Pub Date : 2022-01-01DOI: 10.4236/ojdm.2022.123004
Naoto Morikawa
{"title":"Discrete Exterior Calculus of Proteins and Their Cohomology","authors":"Naoto Morikawa","doi":"10.4236/ojdm.2022.123004","DOIUrl":"https://doi.org/10.4236/ojdm.2022.123004","url":null,"abstract":"","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70627849","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.4236/ojdm.2022.122002
Xiao Feng, Penghao Cao, Z. Chang
In this paper, we prove that the surface of any circular cone can be triangulated into 8 non-obtuse and 20 acute triangles. Furthermore, we also show that the bounds are both the best
{"title":"Acute Triangulations of the Surface of Circular Cone","authors":"Xiao Feng, Penghao Cao, Z. Chang","doi":"10.4236/ojdm.2022.122002","DOIUrl":"https://doi.org/10.4236/ojdm.2022.122002","url":null,"abstract":"In this paper, we prove that the surface of any circular cone can be triangulated into 8 non-obtuse and 20 acute triangles. Furthermore, we also show that the bounds are both the best","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70627710","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-06-08DOI: 10.4236/ojdm.2021.113007
Oladapo Adekunle Ojo, Fatma Salim Ali Al-Kharousi, A. Umar
Let be the partial symmetric semigroup on and let and be its subsemigroups of order-preserving contractions and order-preserving, order-decreasing contractions mappings of , respectively. In this paper we investigate the cardinalities of and , the set idempotents of and , respectively. We also investigate the cardinalities of certain equivalences on and .
{"title":"On the Number of Idempotent Partial Contraction Mappings of a Finite Chain","authors":"Oladapo Adekunle Ojo, Fatma Salim Ali Al-Kharousi, A. Umar","doi":"10.4236/ojdm.2021.113007","DOIUrl":"https://doi.org/10.4236/ojdm.2021.113007","url":null,"abstract":"Let be the partial symmetric semigroup on and let and be its subsemigroups of order-preserving contractions and order-preserving, order-decreasing contractions mappings of , respectively. In this paper we investigate the cardinalities of and , the set idempotents of and , respectively. We also investigate the cardinalities of certain equivalences on and .","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49241084","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-04-15DOI: 10.4236/OJDM.2021.112003
A. Elrokh, S. Nada, E. El-Shafey
A graph is said to be cordial if it has 0 - 1 labeling which satisfies particular conditions. In this paper, we construct the corona between paths and second power of fan graphs and explain the necessary and sufficient conditions for this construction to be cordial.
{"title":"Cordial Labeling of Corona Product of Path Graph and Second Power of Fan Graph","authors":"A. Elrokh, S. Nada, E. El-Shafey","doi":"10.4236/OJDM.2021.112003","DOIUrl":"https://doi.org/10.4236/OJDM.2021.112003","url":null,"abstract":"A graph is said to be cordial if it has 0 - 1 labeling which satisfies particular conditions. In this paper, we construct the corona between paths and second power of fan graphs and explain the necessary and sufficient conditions for this construction to be cordial.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43240719","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-26DOI: 10.4236/ojdm.2021.113006
Viktoriya Bardenova, Vincent Ciarcia, Erik Insko
In this paper we analyze and model three open problems posed by Harris, Insko, Prieto-Langarica, Stoisavljevic, and Sullivan in 2020 concerning the tipsy cop and robber game on graphs. The three different scenarios we model account for different biological scenarios. The first scenario is when the cop and robber have a consistent tipsiness level though the duration of the game; the second is when the cop and robber sober up as a function of time; the third is when the cop and robber sober up as a function of the distance between them. Using Markov chains to model each scenario we calculate the probability of a game persisting through $mathbf{M}$ rounds of the game and the expected game length given different starting positions and tipsiness levels for the cop and robber.
{"title":"Markov Models for the Tipsy Cop and Robber Game on Graph","authors":"Viktoriya Bardenova, Vincent Ciarcia, Erik Insko","doi":"10.4236/ojdm.2021.113006","DOIUrl":"https://doi.org/10.4236/ojdm.2021.113006","url":null,"abstract":"In this paper we analyze and model three open problems posed by Harris, Insko, Prieto-Langarica, Stoisavljevic, and Sullivan in 2020 concerning the tipsy cop and robber game on graphs. The three different scenarios we model account for different biological scenarios. The first scenario is when the cop and robber have a consistent tipsiness level though the duration of the game; the second is when the cop and robber sober up as a function of time; the third is when the cop and robber sober up as a function of the distance between them. Using Markov chains to model each scenario we calculate the probability of a game persisting through $mathbf{M}$ rounds of the game and the expected game length given different starting positions and tipsiness levels for the cop and robber.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45042560","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.4236/OJDM.2021.113005
Roger K. Yeh
{"title":"A Note on n-Set Distance-Labelings of Graphs","authors":"Roger K. Yeh","doi":"10.4236/OJDM.2021.113005","DOIUrl":"https://doi.org/10.4236/OJDM.2021.113005","url":null,"abstract":"","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"11 1","pages":"55-60"},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70627106","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.4236/ojdm.2021.111001
Roxanne Back, A. Castano, Rachel Galindo, J. Finocchiaro
In the field of design theory, the most well-known design is a Steiner Triple System. In general, a G-design on H is an edge-disjoint decomposition of H into isomorphic copies of G. In a Steiner Triple system, a complete graph is decomposed into triangles. In this paper we let H be a complete graph with a hole and G be a complete graph on four vertices minus one edge, also referred to as a . A complete graph with a hole, , consists of a complete graph on d vertices, , and a set of independent vertices of size v, V, where each vertex in V is adjacent to each vertex in . When d is even, we give two constructions for the decomposition of a complete graph with a hole into copies of : the Alpha-Delta Construction, and the Alpha-Beta-Delta Construction. By restricting d and v so that , we are able to resolve both of these cases for a subset of using difference methods and 1-factors.
{"title":"A Decomposition of a Complete Graph with a Hole","authors":"Roxanne Back, A. Castano, Rachel Galindo, J. Finocchiaro","doi":"10.4236/ojdm.2021.111001","DOIUrl":"https://doi.org/10.4236/ojdm.2021.111001","url":null,"abstract":"In the field of design theory, the most well-known design is a Steiner Triple System. In general, a G-design on H is an edge-disjoint decomposition of H into isomorphic copies of G. In a Steiner Triple system, a complete graph is decomposed into triangles. In this paper we let H be a complete graph with a hole and G be a complete graph on four vertices minus one edge, also referred to as a . A complete graph with a hole, , consists of a complete graph on d vertices, , and a set of independent vertices of size v, V, where each vertex in V is adjacent to each vertex in . When d is even, we give two constructions for the decomposition of a complete graph with a hole into copies of : the Alpha-Delta Construction, and the Alpha-Beta-Delta Construction. By restricting d and v so that , we are able to resolve both of these cases for a subset of using difference methods and 1-factors.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"11 1","pages":"1-12"},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70626997","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.4236/OJDM.2021.111002
N. Stojanović
In this paper, tiling a plane with equilateral semi-regular convex polygons is considered, and, that is, tiling with equilateral polygons of the same type. Tiling a plane with semi-regular polygons depends not only on the type of a semi-regular polygon, but also on its interior angles that join at a node. In relation to the interior angles, semi-regular equilateral polygons with the same or different interior angles can be joined in the nodes. Here, we shall first consider tiling a plane with semi-regular equilateral polygons with 2m-sides. The analysis is performed by determining the set of all integer solutions of the corresponding Diophantine equation in the form of , whereare the non-negative integers which are not equal to zero at the same time, and are the interior angles of a semi-regular equilateral polygon from the characteristic angle. It is shown that of all semi-regular equilateral polygons with 2m-sides, a plane can be tiled only with the semi-regular equilateral quadrilaterals and semi-regular equilateral hexagons. Then, the problem of tiling a plane with semi-regular equilateral quadrilaterals is analyzed in detail, and then the one with semi-regular equilateral hexagons. For these semi-regular polygons, all possible solutions of the corresponding Diophantine equations were analyzed and all nodes were determined, and then the problem for different values of characteristic elements was observed. For some of the observed cases of tiling a plane with these semi-regular polygons, some graphical presentations of tiling constructions are also given.
{"title":"Tiling a Plane with Semi-Regular Equilateral Polygons with 2m-Sides","authors":"N. Stojanović","doi":"10.4236/OJDM.2021.111002","DOIUrl":"https://doi.org/10.4236/OJDM.2021.111002","url":null,"abstract":"In this paper, tiling a plane with equilateral semi-regular convex polygons is considered, and, that is, tiling with equilateral polygons of the same type. Tiling a plane with semi-regular polygons depends not only on the type of a semi-regular polygon, but also on its interior angles that join at a node. In relation to the interior angles, semi-regular equilateral polygons with the same or different interior angles can be joined in the nodes. Here, we shall first consider tiling a plane with semi-regular equilateral polygons with 2m-sides. The analysis is performed by determining the set of all integer solutions of the corresponding Diophantine equation in the form of , whereare the non-negative integers which are not equal to zero at the same time, and are the interior angles of a semi-regular equilateral polygon from the characteristic angle. It is shown that of all semi-regular equilateral polygons with 2m-sides, a plane can be tiled only with the semi-regular equilateral quadrilaterals and semi-regular equilateral hexagons. Then, the problem of tiling a plane with semi-regular equilateral quadrilaterals is analyzed in detail, and then the one with semi-regular equilateral hexagons. For these semi-regular polygons, all possible solutions of the corresponding Diophantine equations were analyzed and all nodes were determined, and then the problem for different values of characteristic elements was observed. For some of the observed cases of tiling a plane with these semi-regular polygons, some graphical presentations of tiling constructions are also given.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"11 1","pages":"13-30"},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70627013","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.4236/OJDM.2021.112004
Matthew Davis, Shiv Gupta
In this paper using elementary Galois Theory, we give a detailed explanation of the calculation of the radical expression for π which was first dis-cussed by Vandermonde decades before Galois and we point out and correct a minor correction in his work which was also observed by Lagrange.
{"title":"A Small Correction to a Paper of Vandermonde","authors":"Matthew Davis, Shiv Gupta","doi":"10.4236/OJDM.2021.112004","DOIUrl":"https://doi.org/10.4236/OJDM.2021.112004","url":null,"abstract":"In this paper using elementary Galois Theory, we give a detailed explanation of the calculation of the radical expression for π which was first dis-cussed by Vandermonde decades before Galois and we point out and correct a minor correction in his work which was also observed by Lagrange.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70627070","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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