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

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Asymptotic Notation Analysis Problem

I'm new here. I have some question on asymptotic analysis I am trying to calculate the Big-O of these five functions and rank them up: a: $$2^{log(n)}$$ b: $$2^{2log(n)}$$ c: $$n^{5\over2}$$ d: ...
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0answers
38 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} ...
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4answers
98 views

Limit and infinite sums. Finding $\lim_{x\rightarrow\infty}\sum^{\infty}_{k=1}\frac{1}{k^3 x-k^2}$

Could anyone help me with this problem. Compute $$\lim_{x\rightarrow\infty}\sum^{\infty}_{k=1}\dfrac{1}{k^3 x-k^2}$$ I don't know how to change a limit and a sum. Could you help me with this problem ...
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1answer
34 views

Use recursion tree to give an asymptotically tight solution of T(n)

Assume $T(1) = 3.$ Recurrence is $T(n)=T(n-3)+3n+1$ and I'm showing $\Theta$ bound by computing the exact running time. Starting off: $(Tn-3) + 3n + 1$ $(Tn-9) + 9(n-3) + 3n + 1$ $(Tn-18) + ...
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1answer
41 views

Tight bound for $T(n) = T(n^{1/2}) + 1$ [duplicate]

Can someone help me figure out the big-O for the recurrence relation $T(n) = T(n^{1/2}) + 1$? I didn't think the master theorem would work since it requires $T(n) = T(n/b)$... to have $b$ as a ...
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1answer
33 views

How to give a big O estimate/visualize for these while loop?

This is from Discrete Mathematics and its applications I am currently working on problem 4. I was able to see that for problem 2, that one operation one will run n times for every n(meaning in ...
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1answer
40 views

Growth faster than polynomial, slower than exponential.

Assume $F(n)$ is a positive function. If $F$ is growing faster than a polynomial then is it growing exponentially fast? Is this statement true? Can we find a function $F(n)$ such that ...
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0answers
18 views

Limit of shifted ratios

Let $f$ a be a strictly positive function defined in the positive reals. Additionally suppose that for any $\delta > 0$ we have, as $t \to \infty$, $$ e^{-t^{1+\delta}} \ll f(t) \ll ...
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1answer
23 views

Show the correctness: $\log^3( n)\in o(n^{0.5})$

show the correctness: $\log^3 (n)\in o(n^{0.5})$? I started from this way $$\log \log \log( n) = n^{0.5}$$ then I take $\log$ for two parties $$\log\log\log\log( n) = 0.5 \log( n)$$ ...
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0answers
34 views

Algorithms - Solving the recurrence $T(n) = \sqrt{n} T \left(\sqrt n \right) + n$ [duplicate]

I have been trying to solve the recurrence $T(n) = \sqrt{n} T \left(\sqrt n \right) + n$ for some time now. I only know substitution, recursion trees, and the master method (though it doesn't apply ...
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1answer
46 views

Proving that one function is big o of another?

I'm working through a big-O problem and have the intuition to know the answer, but don't feel comfortable in my proof. I need to prove from definitions (i.e. proving that there exists two constants ...
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0answers
36 views

Effect of differentiation on function growth rate

For sufficiently "nice" functions, the differentiation operator appears to make slow growing functions grow slower and fast growing functions grow faster, with $e^x$ as a fixed point in the middle. ...
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0answers
66 views

Implicit function where the Jacobian determinant is zero

When we have an implicit function defined by $f(x,y)=0$ where Jacobian determinant of $\frac{\partial f}{\partial x}$ is zero Let $x \in \mathbb{R}^n$, $p \in \mathbb{R}$ and $\phi:\mathbb{R}^n \to ...
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1answer
37 views

How to solve this triple summation problem?

For a computer science class we were asked to analyze the run time of an algorithm. The answer was posted. I am not sure the proof is correct. I believe the answer should be kc(n^2) (where k is a ...
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1answer
172 views

Determine the number of paths of length 2 in a complete graph of n nodes

Question: Determine the number of paths of length 2 in a completed graph of n nodes. Give your answer in Big-O notation as a function of n So I started working on this problem however I know im doing ...
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2answers
96 views

Proof $10n = O(n^2)$

As it says in the question name. I want to proof this big-o notation: 10n = O(n^2) Is this way here the correct an proper way to do so? ...
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0answers
14 views

Bound from distinct integer summation

We want to find $r$ positive integers $\{a_i\}_{i=1}^r$ such that of atmost $(s+1)^r$ values obtained from $$\sum_{i=1}s_ia_i$$ where $s_i\in\{0,\dots,s-1,s\}$, we insist on some combination of ...
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1answer
21 views

Showing $\limsup_{h \to {0}}\frac{O(h^2)}{h^2}<\infty$

Let $$y(h)=1-2\sin^{2}(2\pi h) , f(y)=\frac{2}{1+\sqrt(1-y^2)} $$ Justify the statement $$f(y(h))=2-4\sqrt{2}\pi+O(h^2)$$ where $$\limsup_{h \to {0}}\frac{O(h^2)}{h^2}<\infty$$
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1answer
26 views

Big-O of a Function

Given $F(N) = 55N(600 + 50N \log N + 20N) + 20N(30N + 20\sqrt N)(50 + \log N)$. How can one combine multiplication with addition for a Big-O estimate by algebraic means? I'm used to simply taking the ...
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0answers
26 views

Relationship between asymptotic distribution and logarithmic sums of elements of subset of the natural numbers

Consider a subset $A$ of the natural numbers analogous to the primes (but rarer). Let $a_n$ denote the $n$th element of $A$, and $a(n)$ denote the number of elements of $A$ less than or equal to $n$ ...
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2answers
26 views

Big $O$ — $3^n$ vs $n2^n$

I'm trying to compare $f(n) = 3^n$ and $g(n) = n2^n$ to determine whether $f \in O(g)$, $f \in \Omega(g)$, or $f \in \Theta(g)$. My gut is telling me that $g(n) = n2^n$ grows faster, and so $f \in ...
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1answer
58 views

Proof $(\log(n))^{\log(\log(n))} = O(n)$

Can someone provide a proof that $(\log(n))^{\log(\log(n))} = O(n)$? Preferably without calculus, but I'll take what I can get. Just ran into this problem, and I have no way of moving forward, ...
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1answer
49 views

What is the point of big Oh notation when it is used for estimation?

I'm reading a book on number theory at the moment that assumes familiarity with big Oh notation...and while I think I do understand the notation I cannot understand the point of it. For instance let ...
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0answers
13 views

How to solve this kind of difference equation?

How to find $v_k$, $k=0,1,2,\dots$ such that $$v_k + \sum_{n=1}^{k} \frac{\alpha^n}{n}v_{k-n} + \sum_{n=1}^{k}\frac{\beta^n}{n}v_{k+n} = 0,$$ where $\alpha,\beta \in \mathbb{C}$. ($v_i=0$ for ...
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2answers
36 views

Is $\lceil{\lg n}\rceil!$ polynomially bounded?

Is $\lceil{\lg n}\rceil!$ polynomially bounded? I've tried using Stirlings Approximation, and I get that $\lceil{\lg n}\rceil! \approx \sqrt{2\pi}\lceil{\lg n}\rceil^{1/2}\lceil{\lg n}\rceil^{\lceil ...
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1answer
29 views

Big Omega problem : is $n^2\in\Omega (2n^2)$?

Is $n^2\in\Omega (2n^2)$? If we find the limit we can see $\frac{1}{2}>0$, which means it is true, but I haven't learned the limit method. I need to figure out using this definition $\exists ...
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1answer
36 views

Asymptotic formula for sums related to primes

Suppose $0 < \alpha < 1$. What is the asymptotic formula for the sum $$\displaystyle \sum_{p \leq x} \frac{\log p}{p^\alpha}?$$ Thanks for any insights.
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23 views

summation inequality with logarithms

show: $$\sum_{i=1}^n \log_{2}\,i = O(n\log n)$$ Proof by induction: $$\sum_{i=1}^n \log\,i \le n\log n$$ $$\text{Test for n=1: }\sum_{i=1}^1 \log_{2}\,i \le 1\log 1$$ $$0 \le 0\text{ true for ...
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1answer
28 views

Determine whether each pair is $f(n) = O(g(n), f(n) = \Omega(g(n)), or f(n) = \Theta(g(n)).$

For the pair of functions, find whether it's $f(n) = O(g(n), f(n) = \Omega(g(n)), or f(n) = \Theta(g(n)):$ $a) f(n) = 12^n , g(n) = 7^n$ $b) f(n) = log_9(n^4), g(n) = log_9(n^5)$ I understand that: ...
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1answer
19 views

Need explanation on asymptotic running time results for various functions

I did not understand few results from the book problem. Here is the problem: Indicate, for each pair of expressions (A, B) in the table below, whether A is O, o, Ω, ω, Θ of B. Assume that k ≥ 1,  > ...
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2answers
39 views

proof by induction summation inequality

show by induction that: $$\sum_{i=1}^n i^2 = O(n^3)$$ what I have so far: $$\sum_{i=1}^n i^2 <= n^3$$ base case: for n=1 $$\sum_{i=1}^1 i^2 <= 1^3$$ ...
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0answers
23 views

Let g and h be any functions from naturals to (0,infinity)

Let $g$ and $h$ be any functions $\mathbb{N} \to (0,\infty)$. Then $g(n) \in \Omega(h(n))$ implies there is some $N \in \mathbb{N}$ such that $g(n)\ge h(n)$ for all $n \ge N$. Picture of question : ...
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11 views

Am I comparing big-O runtime correctly?

I'm working on some homework regarding runtimes and Big-O notation. I've struggled with this for quite some time, but I think its slowly making more sense. If someone could verify that I'm doing these ...
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2answers
80 views

Higher Order Terms in Stirling's Approximation

Some websites and books give stirling approximation as $$n! = \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n \left( 1 + O \left(\frac{1}{n} \right)\right)$$ However when I check their derivations most ...
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1answer
46 views

If $f(n)$ is not $\Theta (g(n))$ does it follow that $\log f(n)$ is not $\Theta(\log g(n))$?

If $f(n)$ is not $\Theta (g(n))$ does it follow that $\log f(n)$ is not $\Theta(\log g(n))$? We say that $f(n)= \Theta (g(n))$ if there exist some constants $c_1$ and $c_2>0$ and $n_0$, such ...
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1answer
31 views

Solving recurrence relations using master's theorem

Can we solve following recurrence relation using Master's theorem- $T(n)=T(n/2)+\log n$ The thing to notice here is that, do $n (n^{\log b} a)$ and $\log n (f(n))$ have an exponential ...
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1answer
59 views

Asymptotics of $\sum_{\mathfrak{a}}\frac{n^{k-\epsilon}}{\mathfrak{N}\left(\mathfrak{a}\right)^{r\left(k-\epsilon\right)}}$

In this paper by Brian D. Sittinger, the following claim is made: For an algebraic number field $K$ with norm $\mathfrak{N}$, let $\epsilon=\left[K:\mathbb{Q}\right]^{-1}$. Then, taking the sum over ...
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41 views

Compute the asymptotic expansion of the integral by Watson's Lemma

Use Watson's Lemma to find the asymptotic expansion of the following integral as $\lambda \to \infty$ with $\lambda>0.$ Assuming that $\phi (t)$ is infinitely differentiable on $[0,1].$ ...
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0answers
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Calculating Upper bound of a function [duplicate]

If T(N) = T(sqrt(N)) + 1 and T(1) = 1 then what is the upper bound i.e O(N) for this function? sqrt(N) => square root of N
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1answer
55 views

Asymptotics of $\sum_{n\leq x}\tau_{k}\left(n\right)$

We define $\tau_{k}\left(n\right)$ to be the number of ordered $k$-tuples of positive integers with product equal to $n$. It is easily shown that this satisfies the recurrence relation ...
3
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2answers
69 views

Prove that $7n^2 + 2n + 3 = O(n^2)$ using the definition of O notation.

Prove that $7n^2 + 2n + 3 = O(n^2)$ using the definition of O notation. I need to use two constants and prove that they satisfy the O definition. I'm new to big O and want to know whether I am ...
3
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1answer
53 views

What proportion of the positive integers satisfy this number-theoretic inequality?

Let $\sigma(x)$ be the sum of the divisors of the positive integer $x$, and let the abundancy index of $x$ be defined as $$I(x) = \frac{\sigma(x)}{x}.$$ My question is this: What proportion of the ...
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2answers
41 views

Determining order class of $T(n) = nT(n-1) + n$ with $T(1) = 1$

I'm trying to solve the following problem: Define $T(n) = n\cdot T(n-1) + n$ with $T(1) = 1$. Is $T(n) \in \mathcal O(2^n)$? I started by finding the time complexity of $T(n) = n\cdot T(n-1) + ...
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1answer
27 views

Series involving primes

Trying to find an asymptotic bound for the series $$ S(x) =\sum_{p\leq x}\frac{\varphi(p-1)}{(p-1)p} $$ as $x \rightarrow \infty$. Of course $$ \frac{\varphi(p-1)}{p-1} =\prod_{q\mid ...
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2answers
37 views

Big-O math Question

I'm having trouble with this question: Suppose that $f(x), g(x)$ and $h(x)$ are functions such that $f(x)$ is $O(g(x))$ and $g(x)$ is $O(h(x))$. Prove that $f(x)$ is $O(h(x))$. I have tried ...
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3answers
88 views

Proving with Big O Notations

Is there a way I can prove that $O(3^{2n})$ does NOT equal $10^n$? How would that be done? Also, is it okay to simplify $O(3^{2n})$ to $O(9^n)$ to do so?
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1answer
90 views

Big O Proof by Contradiction

Question: Use a proof by contradiction to show that $5^n$ is not $O(3^n)$ NOTE: This is homework, please don't provide an answer, just want to know if I am on the right track. My Attempt: ...
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2answers
22 views

Understanding the logic behind this summation

The following is an excerpt from a proof that $\sum_1^n {i^k} = \theta(n^{k+1})$: $$\sum_1^n{i^k} \ge \sum_{\lceil n/2 \rceil}^n{i^k} \ge \sum_{\lceil n/2 \rceil}^n{\lceil n/2\rceil^k}$$ The first ...
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1answer
30 views

Showing $n^{\log{n}} = o(2^n)$

I would like to show that $n^{log n} = o(2^n)$. Here is my attempt: I see that $\log{(n^{\log{n}})} = (\log{n})^2,$ and $\log{2^n} = n\log{2}$. I also know that $(\log{n})^2=o(n)$, so that for ...
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1answer
9 views

Orders of Asymptotes

We know that $\log(X)^n = o(X^\epsilon)$ for all $n,\epsilon>0$. My questions is, is $\log(X)$ the largest function that is smaller than all (small) powers of $X$. That is, can we find a ...