Generalizations of some irreducibility results by Schur
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Ela Schur Complements of Generally Diagonally Dominant Matrices and a Criterion for Irreducibility of Matrices∗
As is well known, the Schur complements of strictly or irreducibly diagonally dominant matrices are H−matrices; however, the same is not true of generally diagonally dominant matrices. This paper proposes some conditions on the generally diagonally dominant matrix A and the subset α ⊂ {1, 2, . . . , n} so that the Schur complement matrix A/α is an H−matrix. These conditions are then applied to ...
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in which cases either f(x) is irreducible or f(x) is the product of two irreducible polynomials of equal degree. If |an| = n > 1, then for some choice of a1, . . . , an−1 ∈ Z and a0 = ±1, we have that f(x) is reducible. I. Schur (in [8]) obtained this result in the special case that an = ±1. Further results along the nature of Theorem 1 are also discussed in [6]. The purpose of this paper is to...
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An old theorem of Sylvester states that a product of k consecutive positive integers each exceeding k is divisible by a prime greater than k. We shall give a proof of this theorem and apply it to prove a result of Schur on the irreducibility of certain polynomials.
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We introduce a new operation on skew diagrams called composition of transpositions, and use it and a Jacobi-Trudi style formula to derive equalities on skew Schur Q-functions whose indexing shifted skew diagram is an ordinary skew diagram. When this skew diagram is a ribbon, we conjecture necessary and sufficient conditions for equality of ribbon Schur Q-functions. Moreover, we determine all re...
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is irreducible. Irreducibility here and throughout this paper refers to irreducibility over the rationals. Some condition, such as ja0j = janj = 1, on the integers aj is necessary; otherwise, the irreducibility of all polynomials of the form above would imply every polynomial inZ[x] is irreducible (which is clearly not the case). In this paper, we will mainly be interested in relaxing the condi...
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تاریخ انتشار 2009