School of Mathematics University of the Witwatersrand (Wits), Johannesburg 31 August 2015 Submitted in partial fulflment of a Masters degree at Wits / The operation defined on a discrete semigroup S can be extended to the Stone- Cech compactification S of S so that for all a 2 S, the left translation S 3 x 7! ax 2 S is continuous, and for all q 2 S, the right translation S 3 x 7! xq 2 S is continuous. Because any compact right topological semigroup, S contains a smallest two-sided ideal K( S) which is a completely simple semigroup. We give an exposition of some basic results related to the semigroup S and to the semigroup algebra `1( S). In particular, we review the result that `1( N) is semisimple if and only if `1(K( N)) is semisimple. We also review the reduction of the question whether `1(K( N)) is semisimple to a question about K( N).

The results presented here are concerned with questions of decomposability of multiplicative semigroups of matrices with nonnegative entries. Chapter 1 covers some preliminary results which become useful in the remainder of the exposition. Chapters 2 and 3 constitute an exposition of some recent known results on special semigroups. Chapter 2 explores conditions for decomposability of semigroups in terms of conditions derived from linear functionals and in Chapter 3, we give a complete proof of an extension of the celebrated Perron-Frobenius Theorem. No originality is claimed for the results in Chapters 2 and 3. In Chapter 4, we present some new results on sufficient conditions for finiteness of semigroups of matrices.

The results presented here are concerned with questions of decomposability of multiplicative semigroups of matrices with nonnegative entries. Chapter 1 covers some preliminary results which become useful in the remainder of the exposition. Chapters 2 and 3 constitute an exposition of some recent known results on special semigroups. Chapter 2 explores conditions for decomposability of semigroups in terms of conditions derived from linear functionals and in Chapter 3, we give a complete proof of an extension of the celebrated Perron-Frobenius Theorem. No originality is claimed for the results in Chapters 2 and 3. In Chapter 4, we present some new results on sufficient conditions for finiteness of semigroups of matrices.

The word problem of a finitely generated group is commonly defined to be a formal language over a finite generating set. The class of finite groups has been characterised as the class of finitely generated groups that have word problem decidable by a finite state automaton. We give a natural generalisation of the notion of word problem from finitely generated groups to finitely generated semigroups by considering relations of strings. We characterise the class of finite semigroups by the class of finitely generated semigroups whose word problem is decidable by finite state automata. We then examine the class of semigroups with word problem decidable by asynchronous two tape finite state automata. Algebraic properties of semigroups in this class are considered, towards an algebraic characterisation. We take the next natural step to further extend the classes of semigroups under consideration to semigroups that have word problem decidable by a finite collection of asynchronous automata working independently. A central tool used in the derivation of structural results are so-called iteration lemmas. We define a hierarchy of the considered classes of semigroups and connect our original results with previous research.

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The main aim of this thesis is to study the convergence of Lyapounov functions along the trajectories of nonexpansive semigroups in a Hilbert space. The outline of the thesis is as follows. In Chapter 3, it is shown that a regularly Lyapounov function for a semigroup of contractions on a Hilbert space converges to its minimum along the trajectories of the semigroup. In Chapter 4, we show that while a convex Lyapounov function for a semigroup of contractions on a Hilbert space may not converge to its minimum along the trajectories of the semigroup, it converges generically along the trajectories of the semigroups generated by a class of bounded perturbations of the semigroup generator. In Chapter 5, we show that the regularly Lyapounov function nearly converges to its minimum along the trajectories of the semigroups generated by small bounded perturbations of the semigroup generator. Besides that we study a problem of interest in its own right, about the direction of movement of the element of minimal norm in a moving convex set, in Section 4.9. We show that if C is a nonempty closed convex subset of a real Hilbert space H, e is a non-zero arbitrary vector in H, and for each t Є R, z(t) is the closest point in C + te to the origin, then the angle z(t) makes with e is a decreasing function of t while z(t) ≠ 0.

The main aim of this thesis is to study the convergence of Lyapounov functions along the trajectories of nonexpansive semigroups in a Hilbert space. The outline of the thesis is as follows. In Chapter 3, it is shown that a regularly Lyapounov function for a semigroup of contractions on a Hilbert space converges to its minimum along the trajectories of the semigroup. In Chapter 4, we show that while a convex Lyapounov function for a semigroup of contractions on a Hilbert space may not converge to its minimum along the trajectories of the semigroup, it converges generically along the trajectories of the semigroups generated by a class of bounded perturbations of the semigroup generator. In Chapter 5, we show that the regularly Lyapounov function nearly converges to its minimum along the trajectories of the semigroups generated by small bounded perturbations of the semigroup generator. Besides that we study a problem of interest in its own right, about the direction of movement of the element of minimal norm in a moving convex set, in Section 4.9. We show that if C is a nonempty closed convex subset of a real Hilbert space H, e is a non-zero arbitrary vector in H, and for each t Є R, z(t) is the closest point in C + te to the origin, then the angle z(t) makes with e is a decreasing function of t while z(t) ≠ 0.

The main aim of this thesis is to study the convergence of Lyapounov functions along the trajectories of nonexpansive semigroups in a Hilbert space. The outline of the thesis is as follows. In Chapter 3, it is shown that a regularly Lyapounov function for a semigroup of contractions on a Hilbert space converges to its minimum along the trajectories of the semigroup. In Chapter 4, we show that while a convex Lyapounov function for a semigroup of contractions on a Hilbert space may not converge to its minimum along the trajectories of the semigroup, it converges generically along the trajectories of the semigroups generated by a class of bounded perturbations of the semigroup generator. In Chapter 5, we show that the regularly Lyapounov function nearly converges to its minimum along the trajectories of the semigroups generated by small bounded perturbations of the semigroup generator. Besides that we study a problem of interest in its own right, about the direction of movement of the element of minimal norm in a moving convex set, in Section 4.9. We show that if C is a nonempty closed convex subset of a real Hilbert space H, e is a non-zero arbitrary vector in H, and for each t Є R, z(t) is the closest point in C + te to the origin, then the angle z(t) makes with e is a decreasing function of t while z(t) ≠ 0.

The main aim of this thesis is to study the convergence of Lyapounov functions along the trajectories of nonexpansive semigroups in a Hilbert space. The outline of the thesis is as follows. In Chapter 3, it is shown that a regularly Lyapounov function for a semigroup of contractions on a Hilbert space converges to its minimum along the trajectories of the semigroup. In Chapter 4, we show that while a convex Lyapounov function for a semigroup of contractions on a Hilbert space may not converge to its minimum along the trajectories of the semigroup, it converges generically along the trajectories of the semigroups generated by a class of bounded perturbations of the semigroup generator. In Chapter 5, we show that the regularly Lyapounov function nearly converges to its minimum along the trajectories of the semigroups generated by small bounded perturbations of the semigroup generator. Besides that we study a problem of interest in its own right, about the direction of movement of the element of minimal norm in a moving convex set, in Section 4.9. We show that if C is a nonempty closed convex subset of a real Hilbert space H, e is a non-zero arbitrary vector in H, and for each t Є R, z(t) is the closest point in C + te to the origin, then the angle z(t) makes with e is a decreasing function of t while z(t) ≠ 0.

Thesis (M.A.)--Kutztown University of Pennsylvania, 1994. / Source: Masters Abstracts International, Volume: 45-06, page: 3173. Abstract precedes thesis as [2] preliminary leaves. Typescript. Includes bibliographical references (leaves 52-53).