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Some Characterizations of a Normal Subgroup of a Group and Isotopic Classes of Transversals

Algebra Colloquium 2016年第3期23卷 409-422页

作者： vipul kakkar R.P. Shukla Department of Mathematics University of Allahabad Allahabad 211 002 India

Let G be a group and H be a subgroup of G which is either finite or of finite index in G. In this paper, we give some characterizations for the normality of H in G. As a consequence we get a very short and elementary proof of the main theorem of a paper of Lal and Shukla, which avoids the use of the classification of finite simple groups. Further, we study the isotopy between the transversals in some groups and determine the number of isotopy classes of transversals of a subgroup of order 2 in D2p, the dihedral group of order 2p, where p is an odd prime and the isotopism classes are formed with respect to induced right loop structures.

关键词： right loop normalized right transversal right inverse property isotopy

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Global Dynamics of a Multi-group SEIR Epidemic Model with Infection Age

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Chinese Annals of Mathematics,Series B 2021年第6期42卷 833-860页

作者： Vijay Pal BAJIYA Jai Prakash TRIPATHI vipul kakkar Jinshan WANG Guiquan SUN Department of Mathematics Central University of RajasthanBandar Sindri-305817KishangarhAjmerRajasthanIndia Complex Systems Research Center Shanxi UniversityTaiyuan 030006China Department of Mathematics North University of ChinaTaiyuan 030051China Complex Systems Research Center Shanxi UniversityTaiyuan 030006China.

Consider the heterogeneity(e.g.,heterogeneous social behaviour,heterogeneity due to different geography,contrasting contact patterns and different numbers of sexual partners etc.)of host population,in this paper,the authors propose an infection age multigroup SEIR epidemic *** model system also incorporates the feedback variables,where the infectivity of infected individuals may depend on the infection *** the direction of mathematical analysis of model,the basic reproduction number R_0 has been *** global stability of disease-free equilibrium and endemic equilibrium have been established in the term of R_(0).More precisely,for R_(0)≤1,the disease-free equilibrium is globally asymptotically stable and for R_(0)>1,they establish global stability of endemic equilibrium using some graph theoretic techniques to Lyapunov function *** considering a numerical example,they investigate the effects of infection age and feedback on the prevalence of the *** result shows that feedback parameters have different and even opposite effects on different ***,by choosing an appropriate value of feedback parameters,the disease could be eradicated or maintained at endemic ***,the infection age of infected individuals may also change the behaviour of the disease,global stable to damped oscillations or damped oscillations to global stable.

关键词： Multi-group model Infection age Feedback Graph-theoretic approach Lyapunov function

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