Digital Topological Properties of an Alignment of Fixed Point Sets
The present paper investigates digital topological properties of an alignment of fixed point sets which can play an important role in fixed point theory from the viewpoints of computational or digital topology. In digital topology-based fixed point theory, for a digital image <inline-formula>...
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MDPI AG
2020-06-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/6/921 |
| _version_ | 1797566035979665408 |
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| author | Sang-Eon Han |
| author_facet | Sang-Eon Han |
| author_sort | Sang-Eon Han |
| collection | DOAJ |
| description | The present paper investigates digital topological properties of an alignment of fixed point sets which can play an important role in fixed point theory from the viewpoints of computational or digital topology. In digital topology-based fixed point theory, for a digital image <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>X</mi> <mo>,</mo> <mi>k</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, let <inline-formula> <math display="inline"> <semantics> <mrow> <mi>F</mi> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> be the set of cardinalities of the fixed point sets of all <i>k</i>-continuous self-maps of <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>X</mi> <mo>,</mo> <mi>k</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> (see Definition 4). In this paper we call it an alignment of fixed point sets of <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>X</mi> <mo>,</mo> <mi>k</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>. Then we have the following unsolved problem. How many components are there in <inline-formula> <math display="inline"> <semantics> <mrow> <mi>F</mi> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> up to 2-connectedness? In particular, let <inline-formula> <math display="inline"> <semantics> <msubsup> <mi>C</mi> <mi>k</mi> <mrow> <mi>n</mi> <mo>,</mo> <mi>l</mi> </mrow> </msubsup> </semantics> </math> </inline-formula> be a simple closed <i>k</i>-curve with <i>l</i> 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> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>X</mi> <mo>:</mo> <mo>=</mo> <msubsup> <mi>C</mi> <mi>k</mi> <mrow> <mi>n</mi> <mo>,</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> </mrow> </msubsup> <mo>∨</mo> <msubsup> <mi>C</mi> <mi>k</mi> <mrow> <mi>n</mi> <mo>,</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> </mrow> </msubsup> </mrow> </semantics> </math> </inline-formula> be a digital wedge of <inline-formula> <math display="inline"> <semantics> <msubsup> <mi>C</mi> <mi>k</mi> <mrow> <mi>n</mi> <mo>,</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> </mrow> </msubsup> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <msubsup> <mi>C</mi> <mi>k</mi> <mrow> <mi>n</mi> <mo>,</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> </mrow> </msubsup> </semantics> </math> </inline-formula> in <inline-formula> <math display="inline"> <semantics> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>n</mi> </msup> </semantics> </math> </inline-formula>. Then we need to explore both the number of components of <inline-formula> <math display="inline"> <semantics> <mrow> <mi>F</mi> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> up to digital 2-connectivity (see Definition 4) and perfectness of <inline-formula> <math display="inline"> <semantics> <mrow> <mi>F</mi> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> (see Definition 5). The present paper addresses these issues and, furthermore, solves several problems related to the main issues. Indeed, it turns out that the three models <inline-formula> <math display="inline"> <semantics> <msubsup> <mi>C</mi> <mrow> <mn>2</mn> <mi>n</mi> </mrow> <mrow> <mi>n</mi> <mo>,</mo> <mn>4</mn> </mrow> </msubsup> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <msubsup> <mi>C</mi> <mrow> <msup> <mn>3</mn> <mi>n</mi> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>n</mi> <mo>,</mo> <mn>4</mn> </mrow> </msubsup> </semantics> </math> </inline-formula>, and <inline-formula> <math display="inline"> <semantics> <msubsup> <mi>C</mi> <mrow> <mi>k</mi> </mrow> <mrow> <mi>n</mi> <mo>,</mo> <mn>6</mn> </mrow> </msubsup> </semantics> </math> </inline-formula> play important roles in studying these topics because the digital fundamental groups of them have strong relationships with alignments of fixed point sets of them. Moreover, we correct some errors stated by Boxer et al. in their recent work and improve them (see Remark 3). This approach can facilitate the studies of pure and applied topologies, digital geometry, mathematical morphology, and image processing and image classification in computer science. The present paper only deals with <i>k</i>-connected spaces in <i>DTC</i>. Moreover, we will mainly deal with a set <i>X</i> such that <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>X</mi> <mo>♯</mo> </msup> <mo>≥</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula>. |
| first_indexed | 2024-03-10T19:21:10Z |
| format | Article |
| id | doaj.art-b3ced5e839f540a0a0225aa44fe23187 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T19:21:10Z |
| publishDate | 2020-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-b3ced5e839f540a0a0225aa44fe231872023-11-20T03:00:45ZengMDPI AGMathematics2227-73902020-06-018692110.3390/math8060921Digital Topological Properties of an Alignment of Fixed Point SetsSang-Eon Han0Department of Mathematics Education, Institute of Pure and Applied Mathematics, Jeonbuk National University, Jeonju-City Jeonbuk 54896, KoreaThe present paper investigates digital topological properties of an alignment of fixed point sets which can play an important role in fixed point theory from the viewpoints of computational or digital topology. In digital topology-based fixed point theory, for a digital image <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>X</mi> <mo>,</mo> <mi>k</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, let <inline-formula> <math display="inline"> <semantics> <mrow> <mi>F</mi> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> be the set of cardinalities of the fixed point sets of all <i>k</i>-continuous self-maps of <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>X</mi> <mo>,</mo> <mi>k</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> (see Definition 4). In this paper we call it an alignment of fixed point sets of <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>X</mi> <mo>,</mo> <mi>k</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>. Then we have the following unsolved problem. How many components are there in <inline-formula> <math display="inline"> <semantics> <mrow> <mi>F</mi> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> up to 2-connectedness? In particular, let <inline-formula> <math display="inline"> <semantics> <msubsup> <mi>C</mi> <mi>k</mi> <mrow> <mi>n</mi> <mo>,</mo> <mi>l</mi> </mrow> </msubsup> </semantics> </math> </inline-formula> be a simple closed <i>k</i>-curve with <i>l</i> 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> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>X</mi> <mo>:</mo> <mo>=</mo> <msubsup> <mi>C</mi> <mi>k</mi> <mrow> <mi>n</mi> <mo>,</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> </mrow> </msubsup> <mo>∨</mo> <msubsup> <mi>C</mi> <mi>k</mi> <mrow> <mi>n</mi> <mo>,</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> </mrow> </msubsup> </mrow> </semantics> </math> </inline-formula> be a digital wedge of <inline-formula> <math display="inline"> <semantics> <msubsup> <mi>C</mi> <mi>k</mi> <mrow> <mi>n</mi> <mo>,</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> </mrow> </msubsup> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <msubsup> <mi>C</mi> <mi>k</mi> <mrow> <mi>n</mi> <mo>,</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> </mrow> </msubsup> </semantics> </math> </inline-formula> in <inline-formula> <math display="inline"> <semantics> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>n</mi> </msup> </semantics> </math> </inline-formula>. Then we need to explore both the number of components of <inline-formula> <math display="inline"> <semantics> <mrow> <mi>F</mi> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> up to digital 2-connectivity (see Definition 4) and perfectness of <inline-formula> <math display="inline"> <semantics> <mrow> <mi>F</mi> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> (see Definition 5). The present paper addresses these issues and, furthermore, solves several problems related to the main issues. Indeed, it turns out that the three models <inline-formula> <math display="inline"> <semantics> <msubsup> <mi>C</mi> <mrow> <mn>2</mn> <mi>n</mi> </mrow> <mrow> <mi>n</mi> <mo>,</mo> <mn>4</mn> </mrow> </msubsup> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <msubsup> <mi>C</mi> <mrow> <msup> <mn>3</mn> <mi>n</mi> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>n</mi> <mo>,</mo> <mn>4</mn> </mrow> </msubsup> </semantics> </math> </inline-formula>, and <inline-formula> <math display="inline"> <semantics> <msubsup> <mi>C</mi> <mrow> <mi>k</mi> </mrow> <mrow> <mi>n</mi> <mo>,</mo> <mn>6</mn> </mrow> </msubsup> </semantics> </math> </inline-formula> play important roles in studying these topics because the digital fundamental groups of them have strong relationships with alignments of fixed point sets of them. Moreover, we correct some errors stated by Boxer et al. in their recent work and improve them (see Remark 3). This approach can facilitate the studies of pure and applied topologies, digital geometry, mathematical morphology, and image processing and image classification in computer science. The present paper only deals with <i>k</i>-connected spaces in <i>DTC</i>. Moreover, we will mainly deal with a set <i>X</i> such that <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>X</mi> <mo>♯</mo> </msup> <mo>≥</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula>.https://www.mdpi.com/2227-7390/8/6/921digital imagedigital wedgenormal adjacencyk-homotopyk-contractibilityalignment |
| spellingShingle | Sang-Eon Han Digital Topological Properties of an Alignment of Fixed Point Sets Mathematics digital image digital wedge normal adjacency k-homotopy k-contractibility alignment |
| title | Digital Topological Properties of an Alignment of Fixed Point Sets |
| title_full | Digital Topological Properties of an Alignment of Fixed Point Sets |
| title_fullStr | Digital Topological Properties of an Alignment of Fixed Point Sets |
| title_full_unstemmed | Digital Topological Properties of an Alignment of Fixed Point Sets |
| title_short | Digital Topological Properties of an Alignment of Fixed Point Sets |
| title_sort | digital topological properties of an alignment of fixed point sets |
| topic | digital image digital wedge normal adjacency k-homotopy k-contractibility alignment |
| url | https://www.mdpi.com/2227-7390/8/6/921 |
| work_keys_str_mv | AT sangeonhan digitaltopologicalpropertiesofanalignmentoffixedpointsets |