This does not make it look any easier to use; it makes it look even more foreboding. However, it is useful for the following reason. To use the pumping lemma to show that L is not regular, you have to show it does not satisfy the pumping lemma. Therefore you have to show that the NEGATION of the conclusion of the lemma is true. If you can intuitively understand how to form the negation of a log...

in this paper, we study the arens regularity properties of module actions and we extend some proposition from baker, dales, lau and others into general situations. we establish some relationships between the topological centers of module actions and factorization properties of them with some results in group algebras. in 1951 arens shows that the second dual of banach algebra endowed with the e...

We prove that for fixed integer D and positive reals α and γ, there exists a constant C0 such that for all p satisfying p(n) ≥ C0/n, the random graph G(n, p) asymptotically almost surely contains a copy of every tree with maximum degree at most D and at most (1− α)n vertices, even after we delete a (1/2− γ)-fraction of the edges incident to each vertex. The proof uses Szemerédi’s regularity lem...

We give a short proof for the following result on the distribution of three-term arithmetic progressions in sparse subsets of Fp : for every α > 0 there exists a constant C = C(α) such that the following holds for all r ≥ Cp and for almost all sets R of size r of Fp . Let A be any subset of R of size at least αr, then A contains a non-trivial three-term arithmetic progression. This is an analog...

This paper is dedicated to the study of viscous compressible barotropic fluids in dimension N ≥ 2. We address the question of the global existence of strong solutions for initial data close to a constant state having critical Besov regularity. First, this article shows the recent results of [7] and [11] with a new proof. Our result relies on a new a priori estimate for the velocity that we deri...

The celebrated Regularity Lemma of Szemerédi asserts that every sufficiently large graph G can be partitioned in such a way that most pairs of the partition sets span -regular subgraphs. In applications, however, the graph G has to be dense and the partition sets are typically very small. If only one -regular pair is needed, a much bigger one can be found, even if the original graph is sparse. ...

Nash-Williams [21] proved that every graph with n vertices and minimum degree n/2 has at least b5n/224c edge-disjoint Hamiltonian cycles. In [20], he raised the question of determining the maximum number of edge-disjoint Hamiltonian cycles, showing an upper bound of b(n+ 4)/8c. Let α(δ, n) = (δ + √ 2δn− n2)/2. Christofides, Kühn, and Osthus [7] proved that for every > 0, every graph G on a suff...

We show, for any positive integer k, that there exists a graph in which any equitable partition of its vertices into k parts has at least ck/ log∗ k pairs of parts which are not -regular, where c, > 0 are absolute constants. This bound is tight up to the constant c and addresses a question of Gowers on the number of irregular pairs in Szemerédi’s regularity lemma. In order to gain some control ...

Let C (3) n denote the 3-uniform tight cycle, that is the hypergraph with vertices v1, . . . , vn and edges v1v2v3, v2v3v4, . . . , vn−1vnv1, vnv1v2. We prove that the smallest integer N = N(n) for which every red-blue coloring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of C (3) n is asymptotically equal to 4n/3 if n is divisible by 3, and 2n...