All a b c d e f g h i j k l m n o p q r s t u v w x y z Other

Previous Page 3 Next

Displaying 41 – 60 of 158

Showing per page

A note on minimal zero-sum sequences over ℤ

Papa A. Sissokho (2014)

Acta Arithmetica

A zero-sum sequence over ℤ is a sequence with terms in ℤ that sum to 0. It is called minimal if it does not contain a proper zero-sum subsequence. Consider a minimal zero-sum sequence over ℤ with positive terms a ₁ , . . . , a h and negative terms b ₁ , . . . , b k . We prove that h ≤ ⌊σ⁺/k⌋ and k ≤ ⌊σ⁺/h⌋, where σ ⁺ = ∑ i = 1 h a i = - ∑ j = 1 k b j . These bounds are tight and improve upon previous results. We also show a natural partial order structure on the collection of all minimal zero-sum sequences over the set i∈ ℤ : -n ≤ i ≤ n for any positive integer n.

A quantitative aspect of non-unique factorizations: the Narkiewicz constants II

Weidong Gao, Yuanlin Li, Jiangtao Peng (2011)

Colloquium Mathematicae

Let K be an algebraic number field with non-trivial class group G and K be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let F k ( x ) denote the number of non-zero principal ideals a K with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that F k ( x ) behaves, for x → ∞, asymptotically like x ( l o g x ) 1 / | G | - 1 ( l o g l o g x ) k ( G ) . In this article, it is proved that for every prime p, ₁ ( C p ⊕ C p ) = 2 p , and it is also proved that ₁ ( C m p ⊕ C m p ) = 2 m p if ₁ ( C m ⊕ C m ) = 2 m and m is large enough. In particular, it is shown that for...

Currently displaying 41 – 60 of 158