A Higher-Order Calculus for Categories

A calculus for a fragment of category theory is presented. The types in the language denote categories and the expressions functors. The judgements of the calculus systematise categorical arguments such as: an expression is functorial in its free variables; two expressions are naturally isomorphic in their free variables. There are special binders for limits and more general ends. The rules for limits and ends support an algebraic manipulation of universal constructions as opposed to a more traditional diagrammatic approach. Duality within the calculus and applications in proving continuity are discussed with examples. The calculus gives a basis for mechanising a theory of categories in a generic theorem prover like Isabelle

A study in higher education calculus and students' learning styles

This research is devoted to focussing on the influence of different learning style on the performance of undergraduate students in various parts of calculus. In carrying out the study, calculus materials were classified into four main categories (Z4,Z5,Z6,Cals) and, for the Iranian students, the results of their mathematical performance in the university entrance examination is labelled (En) to identify their grounding in high school mathematics at the beginning of the calculus course in higher education. Also, in the present study, students' performance (weakness) in the manipulation of mathematical notation and logical discussion is called (Z1) category and (Cal) indicates students' total achievement in calculus examination which is, in fact, the students' performance on the combination of the categories (Z4,Z5,Z6). These calculus categories are described in Chapter 5. However in short term, multi-conceptual and procedural tasks are classified as (Z4). The (Z5) category is defined as the translation processes between mathematical abstraction (analytic/symbolic) and (pictorial/visual) forms of calculus materials. Moreover, multi-skilled, transferable and procedural skills are labelled as (Z6) category. It should be noted that these categories are interrelated in a scheme to exhibit activities in calculus. 572 students participated in the experimental part of this study and were selected from two Iranian universities (Sabzvar University and Mashhad University) and Glasgow University in Scotland, U.K. During the period of the study, the samples of students were subjected to some psychological tests in order to assign their Field-dependent/Field-independent and Convergent/Divergent learning styles. It was found throughout the study that the most effective combination of learning styles which emerged from the interacting picture of all the psychological factors used in the research, were field-independent/convergent (F1+Con) in Iran, and field-independent/divergent (FI+Div) in Scotland in performing on the calculus. On the other hand, the combination of field-dependent and convergent styles (FD+Con) could lessen achievement in calculus by mathematics/physics students, and field-dependent and divergent styles (FD+Div) would lessen attainment in calculus by engineering students. In addition, when the mean scores in calculus categories were calculated for various groups of students with different learning styles, the convergent thinkers (Con) were found to be best in (Z6), while divergent thinkers (Div) exhibited higher performance in (Z5). These findings demonstrate that the Con/Div way of thinking is the most effective in influencing performance in different areas of calculus, the FI/FD factor takes the second position. All these findings have been combined to form a model which emerges at the end of this thesis. Moreover, in Chapters 3 and 4, a comparison is made between calculus in secondary (high school) and higher education in Iran and Scotland, focussing on content, teaching order, learning objectives and teaching methods

Change Actions: Models of Generalised Differentiation

Cai et al. have recently proposed change structures as a semantic framework for incremental computation. We generalise change structures to arbitrary cartesian categories and propose the notion of change action model as a categorical model for (higher-order) generalised differentiation. Change action models naturally arise from many geometric and computational settings, such as (generalised) cartesian differential categories, group models of discrete calculus, and Kleene algebra of regular expressions. We show how to build canonical change action models on arbitrary cartesian categories, reminiscent of the F\`aa di Bruno construction

Weak Similarity in Higher-Order Mathematical Operational Semantics

Higher-order abstract GSOS is a recent extension of Turi and Plotkin's framework of Mathematical Operational Semantics to higher-order languages. The fundamental well-behavedness property of all specifications within the framework is that coalgebraic strong (bi)similarity on their operational model is a congruence. In the present work, we establish a corresponding congruence theorem for weak similarity, which is shown to instantiate to well-known concepts such as Abramsky's applicative similarity for the lambda-calculus. On the way, we develop several techniques of independent interest at the level of abstract categories, including relation liftings of mixed-variance bifunctors and higher-order GSOS laws, as well as Howe's method

Cartesian Gray-Monoidal Double Categories

In this paper we present cartesian structure for symmetric Gray-monoidal double categories. To do this we first introduce locally cubical Gray categories, which are three-dimensional categorical structures analogous to classical, locally globular, Gray categories. The motivating example comprises double categories themselves, together with their functors, transformations, and modifications. A one-object locally cubical Gray category is a Gray-monoidal double category. Braiding, syllepsis, and symmetry for these is introduced in a manner analogous to that for 2-categories. Adding cartesian structure requires the introduction of doubly-lax functors of double categories to manage the order of copies. The resulting theory is algebraically rather complex, largely due to the bureaucracy of linearizing higher-dimensional boundary constraints. Fortunately, it has a relatively simple and compelling representation in the graphical calculus of surface diagrams, which we present

An Internal Language for Categories Enriched over Generalised Metric Spaces

Programs with a continuous state space or that interact with physical processes often require notions of equivalence going beyond the standard binary setting in which equivalence either holds or does not hold. In this paper we explore the idea of equivalence taking values in a quantale ?, which covers the cases of (in)equations and (ultra)metric equations among others. Our main result is the introduction of a ?-equational deductive system for linear ?-calculus together with a proof that it is sound and complete (in fact, an internal language) for a class of enriched autonomous categories. In the case of inequations, we get an internal language for autonomous categories enriched over partial orders. In the case of (ultra)metric equations, we get an internal language for autonomous categories enriched over (ultra)metric spaces. We use our results to obtain examples of inequational and metric equational systems for higher-order programs that contain real-time and probabilistic behaviour

Fiscal Spending Multiplier Calculations based on Input-Output Tables – with an Application to EU Members

Fiscal spending multiplier calculations have been revived in the aftermath of the global financial crisis. Much of the current literature is based on VAR estimation methods and DSGE models. The aim of this paper is not a further deepening of this literature but rather to implement a calculation method of multipliers which is suitable for open economies like EU member states. To this end, Input-Output tables are used as by this means the import intake of domestic demand components can be isolated in order to get an appropriate base for the calculation of the relevant import quotas. The difference of this method is substantial – on average the calculated multipliers are 15% higher than the conventional GDP fiscal spending multiplier for EU members. Multipliers for specific spending categories are comparably high, ranging between 1.4 and 1.8 for many members of the EU. GDP drops due to budget consolidation might therefore be substantial if monetary policy is not able to react in an expansionary manner.fiscal spending multiplier calculation, Input-Output calculus, income-expenditure model, European Union, stimulus, consolidation