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S. Banerjee,S. Basak,M. R. Adhikari 충청수학회 2006 충청수학회지 Vol.19 No.4
The aim of this paper is to endow a monoid structure on the set S of all oriented knots(links) under the operation , called addition of knots. Moreover, we prove that there exists a homomorphism of monoids between (S d , ) to (N, +), where S d is a subset of S with an extra condition and N is the monoid of non negative integers under usual addition.
Banerjee, S.,Basak, S.,Adhikari, M.R. 충청수학회 2006 충청수학회지 Vol.19 No.4
The aim of this paper is to endow a monoid structure on the set S of all oriented knots(links) under the operation ${\biguplus}$, called addition of knots. Moreover, we prove that there exists a homomorphism of monoids between ($S_d,\;{\biguplus}$) to (N, +), where $S_d$ is a subset of S with an extra condition and N is the monoid of non negative integers under usual addition.
Stange, Daniel E.,Koo, B.K.,Huch, M.,Sibbel, G.,Basak, O.,Lyubimova, A.,Kujala, P.,Bartfeld, S.,Koster, J.,Geahlen, Jessica H.,Peters, Peter J.,van Es, Johan H.,van de Wetering, M.,Mills, Jason C.,Cle Cell Press ; MIT Press 2013 Cell Vol.155 No.2
Proliferation of the self-renewing epithelium of the gastric corpus occurs almost exclusively in the isthmus of the glands, from where cells migrate bidirectionally toward pit and base. The isthmus is therefore generally viewed as the stem cell zone. We find that the stem cell marker Troy is expressed at the gland base by a small subpopulation of fully differentiated chief cells. By lineage tracing with a Troy-eGFP-ires-CreERT2 allele, single marked chief cells are shown to generate entirely labeled gastric units over periods of months. This phenomenon accelerates upon tissue damage. Troy<SUP>+</SUP> chief cells can be cultured to generate long-lived gastric organoids. Troy marks a specific subset of chief cells that display plasticity in that they are capable of replenishing entire gastric units, essentially serving as quiescent ''reserve'' stem cells. These observations challenge the notion that stem cell hierarchies represent a ''one-way street.''