Pub Date : 2021-01-01DOI: 10.1007/978-981-16-0570-3_4
Masum Billal, S. Riasat
{"title":"Lucas Sequences","authors":"Masum Billal, S. Riasat","doi":"10.1007/978-981-16-0570-3_4","DOIUrl":"https://doi.org/10.1007/978-981-16-0570-3_4","url":null,"abstract":"","PeriodicalId":46195,"journal":{"name":"Journal of Integer Sequences","volume":"57 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80493356","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: 10.1007/978-981-16-0570-3_5
Masum Billal, S. Riasat
{"title":"Lehmer Sequences","authors":"Masum Billal, S. Riasat","doi":"10.1007/978-981-16-0570-3_5","DOIUrl":"https://doi.org/10.1007/978-981-16-0570-3_5","url":null,"abstract":"","PeriodicalId":46195,"journal":{"name":"Journal of Integer Sequences","volume":"10 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85242713","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: 10.1007/978-981-16-0570-3_1
Masum Billal, S. Riasat
{"title":"Preliminaries","authors":"Masum Billal, S. Riasat","doi":"10.1007/978-981-16-0570-3_1","DOIUrl":"https://doi.org/10.1007/978-981-16-0570-3_1","url":null,"abstract":"","PeriodicalId":46195,"journal":{"name":"Journal of Integer Sequences","volume":"25 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80150222","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: 10.1007/978-981-16-0570-3_3
Masum Billal, S. Riasat
{"title":"Divisibility Sequences","authors":"Masum Billal, S. Riasat","doi":"10.1007/978-981-16-0570-3_3","DOIUrl":"https://doi.org/10.1007/978-981-16-0570-3_3","url":null,"abstract":"","PeriodicalId":46195,"journal":{"name":"Journal of Integer Sequences","volume":"32 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91109535","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 : 2020-06-30DOI: 10.1017/9781108552332.010
Masum Billal, S. Riasat
Exercise 1: Reduction between Reasoning Tasks We show that A v B if and only if A u ¬B is not satisfiable. If A v B, then in any model I of the TBox, it holds that the AI ⊆ BI . Hence AI ∩ (∆I BI) = ∅. As (A u ¬B)I = AI ∩ (∆I BI), the interpretation of A u ¬B is empty in any model of the TBox, hence A u ¬B is not satisfiable. Conversely, if A is not a subconcept of B there exists a model I of the TBox in which there exists e ∈ AI such that e 6∈ BI . Hence, e ∈ ¬BI , and by definition of conjunction, e ∈ (A ∧ ¬B)I . We have exhibited a model in which (A ∧ ¬B) has a non empty interpretation, and A ∧ ¬B is thus satisfiable. Thus, in order to decide whether A is a subconcept of B, one can check whether A∧¬B is satisfiable. As satisfiability in EL is trivial (every concept is satisfiable) and subsumption is not, there cannot be a reduction from subsumption to satisfiability.
{"title":"Exercises","authors":"Masum Billal, S. Riasat","doi":"10.1017/9781108552332.010","DOIUrl":"https://doi.org/10.1017/9781108552332.010","url":null,"abstract":"Exercise 1: Reduction between Reasoning Tasks We show that A v B if and only if A u ¬B is not satisfiable. If A v B, then in any model I of the TBox, it holds that the AI ⊆ BI . Hence AI ∩ (∆I BI) = ∅. As (A u ¬B)I = AI ∩ (∆I BI), the interpretation of A u ¬B is empty in any model of the TBox, hence A u ¬B is not satisfiable. Conversely, if A is not a subconcept of B there exists a model I of the TBox in which there exists e ∈ AI such that e 6∈ BI . Hence, e ∈ ¬BI , and by definition of conjunction, e ∈ (A ∧ ¬B)I . We have exhibited a model in which (A ∧ ¬B) has a non empty interpretation, and A ∧ ¬B is thus satisfiable. Thus, in order to decide whether A is a subconcept of B, one can check whether A∧¬B is satisfiable. As satisfiability in EL is trivial (every concept is satisfiable) and subsumption is not, there cannot be a reduction from subsumption to satisfiability.","PeriodicalId":46195,"journal":{"name":"Journal of Integer Sequences","volume":"46 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2020-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88948615","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}
In this paper, we look at long arithmetic progressions on conics. By an arithmetic progression on a curve, we mean the existence of rational points on the curve whose x-coordinates are in arithmetic progression. We revisit arithmetic progressions on the unit circle, constructing 3-term progressions of points in the first quadrant containing an arbitrary rational point on the unit circle. We also provide infinite families of three term progressions on the unit hyperbola, as well as conics ax2 + cy2 = 1 containing arithmetic progressions as long as 8 terms.
本文研究圆锥曲线上的长算术级数。所谓曲线上的算术级数，是指曲线上存在 x 坐标在算术级数中的有理点。我们重温了单位圆上的算术级数，构建了包含单位圆上任意有理点的第一象限中点的三项级数。我们还提供了单位双曲线上三项级数的无穷族，以及包含长达 8 项级数的算术级数的圆锥 ax2 + cy2 = 1。
{"title":"Arithmetic Progressions on Conics.","authors":"Abdoul Aziz Ciss, Dustin Moody","doi":"","DOIUrl":"","url":null,"abstract":"<p><p>In this paper, we look at long arithmetic progressions on conics. By an arithmetic progression on a curve, we mean the existence of rational points on the curve whose <i>x</i>-coordinates are in arithmetic progression. We revisit arithmetic progressions on the unit circle, constructing 3-term progressions of points in the first quadrant containing an arbitrary rational point on the unit circle. We also provide infinite families of three term progressions on the unit hyperbola, as well as conics <i>ax</i><sup>2</sup> + <i>cy</i><sup>2</sup> = 1 containing arithmetic progressions as long as 8 terms.</p>","PeriodicalId":46195,"journal":{"name":"Journal of Integer Sequences","volume":"20 ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5535277/pdf/nihms875076.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"35285253","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this note we present some elementary methods for the summation of certain Euler sums with terms involving 1 + 1=3 + 1=5 + ¢ ¢ ¢ + 1=(2k i 1):
本文给出了若干项为1 + 1=3 + 1=5 +¢¢¢+ 1=(2k i 1)的欧拉和求和的一些初等方法:
{"title":"Evaluations of Some Variant Euler Sums","authors":"Hongwei Chen","doi":"10.1090/clrm/035/17","DOIUrl":"https://doi.org/10.1090/clrm/035/17","url":null,"abstract":"In this note we present some elementary methods for the summation of certain Euler sums with terms involving 1 + 1=3 + 1=5 + ¢ ¢ ¢ + 1=(2k i 1):","PeriodicalId":46195,"journal":{"name":"Journal of Integer Sequences","volume":"9 1","pages":"0-0"},"PeriodicalIF":0.5,"publicationDate":"2006-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60550047","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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