Problems
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Mossel and Peres showed that a single-variable function admits a finite-automata Bernoulli factory exactly when it is rational, and their Theorem 2.9 claimed the same extension to multivariable functions. The multivariable claim is false,…
Along the heat flow, do the successive time derivatives of the entropy of X + √t Z alternate in sign, as conjectured by Cheng and Geng? An explicit measure on R has a fifth derivative with the forbidden sign.
Litvak conjectured in 2018 that for every p > 0 the quantity E[min_i ≤ n |g_i|^p], for g ~ N(0,Σ), is minimized over n × n correlation matrices by the Gram matrix of the regular simplex in R^n-1. False: the matrix Σ^cos_ij = cos(π(i-j)/n)…
The Benjamini-Hochberg procedure is known not to control the false discovery rate at its nominal level under arbitrary dependence. A folklore conjecture in the FDR literature held that it must at least control the FDR up to a universal…
The Gaussian Moments Conjecture asks whether, for complex polynomials P,Q in n independent standard real Gaussian variables, E(P^m)=0 for all m≥ 1 forces E(QP^m)=0 for all large m. Explicit counterexamples with E(P^m)=0 and E(QP^m)=m!≠ 0…
Chafai, Dadoun and Youssef asked whether the quadratically penalised logarithmic energy of mean empirical spectral distributions is monotone in the dimension, for Wigner matrices and for matrices with i.i.d. entries. Neither holds: a…
For every smooth positive density f on R^d, must the Fisher information t ↦ I(f * γ_t) be log-convex along the heat flow?
Does the Benjamini-Hochberg procedure always control the false-discovery rate at its nominal level for correlated two-sided Gaussian p-values? A factor model gives FDR > 0.0104 at nominal level α = 0.01.