Designs in finite metric spaces: a probabilistic approach

魔方群直径的概率估计

Probabilistic estimates of the diameters of the Rubik's Cube groups

本文利用概率论中修改版优惠券收集者问题估计魔方群直径，验证了2×2×2和3×3×3魔方在四分之一转度量下的已知直径，并预测4×4×4和5×5×5魔方的直径分别为48（41）和68（58），同时给出直径的近似公式。

小型Transformer计算通用度量嵌入

Small Transformers Compute Universal Metric Embeddings

本研究证明小型概率Transformer能以低失真双Hölder嵌入任意度量空间中的有限数据集，避免维数灾难，并推广到黎曼流形、度量树等结构，实现通用度量嵌入。

Probabilistic Existence of Large Sets of Designs

A new probabilistic technique for establishing the existence of certain regular combinatorial structures has been recentlyintroduced by Kuperberg, Lovett, and Peled (STOC 2012). Using this technique, it can be shown that under certain conditions, a randomly chosen structure has the required properties of a $t$-$(n,k,λ)$ combinatorial design with tiny, yet positive, probability. The proof method of KLP is adapted to show the existence of large sets of designs and similar combinatorial structures as follows. We modify the random choice and the analysis to show that, under the same conditions, no…

精细吸收法：存在性猜想的一种新证明及其在极值与概率设计理论中的应用

Refined Absorption: A New Proof of the Existence Conjecture and its Applications to Extremal and Probabilistic Design Theory

本研究介绍精细吸收法，用其给出组合设计存在性猜想的新证明，该方法可解决极值与概率设计理论的开放问题，提供统一框架，且核心吸收定理可作黑盒使用，免去重复证明。

攻克6/49彩票并用概率方法和组合设计破除常见迷思

Tackling the 6/49 Lottery and Debunking Common Myths with Probabilistic Methods and Combinatorial Designs

本文通过概率方法和组合设计客观分析6/49彩票，构建了达到Schönheim界的(49,6,5)覆盖设计，并证明常见试图战胜赔率的方法会不成比例地降低中奖机会。

Point distributions in compact metric spaces

We consider finite point subsets (distributions) in compact metric spaces. Non-trivial bounds for sums of distances between points of distributions and for discrepancies of distributions in metric balls are given in the case of general rectifiable metric spaces. We generalize Stolarsky's invariance principle to distance-invariant spaces, and for arbitrary metric spaces we prove a probabilistic invariance principle. Furthermore, we construct partitions of general rectifiable compact metric spaces into subsets of equal measure with minimum average diameter.