Extension results for boolean maps and a class of systems of linear inequalities

A Note on Boolean Lattices and Farey Sequences III

We describe monotone maps between subsequences of the Farey sequences.

关于超模函数和亚模函数在超立方体上的塔拉甘型相关不等式

Talagrand-Type Correlation Inequalities for Submodular and Supermodular Functions on the Hypercube

本文研究了在超立方体上亚模和超模函数的相关不等式，证明了在特定条件下协方差的下界，并验证了Friedgut-Kahn-Kalai-Keller谱猜想。

弱映射下的 h-向量不等式

$h$-vector inequalities under weak maps

本文研究拟阵复形的 h-向量在弱映射下的单调性，证明平坦格有序复形的 h-向量逐分量非增，并推广到旗 h-向量及独立复形情形。

On the number of equivalence classes of invertible Boolean functions under action of permutation of variables on domain and range

Let $V_n$ be the number of equivalence classes of invertible maps from $\{0,1\}^n$ to $\{0,1\}^n$, under action of permutation of variables on domain and range. So far, the values $V_n$ have been known for $n\le 6$. This paper describes the procedure by which the values of $V_n$ are calculated for $n\le 30$.

Quantitative Correlation Inequalities via Semigroup Interpolation

Most correlation inequalities for high-dimensional functions in the literature, such as the Fortuin-Kasteleyn-Ginibre (FKG) inequality and the celebrated Gaussian Correlation Inequality of Royen, are qualitative statements which establish that any two functions of a certain type have non-negative correlation. In this work we give a general approach that can be used to bootstrap many qualitative correlation inequalities for functions over product spaces into quantitative statements. The approach combines a new extremal result about power series, proved using complex analysis, with harmonic anal…

由二元关系决定的、在Minion同态下的多重排序布尔克隆

Multisorted Boolean Clones Determined by Binary Relations up to Minion Homomorphisms

该研究刻画了由二元关系决定、两坐标投影至多含两个元素的有限集合克隆类在minion同态下的序结构，将其等价为多重排序布尔克隆类，并提出minion核概念以给出克隆等价类的典范代表。

Berkovich-Uncu type Partition Inequalities Concerning Impermissible Sets and Perfect Power Frequencies

Recently, Rattan and the first author (Ann. Comb. 25 (2021) 697-728) proved a conjectured inequality of Berkovich and Uncu (Ann. Comb. 23 (2019) 263-284) concerning partitions with an impermissible part. In this article, we generalize this inequality upon considering t impermissible parts. We compare these with partitions whose certain parts appear with a frequency which is a perfect t^{th} power. Our inequalities hold after a certain bound, which for given t is a polynomial in s, a major improvement over the previously known bound in the case t=1. To prove these inequalities, our methods invo…

布尔概率界限问题、线性规划聚合与非负二次伪布尔函数

Boole's probability bounding problem, linear programming aggregations, and nonnegative quadratic pseudo-Boolean functions

针对布尔概率界限问题，提出聚合层级与通用分析方法，获得子锥完整多面体描述，显著改进界限质量。

级联图布尔二次多胞形的精确体积计算

Exact volume computation for Boolean quadric polytopes of series-parallel graphs

针对级联图，提出基于级联分解的动态规划算法，精确计算布尔二次多胞形体积，复杂度为 O(d^7)，并分析其与线性松弛的体积比率及奇环不等式的影响。

Improved log-Sobolev inequalities, hypercontractivity and uncertainty principle on the hypercube

Log-Sobolev inequalities (LSIs) upper-bound entropy via a multiple of the Dirichlet form (i.e. norm of a gradient). In this paper we prove a family of entropy-energy inequalities for the binary hypercube which provide a non-linear comparison between the entropy and the Dirichlet form and improve on the usual LSIs for functions with small support. These non-linear LSIs, in turn, imply a new version of the hypercontractivity for such functions. As another consequence, we derive a sharp form of the uncertainty principle for the hypercube: a function whose energy is concentrated on a set of small …