Models of Type Theory Based on Moore Paths

This paper introduces a new family of models of intensional Martin-L\"of type theory. We use constructive ordered algebra in toposes. Identity types in the models are given by a notion of Moore path. By considering a particular gros topos, we show that there is such a model that is non-truncated, i.e. contains non-trivial structure at all dimensions. In other words, in this model a type in a nested sequence of identity types can contain more than one element, no matter how great the degree of nesting. Although inspired by existing non-truncated models of type theory based on simplicial and cubical sets, the notion of model presented here is notable for avoiding any form of Kan filling condition in the semantics of types.Comment: This is a revised and expanded version of a paper with the same name that appeared in the proceedings of the 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017

Cubical Models of Homotopy Type Theory - An Internal Approach

This thesis presents an account of the cubical sets model of homotopy type theory using an internal type theory for elementary topoi. Homotopy type theory is a variant of Martin-Lof type theory where we think of types as spaces, with terms as points in the space and elements of the identity type as paths. We actualise this intuition by extending type theory with Voevodsky's univalence axiom which identifies equalities between types with homotopy equivalences between spaces. Voevodsky showed the univalence axiom to be consistent by giving a model of homotopy type theory in the category of Kan simplicial sets in a paper with Kapulkin and Lumsdaine. However, this construction makes fundamental use of classical logic in order to show certain results. Therefore this model cannot be used to explain the computational content of the univalence axiom, such as how to compute terms involving univalence. This problem was resolved by Cohen, Coquand, Huber and Mortberg, who presented a new model of type theory in Kan cubical sets which validated the univalence axiom using a constructive metatheory. This meant that the model provided an understanding of the computational content of univalence. In fact, the authors present a new type theory, cubical type theory, where univalence is provable using a new "glueing" type former. This type former comes with appropriate definitional equalities which explain how the univalence axiom should compute. In particular, Huber proved that any term of natural number type constructed in this new type theory must reduce to a numeral. This thesis explores models of type theory based on the cubical sets model of Cohen et al. It gives an account of this model using the internal language of toposes, where we present a series of axioms which are sufficient to construct a model of cubical type theory, and hence a model of homotopy type theory. This approach therefore generalises the original model and gives a new and useful method for analysing models of type theory. We also discuss an alternative derivation of the univalence axiom and show how this leads to a potentially simpler proof of univalence in any model satisfying the axioms mentioned above, such as cubical sets. Finally, we discuss some shortcomings of the internal language approach with respect to constructing univalent universes. We overcome these difficulties by extending the internal language with an appropriate modality in order to manipulate global elements of an object.UK EPSRC PhD studentship, funded by grants EP/L504920/1, EP/M506485/1

Internal Universes in Models of Homotopy Type Theory

We begin by recalling the essentially global character of universes in various models of homotopy type theory, which prevents a straightforward axiomatization of their properties using the internal language of the presheaf toposes from which these model are constructed. We get around this problem by extending the internal language with a modal operator for expressing properties of global elements. In this setting we show how to construct a universe that classifies the Cohen-Coquand-Huber-Mörtberg (CCHM) notion of fibration from their cubical sets model, starting from the assumption that the interval is tiny - a property that the interval in cubical sets does indeed have. This leads to an elementary axiomatization of that and related models of homotopy type theory within what we call crisp type theory

A Harmonised Approach to Curating Research-Ready Datasets for Asthma, Chronic Obstructive Pulmonary Disease (COPD) and Interstitial Lung Disease (ILD) in England, Wales and Scotland Using Clinical Practice Research Datalink (CPRD), Secure Anonymised Information Linkage (SAIL) Databank and DataLoch

Background: Electronic healthcare records (EHRs) are an important resource for health research that can be used to improve patient outcomes in chronic respiratory diseases. However, consistent approaches in the analysis of these datasets are needed for coherent messaging, and when undertaking comparative studies across different populations.Methods and results: We developed a harmonised curation approach to generate comparable patient cohorts for asthma, chronic obstructive pulmonary disease (COPD) and interstitial lung disease (ILD) using datasets from within Clinical Practice Research Datalink (CPRD; for England), Secure Anonymised Information Linkage (SAIL; for Wales) and DataLoch (for Scotland) by defining commonly derived variables consistently between the datasets. By working in parallel on the curation methodology used for CPRD, SAIL and DataLoch for asthma, COPD and ILD, we were able to highlight key differences in coding and recording between the databases and identify solutions to enable valid comparisons.Conclusion: Codelists and metadata generated have been made available to help re-create the asthma, COPD and ILD cohorts in CPRD, SAIL and DataLoch for different time periods, and provide a starting point for the curation of respiratory datasets in other EHR databases, expediting further comparable respiratory research

Vesivirus 2117 capsids more closely resemble sapovirus and lagovirus particles than other known vesivirus structures

Vesivirus 2117 is an adventitious agent that in 2009, was identified as a contaminant of CHO cells propagated in bioreactors at a pharmaceutical manufacturing plant belonging to Genzyme. The consequent interruption in supply of Fabrazyme and Cerezyme (drugs used to treat Fabry and Gaucher disease respectively), caused significant economic losses. Vesivirus 2117 is a member of the Caliciviridae; a family of small icosahedral viruses encoding a positive sense RNA genome. We have used cryo-electron microscopy and three dimensional image reconstruction to calculate a structure of vesivirus 2117 virus like particles as well as feline calicivirus and a chimeric sapovirus. We present a structural comparison of several members of the Caliciviridae, showing that the distal P domain of vesivirus 2117 is morphologically distinct from that seen in other known vesivirus structures. Furthermore, at intermediate resolutions we found a high level of structural similarity between vesivirus 2117 and Caliciviridae from other genera, such as sapovirus and rabbit haemorrhagic disease virus. Phylogenetic analysis confirms vesivirus 2117 as a vesivirus closely related to canine vesiviruses. We postulate that morphological differences in virion structure seen between vesivirus clades may reflect differences in receptor usage

Hysteretic thermal spin-crossover in heteroleptic Fe(II) complexes using alkyl chain substituted 2,2’-dipyridylamine ligands

The alkyl chain carrying ligands N,N-di(pyridin-2-yl)butanamide (LC4) and N,N-di(pyridin-2-yl)decanamide (LC10) were combined with NCS− co-ligands to form the neutral heteroleptic Fe(II) complexes trans-[FeII(LC4)2(NCS)2] (1C4) and trans-[FeII(LC10)2(NCS)2] (1C10). Variable temperature crystallographic studies revealed that 1C4 is in the orthorhombic space group Pna21 between 85–200 K whereas 1C10 is in the monoclinic space group P21/c between 85–140 K. The average Fe–N bond lengths suggest that at 85 K 1C4 contains LS Fe(II) centres; however, the ca. 0.18 Å increase in the average Fe–N bond lengths between 85 and 120 K suggests a spin-transition to the HS state occurs within this temperature interval. 1C10 contains LS Fe(II) centres between 85 and 105 K. Upon warming from 105 to 140 K the average Fe–N bond lengths increase by ca. 0.19 Å, which suggests a spin-transition to the HS state. Solid-state magnetic susceptibility measurements showed that 1C4 undergoes semi-abrupt spin-crossover with T1/2 = 127.5 K and a thermal hysteresis of ca. 13 K whereas, 1C10 undergoes an abrupt spin-crossover with T1/2 = 119.0 K, and is also accompanied by thermal hysteresis of ca. 4 K. The crystallographic and magnetic data show that the length of the complex's alkyl chain substituents can have a large impact on the structure of the crystal lattice as well as a subtle effect on the T1/2 value for thermal spin-crossover

Foreign investment in Scotland

A significant feature of the UK economy throughout the post-war period has been the growth in direct foreign investment in manufacturing industries. For the host nation the main benefits are employment creation, income generation and import reduction or export expansion. Scotland has been particularly successful in attracting the lion's inward investment for example in the period 1945- 1965 a total of 108,500 jobs were created by foreign firms setting up manufacturing units in the despite its size, obtained 46,221 (42.6%), whereas the second most SE England, gained only 16,926 (15.6%). The reasons for this success have been attributed primarily to a combination of the availability of labour in Scotland, the financial inducements offered by central government as part of regional policy and, the undoubted attraction of the environment, notably of course golf courses, for foreign businessmen. This brief paper explores the nature of Scotland's direct foreign investment and the reasons for its success

Tourism in Scotland

In 1978 13.2 million tourist and business travellers spent £523 million in Scotland, Foreign visitors, although only 8.5% of the total, contributed £158 million (ie 30.2%) of tourist expenditure. As such tourism is both an "invisible" export, since it is a means of earning foreign exchange, and a significant provider of jobs, many of which are created in the remoter rural areas. Despite its importance statistics relating specifically to tourism are not readily available from published sources. The "industry" actually combines the resources of a wide range of activities, most of which have economic and social links with other sectors of the economy independent from tourism. These include hotels and catering, transport and travel services as well as entertainment, sport and recreation facilities. Consequently the economic impact of tourism is generally assessed not in terms of employment and output but by more indirect measures such as numbers of visitors, or bednights spent in Scotland, as well as tourist expenditure and hotel occupancy data. This brief will examine the wider economic impact of Scotland's tourism industry and its future

Review of the quarter's economic trends [April 1980]

This brief paper surveys recent world and UK economic data and reveals that there are some signs to suggest that the world economy will withstand the 1979 oil price shock better than it did in 1973/74. The main difference is that the 1979 price rise was not super-imposed on as severe an inflation as that which occurred on the previous occasion. The present rate of increase in the world prices should not reach the levels of the last cycle when the twelve month increase in manufacturing prices peaked at 23%. Advance warning by US economists that 1979 was going to be a year of difficulty did not go unnoticed by businessmen in that country who took steps in 1978 to avoid a repetition of the inventory boom and bust cycle which had proved so costly in the recession of 1973/74. Also, consumer expectations are adjusting towards a continuing rise in the price of oil. In the UK real GDP is expected to fall by 2½% from mid-year 1980. Thereafter, it is assumed to grow at an average annual rate of 1% for the next four years