C P ANIL KUMAR
Articles written in Proceedings – Mathematical Sciences
Volume 127 Issue 2 April 2017 pp 295-321 Research Article
Consider a discrete valuation ring $R$ whose residue field is finite of cardinality at least 3. For a finite torsion module, we consider transitive subsets $O$ under the action of the automorphism group of the module. We prove that the associated permutation representation on the complex vector space $C[O]$ is multiplicity free. This is achieved by obtaining a complete description of the transitive subsets of $O$ × $O$ under the diagonal action of the automorphism group.
Volume 129 Issue 3 June 2019 Article ID 0042 Research Article
Onthe gaps inmultiplicatively closed sets generated by atmost two elements
We prove in the the main theorem, Theorem 3.2, that the multiplicatively closed subset of natural numbers, generated by two elements $1$ < $p_{1}$ < $p_{2}$ with $\alpha=\frac{\log\ p_{1}}{\log\ p_{2}}$ irrational, has arbitrarily large gaps by explicitly constructing large integer intervals, with known factorization for the endpoints in terms of generators $p_{1},p_{2}$ obtained from the stabilization sequence of the irrational $\alpha$ (Definition 3.1). Example 5.6 is also illustrated. In the Appendix, for a finitely generated multiplicatively closed subset of natural numbers, we mention another constructive proof (refer to Theorem A.1}) for arbitrarily large gap intervals, where the factorization of the right endpoint is not known in terms of generators unlike in the constructive proof of the main result. The suggested general Question 1.1 remains still open
Volume 130 All articles Published: 20 June 2020 Article ID 0037 Research Article
For two natural numbers $1$ < $p1$ < $p2$, with $\alpha = \frac{log(p1)} {log(p2)}$ irrational, we describe in the Main Theorem $\Omega$ and in Note 1.5, the factorization of two adjacent numbers in the multiplicatively closed subset $S = \{p^{i}_{1} p^{j}_{2}\,|\, i, j \in \mathbb{N}\cup\{0\}\}$ using primary and secondary convergents of $\alpha$. This suggests the general Question 1.2 for more than two generators which is still open.
Volume 131, 2021
All articles
Continuous Article Publishing mode
Click here for Editorial Note on CAP Mode
© 2021-2022 Indian Academy of Sciences, Bengaluru.