Computer Science for Continuous Data:Survey, Vision, Theory, and Practice of a Computer Analysis System

Building on George Boole's work, Logic provides a rigorous foundation for the powerful tools in Computer Science that underlie nowadays ubiquitous processing of discrete data, such as strings or graphs. Concerning continuous data, already Alan Turing had applied "his" machines to formalize and study the processing of real numbers: an aspect of his oeuvre that we transform from theory to practice.The present essay surveys the state of the art and envisions the future of Computer Science for continuous data: natively, beyond brute-force discretization, based on and guided by and extending classical discrete Computer Science, as bridge between Pure and Applied Mathematics

Applications of dynamical systems with symmetry

This thesis examines the application of symmetric dynamical systems theory to two areas in applied mathematics: weakly coupled oscillators with symmetry, and bifurcations in flame front equations. After a general introduction in the first chapter, chapter 2 develops a theoretical framework for the study of identical oscillators with arbitrary symmetry group under an assumption of weak coupling. It focusses on networks with 'all to all' Sn coupling. The structure imposed by the symmetry on the phase space for weakly coupled oscillators with Sn, Zn or Dn symmetries is discussed, and the interaction of internal symmetries and network symmetries is shown to cause decoupling under certain conditions. Chapter 3 discusses what this implies for generic dynamical behaviour of coupled oscillator systems, and concentrates on application to small numbers of oscillators (three or four). We find strong restrictions on bifurcations, and structurally stable heteroclinic cycles. Following this, chapter 4 reports on experimental results from electronic oscillator systems and relates it to results in chapter 3. In a forced oscillator system, breakdown of regular motion is observed to occur through break up of tori followed by a symmetric bifurcation of chaotic attractors to fully symmetric chaos. Chapter 5 discusses reduction of a system of identical coupled oscillators to phase equations in a weakly coupled limit, considering them as weakly dissipative Hamiltonian oscillators with very weakly coupling. This provides a derivation of example phase equations discussed in chapter 2. Applications are shown for two van der Pol-Duffing oscillators in the case of a twin-well potential. Finally, we turn our attention to the Kuramoto-Sivashinsky equation. Chapter 6 starts by discussing flame front equations in general, and non-linear models in particular. The Kuramoto-Sivashinsky equation on a rectangular domain with simple boundary conditions is found to be an example of a large class of systems whose linear behaviour gives rise to arbitrarily high order mode interactions. Chapter 7 presents computation of some of these mode interactions using competerised Liapunov-Schmidt reduction onto the kernel of the linearisation, and investigates the bifurcation diagrams in two parameters

Modelling the time-series of cerebrovascular pressure transmission variation in head injured patients

Cerebral autoregulation is the process by which blood ow is maintained over a changing cerebral perfusion pressure. Clinically autoregulation is an important topic because it directly effects overall patient management strategy. However accurately predicting autoregulatory state or even modelling the underlying general physiological processes is a complex task. There are a number of models published within the literature but there has been no active attempt to compare and classify these models. Starting with the hypothesis that a physiologically based model would be a better predictor of autoregulatory state than a purely statistically based one has led us to investigate approaches to model comparison. Using three different models: a new mathematical arrangement of a physiological model by Ursino, the Highest Model Frequency (HMF) model by Daley and the Pressure reactivity index (PRx) statistical model by Czosnyka, a general comparison was carried out using the Matthews correlation coecient against a known autoregulatory state. This showed that the Ursino model was approximately three times as predictive as both the HMF model and the PRx model. However, in general, all of the models predictive accuracies were relatively poor so a number of optimisation strategies were then assessed. These optimisation strategies ultimately were formed into a generalised modelling framework. This framework draws on the ideas of mathematical topology to underpin and explain any change or optimisation to a model. Within the framework different optimisations can be grouped into four categories, each of which are explored in the text of this thesis: 1) Model Comparison. This is the simplest technique to apply where the number of models under examination are reduced based on the predictive accuracy. 2) Parameter restriction. A classical form of optimisation by constraining a model parameter to cause a better predictive accuracy. In the case of both the HMF and PRx we showed between a two hundred and six hundred percent increase in predictive accuracy over the initial assessment. 3) Parameter alteration. This change allows for related parameters to be substituted into a model. Four different alterations are explored as a surrogate measure for arterial-arteriolar blood volume the most clinically applicable of which is a transcranial impedance technique. This latter technique has the potential to be a non invasive measure correlated with both mean ICP and ICP pulse amplitude. 4) Model alteration. Allows for larger changes to the underlying structure of the model. Two examples are presented: firstly a new asymmetric sigmoid curve to overcome computational issues in the Ursino model and secondly a novel use of fractal characterisation which is applied in a wavelet noise reduction technique. The framework also gives an overview of the autoregulatory research domain as a whole as a result of its abstract nature. This helps to highlight some general issues in the domain including a more standardised way to record autoregulatory status. Finally concluding with research addressing the requirement for easier access to data and the need for the research community to cohesively start to address these issues

Dynamical Systems

Complex systems are pervasive in many areas of science integrated in our daily lives. Examples include financial markets, highway transportation networks, telecommunication networks, world and country economies, social networks, immunological systems, living organisms, computational systems and electrical and mechanical structures. Complex systems are often composed of a large number of interconnected and interacting entities, exhibiting much richer global scale dynamics than the properties and behavior of individual entities. Complex systems are studied in many areas of natural sciences, social sciences, engineering and mathematical sciences. This special issue therefore intends to contribute towards the dissemination of the multifaceted concepts in accepted use by the scientific community. We hope readers enjoy this pertinent selection of papers which represents relevant examples of the state of the art in present day research. [...

Aeronautical engineering: A continuing bibliography with indexes (supplement 253)

This bibliography lists 637 reports, articles, and other documents introduced into the NASA scientific and technical information system in May, 1990. Subject coverage includes: design, construction and testing of aircraft and aircraft engines; aircraft components, equipment and systems; ground support systems; and theoretical and applied aspects of aerodynamics and general fluid dynamics

Fast numerical methods for mixed--integer nonlinear model--predictive control

This thesis aims at the investigation and development of fast numerical methods for nonlinear mixed--integer optimal control and model- predictive control problems. A new algorithm is developed based on the direct multiple shooting method for optimal control and on the idea of real--time iterations, and using a convex reformulation and relaxation of dynamics and constraints of the original predictive control problem. This algorithm relies on theoretical results and is based on a nonconvex SQP method and a new active set method for nonconvex parametric quadratic programming. It achieves real--time capable control feedback though block structured linear algebra for which we develop new matrix updates techniques. The applicability of the developed methods is demonstrated on several applications. This thesis presents novel results and advances over previously established techniques in a number of areas as follows: We develop a new algorithm for mixed--integer nonlinear model- predictive control by combining Bock's direct multiple shooting method, a reformulation based on outer convexification and relaxation of the integer controls, on rounding schemes, and on a real--time iteration scheme. For this new algorithm we establish an interpretation in the framework of inexact Newton-type methods and give a proof of local contractivity assuming an upper bound on the sampling time, implying nominal stability of this new algorithm. We propose a convexification of path constraints directly depending on integer controls that guarantees feasibility after rounding, and investigate the properties of the obtained nonlinear programs. We show that these programs can be treated favorably as MPVCs, a young and challenging class of nonconvex problems. We describe a SQP method and develop a new parametric active set method for the arising nonconvex quadratic subproblems. This method is based on strong stationarity conditions for MPVCs under certain regularity assumptions. We further present a heuristic for improving stationary points of the nonconvex quadratic subproblems to global optimality. The mixed--integer control feedback delay is determined by the computational demand of our active set method. We describe a block structured factorization that is tailored to Bock's direct multiple shooting method. It has favorable run time complexity for problems with long horizons or many controls unknowns, as is the case for mixed- integer optimal control problems after outer convexification. We develop new matrix update techniques for this factorization that reduce the run time complexity of all but the first active set iteration by one order. All developed algorithms are implemented in a software package that allows for the generic, efficient solution of nonlinear mixed-integer optimal control and model-predictive control problems using the developed methods

Topological effects in particle physics phenomenology

This thesis is devoted to the study of topological effects in quantum field theories, with a particular focus on phenomenological applications. We begin by deriving a general classification of topological terms appearing in a non-linear sigma model based on maps from an arbitrary worldvolume manifold to a homogeneous space $G/H$ (where $G$ is an arbitrary Lie group and $H \subset G$). Such models are ubiquitous in phenomenology; in three or more dimensions they cover all cases in which only some subgroup $H$ of a dynamical symmetry group $G$ is linearly realized in vacuo. The classification is based on the observation that, for topological terms, the maps from the worldvolume to $G/H$ may be replaced by singular homology cycles on $G/H$. We find that such terms come in one of two types, which we refer to as `Aharonov-Bohm' (AB) and `Wess-Zumino' (WZ) terms. We derive a new condition for their $G$-invariance, which we call the `Manton condition', which is necessary and sufficient when the Lie group $G$ is connected. Armed with this classification of topological terms, we then apply it to Composite Higgs models based on a variety of coset spaces $G/H$ and discuss their phenomenology. For example, we point out the existence of an AB term in the minimal Composite Higgs model based on $SO(5)/SO(4)$, whose phenomenological effects arise only at the non-perturbative level, and lead to $P$ and $CP$ violation in the Higgs sector. Consideration of the Manton condition leads us to discover a rather subtle anomaly in a non-minimal model based on $SO(5)\times U(1)/SO(4)$ (a model which does, however, feature an AB term not previously noticed in the literature). A particularly rich topological structure, with six distinct terms of various types, is uncovered for the model based on $SO(6)/SO(4)$, which features two Higgs doublets and one singlet. Perhaps most importantly for phenomenology, measuring the coefficients of WZ terms that appear in any of these Composite Higgs models can allow one to probe the gauge group of the underlying microscopic theory. As a further application of our results, we analyse quantum mechanics models featuring such topological terms. In this context, a topological term couples the particle to a background magnetic field. The usual methods for formulating and solving the quantum mechanics of a particle moving in a magnetic field respect neither locality nor any global symmetries which happen to be present. We show how both locality and symmetry can be made manifest, by passing to an otherwise redundant description on a principal bundle over the original configuration space, and by promoting the original symmetry group to a central extension thereof. We then demonstrate how harmonic analysis on the extended symmetry group can be used to solve the Schr{\"o}dinger equation. To conclude our study of topological terms in sigma models, we show that the classification we have proposed may be rigorously justified (and generalised) using differential cohomology theory. In doing so, we introduce the notion of the `$G$-invariant differential characters' of a manifold $M$. Within this language, the Manton condition follows from the homotopy formula for differential characters, and we show that it remains necessary and sufficient under weaker conditions than connectedness of $G$. We prove that the abelian group of $G$-invariant differential characters sits inside various exact sequences and commutative diagrams, which thus provide us with some powerful algebraic tools for classifying topological terms in quantum field theories. In the remainder of the thesis we depart from the topic of sigma models and turn to gauge theories. We analyse anomalies (which may be understood as arising from topological effects) in both the Standard Model (SM) and theories Beyond the Standard Model (BSM). This analysis consists of two parts, in which we consider `local' and `global' anomalies in a gauge symmetry $G$; the former depend only on the Lie algebra of $G$, while the latter are sensitive also to its global structure, {\em i.e.} its topology. We first chart the space of anomaly-free extensions of the SM by a flavour-dependent $U(1)$ gauge symmetry, using arithmetic techniques from Diophantine analysis to cancel all possible local anomalies. We then develop some of these anomaly-free theories into phenomenological models featuring a heavy $Z^\prime$ gauge boson, that can account for a collection of recent measurements involving $b\rightarrow s\mu\mu$ transitions which are discrepant with SM predictions. We discuss how these models might also explain coarse features of the fermion mass problem, such as the heaviness of the third family. We then turn to global anomalies, which we analyse using the Dai-Freed theorem. Our principal tool here is to compute the bordism groups of the classifying spaces of various Lie groups, preserving particular spin structures, using the Atiyah-Hirzebruch spectral sequence. We show that there are no global anomalies (beyond the Witten anomaly associated with the electroweak factor) in four different `versions' of the SM, in which the gauge group is taken to be $G_{\text{SM}}/\Gamma$, with $G_{\text{SM}}=SU(3)\times SU(2) \times U(1)$ and $\Gamma \in \{0,\mathbb{Z}_2,\mathbb{Z}_3,\mathbb{Z}_6\}$. We also show that there are no new global anomalies in $U(1)^m$ extensions of the SM, which feature multiple $Z^\prime$ bosons, or in the Pati-Salam model.Vice-Chancellor's Award (Cambridge Trust