Let K denote a field. Let d denote a nonnegative integer and consider a sequence p = (θi, θ ∗ i , i = 0...d;φj , φj , j = 1...d) consisting of scalars taken from K. We call p a parameter array whenever: (PA1) θi 6= θj, θ ∗ i 6= θ ∗ j if i 6= j, (0 ≤ i, j ≤ d); (PA2) φi 6= 0, φi 6= 0 (1 ≤ i ≤ d); (PA3) φi = φ1 ∑i−1 h=0 θh−θd−h θ0−θd + (θ i − θ ∗ 0)(θi−1 − θd) (1 ≤ i ≤ d); (PA4) φi = φ1 ∑i−1 h=0 ...

Let K denote a field. Let d denote a nonnegative integer and consider a sequence p = (θi, θ ∗ i , i = 0...d;φj , φj , j = 1...d) consisting of scalars taken from K. We call p a parameter array whenever: (PA1) θi 6= θj, θ ∗ i 6= θ ∗ j if i 6= j, (0 ≤ i, j ≤ d); (PA2) φi 6= 0, φi 6= 0 (1 ≤ i ≤ d); (PA3) φi = φ1 ∑i−1 h=0 θh−θd−h θ0−θd + (θ i − θ ∗ 0)(θi−1 − θd) (1 ≤ i ≤ d); (PA4) φi = φ1 ∑i−1 h=0 ...

Let V denote a vector space with finite positive dimension. We consider a pair of linear transformations A : V → V and A : V → V that satisfy (i) and (ii) below: (i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A is diagonal. (ii) There exists a basis for V with respect to which the matrix representing A is irr...

Let K denote a field, and let V denote a vector space over K with finite positive dimension. We consider a pair of linear transformations A : V → V and A∗ : V → V that satisfy (i), (ii) below: (i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A∗ is diagonal. (ii) There exists a basis for V with respect to which ...

Let K denote a field, and let V denote a vector space over K with finite positive dimension. We consider a pair of linear transformations A : V → V and A∗ : V → V that satisfy (i), (ii) below: (i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A∗ is diagonal. (ii) There exists a basis for V with respect to which ...

This wonderful book, with a foreword by Dame Evelyn Glennie is an addition to literature that considers the relationship of music and people who engage in it with the sense of hearing developed in a different way. Evelyn Glennie talks about listening in terms of sharing ideas through physical means such as facial expression and the speed or intensity of movement. She goes further in stating tha...