13 problems

Consider a semilinear elliptic equation with a pure power nonlinearity of exponent $p>1$ on a domain $D$, and suppose solutions are studied through probabilistic representations in…

In the probabilistic group testing model, let the lower and upper bounds refer to the numbers of nonadaptive measurements required to achieve error probability tending to zero in t…

Let $p_{k,n,\delta}(\Gamma_m)$ denote the forwarding probability for a grid network of size $m$, and let $Y$ be a binomial random variable with parameters $n$ and $(\theta^+(p))^2$…

$u$-function-perceptron contiguity conjecture. The random and planted ensembles are contiguous for all $0<K\le K^*$ and $\alpha\in(0,\alpha_c^u(K))$.

Let $\mathscr{T}$ be the tensor space and let … A tensor in $\mathscr{T}^{(R)}$ is sampled from an absolutely continuous probability measure. DCDP convergence conjecture. The DCDP…

The probabilistic determinant formula. For every $D\subseteq[k-1]$,

The delta-process tree probability conjecture. One has

Let $\mathcal{P}_{q}$ be a projective plane of order $q$, and let $t(\mathcal{P}_{q})$ denote the size of its smallest complete arc. For sufficiently large $q$, suppose that … wher…

Interpolated threshold-group-testing conjecture. As the gap between $ell$ and $u$ decreases, the constant in the asymptotic test count decreases from $4K\ln N$ to $2K\ln N$ when…

Let $Y_0,Y_1,\ldots$ be independent uniformly distributed random variables on the unit square, let $\boldsymbol{\Phi}$ be the area-preserving Lambert map from the square to…

Let $G=\operatorname{cl}(\mathcal{O})$ be the ideal class group of an order $\mathcal{O}$ in an imaginary quadratic field, with cardinality $h$. Let $S_k$ be the sequence of ideal…

Let $n$ be the number of people in a room, let $s$ be the number of spies, and let $\ell$ satisfy … A questioning strategy is a procedure for asking questions that determines the i…

Asymptotic sharpness conjecture. When $n$ is large enough, there exist graphs $G$ and $H$ such that