Weighted Homology of Bi-Structures over Certain Discrete Valuation Rings
An RNA bi-structure is a pair of RNA secondary structures that are considered as arc-diagrams. We present a novel weighted homology theory for RNA bi-structures, which was obtained through the intersections of loops. The weighted homology of the intersection complex <i>X</i> features a n...
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| Format: | Article |
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MDPI AG
2021-03-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/7/744 |
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| author | Andrei Bura Qijun He Christian Reidys |
| author_facet | Andrei Bura Qijun He Christian Reidys |
| author_sort | Andrei Bura |
| collection | DOAJ |
| description | An RNA bi-structure is a pair of RNA secondary structures that are considered as arc-diagrams. We present a novel weighted homology theory for RNA bi-structures, which was obtained through the intersections of loops. The weighted homology of the intersection complex <i>X</i> features a new boundary operator and is formulated over a discrete valuation ring, <i>R</i>. We establish basic properties of the weighted complex and show how to deform it in order to eliminate any 3-simplices. We connect the simplicial homology, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mi>i</mi></msub><mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>, and weighted homology, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mrow><mi>i</mi><mo>,</mo><mi>R</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>, in two ways: first, via chain maps, and second, via the relative homology. We compute <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mrow><mn>0</mn><mo>,</mo><mi>R</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> by means of a recursive contraction procedure on a weighted spanning tree and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mrow><mn>1</mn><mo>,</mo><mi>R</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> via an inflation map, by which the simplicial homology of the 1-skeleton allows us to determine the weighted homology <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mrow><mn>1</mn><mo>,</mo><mi>R</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>. The homology module <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mrow><mn>2</mn><mo>,</mo><mi>R</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> is naturally obtained from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mn>2</mn></msub><mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> via chain maps. Furthermore, we show that all weighted homology modules <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mrow><mi>i</mi><mo>,</mo><mi>R</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> are trivial for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><mo>></mo><mn>2</mn></mrow></semantics></math></inline-formula>. The invariant factors of our structure theorems, as well as the weighted Whitehead moves facilitating the removal of filled tetrahedra, are given a combinatorial interpretation. The weighted homology of bi-structures augments the simplicial counterpart by introducing novel torsion submodules and preserving the free submodules that appear in the simplicial homology. |
| first_indexed | 2024-03-10T12:44:29Z |
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| id | doaj.art-60366285aea9480d8c93de5862b2415c |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T12:44:29Z |
| publishDate | 2021-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-60366285aea9480d8c93de5862b2415c2023-11-21T13:34:22ZengMDPI AGMathematics2227-73902021-03-019774410.3390/math9070744Weighted Homology of Bi-Structures over Certain Discrete Valuation RingsAndrei Bura0Qijun He1Christian Reidys2Biocomplexity Institute and Initiative, University of Virginia, Charlottesville, VA 22904-4298, USABiocomplexity Institute and Initiative, University of Virginia, Charlottesville, VA 22904-4298, USABiocomplexity Institute and Initiative, University of Virginia, Charlottesville, VA 22904-4298, USAAn RNA bi-structure is a pair of RNA secondary structures that are considered as arc-diagrams. We present a novel weighted homology theory for RNA bi-structures, which was obtained through the intersections of loops. The weighted homology of the intersection complex <i>X</i> features a new boundary operator and is formulated over a discrete valuation ring, <i>R</i>. We establish basic properties of the weighted complex and show how to deform it in order to eliminate any 3-simplices. We connect the simplicial homology, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mi>i</mi></msub><mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>, and weighted homology, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mrow><mi>i</mi><mo>,</mo><mi>R</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>, in two ways: first, via chain maps, and second, via the relative homology. We compute <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mrow><mn>0</mn><mo>,</mo><mi>R</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> by means of a recursive contraction procedure on a weighted spanning tree and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mrow><mn>1</mn><mo>,</mo><mi>R</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> via an inflation map, by which the simplicial homology of the 1-skeleton allows us to determine the weighted homology <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mrow><mn>1</mn><mo>,</mo><mi>R</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>. The homology module <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mrow><mn>2</mn><mo>,</mo><mi>R</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> is naturally obtained from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mn>2</mn></msub><mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> via chain maps. Furthermore, we show that all weighted homology modules <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mrow><mi>i</mi><mo>,</mo><mi>R</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> are trivial for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><mo>></mo><mn>2</mn></mrow></semantics></math></inline-formula>. The invariant factors of our structure theorems, as well as the weighted Whitehead moves facilitating the removal of filled tetrahedra, are given a combinatorial interpretation. The weighted homology of bi-structures augments the simplicial counterpart by introducing novel torsion submodules and preserving the free submodules that appear in the simplicial homology.https://www.mdpi.com/2227-7390/9/7/744weighted simplicial complexweighted homologymodules over PIDs (Principal Ideal Domain)torsionembeddingspanning sub-trees |
| spellingShingle | Andrei Bura Qijun He Christian Reidys Weighted Homology of Bi-Structures over Certain Discrete Valuation Rings Mathematics weighted simplicial complex weighted homology modules over PIDs (Principal Ideal Domain) torsion embedding spanning sub-trees |
| title | Weighted Homology of Bi-Structures over Certain Discrete Valuation Rings |
| title_full | Weighted Homology of Bi-Structures over Certain Discrete Valuation Rings |
| title_fullStr | Weighted Homology of Bi-Structures over Certain Discrete Valuation Rings |
| title_full_unstemmed | Weighted Homology of Bi-Structures over Certain Discrete Valuation Rings |
| title_short | Weighted Homology of Bi-Structures over Certain Discrete Valuation Rings |
| title_sort | weighted homology of bi structures over certain discrete valuation rings |
| topic | weighted simplicial complex weighted homology modules over PIDs (Principal Ideal Domain) torsion embedding spanning sub-trees |
| url | https://www.mdpi.com/2227-7390/9/7/744 |
| work_keys_str_mv | AT andreibura weightedhomologyofbistructuresovercertaindiscretevaluationrings AT qijunhe weightedhomologyofbistructuresovercertaindiscretevaluationrings AT christianreidys weightedhomologyofbistructuresovercertaindiscretevaluationrings |