26 problems

Scaling-law conjecture. There are more general scaling laws of the critical batch size in relation to $\eta$.

Almost optimal scaling-law conjecture. Reed–Muller codes have an almost optimal scaling-law over BSC channels under ML decoding.

Correlation-length conjecture. The correlation length grows like

Barenblatt's conjecture. The structure-function exponents may depend on the Reynolds number.

Let $N$ be the system size, $p$ the fraction of rewired edges, and $ll(N,p)$ the characteristic path length. Let $N^*$ denote the $p$-dependent crossover length separating the regi…

Let $r$ denote a spatial scale, let $u$ be the kinematic viscosity, let $S{ij}$ be the strain-rate tensor, and let the energy dissipation field be $2nu S{ij}S{ij}$. Let $alpha$ be…

The system consists of reconfigurable intelligent surfaces (RISs) serving multiple operators, with the RIS phases constrained by the non-serving operators. The serving operator's r…

Consider mini-batching with a batch size that remains of constant order as the problem scales. Let the batch size $1$ case be the reference setting, and let compute-optimal exponen…

Let $s\in(0,1)$, $d\in\mathbb{N}$, and let $\mu=\mu_t\otimes dt$ be the measure describing the normal phase in the periodic sample $\mathbb{T}^d_T$. For almost every $t$, write … w…

In an infinite-dimensional linear regression model, suppose only $M$-dimensional sketched covariates are observed, a linear predictor with $M$ trainable parameters is trained by on…

Automorphism-generation conjecture. Having excluded Galilean transformations, $\operatorname{Aut}(A,B,\sigma)$ is generated by the three types of subgroups, the Euclidean groups ac…

Assume the critical parameter conditions considered in the paper, in particular $\gamma+2\lambda=1$ and $\gamma\leq -1$, and consider the associated standard self-similar solutions…

Let $K_+$ and $K_-$ denote the upper and lower fracture thresholds, respectively, for a fixed domain radius $R$. Let $a_1$ and $a_2$ be the model parameters, let $R_*$ denote the r…

Let $N_t(\lambda)$ be the number of transmit antennas in a MISO cellular network with a physically feasible path loss model. Suppose coordinated beamforming perfectly cancels the i…

Let $N_t(\lambda)$ and $N_r(\lambda)$ denote the numbers of transmit and receive antennas, respectively, and suppose … Consider eigenbeamforming with a single data stream and a phy…

Let $N_t(\lambda)$ and $N_r(\lambda)$ be the numbers of transmit and receive antennas, respectively, with … Consider eigenbeamforming with a single data stream and a physically fea…

Let $\tau\geq 0$ denote time, let $xi>0$ denote cluster size, and let $\phi(\tau,\xi)\geq 0$ be the cluster-size density satisfying Smoluchowski's coagulation equation … Assume tha…

Modulated-interferer conjecture. The trends obtained under the two-tone interference model should also hold for modulated interferers.

Grossmann–Lohse's conjecture. The plume shedding frequency scales as

Kraichnan's conjecture. The thermal flux velocity satisfies $U_{flux}=u_\tau/(2Pe_T)$.

Let $E_n$ be the discrete energy, let $x^n_\ast$ denote the discrete configuration under consideration, and let $\rho^n_\ast$ be the empirical measure of the discrete dislocation w…

Under the dual-slope path loss model, let $\tau_{l_2(\alpha_0,\alpha_1;\cdot)}(\lambda,T)$ denote the potential network throughput at BS density $\lambda$ and threshold $T$. The pa…

Let $L$ be the chain length and $M$ the coupling length or block-size parameter of a spatially-coupled low-density parity-check code ensemble. The effect that the ensemble has a va…

Conjecture on the optimal aspect ratio. For large $\text{\textit{Pe}}$,

Conjecture on the transport exponent. For large $\text{\textit{Pe}}$, the exponent $0.58$ is actually $10/17=0.5882\dots$, so that