Winding number of loops and digital category of digital simple closed curves

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications Pub Date : 2025-05-05 DOI:10.1016/j.topol.2025.109410

Samia Ashraf, Amna Amanat Ali

引用次数: 0

Abstract

An analogue of the notion of Lusternik–Schnirelmann category for digital images, named “digital category” is defined to be one less than the number of “subdivision categorical” sets which cover the digital image. We define winding number of loops in digital simple closed 8-curves and use it to compute their digital category. Moreover, by applying this to a specific family of digital simple closed curves, we deduce that the digital category of

S (D, n)

, the n-subdivision of the smallest such curve D consisting of four points (digital circle of radius 1) is equal to the digital category of D itself.

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数字简单闭合曲线的绕组数和数字类别

数字图像的Lusternik-Schnirelmann范畴概念的类似物，称为“数字范畴”，被定义为比覆盖数字图像的“细分范畴”集的数量少一个。定义了数字简单封闭8曲线中环路的圈数，并用它来计算它们的数字范畴。此外，通过将其应用于特定的数字简单封闭曲线族，我们推导出S（D,n）的数字范畴，由四个点组成的最小的这样的曲线D（半径为1的数字圆）的n个细分等于D本身的数字范畴。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Topology and its Applications 数学-数学

CiteScore

1.20

自引率

33.30%

发文量

251

审稿时长

6 months

期刊介绍： Topology and its Applications is primarily concerned with publishing original research papers of moderate length. However, a limited number of carefully selected survey or expository papers are also included. The mathematical focus of the journal is that suggested by the title: Research in Topology. It is felt that it is inadvisable to attempt a definitive description of topology as understood for this journal. Certainly the subject includes the algebraic, general, geometric, and set-theoretic facets of topology as well as areas of interactions between topology and other mathematical disciplines, e.g. topological algebra, topological dynamics, functional analysis, category theory. Since the roles of various aspects of topology continue to change, the non-specific delineation of topics serves to reflect the current state of research in topology. At regular intervals, the journal publishes a section entitled Open Problems in Topology, edited by J. van Mill and G.M. Reed. This is a status report on the 1100 problems listed in the book of the same name published by North-Holland in 1990, edited by van Mill and Reed.