The tag has no wiki summary.

learn more… | top users | synonyms

0
votes
1answer
103 views

Function that dominates everything in little o

I have a function $f(n)$ that satisfies the following property: for any function $g(n) = o(n^{-2})$, we have $f(n) = \Omega(g(n))$ (the implied proportionality constant in the $\Omega$ expression ...
0
votes
0answers
48 views

asymptotic notation with graph colouring

This is my first ever post so I hope this is an appropriate question. Basically I am looking at the paper here: http://homepages.math.uic.edu/~mubayi/papers/biclique.pdf Namely theorem 5. Now, feel ...
11
votes
2answers
358 views

Asymptotics for algebraic numbers of height less than one

The question. Is an asymptotic equivalent known or conjectured for the number $N(d)$ of $\alpha \in \bar{\mathbb{Q}}$ with $h(\alpha) < 1$ and $[\mathbb{Q}(\alpha):\mathbb{Q}] \leq d$? The rather ...
-2
votes
0answers
20 views

Asymptotic expansion on 3 nonlinear ordinary differential equations [closed]

The 3 nonlinear differential equations are as follows \begin{equation} \epsilon \frac{dc}{dt}=\alpha I + \ c (-K_F - K_D-K_N s-K_P(1-q)), \nonumber \end{equation} \begin{equation} \frac{ds}{dt}= ...
3
votes
1answer
142 views

Growth rate of eta related function

Consider the function $$f(x)=\prod\limits_{n=1}^{\infty}(1-x^n)$$ I am interested in an asymptotic formula for $f$ as $x\to 1$. Of course $f\to 0$ but I am interested in how fast.
2
votes
0answers
60 views

Asymptotic formula for restricted partition function

Let $p(n)$ be the partition function. Hardy and Ramanujan - and Uspensky, independently proved the asymptotic formula $$(1) \quad p(n) \sim \frac1{4\sqrt{3}} \frac{e^{c_0\sqrt{n}}}{n} \text{ as } n ...
2
votes
1answer
74 views

M-Wright function asymptotics

Let $M(z;\nu):= \frac{1}{\pi}\sum_{n=1}^{\infty} \frac{(-z)^{n-1}}{(n-1)!}\Gamma(\nu n)\sin(\nu n\pi)=\frac{1}{2\pi i}\int_{\text{H}_a}\exp(\sigma -z\sigma^{\nu})/\sigma^{1-\nu} d\sigma$, ...
0
votes
0answers
66 views

Asymptotic analysis involving a circular multiple integral

Let $t_1,\ldots,t_m>0$, and $m\ge 4$ be an even integer. Consider the function: $$ f(a,b;\mathbf{t})=\int_0^{t_1}\ldots\int_0^{t_m} |x_1-x_m|^a |x_2-x_1|^b |x_3-x_2|^a |x_4-x_3|^b \ldots ...
16
votes
1answer
358 views

Covering a set with geometric progressions

Consider the set $S_n=\{1,2,\cdots ,n\}$. What is the minimum number of distinct geometric progressions that cover $S_n$? Let us call this number $a_n$. I was wondering about this number after doing a ...
3
votes
2answers
140 views

Asymptotics of the number of elements in the intersection of two growing sets

Let $[n]:=\{1,\dots,n\}$ and $0\leq p_n\leq n$. Fix any subset $A_n$ of $[n]$ with $p_n$ elements. The number of subsets $B$ of $[n]$ with $p_n$ elements that are disjoint from $A$ is ...
2
votes
0answers
94 views

Number of k-generated semigroups

Given some $k>1$, I am interested in the number of $k$-generated semigroups of order $n$ (either up to isomorphism or all associative binary operations on an n-element set). At first I thought ...
3
votes
1answer
77 views

Eigenfunctions to 2nd-order Differential Operators: Relation between Frobenius Series Solution and Eigenfunction Normalised to the Delta Function

Consider the 2nd-order linear ODE $x f^{''}(x) + x (\beta - 2 \alpha x) \kappa / \sigma f^{'}(x) - 1 / \sigma \left[ 2 \alpha \kappa - \lambda^2 (\beta - 2 \alpha x)^2 \right] f(x) = 0$, where ...
0
votes
0answers
19 views

Convergence integral in probability

Let $X_1,\dots,X_n$ be i.i.d with distribution function $F$. Let $\hat F_n$ be their modified empirical distribution function, i.e., $$ \hat F_n(x)=\frac1{n+2}\left(1+\sum_{i=1}^n1_{\{X_i\le ...
0
votes
0answers
61 views

On the product of relatively prime number $< N$ [duplicate]

Let $FI(N)$ denote the product of all $\phi(N)$ [relatively prime numbers $<N$] . And define $SFI(N)$ as the product of remaining $N-\phi(N)$ numbers $\le N$ (Which are not relatively prime to $N$) ...
3
votes
0answers
49 views

Limit of a hypergeometric integral

Let $n,N,T$ be positive integers, with $N=\binom{n}{2}$, and $3\leq n\leq T\leq N$. Define: $$P(z):=z^{N+1-T}\int_0^1\frac{(1-t^2)^{n-2}}{(1-(1-z)t)^{N+1}}{}_2F_1\left[-T,N+1,N+2-T; ...
2
votes
0answers
173 views

More information on Kruskal's treatment of Surreal numbers as an asymptotic behavior of a real valued function

The only way that I could think about Surreal numbers is how Conway defined them inductively, with the two axioms and so on. I can't find any information about Kruskal's point of view and would very ...
1
vote
0answers
77 views

References on law of large numbers, CLT and iterated logarithm laws

Having access to those references, accumulating many results in one domain is always a bless, like Feller's book in probability, Dembo-Zeitouni's large deviation, Grimmett's percolation and recent ...
2
votes
1answer
336 views

A generalized urn-ball matching problem; Complicated combinatoric/probabilistic limit

I'm looking for a generalization to the urn-ball matching problem. As a reminder of what I've got in mind, here's the simple version: Randomly assign (with replacement) $N$ balls to $M$ urns. ...
4
votes
1answer
144 views

Asymptotic expansion of the Mordell integral

my question concerns the Mordell integral $$h(z;\tau):=\int_{-\infty}^\infty \frac{e^{\pi i\tau w^2-2\pi zw}}{\cosh(\pi w)}dw,\qquad \Im(\tau)>0,\quad z\in\mathbb{C},$$ which frequently occurs in ...
3
votes
1answer
219 views

leading-order behaviour of riemann zeta function?

Is there any 'guess' as to how the Riemann zeta function $\zeta(\sigma+it)$ (or its modulus) behaves to leading order as $t\rightarrow\infty$, for fixed $\sigma$ in the critical strip? Obviously this ...
3
votes
1answer
96 views

Convergence of the Double Integral of a Polynomial Reciprocal

Let $f \in \mathbb{R}[x,y]$ be a polynomial satisfying the following conditions: (i) $f(\mathbb{R}^2) \subset [a,\infty)$ where $a>0$; (ii) $f$ is non-degenerate, in the sense that there isn't a ...
3
votes
0answers
85 views

Nonlinear Markov process

Consider the following nonlinear $\mathbb{R}$-valued stochastic recursive sequence: $ X_{n+1} = F(X_n) + W_{n+1}, \quad (W_n)_{n\ge1} \stackrel{ \scriptsize \mathrm{i.i.d.} }{ \sim } \phi. $ How can ...
2
votes
1answer
287 views

More asymptotics for trees

This is a follow up to my recent question on the asymptotics of A003238. Lucia gave a fine answer to that question, but as I hinted the 'real' problem I have in mind is slightly different, and I've ...
13
votes
1answer
421 views

Are the asymptotics of A003238 known?

Sequence A003238 of the OEIS counts ``rooted trees with $n$ vertices in which vertices at the same level have the same degree.'' The sequence, $a$, begins 1, 1, 2, 3, 5, 6, 10, 11, 16, ... and it is ...
1
vote
1answer
80 views

The asymptotic behavior of hypergeometric function around -1

Recently, in studing some specific orthogonal polynomials on unit circle, I was lead to study the asymptotic behavior of the following hypergeometric function at the neighberhood of $-1$: $$ ...
6
votes
1answer
423 views

Mean of i.i.d Random Variables With No Expected Value

Let $X$ be an integer-valued random variable and let $X_n$ be the sum of $n$ independent realizations of $X$. I would like to understand the behavior of $X_n/n$ for large $n$ in some cases where $X$ ...
1
vote
1answer
140 views

Asymptotic form of the Gauß Hypergeometric function 2F1 for three parameters approaching infinity

I am trying to find the leading order expression in an expansion for large $\Delta$ of ${}_2F_1\left(\frac{\Delta}{2},\frac{\Delta+1}{2},\Delta,z^{-2}\right)$, where $z\in\mathbb{C}$. The only ...
4
votes
1answer
195 views

Asymptotic behavior of $\{\text{#}(a,x,b,y)\in \mathbb{N^4}| \text{ }ax+by=n\}$ for large $n$

For a fixed natural number $n$ is it possible to obtain an asymptotic for the number of solutions $(a,x,b,y)\in\mathbb{N^4}$ to $ax+by=n$ or equivalently an asymptotic for ...
-1
votes
1answer
118 views

Method of steepest descents for $\int_0^\infty\exp(iz(\frac{1}{3}t^3+t))dt\sim\frac{i}{z}$

Through some calculations I ended up with the integral $\int_0^\infty\exp(iz(\frac{1}{3}t^3+t))dt$. I would like to obtain the result that the behaviour of the integral is $\sim\frac{i}{z}$ as ...
3
votes
3answers
342 views

Estimating a sum involving binomial coefficients [refined]

Having some work done, here is a refined version of my initial question. For integer $m>0$ and $0\le q\le m$, consider the sum $$ S(m,q) = \sum_{i=0}^{m-q} \binom{m}{i} \binom{m-i}{q}^2. $$ I ...
3
votes
1answer
167 views

Asymptotic expansion of modified Bessel function $K_\alpha$

An integral representation for the Bessel function $K_\alpha$ for real $x>0$ is given by $$K_\alpha(x)=\frac{1}{2}\int_{-\infty}^{\infty}e^{\alpha h(t)}dt$$ where ...
1
vote
1answer
275 views

Number of matrices with no repeated columns or rows

If you consider all $m$ by $n$ matrices with entries that are either $0$ or $1$, there are ${2^{n} \choose m}$ with no repeated rows (up to row permutation) and ${2^{m} \choose n}$ with no repeated ...
5
votes
1answer
232 views

Convergence rate of the central limit theorem near the center of the distribution

I'm looking for fast convergence rates for the central limit theorem - when we are not near the tails of the distribution. Specifically, from the general convergence rates stated in the Berry–Esseen ...
2
votes
0answers
110 views

geometric irregularities in pde's

The following question is intended for a person more acquainted with the works of Yves Laurent. see: http://archive.numdam.org/article/ASENS_1987_4_20_3_391_0.pdf (French) ...
2
votes
0answers
167 views

What can be said about a function given its asymptotic expansion?

This is probably not a research level question but I honestly don't know how/where to look for techniques to reconstruct a function from its asymptotic expansion. The expansion I want to know about ...
3
votes
0answers
63 views

Asymptotics of partitions in at most n parts, bounded by r

I posted this question on MathStackexchange (http://math.stackexchange.com/questions/639878/asymptotics-of-partitions-in-at-most-n-parts-bounded-by-r) some time ago, but it did not receive any answer, ...
6
votes
1answer
384 views

Mean time to get $k$ heads for a coin with growing bias

For integers $n,k\geq 1$ we repeatedly toss a coin and count the number of heads that occur. The probabilty of getting a head is $min(t/n,1)$ where $t$ is the current discrete time step. I am trying ...
2
votes
2answers
302 views

Asymptotic behaviour of sequence

I am interested in the sequence $$a(n)=\sum_{k=0}^n {p(n-k) \choose k}$$ where $p(n)$ is a polynomial equation. When $p(n)=n$ this reduces to the Fibonacci sequence, but what about when $p(n)$ is ...
10
votes
3answers
307 views

The intersection of $n$ cylinders in $3$-dimensional space

A standard question in vector calculus is to calculate the volume of the shape carved out by the intersection of $2$ or $3$ perpendicular cylinders of radius $1$ in three dimensional space. Such ...
3
votes
1answer
107 views

Convergence of trajectories and asymptotic stability

Say that an autonomous system $\dot{u} = f(u)$ in $\mathbb{R}^{m}$ has the property that for any two solutions $x(t), y(t)$ corresponding to initial conditions $x(0)$ and $y(0)$ the trajectories are ...
0
votes
1answer
149 views

Approximation of the sum involving binary entropy function

Given the following sum: $S(n) = \sum_{i=1}^{n} \frac{1}{(1-\operatorname{H}(p))^i}$ where $H$ is the binary entropy function defined as: $\operatorname{H}(p) = -p\log p - (1-p)\log (1-p) $. Let ...
0
votes
1answer
262 views

Asymptotic growth for $\sum_{i=1}^{n-1}(n-i)\binom{k}{i}$ [closed]

For positive integers $n,k$, define $$f(n,k):=\sum_{i=1}^{n-1}(n-i)\binom{k}{i}.$$ What are upper and lower bounds of $f(n,k)$ by simpler terms? (e.g. finding bounds which are not a summation like ...
2
votes
0answers
56 views

Associated Legendre/Gegenbauer functions with complex degree at larger order

I am interested in approximating the associated Legendre function (also known as conical function) \begin{equation} P_{-1/2 + i p}^{\frac{2-N}{2}}(x) \end{equation} when $N \to \infty$. The real ...
1
vote
2answers
217 views

Asymptotics of the maximum of binomial random variables

Let $B_i(n,1/2)$ be independent identically distributed binomial random variables. I am interested in the asymptotic growth of the maximum of $n$ such random variables. In ...
5
votes
1answer
229 views

Asymptotic value of a multivariate integral

The following question is a simple case of a type of problem that occurs in combinatorial enumeration problems. Define $$F(x_1,\ldots,x_n) = \frac{1}{(2\pi)^{n/2}}\exp\biggl( -\frac12\sum_{j=1}^n ...
3
votes
1answer
189 views

Asymptotic behavior for the solution of a nonlinear ODE

In a nutshell, if $u$ is a solution to $$ \partial_r^2 u(r)+ \frac{1}{r} \partial_r u(r) - u(r) ( 1- u(r)) = 0, \quad \text{for} \; r>r_0>0\\ \lim_{r \to \infty} u(r) = 0, \quad \text{and} ...
2
votes
0answers
161 views

Sum of binomial coefficients weighted by a lower incomplete regularized gamma function

The following summation turned up in the course of my research: $$S_n=\sum_{k=0}^n {n \choose k}\lambda^k P(k,t)$$ where $P(k,t)=\frac{1}{\Gamma(k)}\int_{0}^t e^{-x}x^{k-1}dx$ is the lower ...
4
votes
0answers
128 views

Nesting big-O with big-Omega $O(g(\Omega(h(n))))$: is it $O$ for all $\Omega$ or for one $\Omega$?

I want to express the following statement about a function $f(n)$: there exists $f_\Omega\in\Omega(h(n))$ such that $f\in O(g(f_\Omega(n))$. What's the correct notation for this? Is it $f\in ...
3
votes
2answers
432 views

A set with not too many integers of the form $\alpha \beta^n + r$

Consider the following (easy) lemma. Lemma. There is a subset $Q$ of the positive integers and a fixed constant $N > 0$ such that 1)$Q$ has positive asymptotic density and 2)for each ...
8
votes
1answer
301 views

Asymptotic behavior of the sequence $u_n = u_{n-1}^2-n$

I am currently interested in the following sequence: $$\begin{cases}u_0 & = & \alpha\\u_n & = & u_{n-1}^2-n\end{cases}$$ where $\alpha > C \approx 1.75793275... $ with $C$ being the ...