Related papers: Locally uniform random permutations with large inc…

In this short note, we study the derivatives of all orders for the random field $$ X_T(h) = \sum_{p \leq T} \frac{\text{Re}(U_p \, p^{-i h})}{p^{1/2}}, \quad h\in [0,1], $$ where $(U_p, \, p ~\text{primes})$ is an i.i.d. sequence of uniform…

We study the space requirements of a sorting algorithm where only items that at the end will be adjacent are kept together. This is equivalent to the following combinatorial problem: Consider a string of fixed length n that starts as a…

We compute the limit distribution for (centered and scaled) length of the longest increasing subsequence of random colored permutations. The limit distribution function is a power of that for usual random permutations computed recently by…

Let $\mathcal{P}(n)$ denote the power set of $[n]$, ordered by inclusion, and let $\mathcal{P}(n,p)$ be obtained from $\mathcal{P}(n)$ by selecting elements from $\mathcal{P}(n)$ independently at random with probability $p$. A classical…

Przytycki has shown that the size $\mathcal{N}_{k}(S)$ of a maximal collection of simple closed curves that pairwise intersect at most $k$ times on a topological surface $S$ grows at most as a polynomial in $|\chi(S)|$ of degree…

A classical theorem of Erdos, Lovasz and Spencer asserts that the densities of connected subgraphs in large graphs are independent. We prove an analogue of this theorem for permutations and we then apply the methods used in the proof to…

Discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. Previous known results show that for any $N$-dimensional subspace of the space of continuous functions it is…

In the subset sum problem we are given n positive integers along with a target integer t. A solution is a subset of these integers summing to t. In this short note we show that for a given subset sum instance there is a proof of size…

Paul Erd\H{o}s and L\'{a}szl\'{o} Lov\'{a}sz proved in a landmark article that, for any positive integer $k$, up to isomorphism there are only finitely many maximal intersecting families of $k-$sets (maximal $k-$cliques). So they posed the…

Let $X_{nr}$ be the $r$th largest of a random sample of size $n$ from a distribution $F (x) = 1 - \sum_{i = 0}^\infty c_i x^{-\alpha - i \beta}$ for $\alpha > 0$ and $\beta > 0$. An inversion theorem is proved and used to derive an…

We calculate the p-the moment of the sum of n independent random variables with respect to symmetric norm in R^n. The order of growth for upper bound p/ln p obtained in ths estimate is optimal. The result extends to generalized Lorentz…

We consider the binomial random graph $G(n,p)$, where $p$ is a constant, and answer the following two questions. First, given $e(k)=p{k\choose 2}+O(k)$, what is the maximum $k$ such that a.a.s.~the binomial random graph $G(n,p)$ has an…

Let $P(n,m)$ be a graph chosen uniformly at random from the class of all planar graphs on vertex set $\left\{1, \ldots, n\right\}$ with $m=m(n)$ edges. We show that in the sparse regime, when $\limsup_{n \to \infty} m/n<1$, with high…

The on-line nearest-neighbour graph on a sequence of $n$ uniform random points in $(0,1)^d$ ($d \in \N$) joins each point after the first to its nearest neighbour amongst its predecessors. For the total power-weighted edge-length of this…

It is shown that the following holds for each $\varepsilon>0$. For $G$ an $n$-vertex graph of maximum degree $D$ and "lists" $L_v$ ($v \in V(G)$) chosen independently and uniformly from the ($(1+\varepsilon)\ln n$)-subsets of $\{1, ...,…

We study the asymptotic behavior of the maximum interpoint distance of random points in a $d$-dimensional set with a unique diameter and a smooth boundary at the poles. Instead of investigating only a fixed number of $n$ points as $n$ tends…

This paper establishes asymptotic results for the maximum likelihood and restricted maximum likelihood (REML) estimators of the parameters in the nested error regression model for clustered data when both of the number of independent…

It is shown that the following holds for each $\varepsilon >0$. For $G$ an $n$-vertex graph of maximum degree $D$, lists $S_v$ of size $D+1$ (for $v\in V(G)$), and $L_v$ chosen uniformly from the ($(1+\varepsilon)\ln n$)-subsets of $S_v$…

We prove that if $(\mathcal{M},d)$ is an $n$-point metric space that embeds quasisymmetrically into a Hilbert space, then for every $\tau>0$ there is a random subset $\mathcal{Z}$ of $\mathcal{M}$ such that for any pair of points $x,y\in…

In a projective plane $\Pi _{q}$ (not necessarily Desarguesian) of order $q,$ a point subset $S$ is saturating (or dense) if any point of $\Pi _{q}\setminus S$ is collinear with two points in$~S$. Using probabilistic methods, the following…