Questions tagged [asymptotics]

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3
votes
0answers
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
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1answer
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
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1answer
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
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0answers
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
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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
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0answers
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
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0answers
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}...
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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
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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
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1answer
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
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1answer
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
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1answer
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
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0answers
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,...
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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
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1answer
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
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1answer
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
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1answer
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
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0answers
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
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1answer
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
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1answer
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
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3answers
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 ...
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0answers
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
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2answers
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)=...
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0answers
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 ...
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0answers
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
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1answer
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
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0answers
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
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1answer
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
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0answers
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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0answers
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
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1answer
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
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1answer
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
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0answers
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
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1answer
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
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1answer
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
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1answer
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
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2answers
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
1answer
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
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0answers
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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0answers
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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0answers
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
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3answers
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
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2answers
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
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0answers
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
1answer
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
1answer
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
1answer
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
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1answer
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
1answer
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
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0answers
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}\...

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