In this work we employ machine learning to understand structured mathematical
data involving finite groups and derive a theorem about necessary properties of
generators of finite simple groups. We create a database of all 2-generated
subgroups of the symmetric group on n-objects and conduct a classification of
finite simple groups among them using shallow feed-forward neural networks. We
show that this neural network classifier can decipher the property of
simplicity with varying accuracies depending on the features. Our neural
network model leads to a natural conjecture concerning the generators of a
finite simple group. We subsequently prove this conjecture. This new toy
theorem comments on the necessary properties of generators of finite simple
groups. We show this explicitly for a class of sporadic groups for which the
result holds. Our work further makes the case for a machine motivated study of
algebraic structures in pure mathematics and highlights the possibility of
generating new conjectures and theorems in mathematics with the aid of machine
learning.

运用机器学习从数学上的结构化数据中了解有限群，并得出关于有限单群生成元必要属性的定理，通过使用浅层前馈神经网络对所有 n 个对象的对称群的二元子群进行分类，验证这种神经网络分类器可以根据特征以不同的准确率识别简单性属性，新的玩具定理关于有限单群生成元的必要属性进行评论，对一类稀有群成立的结果进行了明确展示，表明了纯数学中代数结构机器学习的研究可行性，并凸显出借助于机器学习在数学中生成新的猜想和定理的可能性。

学会简单

Learning to be Simple

Quantifying the similarity between two mathematical structures or datasets
constitutes a particularly interesting and useful operation in several
theoretical and applied problems. Aimed at this specific objective, the Jaccard
index has been extensively used in the most diverse types of problems, also
motivating some respective generalizations. The present work addresses further
generalizations of this index, including its modification into a coincidence
index capable of accounting also for the level of relative interiority between
the two compared entities, as well as respective extensions for sets in
continuous vector spaces, the generalization to multiset addition, densities
and generic scalar fields, as well as a means to quantify the joint
interdependence between two random variables. The also interesting possibility
to take into account more than two sets has also been addressed, including the
description of an index capable of quantifying the level of chaining between
three structures. Several of the described and suggested eneralizations have
been illustrated with respect to numeric case examples. It is also posited that
these indices can play an important role while analyzing and integrating
datasets in modeling approaches and pattern recognition activities, including
as a measurement of clusters similarity or separation and as a resource for
representing and analyzing complex networks.

对 Jaccard 指数及其相关的一些扩展进行理论研究，包括一种新的巧合指数，该指数在比较两个实体集时可以考虑相对内含度的水平，适用于连续向量空间、多重集、密度和通用标量场的扩展，以及量化两个随机变量之间的联合相互依赖性的方法。

Jaccard 指数的进一步推广

Further Generalizations of the Jaccard Index

We discuss quantum non-locality and contextuality, emphasising logical and
structural aspects. We also show how the same mathematical structures arise in
various areas of classical computation.

本研究探讨量子非局域性和上下文性，强调其逻辑和结构方面。同时我们还展示了这些相同的数学结构在各个领域的经典计算中的应用。

上下文语义：从量子力学到逻辑、数据库、约束和复杂性

Contextual Semantics: From Quantum Mechanics to Logic, Databases,  Constraints, and Complexity

We survey some basic mathematical structures, which arguably are more
primitive than the structures taught at school. These structures are orders,
with or without composition, and (symmetric) monoidal categories. We list
several `real life' incarnations of each of these. This paper also serves as an
introduction to these structures and their current and potentially future uses
in linguistics, physics and knowledge representation.

本文介绍了数学结构中的基础内容，包括有序结构、带或不带组合的结构以及（对称的）单调类别。同时列举了这些数学结构在语言学、物理学和知识表示中的应用。