Pub Date : 2023-06-06DOI: 10.1007/s40574-023-00368-6
Nicolò Cangiotti, Marco Capolli, Mattia Sensi, Sara Sottile
In this survey, we propose an overview on Lyapunov functions for a variety of compartmental models in epidemiology. We exhibit the most widely employed functions, and provide a commentary on their use. Our aim is to provide a comprehensive starting point to readers who are attempting to prove global stability of systems of ODEs. The focus is on mathematical epidemiology, however some of the functions and strategies presented in this paper can be adapted to a wider variety of models, such as prey-predator or rumor spreading.
{"title":"A survey on Lyapunov functions for epidemic compartmental models.","authors":"Nicolò Cangiotti, Marco Capolli, Mattia Sensi, Sara Sottile","doi":"10.1007/s40574-023-00368-6","DOIUrl":"10.1007/s40574-023-00368-6","url":null,"abstract":"<p><p>In this survey, we propose an overview on Lyapunov functions for a variety of compartmental models in epidemiology. We exhibit the most widely employed functions, and provide a commentary on their use. Our aim is to provide a comprehensive starting point to readers who are attempting to prove global stability of systems of ODEs. The focus is on mathematical epidemiology, however some of the functions and strategies presented in this paper can be adapted to a wider variety of models, such as prey-predator or rumor spreading.</p>","PeriodicalId":72440,"journal":{"name":"Bollettino della Unione matematica italiana (2008)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10242238/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"10073739","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}
Pub Date : 2023-01-01Epub Date: 2023-02-28DOI: 10.1007/s40574-023-00349-9
Tanja I Schindler, Roland Zweimüller
We study, from the viewpoint of metrical number theory and (infinite) ergodic theory, the probabilistic laws governing the occurrence of prime numbers as digits in continued fraction expansions of real numbers.
我们从度量数论和（无限）遍历理论的角度研究了实数的连分式展开中素数作为数字出现的概率规律。
{"title":"Prime numbers in typical continued fraction expansions.","authors":"Tanja I Schindler, Roland Zweimüller","doi":"10.1007/s40574-023-00349-9","DOIUrl":"10.1007/s40574-023-00349-9","url":null,"abstract":"<p><p>We study, from the viewpoint of metrical number theory and (infinite) ergodic theory, the probabilistic laws governing the occurrence of prime numbers as digits in continued fraction expansions of real numbers.</p>","PeriodicalId":72440,"journal":{"name":"Bollettino della Unione matematica italiana (2008)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10203034/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"9528295","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}
Pub Date : 2021-06-28DOI: 10.1007/s40574-021-00297-2
Michael Lönne
{"title":"documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$pi _1$$end{document}π1 of Miranda moduli spaces of elliptic surfaces","authors":"Michael Lönne","doi":"10.1007/s40574-021-00297-2","DOIUrl":"https://doi.org/10.1007/s40574-021-00297-2","url":null,"abstract":"","PeriodicalId":72440,"journal":{"name":"Bollettino della Unione matematica italiana (2008)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76231870","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 : 2021-01-01Epub Date: 2021-02-22DOI: 10.1007/s40574-021-00279-4
Piermarco Cannarsa, Wei Cheng
The singular set of a viscosity solution to a Hamilton-Jacobi equation is known to propagate, from any noncritical singular point, along singular characteristics which are curves satisfying certain differential inclusions. In the literature, different notions of singular characteristics were introduced. However, a general uniqueness criterion for singular characteristics, not restricted to mechanical systems or problems in one space dimension, is missing at the moment. In this paper, we prove that, for a Tonelli Hamiltonian on , two different notions of singular characteristics coincide up to a bi-Lipschitz reparameterization. As a significant consequence, we obtain a uniqueness result for the class of singular characteristics that was introduced by Khanin and Sobolevski in the paper [On dynamics of Lagrangian trajectories for Hamilton-Jacobi equations. Arch. Ration. Mech. Anal., 219(2):861-885, 2016].
{"title":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">Local singular characteristics on <ns0:math> <ns0:msup><ns0:mrow><ns0:mi>R</ns0:mi></ns0:mrow> <ns0:mn>2</ns0:mn></ns0:msup></ns0:math>.","authors":"Piermarco Cannarsa, Wei Cheng","doi":"10.1007/s40574-021-00279-4","DOIUrl":"https://doi.org/10.1007/s40574-021-00279-4","url":null,"abstract":"<p><p>The singular set of a viscosity solution to a Hamilton-Jacobi equation is known to propagate, from any noncritical singular point, along singular characteristics which are curves satisfying certain differential inclusions. In the literature, different notions of singular characteristics were introduced. However, a general uniqueness criterion for singular characteristics, not restricted to mechanical systems or problems in one space dimension, is missing at the moment. In this paper, we prove that, for a Tonelli Hamiltonian on <math> <msup><mrow><mi>R</mi></mrow> <mn>2</mn></msup> </math> , two different notions of singular characteristics coincide up to a bi-Lipschitz reparameterization. As a significant consequence, we obtain a uniqueness result for the class of singular characteristics that was introduced by Khanin and Sobolevski in the paper [On dynamics of Lagrangian trajectories for Hamilton-Jacobi equations. <i>Arch. Ration. Mech. Anal.</i>, 219(2):861-885, 2016].</p>","PeriodicalId":72440,"journal":{"name":"Bollettino della Unione matematica italiana (2008)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40574-021-00279-4","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"25415646","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}
Pub Date : 2021-01-01Epub Date: 2021-06-28DOI: 10.1007/s40574-021-00301-9
Alfred Geroldinger, Florian Kainrath
Every Krull monoid has a transfer homomorphism onto a monoid of zero-sum sequences over a subset of its class group. This transfer homomorphism is a crucial tool for studying the arithmetic of Krull monoids. In the present paper, we strengthen and refine this tool for Krull monoids with finitely generated class group.
{"title":"On transfer homomorphisms of Krull monoids.","authors":"Alfred Geroldinger, Florian Kainrath","doi":"10.1007/s40574-021-00301-9","DOIUrl":"https://doi.org/10.1007/s40574-021-00301-9","url":null,"abstract":"<p><p>Every Krull monoid has a transfer homomorphism onto a monoid of zero-sum sequences over a subset of its class group. This transfer homomorphism is a crucial tool for studying the arithmetic of Krull monoids. In the present paper, we strengthen and refine this tool for Krull monoids with finitely generated class group.</p>","PeriodicalId":72440,"journal":{"name":"Bollettino della Unione matematica italiana (2008)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40574-021-00301-9","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"39732059","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}
Pub Date : 2020-01-01Epub Date: 2020-08-03DOI: 10.1007/s40574-020-00252-7
Primo Brandi, Rita Ceppitelli, Anna Salvadori
We illustrate a suitable adaptation and modification of classical epidemic evolution models that proves helpful in the study of Covid-19 spread in Italy.
我们举例说明了经典流行病进化模型的适当适应和修改，证明有助于研究Covid-19在意大利的传播。
{"title":"Epidemic evolution models to the test of Covid-19.","authors":"Primo Brandi, Rita Ceppitelli, Anna Salvadori","doi":"10.1007/s40574-020-00252-7","DOIUrl":"https://doi.org/10.1007/s40574-020-00252-7","url":null,"abstract":"<p><p>We illustrate a suitable adaptation and modification of classical epidemic evolution models that proves helpful in the study of Covid-19 spread in Italy.</p>","PeriodicalId":72440,"journal":{"name":"Bollettino della Unione matematica italiana (2008)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40574-020-00252-7","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"38399009","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}
Pub Date : 2019-01-01Epub Date: 2019-02-09DOI: 10.1007/s40574-019-00194-9
Alfred Geroldinger, Qinghai Zhong
It is well-known that a C-monoid is completely integrally closed if and only if its reduced class semigroup is a group and if this holds, then the C-monoid is a Krull monoid and the reduced class semigroup coincides with the usual class group of Krull monoids. We prove that a C-monoid is seminormal if and only if its reduced class semigroup is a union of groups. Based on this characterization we establish a criterion (in terms of the class semigroup) when seminormal C-monoids are half-factorial.
{"title":"A characterization of seminormal C-monoids.","authors":"Alfred Geroldinger, Qinghai Zhong","doi":"10.1007/s40574-019-00194-9","DOIUrl":"https://doi.org/10.1007/s40574-019-00194-9","url":null,"abstract":"<p><p>It is well-known that a C-monoid is completely integrally closed if and only if its reduced class semigroup is a group and if this holds, then the C-monoid is a Krull monoid and the reduced class semigroup coincides with the usual class group of Krull monoids. We prove that a C-monoid is seminormal if and only if its reduced class semigroup is a union of groups. Based on this characterization we establish a criterion (in terms of the class semigroup) when seminormal C-monoids are half-factorial.</p>","PeriodicalId":72440,"journal":{"name":"Bollettino della Unione matematica italiana (2008)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40574-019-00194-9","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"37773499","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}
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