Fixed Point Sets of Digital Curves and Digital Surfaces
Given a digital image (or digital object) <inline-formula><math display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2020-10-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/8/11/1896 |
| _version_ | 1797549242679558144 |
|---|---|
| author | Sang-Eon Han |
| author_facet | Sang-Eon Han |
| author_sort | Sang-Eon Han |
| collection | DOAJ |
| description | Given a digital image (or digital object) <inline-formula><math display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>, we address some unsolved problems related to the study of fixed point sets of <i>k</i>-continuous self-maps of <inline-formula><math display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> from the viewpoints of digital curve and digital surface theory. Consider two simple closed <i>k</i>-curves with <inline-formula><math display="inline"><semantics><msub><mi>l</mi><mi>i</mi></msub></semantics></math></inline-formula> elements in <inline-formula><math display="inline"><semantics><msup><mrow><mi mathvariant="double-struck">Z</mi></mrow><mi>n</mi></msup></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><mrow><mi>i</mi><mo>∈</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>}</mo></mrow><mo>,</mo><msub><mi>l</mi><mn>1</mn></msub><mo>⪈</mo><msub><mi>l</mi><mn>2</mn></msub><mo>≥</mo><mn>4</mn></mrow></semantics></math></inline-formula>. After initially formulating an alignment of fixed point sets of a digital wedge of these curves, we prove that perfectness of it depends on the numbers <inline-formula><math display="inline"><semantics><mrow><msub><mi>l</mi><mi>i</mi></msub><mo>,</mo><mi>i</mi><mspace width="3.33333pt"></mspace><mo>∈</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>}</mo></mrow></mrow></semantics></math></inline-formula>, instead of the <i>k</i>-adjacency. Furthermore, given digital <i>k</i>-surfaces, we also study an alignment of fixed point sets of digital <i>k</i>-surfaces and digital wedges of them. Finally, given a digital image which is not perfect, we explore a certain condition that makes it perfect. In this paper, each digital image <inline-formula><math display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> is assumed to be <i>k</i>-connected and <inline-formula><math display="inline"><semantics><mrow><msup><mi>X</mi><mo>♯</mo></msup><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula> unless stated otherwise. |
| first_indexed | 2024-03-10T15:10:53Z |
| format | Article |
| id | doaj.art-50452bff2bc64d498dfd43db7a013bc2 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T15:10:53Z |
| publishDate | 2020-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-50452bff2bc64d498dfd43db7a013bc22023-11-20T19:18:53ZengMDPI AGMathematics2227-73902020-10-01811189610.3390/math8111896Fixed Point Sets of Digital Curves and Digital SurfacesSang-Eon Han0Department of Mathematics Education, Institute of Pure and Applied Mathematics, Jeonbuk National University, Jeonju-City Jeonbuk 54896, KoreaGiven a digital image (or digital object) <inline-formula><math display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>, we address some unsolved problems related to the study of fixed point sets of <i>k</i>-continuous self-maps of <inline-formula><math display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> from the viewpoints of digital curve and digital surface theory. Consider two simple closed <i>k</i>-curves with <inline-formula><math display="inline"><semantics><msub><mi>l</mi><mi>i</mi></msub></semantics></math></inline-formula> elements in <inline-formula><math display="inline"><semantics><msup><mrow><mi mathvariant="double-struck">Z</mi></mrow><mi>n</mi></msup></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><mrow><mi>i</mi><mo>∈</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>}</mo></mrow><mo>,</mo><msub><mi>l</mi><mn>1</mn></msub><mo>⪈</mo><msub><mi>l</mi><mn>2</mn></msub><mo>≥</mo><mn>4</mn></mrow></semantics></math></inline-formula>. After initially formulating an alignment of fixed point sets of a digital wedge of these curves, we prove that perfectness of it depends on the numbers <inline-formula><math display="inline"><semantics><mrow><msub><mi>l</mi><mi>i</mi></msub><mo>,</mo><mi>i</mi><mspace width="3.33333pt"></mspace><mo>∈</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>}</mo></mrow></mrow></semantics></math></inline-formula>, instead of the <i>k</i>-adjacency. Furthermore, given digital <i>k</i>-surfaces, we also study an alignment of fixed point sets of digital <i>k</i>-surfaces and digital wedges of them. Finally, given a digital image which is not perfect, we explore a certain condition that makes it perfect. In this paper, each digital image <inline-formula><math display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> is assumed to be <i>k</i>-connected and <inline-formula><math display="inline"><semantics><mrow><msup><mi>X</mi><mo>♯</mo></msup><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula> unless stated otherwise.https://www.mdpi.com/2227-7390/8/11/1896digital wedgealignmentperfectfixed point setdigital k-surfacedigital topology |
| spellingShingle | Sang-Eon Han Fixed Point Sets of Digital Curves and Digital Surfaces Mathematics digital wedge alignment perfect fixed point set digital k-surface digital topology |
| title | Fixed Point Sets of Digital Curves and Digital Surfaces |
| title_full | Fixed Point Sets of Digital Curves and Digital Surfaces |
| title_fullStr | Fixed Point Sets of Digital Curves and Digital Surfaces |
| title_full_unstemmed | Fixed Point Sets of Digital Curves and Digital Surfaces |
| title_short | Fixed Point Sets of Digital Curves and Digital Surfaces |
| title_sort | fixed point sets of digital curves and digital surfaces |
| topic | digital wedge alignment perfect fixed point set digital k-surface digital topology |
| url | https://www.mdpi.com/2227-7390/8/11/1896 |
| work_keys_str_mv | AT sangeonhan fixedpointsetsofdigitalcurvesanddigitalsurfaces |