In this paper, some properties of the periodic shadowing are presented. It is shown that a homeomorphism has the periodic shadowing property if and only if so does every lift of it to the universal covering space. Also, it is proved that continuous mappings on a compact metric space with the periodic shadowing and the average shadowing property also have the shadowing property and then are chao...

Let N and M be von Neumann algebras. It is proved that L p (N) does not Banach embed in L p (M) for N infinite, M finite, 1 ≤ p < 2. The following considerably stronger result is obtained (which implies this, since the Schatten p-class Cp embeds in L p (N) for N infinite). Theorem. Let 1 ≤ p < 2 and let X be a Banach space with a spanning set (x ij) so that for some C ≥ 1, (i) any row or column...

We investigate Jarńık’s points for a real function f defined in R, i.e. points x for which lim apy→x |(f(y)− f(x))/(y − x)| = +∞. In 1970, Berman has proved that the set Jf of all Jarńık’s points for a path f of the one-dimensional Brownian motion is the whole R almost surely. We give a simple explicit construction of a continuous function f with Jf = R. The main result of our paper says that f...

competition is against monopoly and intellectual property creates monopoly for intellectual property owner. this article had been paid cooperation between apparently contradictory policies. exhaustion, parallel imports, compulsory license, intellectual property exploitation contract that coincide with competition are methods of cooperation between intellectual property and competition law. comp...

In [kal89], Kaltofen proved the remarkable fact that multivariate polynomial factorization can be done efficiently, in randomized polynomial time. Still, more than twenty years after Kaltofen’s work, many questions remain unanswered regarding the complexity aspects of polynomial factorization, such as the question of whether factors of polynomials efficiently computed by arithmetic formulas als...

Tom Trotter of Arizona State has settled Gra-ham Brightwell's problem about fractional dimension posed in Open Problems #20. Recall that the fractional dimension dim (P) is the minimum total weight assigned to the linear extensions of P so that each ordered incomparable pair is covered with total weight at least 1. Brightwell asked whether there exists a function f such that every n-element pos...

The problem of dynamic connectivity in graphs has been extensively studied in the cell probe model. The task is to design a data structure that supports addition of edges and checks connectivity between arbitrary pair of vertices. Let w, tq, tu denote the word size, expected query time and worst case update time of a data structure for connectivity on graphs of size n. We provide simplified pro...