Limit properties of monotone matrix functions

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...

Cryptographic properties of monotone Boolean functions

by ‘+’.Wedenote the (vector) complement byx = (x 1 + 1, . . . , xn + 1), and the (Boolean function) complement by f (x) = f(x) + 1, for f ∈ Bn. If we de ne the union as (f ∨ g)(x) = f(x) + g(x) + f(x)g(x) then the double complement operation, f(x) Ü→ f (x), has the following properties: f (x) ∨ g(x) = fg(x), f ∨ g(x) = f (x)g(x) (recall that fg(x) = f(x)g(x)). If x = (x 1 , . . . , xn) and y = ...

Definitions and Properties of Monotone Functions

On limit cycles of monotone functions with symmetric connection graph

We study the length of the limit cycles of discrete monotone functions with symmetric connection graph. We construct a family of monotone functions such that the limit cycles are of maximum possible length, which is exponential in the number of variables. Furthermore, we prove for the class of monotone functions with more than two states and connection graph equal to a caterpillar that the leng...

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 ...