An abelian ambient category for behaviors in algebraic systems theory

We describe an abelian category $\mathbf{ab}(M)$ in which the solution sets of finitely many linear equations over an arbitrary ring $R$ with values in an arbitrary left $R$-module $M$ reside as objects. Such solution sets are also called behaviors in algebraic systems theory. We both characterize $\mathbf{ab}(M)$ by a universal property and give a construction of $\mathbf{ab}(M)$ as a Serre quotient of the free abelian category generated by $R$. We discuss features of $\mathbf{ab}(M)$ relevant in the context of algebraic systems theory: if $R$ is left coherent and $M$ is an fp-injective fp-cogenerator, then $\mathbf{ab}(M)$ is antiequivalent to the category of finitely presented left $R$-modules. This provides an alternative point of view to the important module-behavior duality in algebraic systems theory. We also obtain a dual statement: if $R$ is right coherent and $M$ is fp-faithfully flat, then $\mathbf{ab}(M)$ is equivalent to the category of finitely presented right $R$-modules. As an example application, we discuss delay-differential systems with constant coefficients and a polynomial signal space. Moreover, we propose definitions of controllability and observability in our setup.Comment: Fix typos. Add example 7.

New Directions for Contact Integrators

Contact integrators are a family of geometric numerical schemes which guarantee the conservation of the contact structure. In this work we review the construction of both the variational and Hamiltonian versions of these methods. We illustrate some of the advantages of geometric integration in the dissipative setting by focusing on models inspired by recent studies in celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282

Applications of stochastic analysis and algebra to machine learning

In this thesis we consider the application of tools from stochastic analysis and algebra to statistics and machine learning. Most of these tools are different forms of what has become known as signature methods. The signature has been discovered and rediscovered in a few different areas of mathematics in the last 70 years. In short, it maps a path evolving in a vector space to a group enveloping that same space. The reason it is so useful for statistics is twofold: One, its set of invariances is highly desirable in many applications, and two, its image group is highly structured making it particularly amenable to mathematical study using algebraic tools. The primary aim here is to study how one may use the signature to express statistical properties of a given path, and how these properties can be applied to machine learning. This aim manifests itself as - among other things: - a new type of cumulants for signatures that have unique combinatorial properties and can be used to characterise independence of paths, - cumulants on reproducing kernel Hilbert spaces which are related to the signature cumulants, even though signatures are not used explicitly, - a generalisation of the signature to other types of feature maps into non-commutative algebras, - a feature map with an initial topology that captures properties of the filtration of stochastic processes, - and a family of scoring rules with associated divergences, entropies and mutual informations for paths that respect their group structure. These are divided into separate, self contained chapters that can be read independently of one another

Lecturers' tools and strategies in university mathematics teaching: an ethnographic study

The thesis presents the analytical process and the findings of a study on: lecturers teaching practice with first year undergraduate mathematics modules; and lecturers knowledge for teaching with regard to students mathematical meaning making (understanding). Over three academic semesters, I observed and audio-recorded twenty-six lecturers teaching to a small group tutorial of two to eight first year students, and I discussed with the lecturers about their underlying considerations for teaching. The analysis of this thesis focuses on a characterisation of each of three (of the twenty-six) lecturers teaching, which I observed for more than one semester. I chose the teaching of three experienced lecturers, due to diversity in terms of ways of engaging the students with the mathematics, and due to my consideration of their commitment to teaching for students mathematical meaning making. The distinctive nature of the study is concerned with the conceptualisation of university mathematics teaching practice and knowledge within a Vygotskian perspective. In particular, I used for the characterisation of teaching practice and of teaching knowledge the notions tool-mediation and dialectic from Vygotskian theory. I also used a coding process grounded to the data and informed by existing research literature in mathematics education. I conceptualised teaching practice into tools for teaching and actions with tools for teaching (namely strategies). I then conceptualised teaching knowledge as the lecturers reflection on teaching practice. The thesis contributes to the research literature in mathematics education with an analytical framework of teaching knowledge which is revealed in practice, the Teaching Knowledge-in-Practice (TKiP). TKiP analyses specific kinds of lecturer s knowing for teaching: didactical knowing and pedagogical knowing. The framework includes emerging tools for teaching (e.g. graphical representation, rhetorical question, students faces) and emerging strategies for teaching (e.g. creating students positive feelings, explaining), which were common or different among the three lecturers teaching practice. Overall, TKiP is produced to offer a dynamic framework for researcher analysis of university mathematics teaching knowledge. Analysis of teaching knowledge is important for gaining insights into why teaching practice happens in certain ways. The findings of the thesis also suggest teaching strategies for the improvement of students mathematical meaning making in tutorials

Reaction Networks and Population Dynamics

Reaction systems and population dynamics constitute two highly developed areas of research that build on well-defined model classes, both in terms of dynamical systems and stochastic processes. Despite a significant core of common structures, the two fields have largely led separate lives. The workshop brought the communities together and emphasised concepts, methods and results that have, so far, appeared in one area but are potentially useful in the other as well