All questions with tag [math: asymptotics]


Asymptotics of an integral

Consider an integral $$ I(x) = \int\limits_{\mathbb{R}^n} e^{i\xi x } \delta(\xi^2-k^2)\chi( (\xi-k,\gamma)) \, d\xi $$ where $x, k, \gamma \in \mathbb{R}^n$, $|\gamma| = 1$ and $(x,y)\equiv xy = x_1 y_1 + \ldots + x_n y_n$. Here $\delta$ is the Dirac delta, $\chi$ is the Heaviside step function: $$ \chi(t) = 1_{\left\{ t \geqslant 0 \ri...
2022-07-25 17:46:14

Interesting Recurrence Relation $T(n) = T(\sqrt{n}) + T(n-\sqrt{n}) + n$

I found an interesting recurrence that I do not know how to solve. I think this has to do with quicksort with pivots at rank $\sqrt{n}$. I do not know how to approach this problem nor found any helpful resources about it. Here is the recurrence: $$T(n)=T(\sqrt{n})+T(n−\sqrt{n})+n$$ Any help would be much appreciated. Thanks! Let's say the base c...
2022-07-25 17:16:30

How can Big-O be proved using derivatives?

Say we have: $$f(n) \in O(g(n))$$ By definition we require to show that: $$0 \le f(n) \le c\cdot g(n) $$ for some $c>0$ and also for all $n>n_0$. This is generally uncomplicated when sensible and also polynomial features are entailed, yet if features have logarithms and also square origins, I get perplexed and also not exactly sur...
2022-07-25 17:14:03

Next asymptotic term of the average order of sigma

$$ \sum_{k=1}^n\sigma(k)=\frac{\pi^2}{12}n^2+O(n\log n). $$ Is the next asymptotic term recognized? That is, exists a monotonic raising function $f(x)$ such that $$ \sum_{k=1}^n\sigma(k)=\frac{\pi^2}{12}n^2+f(n)+o(f(n)) $$ ? (I would certainly presume something like $f(x)=cx$ if $f$ exists.) At the same time, exist monotonic raising features $f...
2022-07-25 17:13:44

Big O Notation reliability?

Is Big - O symbols constantly trusted? For example: Algorithm A: $n * 10^{100} = \mathcal{O}\left(n\right)$ Algorithm B: $n^{1.001} = \mathcal{O}\left(n^{1.001}\right)$ According to Big - $\mathcal{O}$ symbols, Algorithm $A$ would certainly be extra reliable, yet for all sensible objectives Algorithm $B$ is extra reliable. In a scenario simi...
2022-07-25 17:13:11

WKB approximation question

I read some things on asymptotic evaluation, yet just how do you obtain from the 1st line to the 2nd line? $y \sim \frac{1+x}{2\lambda}\exp\left(\frac{\lambda x}{1+x}\right) - \frac{1+x}{2\lambda}\exp\left(\frac{-\lambda x}{1+x}\right)\\y(1) \sim\frac{1}{\lambda}\exp\left(\frac{\lambda}{2}\right) \text{as } \lambda \rightarrow \infty$ I see th...
2022-07-25 16:39:35

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

How to confirm $\log(n) = O(\sqrt{n})$? Just how do I locate the $c$ and also the $n_0$? I recognize to start, I require to locate something that $\log(n)$ is smaller sized to, yet I m having a tough time thinking of the instance.
2022-07-25 16:31:02

properties of a real analytic function

If there are a distance $r>0$ and also constants $M,C\in\mathbb R$ for all $y\in U$ with $$|\partial^if(x)|\leq M\cdot i!\cdot C^{|i|}\space\space\space\space \forall x\in\mathbb B_r(y),i\in\mathbb N_0^n$$ after that $f\in C^\infty(U)$ is actual analaytic. Yet I do not have any kind of suggestion just how to confirm this. I feel in one's...
2022-07-25 13:19:23

Disproving a big O equation

As a homework assignment I am trying to prove/disprove the next statement: Let $f(x)=O_a(g(x))$, then $\forall A,B\in\mathbbR\rightarrow A\cdot f(x)=O_a(B \cdot g(x))$ Which I think is wrong and thus trying to disprove. So assuming I'm right, my question is logical: we know that there exist a $C_1 \gt 0$ such that $\|f(x)\| \le C_1 \cdot \|g(...
2022-07-25 13:11:21

Big-O notation always holds for this two functions?

For 2 any kind of features $f(n)$ and also $g(n)$ constantly holds: $f(n) = O(g(n))$ or $g(n) = O(f(n))$ Right? Many thanks
2022-07-25 12:34:10

expansion of $\int_0^\infty\left(\frac{\sin t}t\right)^p\mathrm dt$ in inverse powers of $p$

This question relates to this answer I gave to a question about the integral $$\int_0^\infty\left(\frac{\sin t}t\right)^p\mathrm dt\;.$$ I derived an expansion in inverse powers of $p$ and then realized that I don't know how to justify it rigorously or how to determine its radius of convergence. I substituted $u=\sqrt pt$ and applied $$\left(1+...
2022-07-25 12:28:10

asymptotic limit of $\int_0^{\infty}\left(1-\frac{t^2}{2(2k+3)}+\frac{t^4}{2\cdot 4\cdot(2k+3)\cdot(2k+5)}\right)^qdt$

Help me please with the adhering to indispensable. I've asked this inquiry prior to, yet it ends up that it was wrong inquiry. I need to get an asymptotic restriction (with $k$ mosts likely to $\infty$, and also $q$ dealt with) of it. For $q\ge 2, t\ge 0, k \in Z$ $$ \int_0^{\infty}\left(1-\frac{t^2}...
2022-07-25 07:55:44

Asymptotic bounds: $\ll$ vs. $\ll_{\epsilon}$?

I am feeling a bit slow today. In Analytic Number Theory it is usual to express asymptotic bounds by specifying the relation of the constant to a specific variable, i.e. $\log n \ll_\epsilon n^\epsilon$ which means that $\log n \leqslant C_\epsilon n^\epsilon$ for sufficiently large $n$, where the constant $C_\epsilon$ depends only on the consta...
2022-07-25 07:51:22

Big O Notation question

I am attempting to recognize the Big - O and also little - O symbols, so I outlined 2 charts which I have actually uploaded listed below, yet I still do not actually get the principle of it. Just what does the $O\left(\frac{1}{x^6}\right)$ term do?
2022-07-25 07:49:20

Describe growth of $\epsilon n$

For all $\epsilon$ we have that $f(n)\le \epsilon n$ where n is an all-natural number. What can we claim concerning the development of $f(n)$? Plainly $f(n)=O(n)$, can we claim anything sharper?
2022-07-25 07:48:41

Can a function "grow too fast" to be real analytic?

Does there exist a continual function $\: f : \mathbf{R} \to \mathbf{R} \:$ such that for ¢ all real analytic features $\: g : \mathbf{R} \to \mathbf{R} \:$, for all actual numbers $x$, ¢ there exists an actual number $y$ such that $\: x < y \:$ and also $\: g(y) < f(y) \:$?
2022-07-25 07:47:36

asymptotic limit at the integral

I would love to get an asymptotic restriction at the adhering to indispensable: for $p\ge 2, n \in N$, $t \ge 0$ $$ \int_{0}^{\frac 12 \sqrt{(n+1)!}}\left(1-\frac{t^2}{2^2(n+1)!}\right)^p \mathrm{d} t $$ I assume replacement $t=\frac 12 \sqrt{(n+1)!}y$ need to function. Yet after the substittution, I do not recognize what to do. Thanks for your...
2022-07-25 07:37:16

Why is $\log(n!)$ $O(n\log n)$?

I assumed that $\log(n!)$ would certainly be $\Omega(n \log n )$, yet I read someplace that $\log(n!) = O(n\log n)$. Why?
2022-07-24 06:40:44

With probability $o(1)$

I am not exactly sure just how to read little/big O expressions in probability theory: What does a declaration like "with probability $1-o(1)$" suggest? Does it suggest with high probability?
2022-07-24 03:47:12

Compactly supported function whose Fourier transform decays exponentially?

It is popular since a function can not be compactly sustained both on the room side and also the regularity side (so - called unpredictability concept). On the various other hand a function can have rapid degeneration on both sides, as an example guassian function $e^{-x^2}$. My inquiry is whether the intermediate instance exists. Extra specific...
2022-07-24 03:36:51