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Questions tagged [asymptotics]

Asymptotic behavior of functions, asymptotic series and related topics

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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 $d\mu$ ...
Isaac's user avatar
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6 votes
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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 ...
Dan's user avatar
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0 votes
2 answers
142 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 $i=1,2$.
Turbo's user avatar
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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\...
Vincent Granville's user avatar
1 vote
0 answers
109 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 ...
Andrei MF's user avatar
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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 ...
Max Muller's user avatar
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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 ...
53Demonslayer's user avatar
6 votes
1 answer
385 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 $...
Ali's user avatar
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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 \...
Medo's user avatar
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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 ...
Guy Fsone's user avatar
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2 votes
2 answers
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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\...
T. Amdeberhan's user avatar
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1 answer
117 views

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^...
shortfatboy's user avatar
11 votes
2 answers
1k views

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 -...
Charles Bouillaguet's user avatar
2 votes
1 answer
96 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 ...
TheStudent's user avatar
2 votes
0 answers
95 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}+\...
Prof.Hijibiji's user avatar
4 votes
0 answers
120 views

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{...
Tomas's user avatar
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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}}\...
Medo's user avatar
  • 391
1 vote
0 answers
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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) \...
AlbertRapp's user avatar
1 vote
0 answers
40 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]^\...
Seung Hyeon Yu's user avatar
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 ...
TheSimpliFire's user avatar
0 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 ...
dohmatob's user avatar
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6 votes
1 answer
178 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 ...
mick's user avatar
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4 votes
0 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 ...
Colin Defant's user avatar
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 ...
user497696's user avatar
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. ...
Seung Hyeon Yu's user avatar
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....
user1172131's user avatar
6 votes
3 answers
597 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 ...
Abdullah's user avatar
7 votes
2 answers
244 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 ...
usul's user avatar
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4 votes
1 answer
394 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 ...
user avatar
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 ...
Thurmond's user avatar
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-2 votes
1 answer
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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 ...
hello's user avatar
  • 7
4 votes
1 answer
159 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}[...
Josh Lackman's user avatar
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 & \...
Seung Hyeon Yu's user avatar
9 votes
2 answers
632 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 ...
Isaac's user avatar
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0 votes
1 answer
55 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 $\...
Dragomir's user avatar
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 ...
Squala's user avatar
  • 954
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 ...
Felipe Augusto de Figueiredo's user avatar
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(...
Zachary's user avatar
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2 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(...
Zachary's user avatar
  • 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'...
Zachary's user avatar
  • 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\...
Twi's user avatar
  • 2,148
0 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 $n$ ...
KAK's user avatar
  • 179
10 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 ...
Sidharth Ghoshal's user avatar
1 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 (...
Desura's user avatar
  • 201
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^...
Caleb Briggs's user avatar
  • 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_{...
Denis Serre's user avatar
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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{...
Jean Legall's user avatar
5 votes
1 answer
98 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 ...
Testcase's user avatar
  • 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 $d\sigma$ ...
Medo's user avatar
  • 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=...
TheVal's user avatar
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