Questions tagged [asymptotics]
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734
questions
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
votes
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
votes
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
votes
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
votes
0answers
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 $$Y(x)=\frac d{dx}\frac{\...
0
votes
0answers
21 views
Series solution of an ODE with nonpolynomial coefficients
Basically, I have a second-order differential equation for $g(y)$ and I want to obtain a series solution at $y=\infty$ where $g(y)$ should vanish. That would be easy if the ODE contains polynomial ...
9
votes
0answers
260 views
Being even or odd in the product expansion $\prod(1+x^k+x^{k+1})$
Consider the generating function of "partitions with distinct parts"
$$\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}...
0
votes
0answers
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
0answers
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):
$$F(q):=\sum_{k \ge 1} \frac{q^{k^2}}{(q;q)_k^2}\quad \mbox{as } q \to 1^...
1
vote
1answer
75 views
Using $\delta$-method to “estimate” undefined moments of a random variable?
I posted this over on MSE without much luck. Not sure if posting here is considered cross-posting but I can remove it if it is.
Let $X\sim\mathcal N(\sqrt 2,1/x^2)$. The expected value $\mathsf EX^{-1}...
5
votes
1answer
421 views
Is the harmonic series worse than any summable series?
It is well-known that the harmonic series is not summable. In some sense this means that it takes a lot of rather large values.
We define the operator $F_{\varepsilon}: \ell^{\infty}(\mathbb N) \...
1
vote
1answer
29 views
Asymptotics of the right singular vectors as the number of rows diverge [duplicate]
Write $X_m \in \mathbb{R}^{m \times n}$ as a Gaussian ensemble, so that $(X_m)_{ij} \sim \mathcal{N}(0, 1)$ are independent and identically distributed. Assume that $m \geq n$. Write $X_m = U_m \...
4
votes
0answers
126 views
Product of all multinomial coefficients
At some moment I found that for computing a bound on a density, I need to compute (or find a good asymptotic) of the product of all multinomial coefficients, i.e.,
$$
\prod_{\substack{(\alpha_1,\ldots,...
0
votes
0answers
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
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
votes
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
votes
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
votes
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
votes
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
vote
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
votes
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 ...
1
vote
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
votes
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)=...
1
vote
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 ...
0
votes
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
votes
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
votes
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
votes
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
vote
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
votes
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
votes
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
vote
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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}\...
