The classical Szegő—Kolmogorov theorem characterizes the weights ω such that family of exponentials with positive integer frequencies spans whole weighted space \({L^2}(\mathbb{T},w)\) on circle. Kolmogorov’s probabilistic interpretation this result connects it possibility to ‘predict precisely future from past’ for stationary stochastic processes discrete time. We discuss problem whether predi...

In analogy of classical Kolmogorov complexity we develop a theory of the algorithmic information in bits contained in any one of continuously many pure quantum states: quantum Kolmogorov complexity. Classical Kolmogorov complexity coincides with the new quantum Kolmogorov complexity restricted to the classical domain. Quantum Kolmogorov complexity is upper bounded and can be effectively approxi...

aw-d is the set of integer m-tuples (a1 ☺ l **t ",) with o 5 al < .*. < am 5 n , ordered by a < b when ai 5 bi for all i . w-s R. Stanley conjectured that L(m,n) is a symmetric chain order for all (m,n) . We verify this by construction for m = 4 . Research supported in part by NSF grant MCS 77-23738 and Office of Naval Research contract ~00014~76-c-0688. Reproduction in whole or in part is perm...

The proposed thesis research addresses two of the main obstacles to building agents that communicate using natural language: the need for richer representations of linguistic constructions that incorporate aspects of conceptual knowledge, context and goals; and the need for a principled approach to the automatic acquisition of such structures from examples. More generally, it explores the idea ...