Questions tagged [asymptotics]
Asymptotic behavior of functions, asymptotic series and related topics
888
questions
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Estimating the bound of the integral over whole $\mathbb{R}$ of the Taylor remainder term?
Let $f : \mathbb{R} \to \mathbb{R}$ be a smooth function which has a smooth inverse and satisfies the estimate
\begin{equation}
\lvert f(x) \rvert \leq \lvert x \rvert.
\end{equation}
Also, let $d\mu$ ...
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votes
0
answers
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You have $n$ rectangles of area $1$ and variable height. Pack as many as possible in a semicircle of area $n$. How many leftovers will there be?
You have $n$ rectangles of area $1$ and variable height. Pack as many of these rectangles as possible in a semicircle of area $n$. How many leftover rectangles will there be, in terms of $n$?
How to ...
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votes
2
answers
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views
How fast does this summation grow?
$n,i\in\mathbb N$.
The summation in question is
$$\sum_{k=1}^n\prod_{l=1}^k\binom{2^n}{2^l}^i.$$
How fast does this grow? I am specifically looking at $i=1,2$.
- 13.2k
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0
answers
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views
Lowest asymptotic bound to $4^n - 2v_n^2$ where $v$ is an odd integer, $n$ fixed
The general problem is this. I try to find a positive integer $\delta_n$ such $qv_n^2 +\delta_n = p\cdot 4^n$. More precisely, I am looking for a lower bound (depending on $n$) for $\delta_n$ as $n\...
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1
vote
0
answers
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views
Asymptotic location of zeros of of a sequence of analytic functions
Assume we have a sequence of functions $f_n$ analytic in a bounded domain $\Omega \subset \{ |z|\ge 1 \}$ of the complex plane, such that the sequence
$$
g_n(z) = f_n(z) - z^n
$$
converges to an ...
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2
votes
0
answers
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views
Ratio of the number of solutions to unit fraction equations with shifted prime and natural denominators
In a 2018 question posed by Zhi-Wei Sun, he conjectures that for any rational number $r>0$, there are finite sets $P_r^-$ and $P_r^+$ of primes such that
$$r=\sum_{p\in P_r^-}\frac1{p-1}=\sum_{p\in ...
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vote
0
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Asymptotics for $\mathrm{Zi}(x)$ and comparison to $\mathrm{Li}(x)$
Consider a function that attempts to count primes
$$\mathrm{Zi}(x)=\frac{1}{e}\sum_{k=1}^\infty\frac{(\ln x)^k}{kk!\phi(k)}$$
where
$$ \phi(k)= \sum_{n=1}^\infty e^{-n^k} $$
Based on some preliminary ...
6
votes
1
answer
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views
On an asymptotic integral decay
Let $f\in C^{\infty}([-1,0])$ be real-valued and suppose that
$$ \left| \int_{-1}^{0} f(t)\,e^{\lambda t} \,{\rm d} t \right| \leq e^{-\sqrt{\lambda}},$$
for all $\lambda > 0$. Does it follow that $...
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2
votes
1
answer
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The asymptotic behavior of $F(\lambda):=\sum_{k=0}^{\infty}\frac{\Gamma{(a k)}}{\Gamma{(b k)}}{\lambda}^{-k}$, $b>a>0$
Let $b>a>0$. Given $ \lambda>0$, what is the asymptotic behavior of
$$F(\lambda):=\sum_{k=0}^{\infty}\frac{\Gamma{(a k)}}{\Gamma{(b k)}}{\lambda}^{-k}$$
as $\lambda\to 0^{+}$ and as $\lambda \...
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votes
0
answers
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views
When does the Fourier transform of a measure decay?
Let $\mu$ be a Borel measure on $\Bbb R^d$.
It is well known that $\mu= |f|dx$ with $f\in L^1(\Bbb R^d)$ then its Fourier transform satisfies
$$\widehat{\mu}(\xi)\to0,\qquad \xi\to\infty.$$
However if ...
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votes
2
answers
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views
Asymptotics of an integral requested
Given an integer $n\geq2$, consider the following integral
$$I_n:=\int_0^1nx^{n-1}\sqrt{\left\vert \frac{\log(1-x)}{\log n}\right\vert} \, dx.$$
QUESTION. Is this true? It appears to be so.
$$\lim_{n\...
- 40.2k
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votes
1
answer
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Asymptotic approximation of a convolution of binomial coefficients
I would like to find the following limit which is somewhat similar to the usual Vandermonde's convolution for binomial coefficients. Define $L$ as follows.
$$ L \triangleq \lim_{N\to\infty} \frac{1}{2^...
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votes
2
answers
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Mertens-like theorem
Mertens' first theorem states that
$$
\sum_{p \leq n} \frac{\log p}{p} = \log n + O(1).
$$
I read in this paper that the following variant is "classical":
$$
\sum_{p \leq n} \frac{\log p}{p -...
2
votes
1
answer
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views
Proof of Szegö asymptotic theorem
Consider the truncated exponential series
$$P_N(z) = \sum_{n= 0}^N \frac{z^n}{n!}$$
The zeros of this series have been studied by Szëgo and others (see e.g. here). He established an asymptotic for the ...
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votes
0
answers
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views
Asymptotic expansion of Jacobi function
For $\alpha,\beta \in \mathbb{C},\, \alpha$ a non-negative integer, we define $$A_{\alpha,\beta}(t)=(\sinh t)^{2\alpha+1}(\cosh t)^{2\beta+1} $$
and $$ \mathcal{L}_{\alpha,\beta}=\frac{d^2}{dt^2}+\...
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votes
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answers
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Asymptotic analysis for a double integral related to Airy functions
Let $Ai(x,y)$ be the Airy kernel which is given by
\begin{equation}\label{equ2.12}
Ai(x,y)=
\begin{cases}
\dfrac{Ai(x)Ai'(y)-Ai(y)Ai'(x)}{x-y}, & x\ne y, \\
Ai'(x)^2-xAi(x)^2 & x=y. \\
\end{...
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votes
1
answer
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The asymptotic behaviour of a singular integral
Given $0<\alpha, \beta<1$, $a,b>0$, $a^2+b^2<1$.
I am trying to determine the asymptotic behaviour of
$$F(a,b):=\int_{\substack{a/2<x<2a\\\\b/\sqrt{2}<\sqrt{1-x^2}<\sqrt{2}b}}\...
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1
vote
0
answers
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views
Asymptotic behavior of sum of regularly varying function
In the proof of Lemma 4.9 of Beran et al (2013), the authors consider a strictly stationary time series $X = \{X_t, t \in \mathbb{N} \}$ with regularly varying autocovariance function $\gamma_X(k) \...
- 299
1
vote
0
answers
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views
CLT of the left singular vectors of i.i.d. data matrix
Let $\mathbf{X}$ be $(n \times p)$-dimensional random matrix ($n > p$) whose rows $\mathbf{x}_i$ are i.i.d. with some finite moments:
$$
\mathbf{X}^\top = [\mathbf{x}_1, \ldots \mathbf{x}_n]^\...
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5
votes
2
answers
257
views
Signs of Kravchuk matrix asymptotically produce a large circular region with hyperbolic sinks. Why?
The Kravchuk matrix of dimension $n+1$ is such that its entries satisfy $$K_{i,j}^{(n+1)}=[x^i](1+x)^{n-j}(1-x)^j\quad\forall0\le i,j\le n.$$ It enjoys properties such as involution and has various ...
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votes
2
answers
159
views
Asymptotics of the unique root of a polynomial equation defined as a sum of rational expressions
Let $\lambda_1\ge \ldots \ge \lambda_n \gt 0$. Define a function $F:\mathbb R_+ \to \mathbb R_+$ by
$$
F(t) = t^2\sum_{i=1}^n\frac{\lambda_i^2}{(\lambda_i + t)^2}.
$$
It is clear that $F$ is strictly ...
- 6,338
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votes
1
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views
Density of extended Mersenne numbers?
Consider the subset of odd positive integers defined and constructed as follows by these rules :
A) $1$ is in the set.
B) if $x$ is in the set , then $2x + 1$ is in the set.
C) if $x$ and $y$ are in ...
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votes
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answers
102
views
Permutations avoiding a family of consecutive patterns
Let $B=\{1324,14325,154326,1654327,\ldots\}$ be the set of permutation patterns of the form $1(m-1)(m-2)\cdots 2m$ for $m\geq 4$. I'm interested in the set $\mathcal P$ of permutations that avoid all ...
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1
vote
1
answer
76
views
Asymptotic expansion on the following integral of exponential function
I wish to obtain the asymptotic for the following integral
$$
\int_{r: \|r\|\le 1} \exp(M\cdot a^Tr) \, dr,
$$
where $a$ is a given vector in $\mathbb{R}^d$, $\|\cdot\|$ is a general norm function and ...
- 21
1
vote
1
answer
53
views
Asymptotic property of the left singular vectors of i.i.d. data matrix
Let $\mathbf{X}$ be $(n \times p)$-dimensional data matrix ($n > p$) whose rows $\mathbf{x}_i$ are i.i.d. with some finite moments:
$$
\mathbf{X}^\top = [\mathbf{x}_1, \ldots \mathbf{x}_n]^\top.
...
- 282
1
vote
1
answer
101
views
Hypothesis to guarantee Lindeberg's condition
Imagine to have a set of random variables $\{ X_i \}_{i=1}^{n}$ independent (Non identically distributed). In these scenario, if the Lindeberg's condition hold we can extend the result of the CLT, i.e....
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votes
3
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views
How do I solve the following definite integral (preferably by an asymptotic method)?
$$ \int_{1}^{\infty} \frac{\sin^2 (\mu \sqrt{x^2 -1})}{(x+1)^{\frac{9}{2}} (x-1)^{\frac{3}{2}}} \,dx $$
Note: $\mu$ here is an extremely small constant.
I have tried:
Estimating the integral by ...
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votes
2
answers
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views
Does the surface area of the unit Lp ball go to zero for all $p < \infty$?
We know about volume: The $L_{\infty}$ ball of radius one-half, i.e. the hypercube, has volume $1$ in all dimensions. On the other hand, I believe that for every $1 \leq p < \infty$, the volume of ...
- 4,260
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votes
1
answer
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views
Is there a "convolution" of asymptotic growth?
Suppose that I have two asymptotic counts given by
$$
\#\{x \in [0,H] \cap \mathbb Z: f(x) \leq H\} \sim F(H)
$$
and also
$$
\#\{x \in [0,H] \cap \mathbb Z: g(x) \leq H\} \sim G(H).
$$
From these two ...
user494572
3
votes
2
answers
140
views
Precise asymptotics for moments of order statistics of normal distribution
Let $X_1, \cdots, X_n \sim N(0,1)$ be i.i.d. normal random variates. I am interested in understanding the first two moments of the quasi-range $X_{(n)}-X_{(n-1)}$ (i.e., the maximum value minus the ...
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votes
1
answer
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views
applying the watson lemma to an integral [closed]
So i thought about applying the Watson lemma to determine the asymptotic behavior of the integral
$$
I(x)=\int_{0}^{\infty} \frac{e^{-x(t-\ln(t))}}{(1+t^2)} dt,
$$
as $x \rightarrow \infty$.
I think ...
- 7
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votes
1
answer
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views
How to compute the asymptotics of this oscillatory integral?
I posted this on Stackexchange but got no responses or comments.
Consider the following integral, for $\epsilon\ne 0:$
$$\displaystyle\frac{1}{(2\pi)^2\epsilon^4}\int_{\Omega}yb\,e^{\frac{i}{\epsilon}[...
- 885
1
vote
1
answer
72
views
Convergent condition of the high-dimensional submatrix of some orthogonal matrix
Let $\mathbf{V}$ be a $p\times p$ orthogonal matrix (i.e., $\mathbf{V}\mathbf{V}^\top = \mathbf{V}^\top \mathbf{V} = \mathbf{I}$) whose columns are
$$
\mathbf{V} = \begin{bmatrix} \mathbf{v}_1 & \...
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votes
2
answers
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views
Is there a nonpolynomial $C^\infty$ function $f$ such that $\sup_{x \in \mathbb{R}} \lvert f^{(q)}(x) \rvert \leq (\ln q)^{-q}$ for every $q >1$?
The question is as in the title:
Is there a nonpolynomial $C^\infty$ function $f$ on $\mathbb{R}$ such that $\sup_{x \in \mathbb{R}} \lvert f^{(q)}(x) \rvert \leq (\ln q)^{-q}$ for every natural ...
- 1,805
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votes
1
answer
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views
Hamiltonian particle system and its frequency domain
I am interested in the following question.
So let suppose we have finite number of point particles on plane $\mathbb{R}^2$.
We can assume that every $j$ point is represented by Dirac delta function $\...
- 55
8
votes
1
answer
171
views
Asymptotic number of permutation representations of a given group
Let $G$ be a finitely generated group.
I am trying to count the number of permutation representations on $n$ elements, i.e. homomorphisms from $G$ to the symmetric group $S_n$.
Equivalently this is ...
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2
votes
1
answer
134
views
Asymptotic analysis of an expression involving a Fox's H function
One of the performance metrics calculated in the analysis of telecommunications systems is the ergodic channel capacity, $C_{\rm erg}$. During one of my studies, I found the expression below for such ...
1
vote
0
answers
48
views
Necessary and sufficient conditions so that $\sum_{k\ge 0}\frac{x^k}{\Gamma(1+a(k))}\sim\int_0^\infty \frac{x^t}{\Gamma(1+a(t))}\,dt$ as $x\to\infty$?
First, some notation. I'll write $f(x)=o(g(x))$ if $\lim_{x\to\infty} \left|\frac{f(x)}{g(x)}\right|=0$. I'll also write $g(x)=\omega(f(x))$ if $f(x)=o(g(x))$, i.e. $\limsup_{x\to\infty} \left|\frac{g(...
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votes
1
answer
78
views
When is it true that $\sum_{k\ge 0}\frac{x^k}{\Gamma(1+a(k))}\sim\int_0^\infty \frac{x^t}{\Gamma(1+a(t))}\,dt$ as $x\to\infty$?
First, some notation. I'll write $f(x)=o(g(x))$ if $\lim_{x\to\infty} \left|\frac{f(x)}{g(x)}\right|=0$. I'll also write $g(x)=\omega(f(x))$ if $f(x)=o(g(x))$, i.e. $\limsup_{x\to\infty} \left|\frac{g(...
- 655
12
votes
2
answers
456
views
Asymptotics of $\int_0^\infty \frac{x^{2z}}{\Gamma(1+z)}\,dz$ for large $x$
I'm interested in the asymptotics of
$$\int_0^\infty \frac{x^{2z}}{\Gamma(1+z)}\,dz$$
as $x\to\infty$. I expect the results to behave similarly to $e^{x^2}=\sum_{k\ge 0}\frac{x^{2k}}{k!}$. However, I'...
- 655
2
votes
2
answers
304
views
Asymptotic behavior of a hypergeometric function
Can anybody see how to deduce an asymptotic formula for the hypergeometric function
$$ _3F_2\left(\frac{1}{2},x,x;x+\frac{1}{2},x+\frac{1}{2} \hskip2pt\bigg|\hskip2pt 1\right), \quad\mbox{ as } x\to\...
- 2,148
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votes
0
answers
121
views
Asymptotic bound of some number theoretic function
I asked this in stack exchange but did not get anything so I am posting it here.
I am self-studying asymptotic behavior of some number theoretic function and the following question comes up.
Let $n$ ...
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votes
2
answers
426
views
How to determine the asymptotics of $\sum_{n=0}^{\infty} e^{-\frac{2^n}{x}}$
I'm generally interested in being able to find an asymptotic expansion of
$$ \sum_{n=0}^{\infty} \left[ e^{- \frac{f(n)}{x}} \right] $$
As $x \rightarrow \infty$ and $f(n)$ is a smooth monotonically ...
- 1,450
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vote
0
answers
92
views
Converse of transfer theorem : does asymptotic behaviour of coefficients describe the singularity?
I'm interested in Singularity Analysis after reading this Math.Stack post. In the "Analytic Combinatorics" book by P. Flajolet and R. Sedgewick, at p.390, the following transfer theorem (...
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3
votes
0
answers
140
views
The divergent sum $\sum_{n=1}^\infty (-1)^n (n^2)! x^n$
Question
I'm interested in assigning a value to the divergent series $F(x)=\sum_{n=1}^\infty (-1)^n (n^2)! x^n$. I'm hoping that (1) the definition for $F(x)$ has (one-sided) derivatives of $(-1)^n (n^...
- 1,485
3
votes
0
answers
134
views
Spectrum of large Hilbert matrices
Let $x_k>0$ be a increasing sequence of real numbers, such that
$$\sum_0^\infty\frac1{x_k}<+\infty.$$
Let us form the (infinite) Hilbert matrix $A\in{\bf Sym}({\mathbb N};{\mathbb R})$ with
$$a_{...
- 49.5k
0
votes
0
answers
66
views
Stability of a special singular perturbation problem
Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a lower bounded smooth function, i.e., $\inf_{x\in\mathbb{R}^n} f(x)>-\infty$. Consider the following singular perturbation problem:
$$\begin{cases}\dot{...
- 309
5
votes
1
answer
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views
Asymptotic expansion for the number of self-avoiding random walks
This question is cross-posted from https://math.stackexchange.com/questions/4580314/asymptotic-expansion-for-the-number-of-self-avoiding-random-walks.
Let $c_n$ be the number of self-avoiding random ...
- 521
3
votes
1
answer
118
views
What is the optimal asymptotic behavior of this integral over the sphere?
Let $k_{1},\dots, k_{d}>1$ be integers and consider the integral
$$J_{\lambda }=\int_{\mathbb{S}^{d-1}}e^{-\lambda \left(x^{2k_{1}}_{1}+\dots+ x^{2k_{d}}_{d}\right)} d\sigma(x)$$
where $d\sigma$ ...
- 391
2
votes
1
answer
90
views
Asymptotic analysis of a peculiar sum of squares sequence
Let $a,b$ be two positive integers. Let the sequence $\{s_n\}_n$ be the set of all possible sums of squares $a^2+b^2$, such that they are in ascending order
\begin{align*}
& n=1 & s_1=1^2+1^2=...
- 151