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

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22 views

Determing stretching variable in inner expansion of boundary layer problem

I am studying perturbation theory, and I have a problem when reading the book "Introduction to Perturbation Methods" by M.H. Holmes. This is about boundary layer. We know when seeking inner expansion, ...
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0answers
33 views

Limit of monotone decreasing function on generalised inverse.

Consider a right-continuous, monotone decreasing, non-negative function $\bar F(x)$ (its the tail of a probability distribution, but that doesn't matter). Now let \begin{equation} I_{n}=\{x : \bar F(x)...
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0answers
17 views

Question about asympotic behavior of $\frac{1}{s}\int_0^s u(x,t) dt$ .

I am just reading a paper, in the final theorem, the author wants to prove that $u(x,t)$ converges to some $v(x)$ in the $L^2$ norm as $t$ $\to$ $\infty$. But in the proof, he defines a $w(x,s)=\frac{...
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0answers
33 views

Why should $\phi'$ and $\phi''$ be $\mathcal O(1)$?

As Strogatz writes in his book Nonlinear Dynamics And Chaos (p. 64) There are often several ways to nondimensionalize an equation, and the best choice might not be clear at first. Therefore we ...
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1answer
37 views

Why is this big oh $O(n^3)$

Why is this big oh $O(n^3)$? (b) Give a good big-Oh bound on the function $$f(n)=2^{\log_2 n} n^2 + 3n^2 \log_2 n +n -17$$ I am not sure on how to solve this. If someone could help me solve, I ...
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4answers
64 views

Trying to solve recurrence $T(n)=3T(n/3) + 3$

I'm trying to solve the following recurrence without using the Master Theorem: $$T(1)=1;$$ $$T(n)=3T(n/3) + 3$$ My attempt: $T(n) = 3T(n/3) + 3$ $ = 3(3T(n/9) n/3)) + 3)$ $ = 9T(n/9) + 9$ $ = ...
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1answer
33 views

Prove $O(f(n)+g(n)) = O(f(n))$ when $g(n)=O(f(n))$

Given $g(n) = O(f (n))$, how can I prove that the following expression is true: $O(f (n) + g(n)) = O(f (n)) \tag1$ So I just write down what it says: $g(n) = O(f (n)) <=> f(n) \le c_1 g(n)...
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2answers
20 views

Asymptotic upperbound in multiplication

How can someone calculate the asymptotic upperbound of $2^nn^2$? The first term ($2^n$) grows much faster than the second, but saying that as a final result $2^nn^2 = O(2^n)$ would only be true in the ...
1
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1answer
37 views

What is the asymptotic behavior of this integral?

The function $F(x)$ is defined by the following integral $$F(x)=\int_0^x\frac{\left(1-y^3\right)^a}{\sqrt{\left(\dfrac{1-y^3}{1-x^3}\right)^b-\left(\dfrac{y}{x}\right)^4}}\,dy$$ where $a$ and $b$ ...
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0answers
17 views

Find Theta Class of T(n) = T(3n/4) + T(n/6) +5n [duplicate]

I'm not quite sure I can apply the Master Theorem to T(n) = T(3n/4) + T(n/6) + 5n. It is not in the normal form of T(n) = aT(n/b) + f(n). Is it possible to apply the MT to it? If not, can the ...
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0answers
30 views

Finding the inverse of a function involving logarithms

Let $A \asymp B$ mean that there exists universal constants $m,M >0$ such that $mA \leq B \leq MA$. Let $k,n \in \mathbb{N}$ be such that $\log n \leq k \leq n$. I want to prove that $$ k \log(\...
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0answers
36 views

Find the best Big-O estimate

Find the best (i.e., lowest) big-O estimate for the following function: $f(n) = 1 + 3 + 5 + 7 + ...+ (2n-1)$ Since the sum would be $f(n)= \frac{1 + n(2n-1)}2$, that would leave $\frac {2n^2 -n +...
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1answer
72 views

A sum of Stirling numbers of the second kind

Find a formula (either exact or asymptotic in $N$) for $S(N)$, where \begin{equation} S(N) = \sum_{n=N}^\infty \sum_{k=N}^n \sum_{j=0}^k \binom{k}{j} (-1)^{k-j} (1+j)^n \frac{t^n}{n!}. \end{equation} ...
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3answers
53 views

Is $n^\frac{1}{10} \in O((\log n)^{10})$?

This question came up in a recent discussion: is $n^\frac{1}{10} \in O((\log n)^{10})$? First time I've come across a power of a log in a long time, and as far as I recall, there are no identities ...
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1answer
26 views

Big O Notation asymptotic relationship

I cannot prove correctness/incorrectness of the implication of two functions f(n) and g(n) in Big-Oh/asymptotic notation $$g(n) = \Omega(f(n)) ) \implies g(n) = O(n^2f(n))$$ I believe $g(n) = \Omega(...
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0answers
52 views

Can you get the average order of $ \left( 1+|\mu(n)| \right)^{M(n)} $, where $\mu(n)$ and $M(n)$ are the Möbius and Mertens functions, respectively

When yesterday I was interested in do a little study about the arithmetic function $$f(n)=\left( 1+|\mu(n)| \right)^{M(n)},$$ defined for integers $n\geq 1$, which $\mu(n)$ is the Möbius function and $...
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0answers
13 views

Renormalization Group

I am studying singular perturbation technique right now. Can anyone suggest introductory books on singular perturbation using renormalization group method? I have several books on perturbation theory ...
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1answer
39 views

Why does $\frac{\frac12 x+\frac18x^2+O(x^3)}{\frac12x-\frac18x^2+O(x^3)}=1+\frac12x+O(x^2)$?

I was reading the solution to a limit through Taylor expansion but did not understand this passage: $$g(x)=\frac{1-\sqrt{1-x}}{\sqrt{1+x}-1}=\frac{\frac12 x+\frac18x^2+O(x^3)}{\frac12x-\frac18x^2+O(...
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4answers
56 views

How to check if $n!$ is $ O(2^n)$

How can I check if $n! \in O(2^n)$? The definition of $f$ being $O(g)$ is $f(n) \le c g (n)$, where $c>0$. So it would mean $n! \le c 2^n$. What is the clearest way to solve this? (As I am ...
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1answer
42 views

$N$ is approximately linear in $d$ for $N^d=\frac12 e^{N}$

let us look at the function $N^d e^{-N}$, for each $d\in \mathbb{N}$. The graphs of the function for various values of $d$ show a striking phenomenon: the graph look parallel, and with a near-constant ...
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1answer
24 views

For what maximum positive $k$ is $2n \sin^{2} \frac{\pi}{n} > \tan \frac{k\pi}{n}$ true?

I am trying to find the maximum value of $k$ such that the inequality $$2n \sin^{2} \frac{\pi}{n} > \tan \frac{k\pi}{n}$$ is satisfied. I impose restrictions that $n \in \mathbb{Z}$ with $n \geq 5$...
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1answer
78 views

Does “the functions agree at infinity” mean anything?

I want a way to describe how two continuous functions $f,g \colon (X-x) \to Y$ might "share a limit" at the point $x$ when unfortunately neither of $\displaystyle \lim _{y \to x}f(x)$ or $\...
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1answer
29 views

Average of numbers converges, then what happens to the maximum

Let $\{a_n\}$ be a positive sequence of numbers such that $\displaystyle \frac1n\sum_{i=1}^n a_i \to a$ where $a>0$. Then can we say anything about the order of $\displaystyle b_n=\max_{i\in n} a_i$...
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0answers
42 views

10-Subset Sum: Given a set of integers K and an integer M, is there a subset of exactly 10 elements of K whose sum equals M?

I understand that the more general Subset Sum problem is NP-complete, but I am under the assumption that this more specific version of the problem can be solved in polynomial time. However, I can't ...
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0answers
36 views

More elegant derivation of the shift in median bin occupancy

In answering Median of a multinomial variable, I found to my own surprise through a somewhat tedious calculation that the expected value of the median of the ball counts in $3$ bins into which $n$ ...
2
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1answer
60 views

Asymptotic behavior of integrals of Legendre polynomials

By definition $\int_{-1}^1 |P_n(x)|^2 dx = O(n^{-1})$. What about the other powers? Do we know how $\int_{-1}^1 |P_n(x)|^k dx$ behaves for any $k$? Maybe $O(n^{-k/2})$?
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1answer
33 views

Function $(2.2)^n$ — what is it?

The running time of an algorithms is $(2.2)^n$. I have to tell what is the maximum $n$ for reaching 1.000.000 steps. What type of a function is $(2.2)^n$? How its output depends on the input $n$? ...
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0answers
31 views

I am looking a comparison of this computation and Riemann's approach for $lcm(1,2\ldots,x)$

Looking a comparison with a reasoning due to Riemann, I ask to me about the behaviour as $x\to\infty$ of the following arithmetical function $$ \left( \prod_{n\leq x}n^{-\mu(n)}\right)\cdot \left( \...
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2answers
76 views

Approximation of a quotient that involves the Lambert function.

I would like to find an asymptotic upper bound for $$\frac{-\ln n}{W(- \ln^{-c}n)}$$ where $c$ is positive and $W$ is the Lambert function. More precisely, I want something which dominates this ...
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14 views

Is $O(x^2)$ equal to OR a tighter bound for $O(x(x-y))$ if $x, y >0$ and $x>y$ alway hold?

In the question, $O$ is the Big-O notation, please see https://en.wikipedia.org/wiki/Big_O_notation. $x$ and $y$ are variables. Here, let me give you an example showing there exist such questions in ...
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2answers
38 views

Big O for a $\cos$ series

I have to show that $ \sum_1^N \cos(nx) = O(\frac 1{|x|}), [-\pi, \pi] $, x different from 0. I really don't know how to show that. I obviously know that $\cos(nx)$ is bounded by $1$, I know what ...
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0answers
35 views

Products of $k^{\mu(k)}$, where $\mu(n)$ is Möbius function, and the Prime Number Theorem

We can write $$e^{-\Lambda(n)}=\prod_{d\mid n}d^{\mu(d)},$$ where $\mu(n)$ is the Möbius function and thus $\Lambda(n)$ is von Mangoldt's function. Then taking the product from $1$ to $N$ we've for ...
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1answer
20 views

Simple Sigmoid function that levels off at specific points

I need to construct a simple Sigmoid function that levels off at specific values of x, as in this curve: What is the most simple Sigmoid function that I can use ...
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5answers
101 views

Hello. I need to show that $\sqrt n$ grows faster than $(\log n)^{100}$ [closed]

Is there an easy way to show that $$\lim_{n\to \infty}\frac {(\log n)^{100}}{\sqrt n}=0 $$
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1answer
13 views

Which one is asymptotically larger?

The question is to find out which among $n^\sqrt{n}$ or $n^(log_2 n)$ is asymptotically larger? Now as a solution I read somewhere that if we take log on both sides and then compute which one is ...
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1answer
46 views

Determine numerical infinity for Schrodinger equation $−\psi''(z) − (iz)^ N \psi(z) = E\psi(z)$

Consider the following one dimensional Schrodinger equation within the complex plane of $z$ $$ −ψ''(z) − (iz)^ N ψ(z) = Eψ(z). $$ where $N$ can be any real number, the boundary condition is $ψ(z) → 0$ ...
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0answers
17 views

Asymptotic growth of $T(n) =T(n/3) + T(n/6) + n ^ \sqrt{\log_2n}$

Can you give a solution or a hint for finding asymptotic bound for following recurrence relation: $T(n) =T(n/3) + T(n/6) + n ^ \sqrt{\log_2n}$ I know from the source of the problem that it is $\...
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2answers
47 views

If for some $n\in\mathbb{N}$, $\lim\limits_{x\to\infty}\frac{f(x)}{x^n}$ exists, then $f$ is rational

I don't know if this statement is true. Let $F$ be a function and suppose $n>0$, $n\in\mathbb{N}$ is the greatest such that there exists $L\mathbf{\neq 0}$ such that $\lim\limits_{x\to\infty}\frac{...
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0answers
37 views

Asymptotic distribution of zero-drift Geometric Brownian Motion as $t \to \infty$

If we fix the drift at $\mu = 0$, then my geometric brownian motion will have stationary mean, but it seems that the variance will grow without bound. What does the limiting distribution look like for ...
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0answers
9 views

Rate of expected value of $\mathcal{O}_p$

This is certainly very basic but what is the rate of the expected value of a random variable that is bounded in probability. For example, let $X_n = \mathcal{O}_p (a_n)$ is it true that $\mathbb{E} [...
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1answer
14 views

The Big O Notation and the Thetha Notation

I was instructed to find whether $$x*⌈x⌉*⌊x⌋$$ is$$ O(x^3) $$ or $$Big Thetha(x^3)$$ I tried to do a solution by cases, and i got : if x is not an integer, $$x=b+є$$ $$⌈x⌉=b+1$$ $$⌊x⌋=b$$ Then $$x*⌈x⌉*...
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0answers
24 views

Linear combination of asymptotic series

I need to compute an expression of the form \begin{equation}J=\sum_{k=0}^N a_k F_k(z)\end{equation} where z is a large parameter, and a_k are easily computable (k and z-dependent) coefficients. I ...
3
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1answer
42 views

Which singular perturbation method should be used for this system?

Consider the system $$ \varepsilon \dfrac{dx}{dt} = -(x^3 - ax + b)$$ $$ \dfrac{db}{dt} = x - x_a$$ where $\varepsilon \ll 1$. Applying regular perturbation methods isn't suitable because when $\...
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1answer
26 views

Big O notaion O(n) and logaritms [closed]

Can someone explain me the subjects Big O notation and logarithms please? I can't understand those subjects For example if I have a question like this: recall that logan is the power to which you ...
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0answers
22 views

Finding asymptotycs of partition function

I have been stuck in this problem and have no idea of how to solve it. I have a hint from the book but don't really see how to use it. Any suggestion or hint would be really appreciated. Thanks! ...
3
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1answer
47 views

Verifying a step in the prime number theorem

This is an excerpt from Shapiro, "Introduction to the theory of numbers": Suppose that we have an estimate of the form $$|R(x)|\le \alpha x$$ valid for all sufficiently large $x$ (say $x\ge x_2$). ...
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0answers
21 views

Asymptotic upper bound for recurrence relations

The question is to find asymptotic upper bound for recurrence: (1) $T(n)=(T(n/2))^2$ and (2) $T(n)=(T(\sqrt{n}))^2$ with $T(n) = \text{n for n} \leq 2$ I think I will be able to find the ...
0
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1answer
33 views

Asymptotic upper bound $T(n)=(T(n−1))^2$

The question is to find asymptotic upper bound for recurrence: $T(n)=(T(n−1))^2$ $T(n) = \text{n for n} \leq 2$ My attempt: I've tried to use substitution method and getting: $T(n) = ...
0
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0answers
19 views

Big O notation question (conceptual)

In my class, we have defined that $$ f(x) \ll g(x) $$ on $A$ if there exist a strictly positive c such that $$ |f(x)| \le cg(x) $$ for every $x$ on $A$. I'm a bit confused. Say that $ f(x) = x$ ...
4
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2answers
96 views

Prove that the value of the constant $C$ must be $1$

After proving the prime number theorem in class, our professor directs us to a remark by Lagrange that for large values of $x$, $\pi(x)$ is approximately equal to $$ \frac{x}{\log x - B}. $$ (This is ...