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

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2
votes
2answers
68 views

Prove $\frac{-\log(1-x)}{x(1-x)}=1+(1+1/2)x+(1+1/2+1/3)x^3+…$

Let $0<x<1$. How can i prove the following identity: $$\frac{-\log(1-x)}{x(1-x)}=1+(1+1/2)x+(1+1/2+1/3)x^3+...\ \ .$$
5
votes
1answer
111 views

Asymptotic evaluation of integral of algebraic function

I am wondering what techniques exist for the asymptotic evaluation of integrals. Consider the integral $$ I(\lambda) = \int_1^\lambda dx \sqrt{1-\frac 1 x} = \sqrt \lambda \sqrt{\lambda - 1}- ...
1
vote
1answer
76 views

Probability of picking each of m elements at least once after n trials.

Suppose I have 10^9 distinct elements, and an equal probability of picking each one in a given trial. How many trials must be conducted for the probability of having picked every element at least once ...
0
votes
2answers
21 views

How can I prove that Cx will intersect x^2

I want to disprove $ cx \geq x^2 \ \forall \ x $ where c is a real number. (i.e. show that x^2 is not O(x) ) So it seems that I can show that the two must intersect at some point ... if I divide both ...
0
votes
0answers
23 views

Different Upper and Lower Bound

Is there a function or algorithms whose upper bound and lower bound are different? For example f(X) i.e f(X) = O(X^2) and f(X) = Omega(X)
0
votes
2answers
81 views

Asymptotics of $\sum \sqrt{k}$ and $\sum (-1)^k\sqrt{k}$

I was playing around with series recently and asymptotics of $\sum \sqrt{k}$ and $\sum (-1)^k\sqrt{k}$ were required to solve another problem. I have dealt with the first one using an integral ...
0
votes
1answer
41 views

Name of the difference between an asymptote and the curve that approaches it

Consider a function, say a hyperbola, and its asymptote. Is there a specific term for the difference between the two? Answers specific to hyperbola, as well as answers about general terminology, are ...
1
vote
1answer
122 views

Asymptotic Expansion of an Integral involving Modified Bessel Functions

I do not have enough experience with the asymptotic expansion of integrals especially involving Bessel functions. I appreciate any feedback that you guys provide. Here is the problem. Let $a$ and $b$ ...
1
vote
1answer
29 views

Running Time Question

In what situations would a function of $\theta(n^2)$ perform better than $\theta(n \log n)$? I noticed that in comparing the two, they intersect at $n = 4$. After this, $n \log n$ takes over as ...
3
votes
1answer
100 views

How to give an upper bound for a solution of $T(n) = T(0.25n) + T(0.75n) + O(n)$?

We have an algorithm which can be described the recurrence formula: $T(n) = T(\frac{n}{4}) + T(\frac{3n}{4}) + O(n)$ and for $n\le 100$: $T(n) = O(1)$. How to show that $T(n) = O(n \log n)$? ...
0
votes
2answers
24 views

Dominance and Big Oh problem

What is the dominant term in the following expression? 100n + 0.01*(n^2) It is confusing because the power function should be growing faster than the linear function regardless the constants. But ...
3
votes
1answer
44 views

Algebraic number with bounded coefficients

How many algebraic numbers $z$ are there satisfying $P(z)=0$ where $P(z)$ is some polynomial with integer coefficients of degree less than or equal to $n$ such that the absolute value of every ...
2
votes
1answer
46 views

Asymptotic solution to inequality $x < k \ln(1+x)$

What is an upper-bound on $x$, given that $x < k \ln(1+x)$? I believe that the solution is something of the form $\mathcal{O}(k \ln k)$ but I am unable to prove this. This is my first encounter ...
1
vote
2answers
993 views

Find the asymptotes of the Parametric equation?

Consider $$ x(t) = 2 e^{-t} + 3e^{2t}$$ $$y(t) = 5 e^{-t} + 2 e^{2t}$$ which represents a non rectilinear paths Horizontal and Verical Asymptotes : If $t \rightarrow +\infty \ \ or \ \ -\infty$, ...
0
votes
1answer
20 views

Confirm the answer to compute the asymptotic solution to the problem

I have the following problem The solution I derived is $O(g(n))$ where $C = 1, n > 1$. Is this solution correct ?
1
vote
1answer
78 views

Run time/Efficiency of finding Least Common Multiple

The algorithm is: $$\mathrm{lcm}(x,y)=\frac{xy}{\gcd(x,y)}$$ And we can use the Euclidean algorithm for finding $\gcd$. How is the complexity for above method $O(n^3)$, if $x,y$ can at ...
2
votes
0answers
90 views

Asymptotic expansion of integral of $e^{-t}/t^n$.

So we study $$f_{n}(x) = \int_x^{+\infty} \! \frac{e^{-t}}{t^{n}} \, \mathrm{d}t, \quad n \in \mathbb{N^{*}}.$$ I've shown that for every $n$, $f_{n}(x) \sim_{+\infty} \frac{e^{-x}}{x^{n}}$. Now ...
1
vote
1answer
99 views

Infinite sum of asymptotic expansions

I have a question about an infinite sum of asymptotic expansions: Assume that $f_k(x)\sim a_{0k}+\dfrac{a_{1k}}{x}+\dfrac{a_{2k}}{x^2}+\cdots$ with $a_{0k}\leq \dfrac{1}{k^2}$, $a_{1k}\leq ...
0
votes
1answer
39 views

How do limit cycles explain curvilinear asymptotes?

I'm a 17 years old and I have no clue about a concept known as limit cycles. I looked it up and I understand it represents the orbit of functions approaching other A person told me that limit cycles ...
0
votes
1answer
109 views

How to prove a recurrence with multiple terms?

I have to prove that the recursion: $$T(n) = T\left(\frac{n}{3}\right) + T\left(\frac{2n}{3}\right) + n $$ is $$ T(n) = Θ(n*\log n)$$ As you can see, the reccurence has two different terms that ...
0
votes
1answer
34 views

Approximating a binomial sum over a simplex

For partial binomial sums such as $\sum_{k\le\Delta} \binom{n}{k}$ we don't tend to have closed forms. However we still know asymptotic expansions that are easy to work with. Can we do something ...
1
vote
0answers
37 views

Complexity of $T(n) = T(n-10) + \sqrt{n}$

I'm using the iteration method to find the complexity of the following recurrence (I can't use the master theorem because it doesn't match the MT form). $$ T(n) = T(n-10) + \sqrt{n} \text{ and } T(1) ...
1
vote
1answer
61 views

Asymptotic and 3-SAT problem in Algorithm Course

my TA says just one of the following is True, anyone could describe me some detail about following three lines? 1- if $f_i$ be a function of natural numbers to natural numbers and $f_i(n)=O(n)$ then ...
0
votes
2answers
67 views

Big $O$ notation - $n ^ {\log n}$ versus $2^n$

I received an asymptotics question for my homework, which is to compare the orders of growth for $f(n)$ and $g(n)$ where: $f(n) = n^{\log(n)}$ $g(n) = 2^n$ I have an intuition that $f(n) = ...
0
votes
1answer
28 views

Asymptotic form when series form of a real analytic function is known

Given an analytic function $f: \mathbb{R} \to \mathbb{R}$ whose Taylor series converges over all $\mathbb{R}$ and is \begin{equation} f(z) = \sum_{k=0}^{\infty}a_k x^k, \end{equation} and where the ...
2
votes
0answers
68 views

Picking codewords that are close

Let $[n,k,d]$ be a linear code over $\Bbb F_q$ with minimum distance $d$ and number of minimum weight codewords $N_d$. How many ways can you select codewords $c_1,\dots,c_T$ (assume $T\ll q^k$) such ...
2
votes
1answer
78 views

Asymptotic bound of the series $\sum_{n\leq x}\log n / \varphi(n)$

Could someone give me a hint on the computation of the asymptotic bound for the following series $$ \sum_{n\leq x}\frac{\log n }{ \varphi(n)}\,, $$ where $\varphi(n)$ is the Euler totient function? ...
0
votes
0answers
27 views

Differential Equation for Algorithm Time

I'm working on algorithm analysis and time complexity. I've got a homework assignment to calculate a function f(n) at time t and I want to figure out how to write it as a differential equation. ...
2
votes
1answer
102 views

How can we compute best-case and/or average-case and/or worst-case running-time knowing some of them?

Complete the table when it is possible. $$ \begin{array}{c|lcr} \mathrm{Algorithme} & \text{worst-case} & \text{average-case} & \text{best-case} \\ \hline A & O(n) & ... & ...
0
votes
1answer
32 views

How may times should I colour a colour palette to have distinct colours?

Suppose that we have a colour palette, i.e., an array of n elements, which needs to be coloured by distinct numbers. We are only allowed to use 0 or 1 to colour every elements in each colouring step. ...
3
votes
1answer
62 views

Second-order asymptotics for $\pi(n), \theta(n)$

Let $\pi, \vartheta$ be respectively the prime counting function and the first chebyshev function. As you know, $ \pi(x) \sim x/\log x$, and $\vartheta(x) \sim x$, so that, at first order, seems ...
0
votes
1answer
50 views

Question involving summations and the Θ-notation of running times

I think I understand the concept of summations and Θ-notations, however, I don't really understand the question below. If I have understood it correctly, I'm supposed to write out the summations ...
0
votes
1answer
81 views

Elementary proof that $\omega(n)$ is bounded $\frac{\log n}{\log( \log n)}$ in the limit?

I'm trying to show that $\omega(n)$ is less than $\frac{\log n}{\log(\log n)}$ as it's stated without proof in an analytic number theory text. It's a corollary of the PNT, but I want to not use that ...
0
votes
1answer
55 views

Time Complexity for Asymptotic Functions

Here below I have a problem set where I am asked to define the relationship between f(n) and g(n). I have added in my solutions but I wanted to get my answers checked by you guys before I turn this ...
1
vote
1answer
39 views

Confusion about Big O notation

I have a somewhat stupid question regarding the "Big O" notation: Is there any difference between saying $f=O(g)$ and $f\le O(g)$?
0
votes
0answers
41 views

recursive-algorithm problem

I am not to sure were to begin Thanks
0
votes
1answer
13 views

Linking summations with their correct function(s)

Guys can you please guide me step by step on how to link given functions with the functions to choose from. So for example a function $g(n)\in \Theta n^2$ and if there is no match then you say there ...
1
vote
1answer
39 views

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: ...
2
votes
0answers
41 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} ...
2
votes
4answers
139 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 ...
0
votes
1answer
251 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) + ...
0
votes
1answer
137 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 ...
0
votes
1answer
149 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 ...
1
vote
1answer
103 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 ...
0
votes
0answers
21 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 ...
0
votes
2answers
37 views

How to justify adding and multiplying relations with big-O?

I know that it's valid to add and multiply functions in Big O, although I haven't seen a proof why. As such I think this is a valid starting point. However, I have no idea how to progress and any help ...
0
votes
1answer
39 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 $$\log\log\log\log( n) = 0.5 \log( n)$$ ...
3
votes
0answers
47 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 ...
1
vote
1answer
179 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 ...
3
votes
0answers
93 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. ...