A Slightly Improved Bound for the KLS Constant

We refine the recent breakthrough technique of Klartag and Lehec to obtain an improved polylogarithmic bound for the KLS constant.Comment: minor revision fixing typo

Biotransformations Performed by Yeasts on Aromatic Compounds Provided by Hop—A Review

The biodiversity of some Saccharomyces (S.) strains for fermentative activity and metabolic capacities is an important research area in brewing technology. Yeast metabolism can render simple beers very elaborate. In this review, we examine much research addressed to the study of how different yeast strains can influence aroma by chemically interacting with specific aromatic compounds (mainly terpenes) from the hop. These reactions are commonly referred to as biotransformations. Exploiting biotransformations to increase the product’s aroma and use less hop goes exactly in the direction of higher sustainability of the brewing process, as the hop generally represents the highest part of the raw materials cost, and its reduction allows to diminish its environmental impact

Reducing Isotropy and Volume to KLS: An O(n3ψ2)O(n^3\psi^2) Volume Algorithm

We show that the the volume of a convex body in ${\mathbb R}^{n}$ in the general membership oracle model can be computed with $\widetilde{O}(n^{3}\psi^{2}/\varepsilon^{2})$ oracle queries, where $\psi$ is the KLS constant ($\widetilde{O}$ suppresses polylogarithmic terms. $O^{*}$suppresses dependence on error parameters as well as polylogarithmic terms.). With the current bound of $\psi\lesssim n^{\frac{1}{4}}$, this gives an $\widetilde{O}(n^{3.5}/\varepsilon^{2})$ algorithm, the first general improvement on the Lov\'{a}sz-Vempala $\widetilde{O}(n^{4}/\varepsilon^{2})$ algorithm from 2003. The main new ingredient is\emph{ }an $\widetilde{O}(n^{3}\psi^{2})$ algorithm for isotropic transformation, following which we can apply the $\widetilde{O}(n^{3}/\varepsilon^{2})$ volume algorithm of Cousins and Vempala for well-rounded convex bodies. A positive resolution of the KLS conjecture would imply an $\widetilde{O}(n^{3}/\epsilon^{2})$ volume algorithm. We also give an efficient implementation of the new algorithm for convex polytopes defined by $m$ inequalities in ${\mathbb R}^{n}$: polytope volume can be estimated in time $\widetilde{O}(mn^{c}/\varepsilon^{2})$ where $c<3.7$ depends on the current matrix multiplication exponent and improves on the the previous best bound.Comment: 23 pages, 1 figur