凸和非凸码的代数签名
Algebraic signatures of convex and non-convex codes
摘要：
凸码是一种二进制码，它是由一些欧几里得空间中的开凸集集合的交集模式生成的。凸代码与神经科学相关，因为它们来自具有凸感受野的神经元的活动。在本文中，我们使用代数方法来确定是否是凸码。具体来说，我们使用代码的神经理想，这是Stanley Reisner理想的推广。使用神经理想连同其标准生成集，规范形式，我们提供了代数签名的某些家庭的代码是非凸的。我们将这些签名连接到防止代码凸的集合排列的精确条件。最后，我们还提供了代数签名的一些家庭的代码是凸的，包括类的相交完全码。这些结果使我们能够在各种情况下检测凸性和非凸性，并指出一些有趣的开放问题。
英文摘要：
A convex code is a binary code generated by the pattern of intersections of a collection of open convex sets in some Euclidean space. Convex codes are relevant to neuroscience as they arise from the activity of neurons that have convex receptive fields. In this paper, we use algebraic methods to determine if a code is convex. Specifically, we use the neural ideal of a code, which is a generalization of the Stanley-Reisner ideal. Using the neural ideal together with its standard generating set, the canonical form, we provide algebraic signatures of certain families of codes that are non-convex. We connect these signatures to the precise conditions on the arrangement of sets that prevent the codes from being convex. Finally, we also provide algebraic signatures for some families of codes that are convex, including the class of intersection-complete codes. These results allow us to detect convexity and non-convexity in a variety of situations, and point to some interesting open questions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
