Articles of asymptotics

Big Oh Question

I have the following question: Is the following statement true or false? ** All logs to base 2 log2n is a memeber of O(log(n)) My attempt: log2n – clogn <= 0 log2 + logn – clogn <= 0 1 + logn(1-c) <= 0 Now correct me if I’m wrong, but I have to find values […]

gamma function with negative argument

For $k=0,1,2…$ and small $z$ I want to show that $$\Gamma (-k + z) = \frac{ a_k}{z} + b_k + O(z).$$ I understand that the gamma function cannot be expressed as $$\Gamma ( z) = \int_0 ^\infty e^{-x} x^{z-1} dx$$ for negative values of $z$. The graph below shows that for $z \in \mathbb{z^-}$ the […]

Are there conditions for which $\int_{a}^{x} O(f(t)) dt = O\left( \int_{a}^{x} f(t)) dt \right)$ (same for derivatives)

I was wondering if in asymptotics there some conditions for integral/derivatives where I can do something like $$\int_{a}^{x} O(f(t)) dt = O\left( \int_{a}^{x} f(t)) dt \right)$$ Or something like $$ \frac{d O(f(x))}{dx} = O\left( \frac{df(x)}{dx} \right)$$ I’ve been watching some videolectures, and sometimes I just see people using as nothing this stuff, but I wonder […]

Prove that $e^{\sqrt{\log x }}=O(x^n)$

I have to prove the following: Let $n \in \mathbb{N}$. Proove: $$e^{\sqrt{\log x}}=O(x^n) .$$ I just know the definition of $O$: $f(x), g(x)$ are real functions. $f(x)=O(g(x))$ means, that for large $x$, $|f(x)| \leq C \cdot g(x)$ holds. I’m a little bit confused, because I thought $O$ is part of the analysis and not number […]

Help with the integral for the variance of the sample median of Laplace r.v.

When we draw $n$ samples of Laplace-distributed random variable such that $n=2k+1$ and the location parameter is zero, the median $x$ (or the $k$-th order statistic) has the following p.d.f.: $$f_m(x)=\frac{n!}{(k!)^2}2^{-n}\frac{1}{b}e^{-(k+1)|x|/b}(2-e^{-|x|/b})^k$$ where the p.d.f. of the underlying Laplace distribution is given as $f(y)=\frac{1}{2b}e^{-|y|/b}$. The formula for p.d.f. of the median stems from the usual method […]

Solving recurrences of the form $T(n) = aT(n/a) + \Theta(n \log_2 a)$

The time complexity for the merge sort algorithm is $T(n) = 2T(n/2)+\Theta(n)$, and the solution to this recurrence is $\Theta(n\lg n)$. However; assume you are not dividing the array in half for sorting, but instead in, say, $a$ pieces. Then the time would be something like $T(n) = aT(n/a) + \Theta(n \log_2 a)$. (The last […]

Asymptotes and focus of a conic?

This is the conic $$x^2+6xy+y^2+2x+y+\frac{1}{2}=0$$ the matrices associated with the conic are: $$ A’=\left(\begin{array}{cccc} \frac{1}{2} & 1 & \frac{1}{2} \\ 1 & 1 & 3 \\ \frac{1}{2} & 3 & 1 \end{array}\right), $$ $$ A=\left(\begin{array}{cccc} 1 & 3 \\ 3 & 1 \end{array}\right), $$ His characteristic polynomial is: $p_A(\lambda) = \lambda^2-2\lambda-8$ A has eigenvalues discordant […]

Is it possible to identify this sequence?

Interested by this question, $j$ being a positive integer, I tried to work the asymptotics of $$S^{(j)}_n=\sum^{n}_{k=0}\frac{\binom{n}{k}}{n^k(k+j)}=\frac{\, _2F_1\left(j,-n;j+1;-\frac{1}{n}\right)}{j}$$ I quickly noticed (not a proof) that the asymptotics write $$S^{(j)}_n=(-1)^j\left(\left(\alpha_0-\beta_0e\right)-\frac{\left(\alpha_1-\beta_1e\right)}{2n}+\frac{\left(\alpha_2-\beta_2e\right)}{24n^2}\right)+O\left(\frac{1}{n^3}\right)$$ in which the $\alpha_k$’s and $\beta_k$’s are all positive whole numbers depending on $j$. What I found is that $$\alpha_0=(j-1)!\qquad \qquad \beta_0=\text{Subfactorial}[j-1]$$ $$\alpha_1=(j+1)!\qquad \qquad \beta_1=\text{Subfactorial}[j+1]$$ $$\alpha_2=(1+3j)(j+2)!$$ […]

Describing asymptotic behaviour of a function

For question B! x^2+x+1/x^2 = 1+ [x+1/x^2] shouldnt the answer be asymptote at x=0 and y=1 ?? i dont understand the textbook solution

When is a balance assumption consistent?

From Asymptotic analysis and perturbation theory by Paulsen: Find the behavior of the function defined implicitly by $$x^2+xy-y^3=0$$ as $x\to\infty$. […] The final case to try is to assume that $xy$ is the smallest term. Then $x^2 ∼ y^3$, which tells us that $y ∼ x^{2/3}$. To check to see if this is consistent, we […]