منابع مشابه
New results on nonsingular power LCM matrices
Let e and n be positive integers and S = {x1, . . . , xn} be a set of n distinct positive integers. The n × n matrix having eth power [xi, xj ] of the least common multiple of xi and xj as its (i, j)-entry is called the eth power least common multiple (LCM) matrix on S, denoted by ([S]). The set S is said to be gcd closed (respectively, lcm closed) if (xi, xj) ∈ S (respectively, [xi, xj ] ∈ S) ...
متن کاملNotes on the divisibility of GCD and LCM Matrices
Let S = {x1,x2, . . . ,xn} be a set of positive integers, and let f be an arithmetical function. The matrices (S) f = [ f (gcd(xi,xj))] and [S] f = [ f (lcm[xi,xj])] are referred to as the greatest common divisor (GCD) and the least common multiple (LCM) matrices on S with respect to f , respectively. In this paper, we assume that the elements of the matrices (S) f and [S] f are integers and st...
متن کاملEla New Results on Nonsingular Power Lcm Matrices
Let e and n be positive integers and S = {x1, . . . , xn} be a set of n distinct positive integers. The n × n matrix having eth power [xi, xj ] of the least common multiple of xi and xj as its (i, j)-entry is called the eth power least common multiple (LCM) matrix on S, denoted by ([S]). The set S is said to be gcd closed (respectively, lcm closed) if (xi, xj) ∈ S (respectively, [xi, xj ] ∈ S) ...
متن کاملNotes on the Milnor Conjectures
These are four lectures concerning the two Milnor conjectures and their proofs. Voevodsky’s proof of the norm residue symbol conjecture—which is now eight years old—came with an explosion of ideas. The aim of these notes is to help make this explosion a little more accessible to topologists. My intention here is not to give a completely rigorous presentation of this material, but just to discus...
متن کاملNotes on Conjectures of Zhi-wei Sun
Conjecture 1 (1988-04-23). Let a0, . . . , an−1, b0, . . . , bn−1 ∈ N. Suppose that ∑n−1 r=0 are 2πir/n = ∑n−1 r=0 bre , and that the least prime divisor p = p(n) of n is greater than |{0 6 r < n : ar 6= 0}| and |{0 6 r < n : br 6= 0}|. Then ar = br for all r ∈ R(n) = {0, 1, . . . , n− 1}. Remark 1. M. Newman [Math. Ann. 1971] showed that if c0, . . . , cn−1 ∈ Q, ∑n−1 r=0 cre 2πir/n = 0 and |{0...
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ژورنال
عنوان ژورنال: Journal of Algebra
سال: 2007
ISSN: 0021-8693
DOI: 10.1016/j.jalgebra.2007.05.005