Let $Gamma$ be a subgroup of $PSL(2,R)$ generated by three parabolic transformations. The main goal of this paper is to present an algorithm to determine whether or not $Gamma$ is discrete. Historically discreteness algorithms have been considered within several broader mathematical paradigms: the discreteness problem, the construction and deformation of hyperbolic structures on surfaces and notions of automata for groups. Each of these approaches yield equivalent results. The second goal of this paper is to give an exposition of the basic ideas needed to interpret these equivalences, emphasizing related works and future directions of inquiry.
{"title":"A discreteness algorithm for 4-punctured\u0000 sphere groups","authors":"Caleb Ashley","doi":"10.1090/conm/759/15269","DOIUrl":"https://doi.org/10.1090/conm/759/15269","url":null,"abstract":"Let $Gamma$ be a subgroup of $PSL(2,R)$ generated by three parabolic transformations. The main goal of this paper is to present an algorithm to determine whether or not $Gamma$ is discrete. Historically discreteness algorithms have been considered within several broader mathematical paradigms: the discreteness problem, the construction and deformation of hyperbolic structures on surfaces and notions of automata for groups. Each of these approaches yield equivalent results. The second goal of this paper is to give an exposition of the basic ideas needed to interpret these equivalences, emphasizing related works and future directions of inquiry.","PeriodicalId":351002,"journal":{"name":"The Golden Anniversary Celebration of the\n National Association of Mathematicians","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121321503","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}
The Modified Szpiro Conjecture, equivalent to the $abc$ Conjecture, states that for each $epsilon>0$, there are finitely many rational elliptic curves satisfying $N_{E}^{6+epsilon}
{"title":"A constructive proof of Masser’s\u0000 Theorem","authors":"Alexander J. Barrios","doi":"10.1090/conm/759/15265","DOIUrl":"https://doi.org/10.1090/conm/759/15265","url":null,"abstract":"The Modified Szpiro Conjecture, equivalent to the $abc$ Conjecture, states that for each $epsilon>0$, there are finitely many rational elliptic curves satisfying $N_{E}^{6+epsilon}<max!left{ leftvert c_{4}^{3}rightvert,c_{6}^{2}right} $ where $c_{4}$ and $c_{6}$ are the invariants associated to a minimal model of $E$ and $N_{E}$ is the conductor of $E$. We say $E$ is a good elliptic curve if $N_{E}^{6}<max!left{ leftvert c_{4}^{3}rightvert,c_{6}^{2}right} $. Masser showed that there are infinitely many good Frey curves. Here we give a constructive proof of this assertion.","PeriodicalId":351002,"journal":{"name":"The Golden Anniversary Celebration of the\n National Association of Mathematicians","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126042897","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}
We present new proofs of eight integral representations of the Catalan numbers. Then, we create analogous integral representations of the Motzkin numbers and obtain new results. Most integral representations of counting sequences found in the literature are proved by using advanced mathematical techniques. All integral representations in this paper are proved by using standard techniques from integral calculus. Thus, we provide a more simplistic approach of proving integral representations of the Catalan and Motzkin numbers.
{"title":"Catalan and Motzkin integral\u0000 representations","authors":"Peter N. McCalla, A. Nkwanta","doi":"10.1090/conm/759/15270","DOIUrl":"https://doi.org/10.1090/conm/759/15270","url":null,"abstract":"We present new proofs of eight integral representations of the Catalan numbers. Then, we create analogous integral representations of the Motzkin numbers and obtain new results. Most integral representations of counting sequences found in the literature are proved by using advanced mathematical techniques. All integral representations in this paper are proved by using standard techniques from integral calculus. Thus, we provide a more simplistic approach of proving integral representations of the Catalan and Motzkin numbers.","PeriodicalId":351002,"journal":{"name":"The Golden Anniversary Celebration of the\n National Association of Mathematicians","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115834994","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}
For a graph G, the set D ⊆V (G) is a porous exponential dominating set if 1 ≤ ∑ d∈D (2) 1−dist(d ,v) for every v ∈ V (G), where dist(d , v) denotes the length of the shortest d v path. The porous exponential dominating number of G, denoted γe (G), is the minimum cardinality of a porous exponential dominating set. For any graph G, a technique is derived to determine a lower bound for γe (G). Specifically for a grid graph H , linear programing is used to sharpen bound found through the lower bound technique. Lower and upper bounds are determined for the porous exponential domination number of the King Grid Kn , the Slant Grid Sn , and the n-dimensional hypercube Qn . AMS 2010 Subject Classification: Primary 05C69; Secondary 90C05
{"title":"A linear programming method for exponential\u0000 domination","authors":"Michael Dairyko, Michael Young","doi":"10.1090/conm/759/15273","DOIUrl":"https://doi.org/10.1090/conm/759/15273","url":null,"abstract":"For a graph G, the set D ⊆V (G) is a porous exponential dominating set if 1 ≤ ∑ d∈D (2) 1−dist(d ,v) for every v ∈ V (G), where dist(d , v) denotes the length of the shortest d v path. The porous exponential dominating number of G, denoted γe (G), is the minimum cardinality of a porous exponential dominating set. For any graph G, a technique is derived to determine a lower bound for γe (G). Specifically for a grid graph H , linear programing is used to sharpen bound found through the lower bound technique. Lower and upper bounds are determined for the porous exponential domination number of the King Grid Kn , the Slant Grid Sn , and the n-dimensional hypercube Qn . AMS 2010 Subject Classification: Primary 05C69; Secondary 90C05","PeriodicalId":351002,"journal":{"name":"The Golden Anniversary Celebration of the\n National Association of Mathematicians","volume":"81 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128221269","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}
{"title":"An overview of mathematical modeling of\u0000 geometric optics problems involving\u0000 refraction","authors":"H. Mawi","doi":"10.1090/conm/759/15263","DOIUrl":"https://doi.org/10.1090/conm/759/15263","url":null,"abstract":"","PeriodicalId":351002,"journal":{"name":"The Golden Anniversary Celebration of the\n National Association of Mathematicians","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127545166","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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