Taking Rational Numbers at Random
In this article, some prescriptions to define a distribution on the set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>Q</mi><mn>0</mn></msub></semantics></math&...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-09-01
|
| Series: | AppliedMath |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2673-9909/3/3/34 |
| _version_ | 1797581529126273024 |
|---|---|
| author | Nicola Cufaro Petroni |
| author_facet | Nicola Cufaro Petroni |
| author_sort | Nicola Cufaro Petroni |
| collection | DOAJ |
| description | In this article, some prescriptions to define a distribution on the set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>Q</mi><mn>0</mn></msub></semantics></math></inline-formula> of all rational numbers in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula> are outlined. We explored a few properties of these distributions and the possibility of making these rational numbers <i>asymptotically equiprobable</i> in a suitable sense. In particular, it will be shown that in the said limit—albeit no absolutely continuous uniform distribution can be properly defined in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Q</mi><mn>0</mn></msub></semantics></math></inline-formula>—the probability allotted to every single <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>∈</mo><msub><mi mathvariant="double-struck">Q</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula> asymptotically vanishes, while that of the subset of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Q</mi><mn>0</mn></msub></semantics></math></inline-formula> falling in an interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>]</mo></mrow><mo>⊆</mo><msub><mi mathvariant="double-struck">Q</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula> goes to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow></semantics></math></inline-formula>. We finally present some hints to complete sequencing without repeating the numbers in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Q</mi><mn>0</mn></msub></semantics></math></inline-formula> as a prerequisite to laying down more distributions on it. |
| first_indexed | 2024-03-10T23:05:51Z |
| format | Article |
| id | doaj.art-cf4d49658e6b4f3f83a3acc24c61ea2e |
| institution | Directory Open Access Journal |
| issn | 2673-9909 |
| language | English |
| last_indexed | 2024-03-10T23:05:51Z |
| publishDate | 2023-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | AppliedMath |
| spelling | doaj.art-cf4d49658e6b4f3f83a3acc24c61ea2e2023-11-19T09:20:48ZengMDPI AGAppliedMath2673-99092023-09-013364866310.3390/appliedmath3030034Taking Rational Numbers at RandomNicola Cufaro Petroni0Department of Mathematics and TIRES, University of Bari, INFN Sezione di Bari, via E. Orabona 4, 70125 Bari, ItalyIn this article, some prescriptions to define a distribution on the set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>Q</mi><mn>0</mn></msub></semantics></math></inline-formula> of all rational numbers in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula> are outlined. We explored a few properties of these distributions and the possibility of making these rational numbers <i>asymptotically equiprobable</i> in a suitable sense. In particular, it will be shown that in the said limit—albeit no absolutely continuous uniform distribution can be properly defined in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Q</mi><mn>0</mn></msub></semantics></math></inline-formula>—the probability allotted to every single <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>∈</mo><msub><mi mathvariant="double-struck">Q</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula> asymptotically vanishes, while that of the subset of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Q</mi><mn>0</mn></msub></semantics></math></inline-formula> falling in an interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>]</mo></mrow><mo>⊆</mo><msub><mi mathvariant="double-struck">Q</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula> goes to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow></semantics></math></inline-formula>. We finally present some hints to complete sequencing without repeating the numbers in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Q</mi><mn>0</mn></msub></semantics></math></inline-formula> as a prerequisite to laying down more distributions on it.https://www.mdpi.com/2673-9909/3/3/34rational numbersdiscrete distributionsrandomness |
| spellingShingle | Nicola Cufaro Petroni Taking Rational Numbers at Random AppliedMath rational numbers discrete distributions randomness |
| title | Taking Rational Numbers at Random |
| title_full | Taking Rational Numbers at Random |
| title_fullStr | Taking Rational Numbers at Random |
| title_full_unstemmed | Taking Rational Numbers at Random |
| title_short | Taking Rational Numbers at Random |
| title_sort | taking rational numbers at random |
| topic | rational numbers discrete distributions randomness |
| url | https://www.mdpi.com/2673-9909/3/3/34 |
| work_keys_str_mv | AT nicolacufaropetroni takingrationalnumbersatrandom |