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

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2
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1answer
29 views

Multiplying two matrices using Strassen vs squaring identical matrices

I have an assignment question such as follows: when using the Strassen algorithm we have 7 subproblems usually, and I suppose this applies to any two $n*n$ matrices and the run time is ...
0
votes
1answer
22 views

Difference between a convergent series and an asymptotic series?

Can someone let me know the difference between a convergent series and an asymptotic series with an example? Can both the series be the same at some situations? In what situations an asymptotic series ...
0
votes
0answers
12 views

Asymptotic complexities of a conditional function

Let the function $f(n)$ be defined by $$f(n) = \dfrac{n^2}{7}$$, for n even and $$f(n) = 452n$$, for n odd I'm being asked to determine which statements are true and to show validation for those ...
2
votes
0answers
36 views

Asymptotic analysis of coefficients of ordinary generating functions with radius of convergence $1$ seems to always predict polynomial growth rate

Wikipedia gives the following formula for obtaining asymptotic information about the coefficients of an ordinary generating function from information about the generating function itself: if the ...
-1
votes
1answer
33 views

The asymptote of $y=\mathrm{sinc}(t)$ as time increases

Is there any known approximate formula that maps decay percentage of $\mathrm{sinc}(t)$ with decaying time? Or in other words, is there a known asymptote of $y=\mathrm{sinc}(t)$ as time increases?
0
votes
1answer
38 views

Finding a function f(n) such that T(n) = O(f(n))

I need some help understanding how to prove that n log n in the equation below is the dominating term. i.e. Given the equation below, find function f(n) such that T(n) = $\theta$(f(n)): $T(n) = ...
1
vote
1answer
39 views

How many infinite subsets of the Naturals have natural density (asymptotic density) zero?

Are there countably or uncountably many? I know that the set of all primes has density zero. Is there an obvious way of using that result to construct an uncountable family of such sets?
0
votes
0answers
15 views

behavior of function

Im looking at the asymptotic behavior of the function $f(x)=x-c(\lceil \frac{x}{c} \rceil)$ as $x \rightarrow \infty$ for some constant $c>0$. I believe this function is bounded above by $0$ ...
1
vote
2answers
60 views

Notation issue - Asymptotic behaviour: is $\sim$ too restrictive?

As a student I am completely unable to understand unambiguously what is meant by a notation such as $$f \sim g $$ when in Physics the behaviour of two functions at infinity is evaluated. I found a ...
0
votes
1answer
27 views

Understanding Asymptotic Notation of a constant

How can I prove that if $f(n) = O(1)$ leads to $f(n) = \Omega(1)$ as well? I need a Formal definition of the meaning that a function $f(n) = O(1)$
0
votes
1answer
52 views

Prove that$ f(x)=\ln(x)$, where $ x>0$ is of exponential order.

Prove that$ f(x)=\ln(x)$, where $ x>0$ is of exponential order. I know that if there exists a constant a and positive constants $t_0$ and $M$ such that $|f(t)| \leq M e^{at}$ at for all $t > ...
2
votes
1answer
68 views

Asymptotic series of Confluent Hypergeometric function $U(a,1,z) $ as $z \to 0$

Consider the Confluent hypergeometric function $U(a,b,z)$, which is a solution of the Kummer's Equation : $$zw''+(b-z)w'-aw=0$$ it has the following integral representation when $- \pi/2 < \arg ...
1
vote
1answer
32 views

Proof that $\frac{1}{x\sqrt{x}}$ is $O(\frac{1}{x})$

My homework assignment is to proof that $\frac{1}{x\sqrt{x}}$ is $O(\frac{1}{x})$. I've seen different definitions of Big-Oh, but in my book it is defined as $|f(x)| \leq M \cdot|g(x)|$ for all $x ...
0
votes
1answer
39 views

What is this expression in big O notation?

$$2^{n-1} + 2^{n-1} + \ldots + 2$$ pretty basic question, but I'm afraid I don't know if it's $O(2^n)$ or $2^{O(n)}$
0
votes
2answers
32 views

Showing that $\log(\log(n))^{\log(n)}$ is $O(7^{\sqrt n})$

What's a straightforward way to prove that $\log(\log(n))^{\log(n)}$ is $O(7^{\sqrt n})$? (I'm dealing with Big O Notation)
0
votes
1answer
15 views

Does the natural (asymptotic) density of a set A change if a subset of A with natural density zero is subtracted from A?

I know that given two subsets of the Naturals A and B, if the natural density of A equals some non-zero real number a, and the natural density of B is zero, then the natural density of the symmetric ...
10
votes
4answers
204 views

Calculate the infinite sum $\sum_{1}^\infty \frac{\log{n}}{2n-1}$

I would like to calculate an asymptotic expansion for the following infinite sum: $$\displaystyle \sum_{1}^N \frac{\log{n}}{2n-1}$$ when $N$ tends to $\infty$. I found that the asymptotic expansion ...
0
votes
1answer
52 views

Where exactly is $n\log n$ between $n$ and $n^2$?

If I have $n^{1.161}$ and $n^{1.58}$, how do they compare in terms of time complexity relative to $n\log n$? I only know that $n\log n$ is between $n$ and $n^2$. I would probably factor out $n$ ...
2
votes
3answers
35 views

Comparing growth rate of $n^{\log_2{5}}$ vs $n^2 \log{n}$

$\log_2{5}$ is 2.3219.. and thus $n^{\log_2{5}} = n^{2.3219}$. Comparing that with $n^2 \log{n}$ which already has an $n^2$ in front, which one grows faster? I notice that $n\log{n}$ is between ...
0
votes
1answer
30 views

Bound the number of different natural numbers that fit as a sum in $n$ as $n$ increases

Let me explain... I have $n$ integers, with $k$ different values where $k \leq n$. If I sum together the integers with same values I will get a set of different values frequencies. Now if I sum ...
1
vote
1answer
159 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 ...
2
votes
4answers
52 views

Finding growth bounds on Fibonacci Sequence

I've been working on this following problem: Find a constant $c< 1$ such that $F_n \leq 2^{cn}$ for all $n \geq 0$. I honestly have no idea where to begin on this. I've done plenty of proofs ...
2
votes
3answers
34 views

Help formulating a proof showing two lists can be merged with 2n-1 comparisons

I need some help formulating a proof that shows that two lists of size n can be merged in 2n - 1 comparisons. I understand the essence behind it, but have difficulty proving it mathematically. I ...
2
votes
0answers
301 views

Asymptotic Methods - Boundary Layer Problems

I am currently studying a course in Asymptotic and Perturbation Methods and we have recently started discussing "Boundary Layer problems". It is not clear to me, however, exactly what form "Boundary ...
5
votes
2answers
31 views

Big-$O$ for $\frac{1}{1-x}$

I would like to show as $x\rightarrow 0$ $$\frac{1}{1-x}= 1+x^2+x^3+\dots+x^n +O(x^{n+1})$$ My inclination is to multiply by $1-x$ to get: $1=(1-x)(1+x^2+\dots+x^n) +(1-x)O(x^{n+1})$ and then, for ...
0
votes
2answers
23 views

Proving the summation of a function as big theta of another function

Show that $\sum^n_i i^4\log^2i$ = $\Theta(n^5\log^2n)$ I am completely lost on how to solve this problem. I understand that $\Theta$ deals with the upper and lower bounds, so do we prove both big-oh ...
1
vote
0answers
73 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 ...
2
votes
1answer
21 views

Growth of modified binomial recurrence

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 ...
2
votes
1answer
22 views

Using $f=O(g)$ to compare $f^2$ and $g^2$

When we have $f = O(g)$, does this work? $f^2 = O(g^2)$? If I have $n^2 = O(n^3)$, I think that $n^4= O(n^6)$ so I think this is valid. What about $2^f vs 2^g$? Does $f = O(g)$ imply $2^f = O(2^g)$? ...
0
votes
1answer
20 views

Growth rate of $n3^n$ vs $4^n$

Does the latter grow faster? I'm assuming that if we have a^n vs b^n, if b>a then a = O(b), but if there is a n term in front of a does that change it?
0
votes
1answer
22 views

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
votes
0answers
50 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 ...
2
votes
0answers
22 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 ...
2
votes
0answers
28 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} ...
0
votes
2answers
90 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, $$T(n) = 2T(n-1)+^2 $$ $$= 2(2T(n-2)+(n-1)^2+n^2$$ $$=4T(n-2)+2(n-1)^2+n^2$$ $$....$$ ...
1
vote
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, ...
0
votes
1answer
49 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 ...
0
votes
1answer
32 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) ...
1
vote
2answers
72 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 ...
0
votes
2answers
141 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 ...
5
votes
1answer
180 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. ...
1
vote
0answers
92 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 ...
1
vote
3answers
69 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
votes
1answer
20 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
vote
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
vote
1answer
18 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
votes
1answer
92 views

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

How can I understand the sign "$=$" from the following expression: $$\mathcal{o}f((x))=\mathcal{o}f((x))+\mathcal{o}f((x));$$ $$\mathcal{o}(kf((x)))=\mathcal{o}(f(x));$$ ...
0
votes
0answers
11 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 ...
0
votes
2answers
40 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
votes
1answer
31 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 $o([x]^a)$?