Tangled Circuits

The theme of the paper is the use of commutative Frobenius algebras in braided strict monoidal categories in the study of varieties of circuits and communicating systems which occur in Computer Science, including circuits in which the wires are tangled. We indicate also some possible novel geometric interest in such algebras

Stereoselective Olefin Metathesis Processes Using Cyclometalated Ruthenium Alkylidene Complexes

The recent development of a class of Z-selective ruthenium metathesis catalysts containing a crucial cyclometalated N-heterocyclic carbene (NHC) ligand has extended the applicability of ruthenium-mediated olefin metathesis to the production of a variety of useful Z-olefin-containing small molecules, polymers, and natural products. This thesis explores the synthesis and application of a number of novel Z-selective cyclometalated ruthenium alkylidene complexes displaying enhanced activity and selectivity across a range of metathesis transformations. Mechanistic investigations aimed at understanding and controlling stereoselectivity specifically in ring-opening metathesis polymerization (ROMP) are also detailed. Chapter 2 describes the preparation of new Z-selective cyclometalated ruthenium metathesis catalysts via an improved method employing sodium carboxylates. Effects of the cyclometalated NHC ligand on catalyst activity and selectivity in several cross metathesis assays, as well as macrocyclic ring-closing metathesis and other industrially relevant transformations, are investigated. Chapter 3 relates a story in two parts: the first details the synthesis and activity of a series of novel cyclometalated ruthenium alkylidenes displaying unprecedented cis,syndio-selectivity in the ROMP of norbornene- and norbornadiene-derived monomers. The second comprises an extensive study into the origins of stereoselectivity in ROMP in these and related cyclometalated ruthenium initiators. Experimental results are used in conjunction with a computational analysis of propagation transition states to develop a complete stereochemical model for cis,syndio-selctivity in these systems.</p

Delta Lenses and Opfibrations

We compare the delta lenses, also known as d-lenses, of Diskin et al. with the c-lenses, known to be equivalent to opfibrations, already studied by the authors.  Contrary to expectation a c-lens is a d-lens but not conversely. This result is surprising because d-lenses appear to provide the same information as c-lenses, and some more besides, suggesting that the implication would be the reverse -- a d-lens would appear to be a special kind of c-lens. The source of the surprise can be traced to the way the two concepts deal differently with morphisms in a certain base comma category (G,1_\bV).  Both c-lenses and d-lenses are important because they extend the notion of lens to take account of the information available in known transitions between states and this has important implications in practice

Lens put-put laws: monotonic and mixed

Many authors have argued, for good reasons, that in a range of applications the lens put-put law is too strong.  On the other hand, the present authors have shown that very well behaved lenses, which do satisfy the put-put law by definition, are algebras for a certain monad, and that this viewpoint admits fruitful generalisations of the lens concept to a variety of base categories. In the algebra approach to lenses, the put-put law corresponds to the associativity axiom, and so is fundamentally important. Thus we have a dilemma. The put-put law seems inappropriate for many applications, but is fundamental to the mathematical development that can support an extended range of applications. In this paper we resolve this dilemma. We outline monotonic put-put laws and introduce a new mixed put-put law that appears to be immune to many of the objections to the classical put-put law, and still supports a very satisfactory mathematical foundation.

Calculating Colimits Compositionally

We show how finite limits and colimits can be calculated compositionally using the algebras of spans and cospans, and give as an application a proof of the Kleene Theorem on regular languages

Calculating Colimits Compositionally

We show how finite limits and colimits can be calculated compositionally using the algebras of spans and cospans, and give as an application a proof of the Kleene Theorem on regular languages

Synthesis of Highly Cis, Syndiotactic Polymers via Ring-Opening Metathesis Polymerization Using Ruthenium Metathesis Catalysts

The first example of ruthenium-mediated ring-opening metathesis polymerization generating highly cis, highly tactic polymers is reported. While the cis content varied from 62 to >95% depending on the monomer structure, many of the polymers synthesized displayed high tacticity (>95%). Polymerization of an enantiomerically pure 2,3-dicarboalkoxynorbornadiene revealed a syndiotactic microstructure