for any group g, let c(g) denote the set of centralizers of g.we say that a group g has n centralizers (g is a cn-group) if |c(g)| = n.in this note, we prove that every finite cn-group with n ≤ 21 is soluble andthis estimate is sharp. moreover, we prove that every finite cn-group with|g| < 30n+1519 is non-nilpotent soluble. this result gives a partial answer to aconjecture raised by a. ashrafi in ...

Kernel of the symmetric block ciphering methods presented here is the coupling of XOR operations and of invertible substitution tables S with all possible 256**t byte groups (with t=1, 2, 3, ... Bytes, fixed at the beginning) being derived from keys: K(block) := S(S(block) ⊗ Eo) ⊗ Eu with-Eo upper and Eu lower triangular (byte-group-)matrix with (byte-block-length/t)**2 values, value 1 at all n...

This paper proposes an original exchange property of valuations.This property is shown to be equivalent to a property described by Dress and Terhalle in the context of discrete optimization and matroids and shown there to characterize the valuations for which the demand oracle can be implemented by a greedy algorithm. The same exchange property is also equivalent to a property described indepen...

It is shown that every commutative local ring of bounded module type is an almost maximal valuation ring. We say that an associative commutative unitary ring R is of bounded module type if there exists a positive integer n such that every finitely generated R-module is a direct sum of submodules generated by at most n elements. For instance, every Dedekind domain has bounded module type with bo...

Let $R= \oplus_{ \alpha \in G} R_{\alpha}$ be a commutative ring with unity graded by an arbitrary grading monoid $G$. For each positive integer, the notions of graded-n-coherent module and are introduced. In this paper many results generalized from $n$-coherent rings to graded-$n$-coherent rings. last section, we provide necessary sufficient conditions for trivial extension graded-valuation ri...

BACKGROUND Quality of life as an endpoint in a clinical study may be sensitive to the value set used to derive a single score. Focusing on patients' actual valuations in a clinical study, we compare different value sets for the EQ-5D-3L and assess how well they reproduce patients' reported results. METHODS A clinical study comparing inpatient (n = 98) and outpatient (n = 47) rehabilitation of...

We study the word length entropy of automorphisms of residually nilpotent groups, and how the entropy of such group automorphisms relates to the entropy of induced automorphisms on various nilpotent quotients. We show that much like the structure of a nilpotent group is dictated to a large degree by its abelianization, the entropy of an automorphism of a nilpotent group is dictated by its entro...