In this paper, we state the positive real lemma and the strictly positive real lemma (KYP lemma) for non-minimal realization systems. First we show the positive real lemma for stabilizable and observable systems under only the constraint on the regularity of the systems, by using the generalized algebraic Riccati equation. Moreover we show that the solution of the Lyapunov equation in the posit...

Szemerédi’s Regularity Lemma is an important tool for analyzing the structure of dense graphs. There are versions of the Regularity Lemma for sparse graphs, but these only apply when the graph satisfies some local density condition. In this paper, we prove a sparse Regularity Lemma that holds for all graphs. More generally, we give a Regularity Lemma that holds for arbitrary real matrices.

We present a pumping lemma for the class of collapsible pushdown graphs of level 2. This pumping lemma even applies to the ε-contractions of level 2 collapsible pushdown graphs. Our pumping lemma also improves the bounds of Hayashi’s pumping lemma for indexed languages. 1998 ACM Subject Classification F.1.1 Models of Computation

‎In this paper, first we introduce the end of locally finite graphs as an equivalence class of infinite paths in the graph. Then we mention the ends of finitely generated groups using the Cayley graph. It was proved that the number of ends of groups are not depended on the Cayley graph and that the number of ends in the groups is equal to zero, one, two, or infinity. For ...