J.M. Howie, the influential St Andrews semigroupist, claimed that we value an area of pure mathematics to the extent that (a) it gives rise to arguments that are deep and elegant, and (b) it has interesting interconnections with other parts of pure mathematics. This paper surveys some recent results on the transformation semigroup generated by a permutation group G and a single non-permutation ...

We generalize a theorem of Frey by giving sufficient conditions for a subsemigroup T of a cancellative left amenable semigroup S to be left amenable. In particular, we show that if S is left amenable and T does not contain a free subsemigroup on two generators, then T is left amenable as well.

A differentiable semigroup is a topological semigroup (5, *) in which 5 is a differentiable manifold based on a Banach space and the associative multiplication function * is continuously differentiable. If e is an idempotent element of such a semigroup we show that there is an open set U containing e so that there is a C retraction of U into the set of idempotents of S so that (x)<$(y) =...

Let C be a bounded closed convex subset of a uniformly convex Banach space X and let = = {T (t) : t ∈ G} be a commutative semigroup of asymptotically nonexpansive in the intermediate mapping from C into itself. In this paper, we provide the strong mean ergodic convergence theorem for the almost-orbit of =.

We establish some decay properties of the semigroup generated by a linear integro-differential equation in a Hilbert space, which is an abstract version of the equation ut(t)− β∆u(t)− ∫ ∞ 0 k(s)∆u(t− s)ds = 0 describing hereditary heat conduction.

For a non-empty set X denote the full transformation semigroup of by T(X). Let \(\sigma\) be an equivalence relation on and E(X, \(\sigma\)) denotes (under composition) all \(\alpha\) : \(\mapsto\) X, such that \(\subseteq\) ker(\(\alpha\) ). Semigroup transformations with restricted occur when we take whose kernel is contained in some fixed equivalence, \(\sigma\)). First, found disjoint union...

We discuss dual time evolution scenarios which, albeit running according to the same real time clock, in each considered case may be mapped among each other by means of a suitable analytic continuation in time procedure. This dynamical duality is a generic feature of diffusion-type processes. Technically that involves a familiar transformation from a non-Hermitian Fokker-Planck operator to the ...

In this paper, we prove an existence theorem for the pseudo-nonlocal Cauchy problem x′(t) + Ax(t) = f(t, x(t), ∫ t t0 k(t, s, x(s))ds), x0(t0) = x0−g(x), where A is the infinitesimal generator of a C0 semigroup of operator {T (t)}t>0 on a Banach space. The functions f, g are weakly-weakly sequentially continuous and the integral is taken in the sense of Pettis.

Let $(p_t)_{t\geq 0}$ be a convolution semigroup of probability measures on $\mathbb R^d$ defined by $$\int _{\mathbb R^d} e^{i\langle \xi ,x\rangle }\, p_t(\mathrm d x)= e^{-t\psi (\xi )}, \quad \ \in \mathbb R^d, $$ and let $\Omega $ an open subs

Let G be a semitopological semigroup, C a nonempty subset of a real Hilbert space H , and = {Tt : t ∈ G} a representation of G as asymptotically nonexpansive type mappings of C into itself. Let L(x)= {z ∈H : infs∈G supt∈G ‖Ttsx−z‖ = inf t∈G ‖Ttx−z‖} for each x ∈ C and L( )= ⋂x∈C L(x). In this paper, we prove that ⋂s∈G conv{Ttsx : t ∈ G}⋂L( ) is nonempty for each x ∈ C if and only if there exist...