on the unit interval, where n is any natural number. In order to make this map continuous, we think of it as a map on the 1-torus T = R/Z. This system is very well understood, and it has many closed invariant sets and many invariant probability measures. Indeed, let τ : Σ = {0, . . . , n− 1} → R/Z be the map τ(a1, a2, . . . ) = ∑∞ i=1 n ai. Then any shift invariant probability measure ν on Σ, f...

In this paper, for any non-empty subset C of a pseudo BCI-algebra X, the concept of p-closure of C, denoted by C(pc), is introduced and some related properties are investigated. Applying this concept, a characterization of the minimal elements of X is given. It is proved that C(pc) is the least closed pseudo BCI-ideal of X containing C and K(X) for any ideal C of X...

Let $C$ a non-empty, bounded, closed and convex subset of Banach space $X$, denote by $\ell\_{\infty}(C)$ the $\ell\_{\infty}$-sum $C$. In present paper, using degree nondensifiability (DND), we introduce class $r$-$\Delta$-DND-contraction maps $f: \ell\_{\infty}(C)\rightarrow X$ prove that if $f(\ell\_{\infty}(C))\subset C$ then there is some $x^{}\in with $f(x^{},x^{},\ldots,x^{},\ldots)=x^{\...

We prove that the number of near-coherence classes of non-principal ultrafilters on the natural numbers is either finite or 2c. Moreover, in the latter case the Stone-Čech compactification βω of ω contains a closed subset C consisting of 2c pairwise non-nearly-coherent ultrafilters. We obtain some additional information about such closed sets under certain assumptions involving the cardinal cha...

For each p > n we use local oscillations and doubling measures to give intrinsic characterizations of the restriction of the Sobolev space W 1 p (R n) to an arbitrary closed subset of Rn.

We work in set-theory without choice ZF. Given a closed subset F of [0, 1] which is a bounded subset of l(I) (resp. such that F ⊆ l(I)), we show that the countable axiom of choice for finite subsets of I, (resp. the countable axiom of choiceACN) implies that F is compact. This enhances previous results where ACN (resp. the axiom of Dependent Choices DC) was required. Moreover, if I is linearly ...

a watching system in a graph $g=(v, e)$ is a set $w={omega_{1}, omega_{2}, cdots, omega_{k}}$, where $omega_{i}=(v_{i}, z_{i}), v_{i}in v$ and $z_{i}$ is a subset of closed neighborhood of $v_{i}$ such that the sets $l_{w}(v)={omega_{i}: vin omega_{i}}$ are non-empty and distinct, for any $vin v$. in this paper, we study the watching systems of line graph $k_{n}$ which is called triangular grap...

The factorial-additive optimality of primes, i.e., that the sum prime factors is always minimum, implies numbers are a solution to an integer linear programming (ILP) encoding optimization problem. summative primes follows from Goldbach’s conjecture, and viewed as upper efficiency limit for any with fewest possible additions. A consequence above optimally encode—multiplicatively additively—all ...