Related papers: Concerning the bookshelf problem
In the classical coupon collector's problem, every box of breakfast cereal contains one coupon from a collection of n distinct coupons, each equally likely to appear. The goal is to find the expected number of boxes a player needs to…
We consider a generalized version in continuous time of the parking problem of Knuth. Files arrive following a Poisson point process and are stored on a hardware identified with the real line. We specify the distribution of the space of…
We study impartial take away games on 2 unordered piles of finite nonnegative numbers of tokens $(x,y)$. Two players alternate in removing at least one and at most all tokens from the respective piles, according to certain rules, and the…
We prove the law of large numbers for the drift of random walks on the two-dimensional lamplighter group, under the assumption that the random walk has finite $(2+\epsilon)$-moment. This result is in contrast with classical examples of…
Events in mesoscopic systems often take place at first-passage times, as is for instance the case for a colloidal particle that escapes a metastable state. An interesting question is how much work an external agent has done on a particle…
This paper studies a class of $1\frac12$-degree-of-freedom Hamiltonian systems with a slowly varying phase that unfolds a Hamiltonian pitchfork bifurcation. The main result of the paper is that there exists an order of…
We apply to the $n\times n$ chessboard the counting theory from Part I for nonattacking placements of chess pieces with unbounded straight-line moves, such as the queen. Part I showed that the number of ways to place $q$ identical…
A conjecture by Bollob\'as from 1995 (which is a weakenning of the famous Tree Packing Conjecture by Gy\'arf\'as from 1976) states that any set of $k$ trees $T_n,T_{n-1},\dots,T_{n-k+1}$, such that $T_{n-i}$ has $n-i$ vertices, pack into…
The Herman Protocol Conjecture states that the expected time $\mathbb{E}(\mathbf{T})$ of Herman's self-stabilizing algorithm in a system consisting of $N$ identical processes organized in a ring holding several tokens is at most…
Using a bijective proof, we show the number of ways to arrange a maximum number of nonattacking pawns on a $2m\times 2m$ chessboard is ${2m\choose m}^2$, and more generally, the number of ways to arrange a maximum number of nonattacking…
How many edges in an $n$-vertex graph will force the existence of a cycle with as many chords as it has vertices? Almost 30 years ago, Chen, Erd\H{o}s and Staton considered this question and showed that any $n$-vertex graph with $2n^{3/2}$…
The evaluation of variation in oscillation time period of a simple pendulum as its mass varies proves a rich source of discussion in a physics class-room, overcoming erroneous notions carried forward by students as to what constitutes a…
Consider a randomly shuffled deck of $2n$ cards with $n$ red cards and $n$ black cards. We study the average number of moves it takes to go from a randomly shuffled deck to a deck that alternates in color by performing the following move:…
The blocking probability of a finite-source bufferless queue is a fixed point of the Engset formula, for which we prove existence and uniqueness. Numerically, the literature suggests a fixed point iteration. We show that such an iteration…
Consider a string of N+1 damped oscillators moving on the line of which the motion of the first (called the "leader") is independent of the others. Each of the followers `observes' the relative velocity and position of only its nearest…
In the classical secretary problem, one attempts to find the maximum of an unknown and unlearnable distribution through sequential search. In many real-world searches, however, distributions are not entirely unknown and can be learned…
Consider the random polytope, that is given by the convex hull of a Poisson point process on a smooth convex body in $\mathbb{R}^d$. We prove central limit theorems for continuous motion invariant valuations including the Will's functional…
An overview is presented of recent work on some statistical problems on multiparticle random walks. We consider a Euclidean, deterministic fractal or disordered lattice and N >> 1 independent random walkers initially (t=0) placed onto the…
The study of the movement of flocks, whether biological or technological is motivated by the desire to understand the capability of coherent motion of a large number of agents that only receive very limited information. In a biological…
We consider a generalisation of the classical coupon collector's problem, in which at each time step a collector either receives a new copy of a randomly chosen coupon, or loses all their previously collected copies of that coupon. We…