Monotone matrix functions of successive orders

Monotone thematic factorizations of matrix functions

We continue the study of the so-called thematic factorizations of admissible very badly approximable matrix functions. These factorizations were introduced by V.V. Peller and N.J. Young for studying superoptimal approximation by bounded analytic matrix functions. Even though thematic indices associated with a thematic factorization of an admissible very badly approximable matrix function are no...

Concavity of Monotone Matrix Functions of Finite Order

Let f : (0; 1) ! R be a monotone matrix function of order n for some arbitrary but xed value of n. We show that f is a matrix concave function of order bn=2c and that kf(A) ? f(B)k kf(jA ? Bj)k for all n-by-n positive semideenite matrices A and B, and all unitarily invariant norms k k. Because f is not assumed to be a monotone matrix function of all orders, Loewner's integral representation of ...

Majorization and Matrix-Monotone Functions in Wireless Communications

Inverted orders for monotone scoring rules

When each of n judges ranks a set A of m objects from best to worst, and s= (~1, . . . ,sm) is a decreasing sequence of real numbers, the collective ranking determined by s orders the objects in A according to their total scores. The total score of x equals sp times the number of judges w’ho rank x in pth place, summed over p. For normalization purposes, let S, denote the set of all decreasing ...

Differentiation of Monotone Functions

It is clear that limy↓x f(y) ≤ limy↓x f(y). Analogously we can define limy↑x f(y) and limy↑x f(y) and we also have limy↑x f(y) ≤ limy↑x f(y). One can verify as in the sequential case that e.g. (1) limy↓x f(y) ≤ A if and only if for all > 0 there exists a δ > 0 such that f(y) < A+ for all y such that 0 < y − x < δ. (2) limy↓x f(y) ≤ A if and only if for all > 0 and δ > 0 there exists an y with 0...