1,120 problems

Let $n$ be a positive integer and let $j$ be a positive integer. A lucky spot is a parking spot occupied by a car whose preferred spot is that spot. Let $f_j(n)$ denote the polynom…

Let $m\geq 2$ be a fixed integer. For independent exponential random variables $Y_k$ with rates $k$, define the limiting maximizing index … where … and write…

Goldbach-event independence conjecture. The events $A$ and $A'$ satisfy

Let $X^{(n)}_3$ denote the number of distinct adjacent pairs of distinct letters in a geometrically distributed word of length $n$. Gaussian-limit conjecture. The asymptotic distri…

Let $X^{(n)}_1$ be the number of distinct adjacent pairs of identical letters in a geometrically distributed word of length $n$, and let … where the independent random variables…

Mallows unimodality conjecture. If $\pi$ is any consecutive permutation pattern, then the function $p\mapsto\mathbb{E}_{n,p}(\pi)$ is unimodal.

Let $\mathbf{X}_1,\dots,\mathbf{X}_n$ be i.i.d. matrices with common mean matrix $\mathbb{E}\mathbf{X}_1=\mathbf{M}$ and $p$th central moment matrix … For any…

Let $F_m$ be the set of periodic sequences from the Fibonacci recurrence or its parity transform with initial condition $(a_0,a_1)\in(\mathbb Z/m\mathbb Z)^2$, taken modulo $m$. Fo…

Let $m=p_1\cdots p_r$ be a square-free product of $r$ distinct primes, and let $K_{n,d}^m$ denote the number of distinct zeros of a random polynomial in…

Let $X^{(n)}_1$ be the number of distinct adjacent pairs of identical letters in a geometrically distributed word of length $n$, and let $\xi^{(n)}$ be the sum of independent rando…

Let $X\sim\mathcal{N}_{v}(0,\Lambda)$, where $\Lambda=\operatorname{diag}(\lambda)$, and let … be the Euclidean ball of radius $\rho$. Consider the covariance matrix of the vector…

Under Assumption … sigma^Y{i,j}geq sigma^X{i,j}geq 0 … operatorname{Cov}(log SN({mathbf G}),pi({mathbf G})pj({mathbf G}))leq 0. … ; its resolution is not indicated in the supplied…

Strengthened truncated-mean inequality. For every $-1<t<1$,

Let $\mathbf{X}^{(1)},\mathbf{X}^{(2)},\ldots$ be independent and identically distributed $d$-dimensional observations with independent $\operatorname{Exponential}(1)$ coordinates,…

Negative partial-sum conjecture. If there are pairwise different indices $i_1, \dots, i_{n-1}$ such that

Diagonal stability conjecture. There exist positive constants $C_p>0$ such that, if $(r_k)$ is log-concave, then

Asymptotic separation conjecture. For every probability vector $(p,1-p)$, there exists a positive integer $N$ such that, for all $n eg N$, the optimal set $\alpha_n$ contains point…

Optimal-strategy conjecture. After observing $I_{k}=1$, the player whose turn it is should give up his turn to his opponent if and only if $p_{i,k}<\frac{1}{2}$ for all $i>k$. This…

Let $N\ge 3$, and let $l$ and $m$ be positive integers such that $1\le l\le N$, $2\le m\le N-1$, and both $l$ and $m$ are relatively prime to $N$. Let $X_l(m,N)$ be the random vari…

Existence conjecture. Under certain conditions on $W$, there exists a random variable $V$, independent of $W$ and with support in $[0,1]$, such that holds. In terms of…

Let $w$ be a process and let the three assumptions in Definition be the smooth-expectation condition, the relative-fitness-density condition, and the local-change-density condition…

Beta-limit conjecture. As $c\to\infty$, in the two-island case both $X$ and $Y$ converge in distribution to the beta distribution with parameters $(a_1+b_1,a_2+b_2)$; in the seed-b…

Let $d\ge 1$. Suppose $D(x)$ has finite support $\mathcal{U}$. If $D(x)$ is uniformly distributed on $\mathcal{U}$, and if every $x\in \mathcal{U}$ is on the boundary of…

Let $B(k,\mu)=\sum_i {k\brace i}\mu^i$ denote the Bell polynomial, and write $B_x=B(x,1)$. For a Poisson random variable with mean $\mu$, its $k$th raw moment is $B(k,\mu)$. Bell-n…

Consider the Fibonacci recurrence $a_n=a_{n-1}+a_{n-2}$ and its chiral recurrence $a_n=-a_{n-1}+a_{n-2}$ with random integer initialization $(a_0,a_1)\in\mathbb Z^2$. Let $F_n$ and…