On Some Properties of the Limit Points of (<i>z</i>(<i>n</i>)/<i>n</i>)<sub><i>n</i></sub>
Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>F</mi><mi>n</mi></msub><mo>)</mo></mrow><mrow><...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2021-08-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/9/16/1931 |
| _version_ | 1797523017815818240 |
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| author | Eva Trojovská Kandasamy Venkatachalam |
| author_facet | Eva Trojovská Kandasamy Venkatachalam |
| author_sort | Eva Trojovská |
| collection | DOAJ |
| description | Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>F</mi><mi>n</mi></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub></semantics></math></inline-formula> be the sequence of Fibonacci numbers. The order of appearance of an integer <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula> is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>z</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>:</mo><mo>=</mo><mo movablelimits="true" form="prefix">min</mo><mrow><mo>{</mo><mi>k</mi><mo>≥</mo><mn>1</mn><mo>:</mo><mi>n</mi><mo>∣</mo><msub><mi>F</mi><mi>k</mi></msub><mo>}</mo></mrow></mrow></semantics></math></inline-formula>. Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="script">Z</mi><mo>′</mo></msup></semantics></math></inline-formula> be the set of all limit points of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><mi>z</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>/</mo><mi>n</mi><mo>:</mo><mi>n</mi><mo>≥</mo><mn>1</mn><mo>}</mo></mrow></semantics></math></inline-formula>. By some theoretical results on the growth of the sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><mi>z</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>/</mo><mi>n</mi><mo>)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></msub></semantics></math></inline-formula>, we gain a better understanding of the topological structure of the derived set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="script">Z</mi><mo>′</mo></msup></semantics></math></inline-formula>. For instance, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>,</mo><mn>2</mn><mo>}</mo></mrow><mo>⊆</mo><msup><mi mathvariant="script">Z</mi><mo>′</mo></msup><mo>⊆</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>]</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="script">Z</mi><mo>′</mo></msup></semantics></math></inline-formula> does not have any interior points. A recent result of Trojovská implies the existence of a positive real number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><mo><</mo><mn>2</mn></mrow></semantics></math></inline-formula> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="script">Z</mi><mo>′</mo></msup><mo>∩</mo><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is the empty set. In this paper, we improve this result by proving that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mfrac><mn>12</mn><mn>7</mn></mfrac><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula> is the largest subinterval of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>]</mo></mrow></semantics></math></inline-formula> which does not intersect <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="script">Z</mi><mo>′</mo></msup></semantics></math></inline-formula>. In addition, we show a connection between the sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>x</mi><mi>n</mi></msub><mo>)</mo></mrow><mi>n</mi></msub></semantics></math></inline-formula>, for which <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>z</mi><mrow><mo>(</mo><msub><mi>x</mi><mi>n</mi></msub><mo>)</mo></mrow><mo>/</mo><msub><mi>x</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula> tends to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> (as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></semantics></math></inline-formula>), and the number of preimages of <i>r</i> under the map <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>↦</mo><mi>z</mi><mo>(</mo><mi>m</mi><mo>)</mo><mo>/</mo><mi>m</mi></mrow></semantics></math></inline-formula>. |
| first_indexed | 2024-03-10T08:37:27Z |
| format | Article |
| id | doaj.art-c748cfa3960743cb8221264edb514dfc |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T08:37:27Z |
| publishDate | 2021-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-c748cfa3960743cb8221264edb514dfc2023-11-22T08:34:03ZengMDPI AGMathematics2227-73902021-08-01916193110.3390/math9161931On Some Properties of the Limit Points of (<i>z</i>(<i>n</i>)/<i>n</i>)<sub><i>n</i></sub>Eva Trojovská0Kandasamy Venkatachalam1Department of Mathematics, Faculty of Science, University of Hradec Králové, 500 03 Hradec Králové, Czech RepublicDepartment of Mathematics, Faculty of Science, University of Hradec Králové, 500 03 Hradec Králové, Czech RepublicLet <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>F</mi><mi>n</mi></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub></semantics></math></inline-formula> be the sequence of Fibonacci numbers. The order of appearance of an integer <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula> is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>z</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>:</mo><mo>=</mo><mo movablelimits="true" form="prefix">min</mo><mrow><mo>{</mo><mi>k</mi><mo>≥</mo><mn>1</mn><mo>:</mo><mi>n</mi><mo>∣</mo><msub><mi>F</mi><mi>k</mi></msub><mo>}</mo></mrow></mrow></semantics></math></inline-formula>. Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="script">Z</mi><mo>′</mo></msup></semantics></math></inline-formula> be the set of all limit points of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><mi>z</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>/</mo><mi>n</mi><mo>:</mo><mi>n</mi><mo>≥</mo><mn>1</mn><mo>}</mo></mrow></semantics></math></inline-formula>. By some theoretical results on the growth of the sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><mi>z</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>/</mo><mi>n</mi><mo>)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></msub></semantics></math></inline-formula>, we gain a better understanding of the topological structure of the derived set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="script">Z</mi><mo>′</mo></msup></semantics></math></inline-formula>. For instance, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>,</mo><mn>2</mn><mo>}</mo></mrow><mo>⊆</mo><msup><mi mathvariant="script">Z</mi><mo>′</mo></msup><mo>⊆</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>]</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="script">Z</mi><mo>′</mo></msup></semantics></math></inline-formula> does not have any interior points. A recent result of Trojovská implies the existence of a positive real number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><mo><</mo><mn>2</mn></mrow></semantics></math></inline-formula> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="script">Z</mi><mo>′</mo></msup><mo>∩</mo><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is the empty set. In this paper, we improve this result by proving that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mfrac><mn>12</mn><mn>7</mn></mfrac><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula> is the largest subinterval of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>]</mo></mrow></semantics></math></inline-formula> which does not intersect <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="script">Z</mi><mo>′</mo></msup></semantics></math></inline-formula>. In addition, we show a connection between the sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>x</mi><mi>n</mi></msub><mo>)</mo></mrow><mi>n</mi></msub></semantics></math></inline-formula>, for which <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>z</mi><mrow><mo>(</mo><msub><mi>x</mi><mi>n</mi></msub><mo>)</mo></mrow><mo>/</mo><msub><mi>x</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula> tends to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> (as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></semantics></math></inline-formula>), and the number of preimages of <i>r</i> under the map <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>↦</mo><mi>z</mi><mo>(</mo><mi>m</mi><mo>)</mo><mo>/</mo><mi>m</mi></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2227-7390/9/16/1931order of appearancefibonacci numbersderived setgreatest prime factornatural density |
| spellingShingle | Eva Trojovská Kandasamy Venkatachalam On Some Properties of the Limit Points of (<i>z</i>(<i>n</i>)/<i>n</i>)<sub><i>n</i></sub> Mathematics order of appearance fibonacci numbers derived set greatest prime factor natural density |
| title | On Some Properties of the Limit Points of (<i>z</i>(<i>n</i>)/<i>n</i>)<sub><i>n</i></sub> |
| title_full | On Some Properties of the Limit Points of (<i>z</i>(<i>n</i>)/<i>n</i>)<sub><i>n</i></sub> |
| title_fullStr | On Some Properties of the Limit Points of (<i>z</i>(<i>n</i>)/<i>n</i>)<sub><i>n</i></sub> |
| title_full_unstemmed | On Some Properties of the Limit Points of (<i>z</i>(<i>n</i>)/<i>n</i>)<sub><i>n</i></sub> |
| title_short | On Some Properties of the Limit Points of (<i>z</i>(<i>n</i>)/<i>n</i>)<sub><i>n</i></sub> |
| title_sort | on some properties of the limit points of i z i i n i i n i sub i n i sub |
| topic | order of appearance fibonacci numbers derived set greatest prime factor natural density |
| url | https://www.mdpi.com/2227-7390/9/16/1931 |
| work_keys_str_mv | AT evatrojovska onsomepropertiesofthelimitpointsofiziiniinisubinisub AT kandasamyvenkatachalam onsomepropertiesofthelimitpointsofiziiniinisubinisub |