Aug 2026· arXiv.org· Vol abs/2608.30275· 0 citations· 12 references
Computer ScienceMathematics
Abstract
We construct smooth, strictly log-concave counterexamples to the Gaussian completely monotone conjecture in every dimension. In one dimension, they form an explicit family $f_m$ whose signed $m$th entropy derivative at time zero is negative for every sufficiently large $m$; the inequality persists for all sufficiently small positive times. Tensorization with a broad Gaussian factor gives the higher-dimensional examples. The argument is analytic and self-contained. It reduces the sign to a two-frequency entropy calculation on the circle and transfers the resulting asymptotic to the real line through an exact heat-flow formula for Gaussian-windowed Fourier modes. The proof was developed by GPT-5.6 Sol Pro under the authors'guidance.
We study logarithmic complete monotonicity of a power family built from the digamma function. An explicit Laplace representation of its negative logarithmic derivative reduces the question to positivity of a continuous kernel. We prove a strict kernel bound on an explicit parameter range and obtain uniform lower bounds...
We prove a dimension-free Poincar\'e inequality for isotropic unconditional log-concave probability measures, establishing the Kannan-Lov\'asz-Simonovits (KLS) conjecture in the unconditional setting. The proof combines a new application of Dunkl operator theory combined with an appropriate transformation of the log-co...
We study the LOV\'ASZ number of dense random circulant graphs generated by independent Bernoulli bits indexed by cyclic distances. Although only $O(n)$ independent random inputs determine $O(n^2)$ globally dependent edge indicators, we prove sharp expectation asymptotics along all odd orders, including composite orders...
Xin Chen, Hong-Wei Hu, Yan-Qing Hu et al.· 0 citations
We study optimal polynomial approximants in spaces of Dirichlet-type over the bidisc. Given a nonzero function f, the linear optimal polynomial approximant (opa) to its reciprocal is the affine polynomial p for which pf is closest to the constant function 1 in the space norm. We prove that, for every positive value of...
For the empirical measure of the first $n$ points of one infinite sequence of independent uniform samples in the unit square, we identify the leading constant in the bounded-Lipschitz distance to area measure, equivalently in $W_1$, both in expectation and almost surely along all sample sizes. The constant is $c_*=\kap...
Wei-Tao Huang, Ren-Zhen Wang, Yichen Wu et al.· 0 citations
Using a dioid algebraic structure, we show that there exist spherically symmetric trees $T(2^m 1^n)$ whose independence polynomials exhibit multiple breaks in log-concavity, a result established by estimating the asymptotic growth of the coefficients of these polynomials. Provided the parameter $n$ is a sufficiently la...
César Bautista-Ramos· 2 citations
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