23 problems

John Thom measured the diameters of 33 Druid circles in Western Scotland and considered their integer-valued measurements in feet. Thom's conjecture. The diameters were intended to…

Coddington–Baillie's conjecture. For these Ising models,

Chunk-energy sufficiency conjecture. Restricting weights to depend on $\mathbf{x}$ only through the chunk-energy vector is sufficient without loss of optimality for the step-$d$ sl…

Let $\Delta_k$ denote the optimal asymptotic MSE constant within the generalized grouping (GG) family for groups of size $k$. Monotonicity conjecture. … i.e., the optimal asymptoti…

The Heston model is simulated using the Almost Exact Simulation (AES) scheme, which samples the variance process from its non-negative marginal distribution. European option-pricin…

Let Algorithm 1 assign values $\langle \varphi^{i,i+1}_{x}\rangle$ to pairs of bits $x$, where some pairs are parts of the rules and others are not. Higher-value pair conjecture. S…

Simultaneous non-monotonicity likelihood conjecture. The fraction of election outcomes satisfying all conditions simultaneously lies between $0.003$ and $0.004$:

Higher-order TRMC convergence conjecture. Convergence results of type can be extended to these higher-order time-relaxed Monte Carlo methods by iterating the same proof argument fo…

Let $X$ be a radial Riemannian manifold with stationary probability distribution $p(x)\propto{\mathrm{e}^{-V(x)}}$. Suppose an update scheme changes the radial component in a non-t…

Convergence conjecture. This randomized two-stage stochastic gradient descent method possesses better convergence properties than the gradient descent method.

A piecewise-deterministic Markov process (PDMP) sampler is a Monte Carlo method whose dynamics include deterministic motion interrupted by random events, and a target density may h…

Let an optimal stopping problem be given, with utility estimated using Monte Carlo simulation of the underlying process. Broadie and Glasserman's conjecture. There are no general u…

Let $K$ be the number of temperatures, let $\alpha_K$ denote the highest-temperature parameter in the Markov case, and let $(1/2)^{K-1}$ be its value in the static setting. The low…

Successful-pivot probability conjecture. The probability of a successful pivot move satisfies

Successful-pivot exponent conjecture. The exponent satisfies

Let $B$ be the intersection indicator observable for a concatenated pair of self-avoiding walks of total length $N$, and let $\tau_{\mathrm{int}}(B)$ denote its integrated autocorr…

Consider the test of $H_0:\alpha=\alpha_0$ against $H_1:\alpha\neq\alpha_0$ for $alpha_0\in(-1/2,1/4)$, with empirical rejection rates evaluated by Monte Carlo simulation at nomina…

Consider the Monte-Carlo approximation of the semi-Lagrangian scheme for the Dirichlet problem, with time and space discretization parameters satisfying $\delta t=\delta x$. The Mo…

Pivot time-scale conjecture. The integrated autocorrelation time of this observable is of the same order as the time required to make successful pivots at all length scales with re…

Terminal-impurity maximization conjecture. The number of perfect matchings is maximized when all the impurities are on the terminals.

Dilation-ensemble scaling-limit conjecture. The scaling limit of the dilation ensemble is related to the limit

Let $T\in(0,\infty)$, let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space with a normal filtration $(\mathcal{F}_t)_{t\in[0,T]}$, let $W^{l,k}$ be a family of independent…

Let $\left(\overline{\mathcal R_{[L]}}\right)_{L=1,2,\dots}$ be a sequence of restriction operators whose macroscopic state variables are the first $L$ centralized moments, with…