BOLETIN DE LA SOCIEDAD MATEMATICA MEXICANA最新文献

{"title":"On a variant of Pillai's problem involving <i>S</i>-units and Fibonacci numbers.","authors":"Volker Ziegler","doi":"10.1007/s40590-022-00450-7","DOIUrl":"https://doi.org/10.1007/s40590-022-00450-7","url":null,"abstract":"<p><p>Let us denote by <math><msub><mi>F</mi> <mi>n</mi></msub> </math> the <i>n</i>-th Fibonacci number. In this paper we show that there exist at most finitely many integers <i>c</i> such that the exponential Diophantine equation <math> <mrow><msub><mi>F</mi> <mi>n</mi></msub> <mo>-</mo> <msup><mn>2</mn> <mi>x</mi></msup> <msup><mn>3</mn> <mi>y</mi></msup> <mo>=</mo> <mi>c</mi></mrow> </math> has more than one solution <math> <mrow><mrow><mo>(</mo> <mi>n</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo></mrow> <mo>∈</mo> <msup><mrow><mi>N</mi></mrow> <mn>3</mn></msup> </mrow> </math> with <math><mrow><mi>n</mi> <mo>></mo> <mn>1</mn></mrow> </math> . Moreover, in the case that <math><mrow><mi>c</mi> <mo>></mo> <mn>0</mn></mrow> </math> we find all integers <i>c</i> such that the Diophantine equation has at least three solutions and in the case that <math><mrow><mi>c</mi> <mo><</mo> <mn>0</mn></mrow> </math> we find all integers <i>c</i> such that the Diophantine equation has at least four solutions.</p>","PeriodicalId":45404,"journal":{"name":"BOLETIN DE LA SOCIEDAD MATEMATICA MEXICANA","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9287265/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"40607925","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}

{"title":"Modeling and analysis of a within-host HIV/HTLV-I co-infection.","authors":"A M Elaiw,&nbsp;N H AlShamrani","doi":"10.1007/s40590-021-00330-6","DOIUrl":"https://doi.org/10.1007/s40590-021-00330-6","url":null,"abstract":"<p><p>Human immunodeficiency virus (HIV) and human T-lymphotropic virus type I (HTLV-I) are two retroviruses that attack the <math> <msup><mrow><mtext>CD4</mtext></mrow> <mo>+</mo></msup> </math> T cells and impair their functions. Both HIV and HTLV-I can be transmitted between individuals through direct contact with certain body fluids from infected individuals. Therefore, a person can be co-infected with both viruses. HIV causes acquired immunodeficiency syndrome (AIDS), while HTLV-I is the causative agent for adult T-cell leukemia (ATL) and HTLV-I-associated myelopathy/tropical spastic paraparesis (HAM/TSP). Several mathematical models have been developed in the literature to describe the within-host dynamics of HIV and HTLV-I mono-infections. However, modeling a within-host dynamics of HIV/HTLV-I co-infection has not been involved. The present paper is concerned with the formulation and investigation of a new HIV/HTLV-I co-infection model under the effect of Cytotoxic T lymphocytes (CTLs) immune response. The model describes the interaction between susceptible <math> <msup><mrow><mtext>CD4</mtext></mrow> <mo>+</mo></msup> </math> T cells, silent HIV-infected cells, active HIV-infected cells, silent HTLV-infected cells, Tax-expressing HTLV-infected cells, free HIV particles, HIV-specific CTLs and HTLV-specific CTLs. The HIV can spread by virus-to-cell transmission. On the other side, HTLV-I has two modes of transmission, (i) horizontal transmission via direct cell-to-cell contact through the virological synapse, and (ii) vertical transmission through the mitotic division of Tax-expressing HTLV-infected cells. The well-posedness of the model is established by showing that the solutions of the model are nonnegative and bounded. We define a set of threshold parameters which govern the existence and stability of all equilibria of the model. We explore the global asymptotic stability of all equilibria by utilizing Lyapunov function and Lyapunov-LaSalle asymptotic stability theorem. We have presented numerical simulations to justify the applicability and effectiveness of the theoretical results. In addition, we evaluate the effect of HTLV-I infection on the HIV dynamics and vice versa.</p>","PeriodicalId":45404,"journal":{"name":"BOLETIN DE LA SOCIEDAD MATEMATICA MEXICANA","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40590-021-00330-6","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"25568391","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}

{"title":"On the <i>x</i>-coordinates of Pell equations that are sums of two Padovan numbers.","authors":"Mahadi Ddamulira","doi":"10.1007/s40590-021-00312-8","DOIUrl":"https://doi.org/10.1007/s40590-021-00312-8","url":null,"abstract":"<p><p>Let <math> <msub><mrow><mo>(</mo> <msub><mi>P</mi> <mi>n</mi></msub> <mo>)</mo></mrow> <mrow><mi>n</mi> <mo>≥</mo> <mn>0</mn></mrow> </msub> </math> be the sequence of Padovan numbers defined by <math> <mrow><msub><mi>P</mi> <mn>0</mn></msub> <mo>=</mo> <mn>0</mn></mrow> </math> , <math> <mrow><msub><mi>P</mi> <mn>1</mn></msub> <mo>=</mo> <msub><mi>P</mi> <mn>2</mn></msub> <mo>=</mo> <mn>1</mn></mrow> </math> , and <math> <mrow><msub><mi>P</mi> <mrow><mi>n</mi> <mo>+</mo> <mn>3</mn></mrow> </msub> <mo>=</mo> <msub><mi>P</mi> <mrow><mi>n</mi> <mo>+</mo> <mn>1</mn></mrow> </msub> <mo>+</mo> <msub><mi>P</mi> <mi>n</mi></msub> </mrow> </math> for all <math><mrow><mi>n</mi> <mo>≥</mo> <mn>0</mn></mrow> </math> . In this paper, we find all positive square-free integers <i>d</i> such that the Pell equations <math> <mrow><msup><mi>x</mi> <mn>2</mn></msup> <mo>-</mo> <mi>d</mi> <msup><mi>y</mi> <mn>2</mn></msup> <mo>=</mo> <mi>N</mi></mrow> </math> with <math><mrow><mi>N</mi> <mo>∈</mo> <mo>{</mo> <mo>±</mo> <mn>1</mn> <mo>,</mo> <mo>±</mo> <mn>4</mn> <mo>}</mo></mrow> </math> , have at least two positive integer solutions (<i>x</i>, <i>y</i>) and <math><mrow><mo>(</mo> <msup><mi>x</mi> <mo>'</mo></msup> <mo>,</mo> <msup><mi>y</mi> <mo>'</mo></msup> <mo>)</mo></mrow> </math> such that both <i>x</i> and <math><msup><mi>x</mi> <mo>'</mo></msup> </math> are sums of two Padovan numbers.</p>","PeriodicalId":45404,"journal":{"name":"BOLETIN DE LA SOCIEDAD MATEMATICA MEXICANA","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40590-021-00312-8","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"25452629","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}

{"title":"On a problem of Pillai with Fibonacci numbers and powers of 3.","authors":"Mahadi Ddamulira","doi":"10.1007/s40590-019-00263-1","DOIUrl":"https://doi.org/10.1007/s40590-019-00263-1","url":null,"abstract":"<p><p>Consider the sequence <math> <msub><mrow><mo>{</mo> <msub><mi>F</mi> <mi>n</mi></msub> <mo>}</mo></mrow> <mrow><mi>n</mi> <mo>≥</mo> <mn>0</mn></mrow> </msub> </math> of Fibonacci numbers defined by <math> <mrow><msub><mi>F</mi> <mn>0</mn></msub> <mo>=</mo> <mn>0</mn></mrow> </math> , <math> <mrow><msub><mi>F</mi> <mn>1</mn></msub> <mo>=</mo> <mn>1</mn></mrow> </math> , and <math> <mrow><msub><mi>F</mi> <mrow><mi>n</mi> <mo>+</mo> <mn>2</mn></mrow> </msub> <mo>=</mo> <msub><mi>F</mi> <mrow><mi>n</mi> <mo>+</mo> <mn>1</mn></mrow> </msub> <mo>+</mo> <msub><mi>F</mi> <mi>n</mi></msub> </mrow> </math> for all <math><mrow><mi>n</mi> <mo>≥</mo> <mn>0</mn></mrow> </math> . In this paper, we find all integers <i>c</i> having at least two representations as a difference between a Fibonacci number and a power of 3.</p>","PeriodicalId":45404,"journal":{"name":"BOLETIN DE LA SOCIEDAD MATEMATICA MEXICANA","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40590-019-00263-1","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"38120565","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}

{"title":"Repdigits as sums of three Padovan numbers.","authors":"Mahadi Ddamulira","doi":"10.1007/s40590-019-00269-9","DOIUrl":"https://doi.org/10.1007/s40590-019-00269-9","url":null,"abstract":"<p><p>Let <math> <msub><mrow><mo>{</mo> <msub><mi>P</mi> <mi>n</mi></msub> <mo>}</mo></mrow> <mrow><mi>n</mi> <mo>≥</mo> <mn>0</mn></mrow> </msub> </math> be the sequence of Padovan numbers defined by <math> <mrow><msub><mi>P</mi> <mn>0</mn></msub> <mo>=</mo> <mn>0</mn></mrow> </math> , <math> <mrow><msub><mi>P</mi> <mn>1</mn></msub> <mo>=</mo> <mn>1</mn> <mo>=</mo> <msub><mi>P</mi> <mn>2</mn></msub> </mrow> </math> , and <math> <mrow><msub><mi>P</mi> <mrow><mi>n</mi> <mo>+</mo> <mn>3</mn></mrow> </msub> <mo>=</mo> <msub><mi>P</mi> <mrow><mi>n</mi> <mo>+</mo> <mn>1</mn></mrow> </msub> <mo>+</mo> <msub><mi>P</mi> <mi>n</mi></msub> </mrow> </math> for all <math><mrow><mi>n</mi> <mo>≥</mo> <mn>0</mn></mrow> </math> . In this paper, we find all repdigits in base 10 which can be written as a sum of three Padovan numbers.</p>","PeriodicalId":45404,"journal":{"name":"BOLETIN DE LA SOCIEDAD MATEMATICA MEXICANA","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40590-019-00269-9","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"38054616","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}