How distributive are the bad Laver tables?

Suppose that $n\in\omega\setminus\{0\}$. Then define $(S_{n},*)$ to be the algebra where $S_{n}=\{1,…,n\}$ and $*$ is the unique operation on $S_{n}$ where $n*x=x$ $x*1=x+1\,\text{mod}\, n$ and if $y<n$, then $x*(y+1)=(x*y)*(x*1)$. The algebra $(S_{n},*)$ satisfies the self-distributivity law $x*(y*z)=(x*y)*(x*z)$ if and only if $n$ is a power of $2$, and if $n$ is a power of $2$, … Read more

Table of LCM’s vs. table of products

In 2004 Kevin Ford established sharp asymptotics on Erdős’ problem on the number of different products a⋅b, a,b∈{1,…,n}. (, see also discussion here: Number of elements in the set {1,⋯,n}⋅{1,⋯,n}) My naive question is whether there are much less different numbers of the form lcm(a,b), where a,b∈{1,…,n}. Answer AttributionSource : Link , Question Author : … Read more

Asymptotic expansion of Mellin transform of products of modified Bessel function K

Let n≥1 be an integer, let F(x,y)=∫∞0un(x+y)(Kx−y(u))ndu for x,y≥0. When n=1, this is just Mellin transform of the Bessel K function. When n=2, F(x,y) has an explicit form in product of Gamma functions, given by the Parseval formula for Mellin transform. For general n, I expect some Stirling formula type estimation for F(x,y). I tried … Read more

Effective prime number theorem

The prime number theorem implies that for every ϵ>0, there is n_\epsilon such that for all n≥n_\epsilon the number of primes in [n,cn] is at least \frac{(c−1−\epsilon)n}{\log n} and at most \frac{(c−1+\epsilon)n}{\log n} for every c>1. What is the precise value of n_\epsilon as a function of \epsilon known unconditionally and conditionally on reasonable conjectures? … Read more

The asymptotic behavior of the ratio between the largest two of nn i.i.d. chi-square random variables

My question is about the asymptotic behavior of the ratio between the largest and second largest values of n independent chi-square random variables. Let X1,…,Xn be n independent and identically distributed random variables with distribution χ21. Let X(n) be the largest and X(n−1) be the second largest of these n random variables. I was wondering … Read more

Asymptotic behavior of sums with binomial coefficients

Let f(n) be defined on non-negative integers and define S(m) = \sum_{n=0}^{m} \binom{m}{n} f(n). Depending on the choice of f, S(m) may have a closed form; for example, when f(n) = n, then S(m) = m 2^{m-1}. My question is, does a general theory exist which relates the asymptotic behaviors of f and S, and … Read more

Asymptotic form of the eigenvalues / eigenfunctions of a Fredholm equation

Consider the kernel: $$k(x,y,\xi) = \sqrt{\frac{1}{\pi} \frac{1}{y+a}} \exp \left(-\frac{\xi^2}{y+a }\right)$$ I am trying to find the asymptotic form of the solutions to the following homogeneous Fredholm equation of the second kind: $$K_\epsilon [u_\epsilon] (x) = \frac{1}{\epsilon} \int_0^1 k\left(x,y, \frac{x-y}{\epsilon} \right) \mathrm d y = \lambda_\epsilon u_\epsilon (x)$$ for small $\epsilon > 0$. That is, I … Read more

Asymptotic expansion of a Gaussian integral and heat kernel

When considering the heat kernel of a Schr\”odinger operator −Δ+V(x) where Δ is the standard Laplacian on Rn and V is a nonnegative potential function that has nice behavior at infinity (proper, grows polynomially), one usually sees the term e−tV(x) and the asymptotic expansion of the Gaussian integral ∫Rne−tV(x), t→0. If V(x) is homogeneous, namely, V(rx)=rαV(x), … Read more

Bound on number of steps needed for points to meet enclosing convex polygon

Let P be the set of equidistant points on the unit circle which are then randomly shuffled. They then take discrete steps towards the midpoint between the 2 points that they were originally adjacent to along a curve analogous to a pursuit curve. The illustration below shows a black line joining each point with its … Read more

Dividing a finite arithmetic progression into two sets of same sum: always the same asymptotics?

This is inspired by the recent question How many solutions ±1±2±3…±n=0. The oeis entries A063865 linked to this question and A292476/A156700 for the related one “How many solutions ±1±3±5…±n=0” give the asymptotics as n→∞, respectively √6π2nn3/2 and 2√6π2n/2n3/2 (the first one only applies for n≡0 or 3 \pmod 4, the second one only for n \equiv 3 … Read more