26 views

Growth Rates of F(n) vs. F(n) + F(n-1) + … F(1)

I am trying to understand growth rates between a function and its sum recursively. For example I understand that if: $F(n) = n$ Then the sum $n + (n - 1) + ... 2 + 1 = \frac{n(n-1)}{2}$ which is ...
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0answers
52 views

Algorithm for matrix addition and multiplication

Let $m$, $n$ be integers such that $0 \leq m,n < N$. Define: Algorithm A: Computes $m + n$ in time $O(A(N))$ Algorithm B: Computes $m \cdot n$ in time $O(B(N))$ Algorithm C: Computes $m\bmod n$ ...
0
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2answers
35 views

Please provide additional information for a Big-O problem solution

I am studying a Big-O example but I just do not get the idea. I have already seen that this question was asked in this forum but I am still confused. Can someone please provide another explanation so ...
1
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1answer
31 views

Why do O(logn) & O(exp(n)) Have Polynomial & Non-Polynomial Running Time Complexities Respectively Despite Their Taylor Series?

I understand that a function, say $f(x)$, belongs to a class $O(g(x))$ iff: $$ \exists k > 0 \ \ \exists \ \forall n > n_0: |f(n)| \leq |g(n) \cdot k| $$ I also know that $log(x)$ is has ...