We consider the special Jin-Xin relaxation model (0.1) ut + A(u)ux = 2(uxx − utt), We assume that the initial data (u0, 2u0,t) are sufficiently smooth and close to (ū, 0) in L∞ and have small total variation. Then we prove that there exists a solution (u2(t), 2ut(t)) with uniformly small total variation for all t ≥ 0, and this solution depends Lipschitz continuously in the L1 norm w.r.t. the in...

A variety V has an alg-universal n-expansion if the addition of n nullary operations to algebras from V produces an alg-universal category. It is proved that any semigroup variety V containing a semigroup that is neither an inflation of a completely simple semigroup nor an inflation of a semilattice of groups has an alg-universal 3-expansion. We say that a variety V is var-relatively alg-univer...

1 Let (X , .) be a Banach space. In general, for a C 0-semigroup {T (t)} t≥0 on (X , .), its adjoint semigroup {T * (t)} t≥0 is no longer strongly continuous on the dual space (X * ,. which the usual semigroups in literature becomes C 0-semigroups. The main purpose of this paper is to prove that only a core can be the domain of uniqueness for a C 0-semigroup on (X * , C(X * , X)). As applicatio...

Exactly as in semigroups, Green’s relations play an important role in the theory of ordered semigroups—especially for decompositions of such semigroups. In this paper we deal with the -trivial ordered semigroups which are defined via the Green’s relation , and with the nil and Δ-ordered semigroups. We prove that every nil ordered semigroup is -trivial which means that there is no ordered semigr...

For every Banach space (Y, ‖ · ‖Y ) that admits an equivalent uniformly convex norm we prove that there exists c = c(Y ) ∈ (0,∞) with the following property. Suppose that n ∈ N and that X is an n-dimensional normed space with unit ball BX . Then for every 1-Lipschitz function f : BX → Y and for every ε ∈ (0, 1/2] there exists a radius r > exp(−1/ε), a point x ∈ BX with x + rBX ⊆ BX , and an aff...

In this paper, varieties of completely regular semigroups are studied. This paper is divided into six sections. Section 1 contains an introduction to varieties of completely regular semigroups and preliminaries. Most of the notation needed in this paper is given. In Section 2, the operators ←− ( ) and −→ ( ) on the lattice of subvarieties of varieties of completely regular semigroups are invest...

where 0 ≤ t0 < t1 < ··· < tp ≤ a, a > 0, −A is the infinitesimal generator of aC0-semigroup in a Banach space X , u0 ∈ X , and f : [0,a]×X → X , g : [0,a]p ×X → X are given functions. Subsequently he has investigated the same type of problem for different kinds of evolution equations in Banach spaces [10, 11, 12, 13, 14]. Ntouyas and Tsamatos [31] have established the global existence of soluti...

To the Farey tesselation of the upper half-plane we associate an AF algebra A encoding the ‘cutting sequences’ that define vertical geodesics. The Effros-Shen AF algebras arise as quotients of A. We define a unital completely positive map E : A → A and an endomorphism ρ : A→ A such that Eρ = idA , and which can be regarded as noncommutative analogs of the Farey map F : [0, 1] → [0, 1] acting on...

We consider a two-dimensional quantum control system evolving under an entropyincreasing irreversible dynamics in the semigroup form. Considering a phenomenological approach to the dynamics, we show that the accessibility property of the system depends on whether its evolution is assumed to be positive or completely positive. In particular, we characterize the family of maps having different ac...