For a word S, let f(S) be the largest integer m such that there are two disjoint identical (scattered) subwords of length m. Let f(n,Σ) = min{f(S) : S is of length n, over alphabet Σ}. Here, it is shown that 2f(n, {0, 1}) = n− o(n) using the regularity lemma for words. In other words, any binary word of length n can be split into two identical subwords (referred to as twins) and, perhaps, a rem...

In this paper we analyze the practical implications of Szemerédi’s regularity lemma in the preservation of metric information contained in large graphs. To this end, we present a heuristic algorithm to find regular partitions. Our experiments show that this method is quite robust to the natural sparsification of proximity graphs. In addition, this robustness can be enforced by graph densification.

We characterize strong F-regularity, a property associated with tight closure, in a large class of rings. A special case of our results is a workable criterion in complete intersection rings. Tight closure is a recently introduced operation linked to a variety of results in commutative algebra. Among the major results achieved with its use are a generalization of the Direct Summand Conjecture f...

Szemerédi’s regularity lemma is a deep result from extremal graph theory which states that every graph can be well-approximated by the union of a constant number of random-like bipartite graphs, called regular pairs. Although the original proof was non-constructive, efficient (i.e., polynomial-time) algorithms have been developed to determine regular partitions for arbitrary graphs. This paper ...

Pontryagin’s maximum principle in its infinite dimensional version provides (separate) necessary and su cient conditions for both time and norm optimality for the system y0 = Ay + u (A the infinitesimal generator of a strongly continuous semigroup); in particular it provides a costate z(t) for every time or norm optimal control ū(t) hitting a target ȳ 2 D(A). This paper shows that for the right...

Let Y = (G, @, I ) be a linearly ordered, commutative group and u@u = max(u, t’) for all u, IJEG. Extend 0, @ in the usual way on matrices over G. An m x n matrix A is said to have strongly linearly independent (SLI) columns, if for some b the system of equations A@x = b has a unique solution. If, moreover, m = n then A is said to be strongly regular (SR). This paper is a survey of results conc...

The regularity lemma of Szemerédi asserts that one can partition every graph into a bounded number of quasi-random bipartite graphs. In some applications however, one would like to have a strong control on how quasi-random these bipartite graphs are. Alon, Fischer, Krivelevich and Szegedy obtained a powerful variant of the regularity lemma, which allows one to have an arbitrary control on this ...

As the main result of this paper it is proved that an interval matrix [Ac −∆, Ac +∆] is strongly regular if and only if the matrix inequality M(I − |I − RAc| − |R|∆) ≥ I has a solution, where M and R are real square matrices and M is nonnegative. Several consequences of this result are drawn.