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This page is an electronic archive of the preprints of Statistics & Mathematics Unit, Indian Statistical Institute, Bangalore Centre. | |||
Preprints (2016)Geometry of the Multiplicatively Closed Sets Generated by at most two Elements and Arbitrarily Large Gaps [Full article pdf] Abstract: We prove that the multiplicatively closed subset generated by at most two elements in the set of natural numbers $\mathbb{N}$ has arbitrarily large gaps by explicitly constructing large integer intervals which do not contain any element from the multiplicatively closed set. We also give a criterion by using a geometric correspondence between maximal singly generated multiplicatively closed sets and points of the space $\mathbb{PF}^{\infty}_{\mathbb{Q}\geq 0}$ as to when a finitely generated multiplicatively closed set gives rise to a doubly multiplicatively closed line. In the appendix section we discuss another constructive proof for arbitrarily large gap intervals where the prime factorization is not known for the right end-point unlike the constructive proof of the main result of the article in the case of multiplicatively closed set $\{p_1^ip_2^j\mid i,j \in \mathbb{N}\cup\{0\}\}$ with $p_1< p_2,Log_{p_1}(p_2)$ irrational for which the prime factorization is known for both the end-points of the gap interval via the stabilization sequence of the irrational $\frac{1}{Log_{p_1}(p_2)}$. March 10, 2016 isibc/ms/2016/1 On the Surjectivity of Certain Maps [Full article pdf] Abstract: We prove in this article the surjectivity of three maps. We prove the surjectivity of the chinese remainder reduction map associated to projective space of an ideal with a given factorization into ideals whose radicals are pairwise distinct maximal ideals. We prove the surjectivity of the reduction map of the strong approximation type for a ring quotiented by an ideal which satisfies unital set condition. We prove for a dedekind domain, for $k \geq 2$, the map from k-dimensional special linear group to the product of projective spaces of $k-$mutually comaximal ideals associating the $k-$rows or $k-$columns is surjective. Finally this article leads to three interesting questions mentioned in the introduction section. March 14, 2016 isibc/ms/2016/2 | |||
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Preprint Series: Delhi Centre; Kolkata Centre. | |||
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