Displaying similar documents to “Intertwining numbers; the n -rowed shapes”

Singular eigenvalue problems for second order linear ordinary differential equations

Árpád Elbert, Takaŝi Kusano, Manabu Naito (1998)

Archivum Mathematicum

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We consider linear differential equations of the form ( p ( t ) x ' ) ' + λ q ( t ) x = 0 ( p ( t ) > 0 , q ( t ) > 0 ) ( A ) on an infinite interval [ a , ∞ ) and study the problem of finding those values of λ for which () has principal solutions x 0 ( t ; λ ) vanishing at t = a . This problem may well be called a singular eigenvalue problem, since requiring x 0 ( t ; λ ) to be a principal solution can be considered as a boundary condition at t = ∞ . Similarly to the regular eigenvalue problems for () on compact intervals, we can prove a theorem asserting that there exists a sequence { λ n } of eigenvalues...

An improvement of an inequality of Fiedler leading to a new conjecture on nonnegative matrices

Assaf Goldberger, Neumann, Michael (2004)

Czechoslovak Mathematical Journal

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Suppose that A is an n × n nonnegative matrix whose eigenvalues are λ = ρ ( A ) , λ 2 , ... , λ n . Fiedler and others have shown that det ( λ I - A ) ≤ λ n - ρ n , for all λ > ρ , with equality for any such λ if and only if A is the simple cycle matrix. Let a i be the signed sum of the determinants of the principal submatrices of A of order i × i , i = 1 , ... , n - 1 . We use similar techniques to Fiedler to show that Fiedler’s inequality can be strengthened to: det ( λ I - A ) + ∑ i = 1 n - 1 ρ n - 2 i | a i | ( λ - ρ ) i ≤ λ n - ρ n , for all λ ≥ ρ . We use this inequality to derive the inequality that: ∏ 2 n ( ρ - λ i ) ≤ ρ n - 2 ∑ i = 2 n ( ρ - λ i ) . In the spirit of a celebrated conjecture...