We introduce and study some concepts of sensitivity via Furstenberg families. A dynamical system X, f is F-sensitive if there exists a positive ε such that for every x ∈ X and every open neighborhood U of x there exists y ∈ U such that the pair x, y is not F-ε-asymptotic; that is, the time set {n : d f x , f y > ε} belongs to F, where F is a Furstenberg family. A dynamical system X, f is F1, F2...

Lemma 17 Suppose that Θ(ā, h̄ ≥ θ > 0 for all ā > 0. (i) For any initial conditions F (w) and F ′(w) the HJB equation (3) has a unique solution F in any interval [w, w̄] ⊂ R. (ii) F is twice continuously differentiable and (F, F ′) depends continuously on the initial conditions. (iii) F ′ is monotone with respect to F ′(w). That is, if F1 and F2 are two solutions of the HJB equation in an interva...

Bent functions (Dillon 1974; Rothaus 1976) are extremal objects in combinatorics and Boolean function theory. They have been studied for about 40 years; even more, under the name of difference sets in elementary Abelian 2-groups. The motivation for the study of these particular difference sets is mainly cryptographic (but bent functions play also a role in coding theory and sequences; and as di...

For n ∈ N consider the n-dimensional hypercube as equal to the vector space F2 , where F2 is the field of size two. Endow F2 with the Hamming metric, i.e., with the metric induced by the `1 norm when one identifies F2 with {0, 1} ⊆ R. Denote by `2 (F2 ) the n-fold Pythagorean product of F2 , i.e., the space of all x = (x1, . . . , xn) ∈ ∏n j=1 F n 2 , equipped with the metric ∀x, y ∈ n ∏ j=1 F2...

Let F be an n-uniform hypergraph on 2n vertices. Suppose that |F1∩F2∩F3∩F4| ≥ 2 and |F1∪F2∪F3∪ F4| ≤ n−2 holds for all F1,F2,F3,F4 ∈F . We prove that the size of F is at most (2n−4 n−2 ) for n sufficiently large.

Let F be an n-uniform hypergraph on 2n vertices. Suppose that |F1 ∩ F2 ∩ F3| ≥ 1 and |F1 ∪ F2 ∪ F3| ≤ 2n− 1 holds for all F1, F2, F3 ∈ F . We prove that the size of F is at most ( 2n−2 n−1 ) .

Let V be an n-dimensional vector space over the finite field Fq, and [Vk] denote family of all k-dimensional subspaces V. The families F1⊆[Vk1],F2⊆[Vk2],…,Fr⊆[Vkr] are said to r-cross t-intersecting if dim⁡(F1∩F2∩⋯∩Fr)≥t for Fi∈Fi,1≤i≤r. F1, F2,…,Fr non-trivial dim⁡(∩1≤i≤r∩F∈FiF)<t. In this paper, we first determine structure with maximum product their sizes. As a consequence, partially prove o...

For any time bound f , let H(f) denote the hierarchy conjecture which means that the restriction of the numbers of work tapes of deterministic Turing machines to b ∈ N generates an infinite hierarchy of proper subclasses DTIMEb(f) ⊂ DTIME(f). We show that H(f) implies separations of deterministic from nondeterministic time classes. H(f) follows from the gap property, G(f), which says that there...