In this note we prove that bent functions can be precisely characterized in terms of a special class of strongly regular graphs, thus providing a positive answer to a question raised in the paper Spectral Analysis of Boolean Functions as a Graph Eigenvalue Problem. (After publication of the above paper by the rst two authors, the third sent them a remark showing that bent functions are exactly ...

In this paper, we completely determine the connectivity of every infinite circulant digraphs and prove that almost all infinite circulant digraphs are infinitely strongly connected and therefore have both oneand two-way infinite Hamiltonian paths.

One of our main results is the following convergence theorem for one-parameter nonexpansive semigroups: let C be a bounded closed convex subset of a Hilbert space E, and let {T(t) : t ∈ R+} be a strongly continuous semigroup of nonexpansive mappings on C. Fix u ∈ C and t1, t2 ∈ R+ with t1 < t2. Define a sequence {xn} in C by xn = (1− αn)/ (t2 − t1) ∫ t2 t1 T(s)xnds+ αnu for n ∈N, where {αn} is ...

We give a randomized (Las-Vegas) parallel algorithm for computing strongly connected components of a graph with n vertices andm edges. The runtime is dominated byO(log n) multi-source parallel reachability queries; i.e. O(log n) calls to a subroutine that computes the union of the descendants of a given set of vertices in a given digraph. Our algorithm also topologically sorts the strongly conn...

We give novel algorithms for stochastic strongly-convex optimization in the gradient oracle model which return a O( 1 T )-approximate solution after T iterations. The first algorithm is deterministic, and achieves this rate via gradient updates and historical averaging. The second algorithm is randomized, and is based on pure gradient steps with a random step size. This rate of convergence is o...

To a strongly continuous semigroup ? T(t) t0 on a Banach space X we will associate semigroups ? T n (t) t0 on new Banach spaces X n for each n 2 Z. This construction is inspired by the classical Sobolev spaces and, due to its simplicity, of great help in understanding abstract and concrete semigroups. We start with a strongly continuous semigroup ? T(t) t0 on a Banach space X for which we assum...

In this article, we establish a weak stability theorem for Ishikawa iterations in Hilbert spaces. Moreover, strongly demicontractive mapping is presented to illustrate that the iteration of T weakly T-stable but not T-stable. Our results are new and extend several known results.