we show that the variational inequality $vi(c,a)$ has aunique solution for a relaxed $(gamma , r)$-cocoercive,$mu$-lipschitzian mapping $a: cto h$ with $r>gamma mu^2$, where$c$ is a nonempty closed convex subset of a hilbert space $h$. fromthis result, it can be derived that, for example, the recentalgorithms given in the references of this paper, despite theirbecoming more complicated, are not...

In the transformation semigroup (X, S) we introduce the height of a closed nonempty invariant subset of X, define the transformed dimension of nonempty subset S of X and obtain some results and relations.

If given n ∈ N and Γ, a multiplicatively closed subset of Zn, then the set HΓ = {n ∈ Z : x ∈ N : x + nZ ∈ Γ} ∪ {1} is a multiplicative submonoid of N0 known as a congruence monoid. Much work has been done to characterize the factorial (every element has unique factorization) and half-factorial (lengths of irreducible factorizations of an element remain constant) properties of such objects. Our ...

Let $R\ $be a commutative ring with $1\neq0$ and $M$ be an $R$-module. Suppose that $S\subseteq R\ $is multiplicatively closed set of $R.\ $Recently Sevim et al. in \cite{SenArTeKo} introduced the notion $S$-prime submodule which is generalization prime used them to characterize certain classes rings/modules such as submodules, simple modules, torsion free modules,\ $S$-Noetherian modules etc. ...

Let r ≥ 2 and s ≥ 2 be distinct integers. We establish that, if r and s are multiplicatively independent, then the r-ary expansion and the s-ary expansion of an irrational real number, viewed as infinite words on {0, 1, . . . , r− 1} and {0, 1, . . . , s− 1}, respectively, cannot have simultaneously a low block complexity. In particular, they cannot be both Sturmian words. We also discuss the c...

(5) For every real number r and for every subset P of E2 T such that P = {p; p ranges over points of E2 T: p1 ≥ r} holds P is closed. (6) For every real number r and for every subset P of E2 T such that P = {p; p ranges over points of E2 T: p1 ≤ r} holds P is closed. (7) For every real number r and for every subset P of E2 T such that P = {p; p ranges over points of E2 T: p1 = r} holds P is clo...

Let R be a ring and n a fixed positive integer, we investigate the properties of n-strongly Gorenstein projective, injective and flat modules. Using the homological theory , we prove that the tensor product of an n-strongly Gorenstein projective (flat) right R -module and projective (flat) left R-module is also n-strongly Gorenstein projective (flat). Let R be a coherent ring ,we prove that the...

in the transformation semigroup (x, s) we introduce the height of a closed nonempty invariant subset of x, define the transformed dimension of nonempty subset s of x and obtain some results and relations.

For simplicity, we adopt the following rules: a, a1, a2, b, b1, b2 are real numbers, p is a finite sequence, F , G, H are finite sequences of elements of R, i, j, k are natural numbers, f is a function from R into R, and x1 is a set. Let I1 be a subset of R. We say that I1 is closed-interval if and only if: (Def. 1) There exist real numbers a, b such that a ¬ b and I1 = [a, b]. Let us mention t...