The main contribution of this thesis is a study of the dynamic programming and greedy strategies for solving combinatorial optimization problems. The study is carried out in the context of a calculus of relations, and generalises previous work by using a loop operator in the imperative programming style for generating feasible solutions, rather than the fold and unfold operators of the functional programming style. The relationship between fold operators and loop operators is explored, and it is shown how to convert from the former to the latter. This fresh approach provides additional insights into the relationship between dynamic programming and greedy algorithms, and helps to unify previously distinct approaches to solving combinatorial optimization problems. Some of the solutions discovered are new and solve problems which had previously proved difficult. The material is illustrated with a selection of problems and solutions that is a mixture of old and new. Another contribution is the invention of a new calculus, called the graph calculus, which is a useful tool for reasoning in the relational calculus and other non-relational calculi. The graph

Consider two spatially separated agents, Alice and Bob, each of whom is able to make local quantum measurements, and who can communicate with each other over a purely classical channel. As has been pointed out by a number of authors, the set of mathematically consistent joint probability assignments (“states”) for such a system is properly larger than the set of quantum-mechanical mixed states for the joint Alice-Bob system. Indeed, it is canonically isomorphic to the set of positive (but not necessarily completely positive) linear maps L(HA) → L(HB) from the bounded linear operators on Alice’s Hilbert space to those on Bob’s, satisfying Tr (φ(1)) = 1. The present paper explores the properties of these states. We review what is known, including the fact that allowing classical communication between parties is equivalent to enforcing “noinstantaneous-signalling” (“no–influence”) in the direction opposite the communication. We establish that in the subclass of “decomposable”

Abstract. We examine the possibility to extend measures and signed measures on a concrete logic on a finite set to those on all its subsets. 1.

We consider sets of coin denominations which permit change to be made using as few coins as possible, on average, and explain why the United States should adopt an 18 ¡ piece. 1

On the first hand, this report studies the class of Greedy Algorithms in order to find an as systematic as possible strategy that could be applied to the specification of some problems to lead to a correct program solving that problem. On the other hand, the standard formalisms underlying the Greedy Algorithms (matroid, greedoid and matroid embedding) which are dependent on the particular type set are generalized to a formalism independent of any data type based on an algebraic specification setting.

We consider sets of coin denominations which permit change to be made using as few coins as possible, on average, and explain why the United States should adopt an 18 piece.

Abstract. Let H be a separable infinite dimensional complex Hilbert space. We prove that every continuous 2-local automorphism of the poset (that is, partially ordered set) of all idempotents on H is an automorphism. Similar results concerning the orthomodular poset of all projections and the Jordan ring of all selfadjoint operators on H without the assumption on continuity are also presented.

Abstract. In this paper we describe the structure of all sequential isomorphisms between the sets of von Neumann algebra effects. It turns out that if the underlying algebras have no commutative direct summands, then every sequential isomorphism between the sets of their effects extends to the direct sum of a *-isomorphism and a *-antiisomorphism between the underlying von Neumann algebras. 1. Introduction and Statement

Let H be a Hilbert C∗-module over a matrix algebra A. It is proved that any function T: H → H which preserves the absolute value of the (generalized) inner product is of the form Tf = ϕ(f)Uf (f ∈ H), where ϕ is a phase-function and U is an A-linear isometry. The result gives a natural extension of Wigner’s classical unitary-antiunitary theorem for Hilbert modules.

Dedicated to the memory of Professor Béla Szőkefalvi-Nagy Abstract. Let H be a complex separable infinite dimensional Hilbert space. We describe the form of all *-semigroup endomorphisms φ of B(H) which are uniformly continuous on every commutative C ∗-subalgebra. In particular, we obtain that if φ satisfies φ(0) = 0, then φ is additive. 1.