# Questions tagged [asymptotics]

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734
questions

**3**

votes

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95 views

### Will an integer combination of some number of copies of the set of powers of 2 and the set of powers of 3 always have natural density 0?

Consider a Diophantine equation of the form
$$(c_1 2^{x_1} + \dots + c_n 2^{x_n}) + (c_{n+1} 3^{x_{n+1}} + \dots + c_m 3^{x_m}) = y$$
where $x_1, \dots, x_m, y$ are our variables (here $x_1, \dots, ...

**4**

votes

**1**answer

84 views

### Estimate of $\frac{\int x^{2p}\,e^{-x^{2n}\,+\,\omega(x,y)}\;dx}{\int e^{-x^{2n}\,+\,\omega(x,y)}\;dx}$

For every $x,y\in\mathbb R$ let
$$ V(x,y) \,\equiv\, a\,x^{2n} + b\,y^{2m} - \omega(x,y)\,$$
where $a,b>0$, $n,m\in\mathbb N$, $n\geq m\geq1$, and $\omega$ is such that $\omega(x,y)/(x^{2n}+y^{2m})...

**2**

votes

**1**answer

135 views

### Existence of function $f$ such that $f(x) \sim \sum_{j \in \mathbb{N}} x^{1 - \frac{1}{j}}$

Is there a function $f$ on $\mathbb{R}$ such that as $x \to 0$,
$$
f(x) = \sum_{j=0}^N x^{1 - \frac{1}{j}} + o(x^{1- \frac{1}{N}}),
$$
for every $N \in \mathbb{N}$?
Heuristically there shouldn't be ...

**0**

votes

**0**answers

33 views

### Asymptotic optimal sphericity

How quickly does maximum sphericity of polyhedra with $n$ faces approach 1 as $n→∞$? I can show that sphericity $1 - \frac{5 \sqrt{3} π}{27n} - O(n^{-3/2})$ is possible. Is this, especially $O(n^{-3/...

**0**

votes

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33 views

### Almost-differential functional equations

The ODE $y'(x)+P(x)y(x)=Q(x)$ has solution $$I(x)y(x)=\int I(x)Q(x)\,dx$$ where $I(x)=\exp\int P(x)\,dx$. Equivalently, $$Y(x)+P(x)\int_0^xY(t)\,dt=Q(x)\tag1$$ has solution $$Y(x)=\frac d{dx}\frac{\...

**0**

votes

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21 views

### Series solution of an ODE with nonpolynomial coefficients

Basically, I have a second-order differential equation for $g(y)$ and I want to obtain a series solution at $y=\infty$ where $g(y)$ should vanish. That would be easy if the ODE contains polynomial ...

**9**

votes

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260 views

### Being even or odd in the product expansion $\prod(1+x^k+x^{k+1})$

Consider the generating function of "partitions with distinct parts"
$$\sum_nQ(n)x^n=\prod_k(1+x^k).$$
It's known that
$$\left[\prod_k(1+x^k)\right] \mod 2=\prod_m(1-x^m)=\sum_{j\in\mathbb{Z}...

**0**

votes

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21 views

### Assigning negative integer moments to random variables with Hadamard regularization

Let $X\sim F_X$ denote a continuous random variable that admits a density $f_X$ with support $\mathcal S=\operatorname{supp}(X)\ni 0$ and assume $f_X(0)>0$. I am interested in defining a ...

**3**

votes

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99 views

### Asymptotics of a combinatorial series

I am interested in the exact asymptotics of the following combinatorial series (which arises from the study of a Markov chain):
$$F(q):=\sum_{k \ge 1} \frac{q^{k^2}}{(q;q)_k^2}\quad \mbox{as } q \to 1^...

**1**

vote

**1**answer

75 views

### Using $\delta$-method to “estimate” undefined moments of a random variable?

I posted this over on MSE without much luck. Not sure if posting here is considered cross-posting but I can remove it if it is.
Let $X\sim\mathcal N(\sqrt 2,1/x^2)$. The expected value $\mathsf EX^{-1}...

**5**

votes

**1**answer

421 views

### Is the harmonic series worse than any summable series?

It is well-known that the harmonic series is not summable. In some sense this means that it takes a lot of rather large values.
We define the operator $F_{\varepsilon}: \ell^{\infty}(\mathbb N) \...

**1**

vote

**1**answer

29 views

### Asymptotics of the right singular vectors as the number of rows diverge [duplicate]

Write $X_m \in \mathbb{R}^{m \times n}$ as a Gaussian ensemble, so that $(X_m)_{ij} \sim \mathcal{N}(0, 1)$ are independent and identically distributed. Assume that $m \geq n$. Write $X_m = U_m \...

**4**

votes

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126 views

### Product of all multinomial coefficients

At some moment I found that for computing a bound on a density, I need to compute (or find a good asymptotic) of the product of all multinomial coefficients, i.e.,
$$
\prod_{\substack{(\alpha_1,\ldots,...

**0**

votes

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29 views

### Asymptotics of a sum involving multiplicative partitions of an integer $n$ into $k$ possibly non-distinct parts $≥2$

Let $x\in\left(0,1\right)$. For each integer $n\geq2$, let $\Omega\left(n\right)$ denote the number of prime factors of $n$, counted according to multiplicities; thus $\Omega\left(2\right)=1$, $\Omega\...

**3**

votes

**1**answer

106 views

### Randomized version of Turán's theorem II

$\newcommand{\om}{\omega}$Let $\om(G)$ denote the number of vertices in a largest clique of an (undirected) graph $G$ with the set $[n]:=\{1,\dots,n\}$ of vertices. Then
\begin{equation}
\om(G)\ge\...

**5**

votes

**1**answer

170 views

### Randomized version of Turán's theorem

Turán's theorem says the following.
Take any natural $n$ and $r$. Suppose that
\begin{equation*}
|G|>\Big(1-\frac1r\Big)\frac{n^2}2, \tag{0}
\end{equation*}
where $|G|$ is the number of edges of ...

**0**

votes

**1**answer

114 views

### Ratio limit results for restricted partition functions

This concerns difference/limit ratio results for special restricted partitions.
Let $r,a, b$ be nonnegative integers; define $p(r,a,b)$ to be the number of partitions of the integer $r$ using at most $...

**2**

votes

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119 views

### Is there an elementary subexponential upper bound on the size of the stable stems?

This is a question in stable homotopy theory which I will boil down to a pure combinatorics question. If you're not interested in the homotopy theory, feel free to skip to the end for the ...

**0**

votes

**1**answer

46 views

### Perturbative approach starting from a probability distribution approximated form

I approximate a probability distribution $P_x(x)$ with a $P_x^{app}(x)$,
such that $P_x(x)-P_x^{app}(x) = O(\epsilon)$ uniformly in x, where epsilon is a small positive quantity.
Consider the generic ...

**1**

vote

**1**answer

47 views

### Critical point of saddle point equation

Consider the following integral:
\begin{equation}
\int \mathrm{d}\rho \frac{1}{\rho} e^{N f(\rho)}
\end{equation}
Where:
\begin{equation}
f(\rho)=\ln \rho-\frac{1}{2} \rho^{2}+\frac{1}{2 p w^{2}} \rho^...

**10**

votes

**3**answers

417 views

### Asymptotic analysis of $x_{n+1} = \frac{x_n}{n^2} + \frac{n^2}{x_n} + 2$

Problem: Let $x_1 = 1$ and $x_{n+1} = \frac{x_n}{n^2} + \frac{n^2}{x_n} + 2, \ n\ge 1$.
Find the third term in the asymptotic expansion of $x_n$.
I have posted it in MSE six months ago without ...

**1**

vote

**0**answers

33 views

### Entropy per site of quantum spin chain

It’s fork lore that von Neumann entropy (and free energy) grows linearly with respect to the size of a quantum system. Is there a rigorous demonstration in the toy model of a quantum spin chain with (...

**2**

votes

**2**answers

227 views

### Asymptotic of an improper integral

I found myself stuck with an "elementary" claim in some article. A simplified version of the problem is:
Let $p : [0,1]\to [0,1]$ be a continuous and non decreasing function such that $p(0)=...

**1**

vote

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60 views

### Approximating the partial sum of remainders function

This is a question related to the one I posted here, but I have found some more interesting and general results and thought here might be a better place to ask.
Let $R_{k,N}$ denote the remainder of ...

**0**

votes

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42 views

### Can the Bessel functions tend to a plane wave?

Can the Bessel functions tend to a plane wave?
If I have this function:
$$
y(u)= c_1J_{-\sqrt{b}/2}(e^{2u}/6)+c_2J_{\sqrt{b}/2}(e^{2u}/6)+c_1J_{-i\sqrt{b}/2}(e^{2u}/6)+c_2J_{i\sqrt{b}/2}(e^{2u}/6)
$$
...

**9**

votes

**1**answer

283 views

### Two-term recurrence relation

We consider the following system of recurrence relations for $n \in \mathbb Z$ and $\vert \lambda \vert=1$ with $\lambda \in \mathbb{C}$
$$a_{n+1} = \lambda a_{n-1}+ \lambda^* a_n + \lambda^* n b_n $$
...

**2**

votes

**0**answers

108 views

### Asking for results on critical points and similar properties of solutions of nonlinear Volterra integral equations - Physically coherent solutions

I have a system of nonlinear Volterra integral equations of form
$$x(t)=x_0+\int_0^t K(t,s)F(x(s))ds$$
and I am interested on the critical points of $x(t)$, I mean maximum, minimum, increasing and ...

**0**

votes

**1**answer

47 views

### Asymptotic behavior of the Student's t-quantile function of Student's t-cumulative distribution function

Let's denote
$F_{t_u}^{-1}(x)$ the quantile function of the Student's t-distribution $t_u$ with $u$ degrees of freedom and
$F_{t_v}(x)$ the cumulative distribution function of the t-distribution $t_v$...

**1**

vote

**0**answers

35 views

### Convergence result on Cornish Fisher expansion of binomial distribution

Since it is known that Cornish Fisher expansion of quantiles does not have guaranteed convergence for all distribution, I wonder specifically if any convergence result is known in literature for CF ...

**2**

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263 views

### For the following class of matrices, are the determinants invariant under permutations?

I want to ask a question regarding the invariance of determinants under permutation. The following matrix is the one I want to discuss here. (It's just a symmetric block tridiagonal matrix with non-...

**0**

votes

**1**answer

57 views

### lower bound for sum of the n factors of the inclusion exclusion principle

Suppose the following relation is established:
$P\Bigl(A\cup B \cup C\cup D\Bigr) < P\Bigl(E\cup F\cup G\Bigr)$
based on boole's inequality, for each of the above probabilities we can have the ...

**1**

vote

**1**answer

64 views

### Tail bounds on random series in Hilbert space

Tail bounds on random series in Hilbert space
Let $X_n$, $n \in \mathbb {N}$, be independent $\pm 1$ symmetric random variables, and $a_n$,
$n \in \mathbb {N}$, be a sequence in a Hilbert space $H$ ...

**4**

votes

**0**answers

129 views

### Asymptotics of ratios of polynomially recursive sequences

A sequence $a_n$ is said to be polynomially recursive (P-recursive) if it satisfies:
$$p^{[r]}(n)a_{n+r}+\cdots+p^{[1]}(n)a_{n+1}+\cdots + p^{[0]}(n)a_n=0$$
where $p^{[i]}(t)\in \mathbb{Q}[t]$ are ...

**6**

votes

**1**answer

248 views

### Prove or disprove that $\sup_{n\in\mathbb{N}}\left|\sum_{\substack{d|n \\d<Q}}\mu(d)\right|\sim\pi(Q)$

To begin, let us set
$$A_Q(n):=\sum_{d|n \\ d<Q}\mu(d)$$
If we fix $Q$ and let $n$ vary, we get a very surprising amount of cancellation. For instance, the trivial bound
\begin{align*}
\mathbb{E}_{...

**5**

votes

**1**answer

171 views

### A simple oscillatory integral with a non-smooth phase

Let $\phi\in C_c^\infty(\mathbb{R})$ be an even function such that $\chi_{(-1/2,1/2)}\le\phi\le \chi_{(-1,1)}$, where $\chi_{(a,b)}$ stands for the indicator function of the interval $(a,b)$. For $\...

**2**

votes

**1**answer

56 views

### If signed measures $\mu_n$ are such that $\mu_n\to\mu$ and $\|\mu_n\|\to c\in(0,\infty)$, does $\exp^*(\mu_n)/\|\exp^*(\mu_n)\|$ necessarily converge?

$\newcommand{\R}{\mathbb R}$Let $M$ denote the set of all finite signed measures on a separable Banach space $B$. For any $\mu\in M$, let
\begin{equation*}
\exp^*(\mu):=\sum_{k=0}^\infty\frac{\mu^{...

**3**

votes

**2**answers

553 views

### Density of the set of numbers whose sum of digits is prime

Let $A$ be the set of numbers whose sum of digits is prime (http://oeis.org/A028834).
I would like to know if $A$ has zero natural density, that is, if $$\lim_{n \to +\infty} \frac{A(n)}{n} = 0,$$ ...

**1**

vote

**1**answer

168 views

### Enumerating binary matrices by $X$-ray sequences

Consider all $n\times n$ binary (entries are either $0$ or $1$) matrices, denoted $\mathcal{B}_n$.
Define the $X$-ray sequence of $A=(a_{ij})\in\mathcal{B}$ by $X(A)=x(1)x(2)\cdots x(2n-1)$ where
$x(k)...

**2**

votes

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114 views

### Weaker version of the Borel lemma for vector-valued functions

Borel's lemma for Frechét-spaces $V$ says:
(i) For every $(v_j)_{j \in \mathbb{N}} \in V^\mathbb{N}$ there exists an interval $I \subseteq \mathbb{R}$ containing $0$ and a smooth $f: I \to V$ such ...

**6**

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71 views

### Clarification for a statment from ArXiv:1812.07690 “Asymptotics of Nahm sums at roots of unity”

This is a cross-post of the same math.SE question to MO, thinking that is better suited here.
My question is about Lemma 2.1 from the ArXiv:1812.07690 by D. Zagier and S. Garoufalidis which concerns q-...

**2**

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**0**answers

95 views

### The uniform “probability” on $\mathbf{N}$: What occurs beyond logarithmic density?

This is a follow-up to Question #47134. There is obviously no uniform probability distribution on $\mathbf{N}$ (or $\mathbf{Z}$); however, using the notion of amenability, you can show that any ...

**4**

votes

**3**answers

304 views

### How to obtain the asymptotics of Legendre polynomials directly from their generating function

I'm reading about Legendre polynomials for additional information since it is interesting to know! Moreover it would help me with
a task I am working on. See
https://math.stackexchange.com/questions/...

**3**

votes

**2**answers

168 views

### Asymptotic bound for $\sum_{x=0}^\infty \sum_{y=0}^\infty (x+y)^m e^{-\frac{x^2}{2i} - \frac{y^2}{2j}}$ for $i$ and $j$ large

Note: This question relates to two previous questions on math.stackexchange (1 and 2), neither of which had satisfactory answers after posting bounties.
Whilst trying to count certain types of ...

**0**

votes

**0**answers

23 views

### Minimization of a palindromic-like sequence and asymptotics

Suppose that I have a sequence of $n$ numbers $x_1, \ldots, x_n$ all taken from
the real interval $[0,1]$.
I am interested in minimizing the infinity norm of the vector
$$ v = \left( \frac{x_{1}}{x_2},...

**4**

votes

**1**answer

172 views

### An asymptotic expansion of a infinite sum

I am interested in the asymptotic expansion in $t$($t>0$) when $t\to 0^+$ of the following series
$$
\sum_{k\ge 0}e^{-k^{2/n}t}
$$
for integer $n>2$ (n=1 follows from Poisson summation formula ...

**2**

votes

**1**answer

130 views

### Existence of an asymptote for $g(x)=\frac{f(x)f'(x)+f(1)f'(1)}{f'(x)+f'(1)}-f\left(\frac{xf'(x)+f'(1)}{f'(x)+f'(1)}\right)$

Working with the Slater's inequality (compagnion of Jensen's inequality) I find this statement :
Let $f(x)$ be a continuous,twice differentiable function ,convex or concave and non constant on $(0,\...

**1**

vote

**1**answer

77 views

### Saddle point approximation of terms in a sum

(asked in MSE, but received no attention)
Suppose I need to compute a sum,
$$ \sum_{n=0}^N a_n,$$
each term of which involves an integral,
$$a_n=\int e^{Nf(x)+ng(x)}dx.$$
I am interested in the large-$...

**0**

votes

**1**answer

55 views

### Distribution of line segment intersections in random pointsets

let $P$ be a set of $n$ points that are uniformly distributet inside the unit square ore unit circle, and $L=\lbrace\ell_{ij}\rbrace := \lbrace \lbrace \alpha p+ (1-\alpha q)\rbrace\,|\,0\le\alpha\le ...

**9**

votes

**1**answer

231 views

### Concentration inequalities for very rare events on a multiplicative scale

Let $E_1, \dots, E_N$ be independent events, each of probability $p$, where $p$ is very close to $0$. Let $A_N = \frac{1}{N} ( 1_{E_1} + \dots + 1_{E_N} )$ be the proportion of the events $E_i$ that ...

**0**

votes

**0**answers

22 views

### Saddle point approximation when $-N$ is not extractable

I am trying to evaluate the following integral using the saddle point method:
\begin{equation}
\int_0^\infty \exp\left\{-N\ln\left[\frac{\epsilon}{\sigma}+\ln (\sigma+1)+\frac{y^{2}}{\sigma+1-\tau}\...