36 problems

Private-message extension conjecture. If the capacity of the $K$-user relay network can be found for any $K$, then the capacity of the $K$-user relay network in which the relay exc…

Let $gap$ denote the gap to channel capacity, let $\langle P_e\rangle$ denote the average probability of error, and consider sparse-graph codes with iterative decoding. Khandekar–M…

Capacity conjecture. The existence of minimizers of $E_\kappa$ in $\mathcal{J}$ is completely determined by the value of $\operatorname{cap}(A)$ and the value of $\kappa$.

Mattingly et al.'s support-size conjecture. Using numerical evidence, the number of support points should satisfy

Let $\Omega_j$ be the isoperimetric regions associated with a sequence satisfying the hypotheses of the zero mass stability conjecture, and let $B_0(R)\subset{\mathbb E}^3$ be the…

Strip energy ball comparison conjecture. If

The paper considers characterizations of heat kernel estimates and Harnack inequalities for diffusions, pure jump processes, and processes with both diffusions and jumps, where cut…

Let $(X,d,m)$ be a metric measure space satisfying volume doubling, denoted by. Let denote the relevant Poincaré inequality, the uniform capacity condition, and the two-sided heat-…

Let $a_{ij}(\cdot)$ be the coefficient matrix in the framework above, let $\overline{a}_{ij}$ denote its homogenized coefficients, and let $\beta_0$ be the critical value. Write…

Optimal-order conjecture. Assuming that the optimal order of the moderate-deviation scale is the same as that for the volume of random walks in dimension two, the optimal order of…

Let a discrete-time Poisson channel (DTPC) operate under an average power constraint, and let a capacity-achieving input distribution be a distribution attaining the channel capaci…

Let $n\geq 2$ and $0<p<n$. Let $K\subset\mathbb{R}^n$ be a compact set with an irreducible group of isometries, let $M$ be a volume-preserving invertible linear map, and write…

Let a finitely generated group have a symmetric set of generators, and let $\mathcal{C}_n$ denote the capacity of the range of the associated random walk. In dimension $d=4$, the r…

Low-dimensional capacity-limit conjecture. The normalized capacity converges in distribution:

Low-dimensional capacity-order conjecture. The capacity of the range satisfies

Let $(M,g)$ be an asymptotically flat 3-manifold with nonnegative scalar curvature and positive ADM mass, and assume that $partial M$ is empty or minimal. For $V>0$ sufficiently la…

Let $(M,g)$ be an asymptotically flat 3-manifold with nonnegative scalar curvature and positive ADM mass. A Bernoulli surface is a surface $S=partial A$ for a compact set $A$ such…

Let $h$ be a continuous, increasing, concave function defined and positive in some right neighborhood of $0$. For a set $E\subset\mathbb{R}$, define its $h$-volume by … where … Ass…

Let $B_{R_2}^+$ be the relevant half-ball, let $K_\delta^+$ be the shrinking perturbation, and let $u$ be the normalized eigenfunction associated with the relevant eigenvalue. Defi…

Let $N\ge 3$, let $\Omega$ be an open set, and let $B_r$ be a ball with $|B_r|=|\Omega|$. The Fraenkel asymmetry of $\Omega$ is … Here $\operatorname{cap}$ denotes the Newtonian ca…

Let $p,q \in \mathbb{R}_+[x_1,\ldots,x_n]$ be homogeneous real stable polynomials of total degree $d$. For a multi-index $\mu$ with $|\mu|=d$, write $p_\mu$ and $q_\mu$ for the cor…

Let $s>0$, $\beta\geq 0$, $p>1$, and let $K\subset\mathbb R^N$ be compact. Denote by $\operatorname{cap}^{\mathbb R^N}_{\frac{2s(1+\beta)}{p},p'}(K)$ the corresponding Bessel capac…

Consider the multidimensional semiconstrained system $[?]$ described above, with its limiting capacity bounded below by the product-measure construction and illustrated in Figure r…

Let $\Phi_n$ and $\Psi_n$ denote sequences of random quantum channels, with the paper distinguishing orthogonal and unitary random channels, and let the Holevo capacity measure the…

Let $X[0,n]$ be the range up to time $n$ of a simple random walk on $\mathbb{Z}^3$, and let $\operatorname{Cap}(X[0,n])$ denote its capacity. Random-limit conjecture. The normalize…