Let the mn edges of K_(m,n) carry independent mean-one exponential costs and let C_(k,m,n) be the minimum cost of a k-matching. Wästlund showed that the moment generating function F_(k,m,n)(t)=E e^(tC_(k,m,n)) is a rational function of t whose first pole si…
Rapaport, Tkocz and Wu (arXiv:2512.09077) determine the sharp constant Aₚ in the Khinchin-type inequality Aₚ|S|₂ ≤ |S|ₚ for Steinhaus sums S=Σⱼ zⱼξⱼ in the last open range -1<p<0. The technical core of their proof, their Lemma 10, bounds ∫₀^∞|J₀(t)|³tᵖ⁻¹ d…
Let varepsilon₁,…,varepsilonₙ be independent Rademacher signs, let a ∈ ℝⁿ satisfy Σᵢaᵢ²=1, write overline Sₙ=n^(-1/2)(varepsilon₁+…+varepsilonₙ) and Δₙ(a)=Σᵢ(aᵢ²-frac1n)². Gao and Qian introduce the optimal dimension-free third-moment stability constant C₃ᵒ…
Fix k ≥ 2 and bond uniformly chosen nearest-neighbour k-tuples in a row of n molecules until none is left. Two limiting densities describe the jammed configuration: the density mₖ of bonded molecules and the density g_(k;l) of maximal gaps of exactly l unbo…
For a real random variable X with E X=0, E X²=1 and E X⁴ ≤ κ, the exact value of supP(X ≥ t) is a classical Chebyshev-type extremal problem, solved in closed form on most of its parameter range by Li, Han, Jiang and Gao. We impose one further hypothesis — t…
For a probability vector p on K outcomes let G(p)=1-|p|₂² be the Gini impurity and H(p)=-Σᵢ pᵢlog pᵢ the Shannon entropy in nats. A recent paper on continual learning displays the two-sided bound (1-1/K)/((log K)²)H(p)² ≤ G(p) ≤ (1)/(2log 2)H(p), calls both…
Let φ and Φ be the density and the distribution function of the standard Gaussian law, and let I(p)=φ(Φ⁻¹(p)) be the Gaussian isoperimetric function. In a recent study of one-dimensional sub-Gaussian comparison in convex order, Zhang proves the elementary l…
The finite free additive convolution boxplusₙ of Marcus, Spielman and Srivastava acts on the root vectors of monic real-rooted polynomials of degree n by a map Ω that is smooth at a simple-root configuration. Hashemi's coupling matrix Eₙ is the partial Jaco…
In an appendix to their study of positive autocorrelation at unit lag for stationary random walk Metropolis–Hastings, Fill and Janson introduce the two-variable polynomial h(t,a;r) &= [1+(3+a)t]ʳ([1+(2+a)t]ʳ⁺¹-[1+at]ʳ⁺¹) &qquad()- [1+(1+a)t]ʳ([1+(4+a)t]ʳ⁺¹-…
Fill's notes [fill] study the Markov chain on the permutations of [n] that picks one of the n-1 adjacent positions uniformly and swaps the two labels there with a probability depending only on the labels. Four conjectures are stated there; three are now the…
Brazitikos and Pandis (arXiv:2512.12254) prove Hunter's conjecture of 1977: for even n the minimum of the complete homogeneous symmetric polynomial h₂ₖ on the unit sphere of ℝⁿ is attained at the half-plus/half-minus vector. The base case k=2 of their induc…
Let H ∼ Hyp(n,i,k) count the black marbles in a sample without replacement of size k drawn from an urn holding i black and n-i white marbles, so that E H=ik/n. In a study of lower bounds for ℙ(H ≥ E H), Ai and Pelekis record a numerical observation: under t…
In a round-robin tournament of n equally strong players, let s_((n)) and s_((n-1)) be the largest and the second largest score. Both are named as objects of study in the recent literature, which calls the exact distribution computationally intractable even…
For a probability measure with all moments finite, free infinite divisibility forces every shifted free-cumulant Hankel matrix H_N=(κᵢ₊ⱼ₊₂)_(i,j=0)^(N) to be positive semidefinite. A single vector on which one of those matrices is negative therefore certifi…
